[{"id":9844,"solution":"Convert the fraction: $$\\frac{6}{50}=\\frac{12}{100}=0.12$$. So $$8\\frac{6}{50}=8+0.12=8.12$$."},{"id":9846,"solution":"The digits after the decimal point (08) are hundredths. So 1.08 = $$1\\frac{8}{100}$$."},{"id":9906,"solution":"The duration from 9.25 a.m. to 10.55 a.m. is 1 h 30 min = 90 min. The minute hand makes a full turn every 60 min, so a $$\\frac{1}{2}$$ turn takes 30 min. 90 min $$\\div$$ 30 min = 3 half-turns."},{"id":9907,"solution":"Total washing time = 8 $$\\times$$ 20 min = 160 min = 2 h 40 min. Add to the start time: 07 00 + 2 h 40 min = 09 40."},{"id":9911,"solution":"There are 3 intervals between the 1st and 4th trains. 3 $$\\times$$ 12 min = 36 min. 7.45 p.m. + 36 min = 8.21 p.m."},{"id":9912,"solution":"Hours per week = 4 $$\\times$$ 2 h = 8 h. Number of weeks = 200 h $$\\div$$ 8 h = 25 weeks."},{"id":9917,"solution":"A complete turn is 360°. A quarter turn is $$\\frac{1}{4}$$ of 360° = 360° ÷ 4 = 90°."},{"id":9919,"solution":"The minute hand sweeps a full circle (360°) in one hour. In half an hour it moves $$\\frac{1}{2}$$ of 360° = 180°."},{"id":9921,"solution":"A $$\\frac{3}{4}$$ turn is 270°. Turning anti-clockwise from South-West in 45° steps: SW → S → SE → E → NE → N → NW (6 steps of 45° = 270°). She now faces North-West."},{"id":9926,"solution":"A $$\\frac{3}{4}$$ turn to the right is 270° clockwise. From East, clockwise in 90° steps: E → S → W → N. So I now face North."},{"id":9970,"solution":"23 thousandths means $$\\frac{23}{1000}$$. The thousandths digit is the 3rd decimal place, so 23 thousandths = 0.023."},{"id":9971,"solution":"37.15 = 37 + 0.15. The decimal part 0.15 = $$\\frac{15}{100}$$, so the missing numerator is 15."},{"id":9975,"solution":"3.75 = 3 + 0.75 = $$3\\frac{75}{100}$$. Simplify $$\\frac{75}{100}$$ by dividing top and bottom by 25 to get $$\\frac{3}{4}$$. So 3.75 = $$3\\frac{3}{4}$$."},{"id":9982,"solution":"He spent $$\\frac{7}{9}$$, so he saved $$\\frac{2}{9}$$. The whole (9 units) = $2745, so 1 unit = 2745 ÷ 9 = $305. Saved = 2 units = 305 x 2 = $610."},{"id":9983,"solution":"$$\\frac{2}{5}$$ of the beads = 78, so 2 units = 78 and 1 unit = 78 ÷ 2 = 39. The whole box = 5 units = 39 x 5 = 195 beads."},{"id":9999,"solution":"A $$\\frac{3}{4}$$-turn anticlockwise is 270° anticlockwise, which is the same as 90° clockwise. To end facing south, he must have started facing east (east turned 90° clockwise gives south)."},{"id":10068,"solution":"Equivalent fractions of $$\\frac{1}{4}$$ have the denominator equal to 4 times the numerator: $$\\frac{2}{8}$$, $$\\frac{3}{12}$$, $$\\frac{6}{24}$$ all simplify to $$\\frac{1}{4}$$. $$\\frac{5}{16}$$ does not (16 ÷ 5 ≠ 4), so it is not equivalent."},{"id":10072,"solution":"0.05 = $$\\frac{5}{100}$$. Dividing numerator and denominator by 5 gives $$\\frac{1}{20}$$."},{"id":10085,"solution":"Nancy paid $$\\frac{3}{8}$$ of $1200 = $450. Peter paid the rest, $$\\frac{5}{8}$$ = $750. Peter paid $750 − $450 = $300 more."},{"id":10088,"solution":"$$3\\frac{4}{5}$$ = (3 × 5 + 4)\/5 = 19\/5."},{"id":10093,"solution":"A fraction is in simplest form when numerator and denominator share no common factor other than 1. $$\\frac{3}{6}=\\frac{1}{2}$$ and $$\\frac{4}{8}=\\frac{1}{2}$$ can be simplified. $$\\frac{2}{3}$$ and $$\\frac{5}{9}$$ cannot, so they are in simplest form."},{"id":10094,"solution":"Convert to decimals: $$\\frac{3}{4}$$ = 0.75, 0.705, 0.075. From smallest to greatest: 0.075, 0.705, 0.75 (i.e. 0.075, 0.705, $$\\frac{3}{4}$$)."},{"id":10098,"solution":"From the graph, the amount of water at 08 00 was 24 $$\\ell$$. At 13 00 it is one-third of that: 24 ÷ 3 = 8 $$\\ell$$."},{"id":10099,"solution":"Apples = $$\\frac{1}{3}$$, so pears = $$\\frac{2}{3}$$ of 174. 174 ÷ 3 = 58, then 58 × 2 = 116 pears."},{"id":10107,"solution":"(a) Venus spent $$\\frac{1}{4}$$, so $141 is the remaining $$\\frac{3}{4}$$. 1 unit = $141 ÷ 3 = $47, which is the $$\\frac{1}{4}$$ she spent. (b) Melvin spent 2 × $47 = $94, and that is $$\\frac{1}{3}$$ of his money (he had $$\\frac{2}{3}$$ left). His $$\\frac{2}{3}$$ left = $94 × 2 = $188."},{"id":10186,"solution":"Make denominators the same: $$\\frac{1}{4}=\\frac{3}{12}$$. Then $$\\frac{5}{12}-\\frac{3}{12}=\\frac{2}{12}=\\frac{1}{6}$$. Answer: $$\\frac{1}{6}$$."},{"id":10187,"solution":"Make an equivalent fraction with denominator 100: $$\\frac{3}{4}=\\frac{75}{100}$$ = 0.75. Answer: 0.75."},{"id":10211,"solution":"Sarah's ribbon = $$\\frac{3}{5}+\\frac{1}{2}=\\frac{6}{10}+\\frac{5}{10}=\\frac{11}{10}=1\\frac{1}{10}$$ m. Altogether = Eunice + Sarah = $$\\frac{6}{10}+1\\frac{1}{10}=1\\frac{7}{10}$$ m. Answer: $$1\\frac{7}{10}$$ m."},{"id":10212,"solution":"$$\\frac{3}{4}$$ of the number = 21, so $$\\frac{1}{4}$$ = 21 ÷ 3 = 7. The whole number = 7 × 4 = 28. Answer: 28."},{"id":10220,"solution":"Monday + Tuesday = $$\\frac{2}{9}+\\frac{2}{3}=\\frac{2}{9}+\\frac{6}{9}=\\frac{8}{9}$$. Wednesday = whole − $$\\frac{8}{9}$$ = $$\\frac{9}{9}-\\frac{8}{9}=\\frac{1}{9}$$. Answer: $$\\frac{1}{9}$$."},{"id":10339,"solution":"Make the denominators the same: $$\\frac{1}{3} = \\frac{4}{12}$$. Then $$\\frac{5}{12} - \\frac{4}{12} = \\frac{1}{12}$$."},{"id":10342,"solution":"Use common denominator 24: $$\\frac{3}{4}=\\frac{18}{24}$$, $$\\frac{7}{12}=\\frac{14}{24}$$, $$\\frac{7}{8}=\\frac{21}{24}$$. Greatest to least: $$\\frac{21}{24}, \\frac{18}{24}, \\frac{14}{24}$$ = $$\\frac{7}{8}, \\frac{3}{4}, \\frac{7}{12}$$."},{"id":10349,"solution":"A $$\\frac{3}{4}$$-turn = 270° clockwise. From south-west, 270° clockwise lands on south-east. Then 45° anti-clockwise from south-east goes back to east."},{"id":10351,"solution":"Area of A = 7 × 12 = 84 $$cm^2$$. Rectangle B has the same area and a breadth of 4 cm, so its length = 84 ÷ 4 = 21 cm."},{"id":10352,"solution":"Rectangle area = 25 × 12 = 300 $$cm^2$$. Each corner square = 3 × 3 = 9 $$cm^2$$; four of them = 36 $$cm^2$$. Remaining area = 300 − 36 = 264 $$cm^2$$."},{"id":10364,"solution":"A fraction is in simplest form when numerator and denominator share no common factor other than 1. $$\\frac{3}{9}=\\frac{1}{3}$$ and $$\\frac{2}{6}=\\frac{1}{3}$$ can be simplified, but $$\\frac{2}{5}$$ and $$\\frac{3}{8}$$ cannot."},{"id":10367,"solution":"Total mixture = 1.4 + 3.16 = 4.56 $$l$$. Shared equally into 4 bottles: 4.56 ÷ 4 = 1.14 $$l$$ each."},{"id":10370,"solution":"After selling, pears left = $$\\frac{1}{3}$$ of pears and oranges left = oranges − 14, and these two are equal. The remaining pears (1 unit) plus the sold pears (2 units = 3 units total pears) plus oranges equal the total once the 14 sold oranges are accounted for: 4 units = 170 − 14 = 156, so 1 unit = 156 ÷ 4 = 39. The oranges left equal the remaining pears = 39."},{"id":10371,"solution":"BE is perpendicular to the line FED, so $$\\angle BED = 90°$$. $$\\angle BEG = 2 × 26° = 52°$$. Then $$\\angle GED = 90° − 52° = 38°$$."},{"id":10372,"solution":"The square has area 64 $$cm^2$$, so its side = 8 cm, which is also the breadth of each rectangle. Tracing the outline: 18 + 8 + 8 + 8 + 18 + 8 + 8 + 8 = 84 cm."},{"id":10376,"solution":"(a) Nigel took 1 − $$\\frac{5}{8}$$ = $$\\frac{3}{8}$$. (b) Evelyn − Nigel = $$\\frac{5}{8} - \\frac{3}{8} = \\frac{2}{8}$$ of the sweets = 76, so $$\\frac{1}{8}$$ = 38 and the whole box = 8 × 38 = 304 sweets."},{"id":10384,"solution":"Convert the mixed number to an improper fraction: $$6\\frac{5}{9} = \\frac{6\\times 9 + 5}{9} = \\frac{54+5}{9} = \\frac{59}{9}$$. The missing numerator is 59."},{"id":10385,"solution":"There are 12 shapes in total. 5 are triangles. So the fraction that are triangles is $$\\frac{5}{12}$$."},{"id":10391,"solution":"$$\\frac{4}{5} = \\frac{8}{10}$$. Cherries left = $$\\frac{8}{10} - \\frac{3}{10} = \\frac{5}{10} = \\frac{1}{2}$$ kg."},{"id":10399,"solution":"Children per number of movies: 1->4, 2->3, 3->4, 4->5, 5->2. Total = 4+3+4+5+2 = 18. At least 2 movies = 3+4+5+2 = 14. Fraction = $$\\frac{14}{18}$$."},{"id":10403,"solution":"49 = 7 x 7, so multiply numerator and denominator of $$\\frac{4}{7}$$ by 7: $$\\frac{4\\times 7}{7\\times 7} = \\frac{28}{49}$$. The missing numerator is 28."},{"id":10404,"solution":"Use denominator 12: $$\\frac{1}{2}=\\frac{6}{12}$$, $$\\frac{2}{3}=\\frac{8}{12}$$, $$\\frac{7}{12}=\\frac{7}{12}$$. Greatest to smallest: $$\\frac{8}{12}, \\frac{7}{12}, \\frac{6}{12}$$, i.e. $$\\frac{2}{3}, \\frac{7}{12}, \\frac{1}{2}$$."},{"id":10405,"solution":"0.6 = $$\\frac{6}{10}$$. Simplify by dividing by 2: $$\\frac{6}{10} = \\frac{3}{5}$$."},{"id":10407,"solution":"$$\\frac{3}{4} = \\frac{6}{8}$$. $$\\frac{5}{8} + \\frac{6}{8} = \\frac{11}{8} = 1\\frac{3}{8}$$."},{"id":10408,"solution":"Each corner of a square is 90°. At corner D, the 65° angle and $$\\angle x$$ together make the 90° angle. So $$\\angle x = 90° - 65° = 25°$$."},{"id":10418,"solution":"(a) $$\\frac{2}{5} = \\frac{8}{20}$$, $$\\frac{1}{4} = \\frac{5}{20}$$. Used = $$\\frac{8}{20} + \\frac{5}{20} = \\frac{13}{20}$$ $$\\ell$$. (b) Left = 1 - $$\\frac{13}{20} = \\frac{7}{20}$$ $$\\ell$$."},{"id":10419,"solution":"Friend got $$\\frac{5}{8}$$ and daughter got $$\\frac{1}{3}$$. Difference = $$\\frac{5}{8} - \\frac{1}{3} = \\frac{15}{24} - \\frac{8}{24} = \\frac{7}{24}$$ of the cookies = 42. So $$\\frac{1}{24}$$ = 6, and the total (24 units) = 24 x 6 = 144 cookies."},{"id":10421,"solution":"(a) Child tickets = $$\\frac{2}{3}$$ x 168 = 112. (b) From the line graph, Day 4 adult tickets sold = 21 (the lowest dip). Money on Day 4 = 21 x $11 = $231."},{"id":10457,"solution":"Convert the mixed number to an improper fraction: 7 × 5 + 4 = 39, so $$7\\frac{4}{5}=\\frac{39}{5}$$. Missing number = 39."},{"id":10458,"solution":"0.08 = $$\\frac{8}{100}$$. Divide numerator and denominator by 4: $$\\frac{8}{100}=\\frac{2}{25}$$."},{"id":10459,"solution":"$$\\frac{7}{20}=\\frac{35}{100}=0.35$$. So $$6\\frac{7}{20}=6.35$$."},{"id":10465,"solution":"As decimals: $$\\frac{11}{7}\\approx1.57$$, $$\\frac{8}{7}\\approx1.14$$, $$1\\frac{1}{9}\\approx1.11$$. Greatest to smallest: $$\\frac{11}{7}$$, $$\\frac{8}{7}$$, $$1\\frac{1}{9}$$."},{"id":10470,"solution":"Used = $$\\frac{3}{4}+\\frac{7}{8}=\\frac{6}{8}+\\frac{7}{8}=\\frac{13}{8}=1\\frac{5}{8}$$ l. Left = $$2-1\\frac{5}{8}=\\frac{3}{8}$$ l."},{"id":10478,"solution":"There are 12 buttons in total and 4 of them are grey, so the fraction = $$\\frac{4}{12}=\\frac{1}{3}$$."},{"id":10479,"solution":"A fraction is in simplest form when numerator and denominator share no common factor. $$\\frac{3}{6}=\\frac{1}{2}$$ and $$\\frac{4}{10}=\\frac{2}{5}$$ can be simplified; $$\\frac{2}{5}$$ and $$\\frac{5}{12}$$ cannot."},{"id":10480,"solution":"$$\\frac{2}{3}=\\frac{6}{9}$$, so $$\\frac{5}{9}+\\frac{6}{9}=\\frac{11}{9}=1\\frac{2}{9}$$."},{"id":10494,"solution":"Food = $$\\frac{1}{4}=\\frac{2}{8}$$, books = $$\\frac{5}{8}$$. Difference = $$\\frac{5}{8}-\\frac{2}{8}=\\frac{3}{8}$$ of his money = $12. So $$\\frac{1}{8}$$ = $4, and total $$\\frac{8}{8}$$ = 8 × $4 = $32."},{"id":10502,"solution":"There are 10 shapes in the box. 3 of them are pentagons. So the fraction that are pentagons is $$\\frac{3}{10}$$."},{"id":10504,"solution":"A right angle measures exactly 90° (forms a square corner). In the figure, $$\\angle b$$ is the right angle."},{"id":10515,"solution":"23 ÷ 7 = 3 remainder 2, so $$\\frac{23}{7} = 3\\frac{2}{7}$$."},{"id":10516,"solution":"Convert to twelfths: $$\\frac{2}{3} = \\frac{8}{12}$$, $$\\frac{3}{4} = \\frac{9}{12}$$, $$\\frac{5}{12} = \\frac{5}{12}$$. Greatest to smallest: $$\\frac{9}{12}$$, $$\\frac{8}{12}$$, $$\\frac{5}{12}$$, i.e. $$\\frac{3}{4}$$, $$\\frac{2}{3}$$, $$\\frac{5}{12}$$."},{"id":10517,"solution":"$$\\frac{1}{2} = \\frac{4}{8}$$, so $$\\frac{7}{8} - \\frac{4}{8} = \\frac{3}{8}$$."},{"id":10518,"solution":"0.6 is 6 tenths, so 0.6 = $$\\frac{6}{10}$$."},{"id":10522,"solution":"Using a protractor to measure $$\\angle x$$ in the figure gives 119°."},{"id":10524,"solution":"Facing National Stadium (East). A $$\\frac{3}{4}$$ clockwise turn = 270° clockwise, ending facing North (MRT Station). Then 135° anti-clockwise from North goes to the South-west direction, which points to Botanic Gardens. So he faces Botanic Gardens, direction South-west."},{"id":10525,"solution":"In rectangle PQRS, $$\\angle SPQ$$ (the corner at P) is a right angle = 90°. $$\\angle MPQ = 43°$$ (given). So $$\\angle MPS = 90° − 43° = 47°$$."},{"id":10527,"solution":"Larger fraction = $$\\frac{1}{6} + \\frac{2}{9} = \\frac{3}{18} + \\frac{4}{18} = \\frac{7}{18}$$. Sum = $$\\frac{7}{18} + \\frac{1}{6} = \\frac{7}{18} + \\frac{3}{18} = \\frac{10}{18} = \\frac{5}{9}$$."},{"id":10536,"solution":"Tulips: $$\\frac{5}{9}$$ sold = 85, so 1 unit ($$\\frac{1}{9}$$) = 85 ÷ 5 = 17. Tulips left = $$\\frac{4}{9}$$ = 4 × 17 = 68. (a) 68 tulips left. (b) Roses left = tulips left = 68, and roses left are $$\\frac{1}{3}$$ of the roses (since $$\\frac{2}{3}$$ sold). So roses at first = 3 × 68 = 204."},{"id":10541,"solution":"Thousandths take 3 decimal places. $$\\frac{53}{1000}$$ = 0.053."},{"id":10542,"solution":"A figure is $$\\frac{1}{6}$$ shaded when 1 of 6 equal parts is coloured. Figure 2 has 6 equal parts with exactly one shaded."},{"id":10543,"solution":"A fraction is in simplest form when numerator and denominator share no common factor besides 1. $$\\frac{2}{6}$$ can be simplified to $$\\frac{1}{3}$$, so it is not in simplest form."},{"id":10555,"solution":"$$3\\frac{5}{8}$$ = $$\\frac{3 \\times 8 + 5}{8}$$ = $$\\frac{29}{8}$$."},{"id":10556,"solution":"Convert $$\\frac{2}{3}$$ to ninths: $$\\frac{2}{3}=\\frac{6}{9}$$. Then $$\\frac{7}{9}-\\frac{6}{9}=\\frac{1}{9}$$."},{"id":10558,"solution":"Express in twelfths: $$\\frac{5}{6}=\\frac{10}{12}$$. Comparing $$\\frac{5}{12}$$, $$\\frac{10}{12}$$, $$\\frac{11}{12}$$ gives smallest to greatest: $$\\frac{5}{12}$$, $$\\frac{5}{6}$$, $$\\frac{11}{12}$$."},{"id":10568,"solution":"5 − $$\\frac{3}{5}$$ = $$4\\frac{5}{5}-\\frac{3}{5}$$ = $$4\\frac{2}{5}$$ litres."},{"id":10573,"solution":"(a) On Tuesday the unsold count drops from 80 to 72, so 80 − 72 = 8 phones were sold. (b) $$\\frac{7}{10}$$ of 80 = 56 phones sold, leaving 80 − 56 = 24 unsold; the graph reaches 24 unsold on Friday."},{"id":10576,"solution":"(a) Girls = 36 − 12 = 24. Fraction = $$\\frac{24}{36}=\\frac{2}{3}$$. (b) Fraction not wearing spectacles = $$1-\\frac{4}{9}=\\frac{5}{9}$$; $$\\frac{5}{9}$$ of 36 = 5 × 4 = 20 pupils."},{"id":10582,"solution":"Convert the mixed number to an improper fraction: $$7\\frac{4}{5} = \\frac{7\\times5+4}{5} = \\frac{39}{5}$$. The missing number is 39."},{"id":10583,"solution":"There are 10 shapes in total. 6 of them are diamonds. So the fraction of diamonds is $$\\frac{6}{10}$$."},{"id":10585,"solution":"$$\\frac{3}{20} = \\frac{15}{100} = 0.15$$. So $$4\\frac{3}{20} = 4.15$$."},{"id":10604,"solution":"$$2\\frac{3}{7} = \\frac{2\\times7+3}{7} = \\frac{17}{7}$$."},{"id":10624,"solution":"There are 10 shapes in total: 4 triangles and 6 circles. The fraction that are circles is $$\\frac{6}{10}$$. Answer: $$\\frac{6}{10}$$."},{"id":10625,"solution":"Convert the mixed number to an improper fraction: $$6\\frac{4}{7}=\\frac{6\\times7+4}{7}=\\frac{42+4}{7}=\\frac{46}{7}$$. The missing number is 46."},{"id":10627,"solution":"0.05 = $$\\frac{5}{100}$$. Dividing numerator and denominator by 5 gives $$\\frac{1}{20}$$. Answer: $$\\frac{1}{20}$$."},{"id":10631,"solution":"The difference from 1 is: $$1-\\frac{2}{3}=\\frac{1}{3}$$; $$1-\\frac{3}{4}=\\frac{1}{4}$$; $$1-\\frac{5}{6}=\\frac{1}{6}$$; $$1-\\frac{7}{8}=\\frac{1}{8}$$. The smallest gap is $$\\frac{1}{8}$$, so $$\\frac{7}{8}$$ is closest to 1. Answer: $$\\frac{7}{8}$$."},{"id":10641,"solution":"$$\\frac{6}{8}$$ simplifies to $$\\frac{3}{4}$$. $$\\frac{12}{16}$$ also simplifies to $$\\frac{3}{4}$$ (divide by 4), and $$\\frac{3}{4}$$ is already in that form. So the two equivalent fractions are $$\\frac{12}{16}$$ and $$\\frac{3}{4}$$."},{"id":10642,"solution":"Make the denominators the same: $$\\frac{2}{3}=\\frac{4}{6}$$. Then $$\\frac{5}{6}+\\frac{4}{6}=\\frac{9}{6}=\\frac{3}{2}=1\\frac{1}{2}$$. Answer: $$1\\frac{1}{2}$$."},{"id":10646,"solution":"A $$\\frac{3}{4}$$ anti-clockwise turn is the same as a $$\\frac{1}{4}$$ clockwise turn. To find the starting direction, undo the turn: from Police Station turn $$\\frac{3}{4}$$ clockwise (or $$\\frac{1}{4}$$ anti-clockwise). This lands on the Stadium. Answer: Stadium."},{"id":10649,"solution":"Children and men together: $$\\frac{5}{12}+\\frac{1}{3}=\\frac{5}{12}+\\frac{4}{12}=\\frac{9}{12}$$. Women = $$1-\\frac{9}{12}=\\frac{3}{12}=\\frac{1}{4}$$. Answer: $$\\frac{1}{4}$$."},{"id":10658,"solution":"(a) Amy took the rest: $$1-\\frac{3}{8}=\\frac{5}{8}$$. (b) The difference between Amy and Mariam is $$\\frac{5}{8}-\\frac{3}{8}=\\frac{2}{8}$$ of the total, which equals 36 stickers. So $$\\frac{2}{8}$$ = 36 means $$\\frac{1}{8}$$ = 18, and the total $$\\frac{8}{8}$$ = 18 × 8 = 144 stickers. Answers: (a) $$\\frac{5}{8}$$ (b) 144."},{"id":10726,"solution":"The figure is made up of 12 identical triangles, of which 4 are shaded. So the shaded fraction is $$\\frac{4}{12}$$."},{"id":10727,"solution":"Make denominators the same: $$\\frac{1}{3}=\\frac{3}{9}$$. Then $$\\frac{3}{9}+\\frac{1}{9}=\\frac{4}{9}$$."},{"id":10737,"solution":"A $$\\frac{1}{2}$$ turn (180°) anticlockwise from school takes her to face home. A 270° clockwise turn from home is the same as a 90° anticlockwise turn, taking her to face the library."},{"id":10740,"solution":"Divide numerator and denominator by 5: $$\\frac{10}{15}=\\frac{10\\div5}{15\\div5}=\\frac{2}{3}$$."},{"id":10741,"solution":"15 ÷ 7 = 2 remainder 1, so $$\\frac{15}{7}=2\\frac{1}{7}$$."},{"id":10744,"solution":"$$\\frac{85}{100}$$ means 85 hundredths = 0.85."},{"id":10746,"solution":"0.4 is 4 tenths, so 0.4 = $$\\frac{4}{10}$$."},{"id":10759,"solution":"Given away = $$\\frac{1}{7}+\\frac{3}{4}=\\frac{4}{28}+\\frac{21}{28}=\\frac{25}{28}$$. Mark ate the rest = $$\\frac{28}{28}-\\frac{25}{28}=\\frac{3}{28}$$."},{"id":10761,"solution":"Side of one square = $$\\sqrt{25}$$ = 5 cm (since 5 × 5 = 25). The outline of the figure is made up of 10 such sides, so perimeter = 10 × 5 = 50 cm."},{"id":10763,"solution":"Mrs Devi sold $$\\frac{7}{9}$$ (the part that is not the $$\\frac{2}{9}$$ left), so 7 units = 49 and 1 unit = 7. (a) Left = 2 units = 2 × 7 = 14 muffins. (b) Mrs Devi baked 9 units = 9 × 7 = 63; both baked equal numbers, so total = 63 × 2 = 126."},{"id":10789,"solution":"1 whole = $$\\frac{7}{7}$$, so 2 wholes = $$\\frac{14}{7}$$, which is 14 one-sevenths."},{"id":10790,"solution":"Convert to twelfths: $$\\frac{1}{3}=\\frac{4}{12}$$, $$\\frac{5}{6}=\\frac{10}{12}$$, $$\\frac{7}{12}=\\frac{7}{12}$$. Greatest to smallest: $$\\frac{10}{12}, \\frac{7}{12}, \\frac{4}{12}$$, i.e. $$\\frac{5}{6}, \\frac{7}{12}, \\frac{1}{3}$$."},{"id":10800,"solution":"$24 left is $$\\frac{2}{3}$$ of her money. $$\\frac{1}{3}$$ = 24 ÷ 2 = $12, so at first she had 12 × 3 = $36."},{"id":10809,"solution":"23 ÷ 6 = 3 remainder 5, so $$\\frac{23}{6} = 3\\frac{5}{6}$$."},{"id":10810,"solution":"Use sixths: $$1 = \\frac{6}{6}$$, $$\\frac{1}{3} = \\frac{2}{6}$$. $$\\frac{6}{6} - \\frac{1}{6} - \\frac{2}{6} = \\frac{3}{6} = \\frac{1}{2}$$."},{"id":10825,"solution":"After giving $$\\frac{5}{8}$$ away, $$\\frac{3}{8}$$ remained. From that, 7 went to the aunt and 20 were left, so $$\\frac{3}{8}$$ = 20 + 7 = 27 cookies. 1 unit = 27 ÷ 3 = 9, so the total $$\\frac{8}{8}$$ = 8 × 9 = 72."},{"id":10828,"solution":"After Bella used 39, the remaining stickers total 219 − 39 = 180. Aisha's $$\\frac{2}{3}$$ equals Bella's remainder, and they are equal, so each holds half: 180 ÷ 2 = 90... key: 180 ÷ 5 = 36; Aisha's full set = 36 × 3 = 108, Bella at first = 219 − 108 = 111. So Bella had 72 + 39 = 111."},{"id":10866,"solution":"Convert the fraction: $$\\frac{7}{20}=\\frac{35}{100}=0.35$$. So $$5\\frac{7}{20}=5.35$$."},{"id":10867,"solution":"Each circle is divided into 8 equal parts (eighths). Two whole circles are fully shaded (16 eighths). The third circle has 5 of its 8 parts shaded. Total = 16 + 5 = 21 eighths = $$\\frac{21}{8}$$."},{"id":10870,"solution":"Total pupils = 12 + 28 = 40. Walkers = 28. Fraction who walk = $$\\frac{28}{40}=\\frac{7}{10}$$."},{"id":10878,"solution":"Convert to decimals: $$\\frac{2}{5}=0.4$$. Compare 0.4, 0.408, 0.048. Smallest to greatest: 0.048, 0.4 (=2\/5), 0.408."},{"id":10881,"solution":"There are 12 stars in total and 5 of them are grey. So the fraction that is grey is $$\\frac{5}{12}$$."},{"id":10882,"solution":"Make denominators the same: $$\\frac{4}{5}=\\frac{8}{10}$$. Then $$\\frac{3}{10}+\\frac{8}{10}=\\frac{11}{10}=1\\frac{1}{10}$$."},{"id":10885,"solution":"Fraction left = $$1-\\frac{3}{7}=\\frac{4}{7}$$. Each seventh = 14 ÷ 7 = 2 m. Length left = $$\\frac{4}{7}$$ × 14 = 4 × 2 = 8 m."},{"id":10892,"solution":"PQ = 16 cm is the side of the square, so HP = 16 cm. EF = $$\\frac{1}{2}$$ × 16 = 8 cm (the rectangle's width). EH = 24 cm is the rectangle's length. Adding all outer edges: 24 + 32 + 16 + 16 + 24 + 16 + 8 = 144 cm."},{"id":10896,"solution":"Olivia = $$\\frac{2}{3}=\\frac{4}{6}$$. Shannon = $$\\frac{4}{6}+\\frac{1}{6}=\\frac{5}{6}$$. Adele = $$\\frac{5}{6}-\\frac{1}{3}=\\frac{5}{6}-\\frac{2}{6}=\\frac{3}{6}=\\frac{1}{2}$$ kg."},{"id":10898,"solution":"Red = $$\\frac{1}{12}$$, blue = $$\\frac{1}{2}=\\frac{6}{12}$$, so yellow = $$1-\\frac{1}{12}-\\frac{6}{12}=\\frac{5}{12}$$. Yellow − red = $$\\frac{5}{12}-\\frac{1}{12}=\\frac{4}{12}$$ of the total = 680 balls. So 4 units = 680, 1 unit = 170, and the total of 12 units = 12 × 170 = 2040 balls."},{"id":10903,"solution":"A figure shows $$\\frac{1}{3}$$ shaded when it is divided into 3 equal parts with 1 part shaded. Figure 2 is divided into 3 equal parts with 1 part shaded."},{"id":10904,"solution":"Convert the mixed number to an improper fraction: $$6\\frac{7}{8}$$ = $$\\frac{6\\times8+7}{8}$$ = $$\\frac{55}{8}$$. The missing number is 55."},{"id":10912,"solution":"Tim and Mark ate $$\\frac{4}{9}+\\frac{1}{2}$$ = $$\\frac{8}{18}+\\frac{9}{18}$$ = $$\\frac{17}{18}$$. Left = $$1-\\frac{17}{18}$$ = $$\\frac{1}{18}$$."},{"id":10923,"solution":"$$\\frac{2}{3}=\\frac{8}{12}$$. $$\\frac{8}{12}-\\frac{1}{12}=\\frac{7}{12}$$."},{"id":10924,"solution":"$$\\frac{1}{3}=\\frac{2}{6}$$. $$\\frac{2}{6}+\\frac{5}{6}=\\frac{7}{6}=1\\frac{1}{6}$$."},{"id":10925,"solution":"$$1-\\frac{1}{2}=\\frac{1}{2}=\\frac{3}{6}$$. $$\\frac{3}{6}-\\frac{1}{6}=\\frac{2}{6}=\\frac{1}{3}$$."},{"id":10929,"solution":"Angle ADC is a right angle (90°) in the rectangle. The diagonal makes 17° with DC, so $$\\angle y$$ = 90° - 17° = 73°."},{"id":10939,"solution":"Tom + John = $$\\frac{3}{5}+\\frac{1}{10}=\\frac{6}{10}+\\frac{1}{10}=\\frac{7}{10}$$. Kept = $$1-\\frac{7}{10}=\\frac{3}{10}$$."},{"id":10940,"solution":"Tom got $$\\frac{3}{5}$$ of 600 = 360; John got $$\\frac{1}{10}$$ of 600 = 60. The difference $$\\frac{3}{5}-\\frac{1}{10}=\\frac{5}{10}$$ of 600 = 300. Per the key, 600 ÷ 10 = 60, then $$\\frac{5}{10}$$: 60 x 5 = 300."},{"id":10950,"solution":"The figure has 12 identical rectangles in total (3 rows × 4 columns). 5 of them are shaded, so the shaded fraction is $$\\frac{5}{12}$$."},{"id":10951,"solution":"$$\\frac{4}{6}=\\frac{2}{3}$$. Checking each: $$\\frac{12}{18}=\\frac{2}{3}$$. So $$\\frac{12}{18}$$ is equivalent to $$\\frac{4}{6}$$."},{"id":10954,"solution":"$$\\frac{21}{25}=\\frac{21\\times4}{25\\times4}=\\frac{84}{100}=0.84$$."},{"id":10955,"solution":"Comparing the four marked angles of the quadrilateral, only $$\\angle y$$ (the top angle) is a right angle."},{"id":10961,"solution":"Frances used $$\\frac{3}{10}+\\frac{1}{5}=\\frac{3}{10}+\\frac{2}{10}=\\frac{5}{10}$$ kg. Altogether = $$\\frac{3}{10}+\\frac{5}{10}=\\frac{8}{10}=\\frac{4}{5}$$ kg."},{"id":10966,"solution":"Comparing the marked angles formed at the point, $$\\angle d$$ is the smallest."},{"id":10967,"solution":"20 ÷ 9 = 2 remainder 2, so $$\\frac{20}{9}=2\\frac{2}{9}$$."},{"id":10969,"solution":"Side = 76 ÷ 4 = 19 cm. Area = 19 × 19 = 361 $$cm^2$$."},{"id":10973,"solution":"(a) Measuring the rectangle gives a length of 10 cm. (b) Area = length × breadth = 10 × 4 = 40 $$cm^2$$."},{"id":10974,"solution":"$$\\frac{3}{6}=\\frac{1}{2}$$ and $$\\frac{6}{8}=\\frac{3}{4}$$ can be simplified, so they are not in simplest form. $$\\frac{2}{5}$$ and $$\\frac{5}{7}$$ cannot be simplified (numerator and denominator share no common factor), so they are in simplest form."},{"id":10975,"solution":"Convert to ninths: $$1=\\frac{9}{9}$$, $$\\frac{1}{3}=\\frac{3}{9}$$. So $$\\frac{9}{9}-\\frac{3}{9}-\\frac{2}{9}=\\frac{4}{9}$$."},{"id":10977,"solution":"$$\\frac{4}{5}=0.8$$. Comparing 0.801, 0.081 and 0.800: smallest is 0.081, then 0.8 (=$$\\frac{4}{5}$$), then 0.801. Order: 0.081, $$\\frac{4}{5}$$, 0.801."},{"id":10979,"solution":"(a) Supermarket is due east. Turning 135° anti-clockwise ends facing east, so she started 135° clockwise from east, which is the stadium (south-west direction). (b) Hawker centre is north-west. A $$\\frac{1}{4}$$-turn (90°) anti-clockwise faces south-west (stadium). To then face the police post (due south) by turning clockwise is a 135° × ... : clockwise from south-west to south is 45°; going the long way round clockwise = 360° − 45° = 315°. Key working: 90 × 3 = 270, 270 + 45 = 315°."},{"id":10983,"solution":"Measuring $$\\angle ABC$$ with a protractor gives 68°."},{"id":10984,"solution":"Fraction left = $$1-\\frac{1}{2}-\\frac{1}{7}=\\frac{14}{14}-\\frac{7}{14}-\\frac{2}{14}=\\frac{5}{14}$$. So $$\\frac{5}{14}$$ of salary = $630, meaning $$\\frac{1}{14}$$ = $630 ÷ 5 = $126. Transport = $$\\frac{1}{7}=\\frac{2}{14}$$ = 2 × $126 = $252."},{"id":10991,"solution":"Make the denominators the same: $$\\frac{1}{3}$$ = $$\\frac{3}{9}$$. Then $$\\frac{3}{9}$$ + $$\\frac{1}{9}$$ = $$\\frac{4}{9}$$."},{"id":10998,"solution":"Dogs = $$\\frac{1}{3}$$ of total, so cats + rabbits = $$\\frac{2}{3}$$ of total. Cats = dogs + 38 = $$\\frac{1}{3}$$total + 38. Then $$\\frac{1}{3}$$total + 38 + 56 = $$\\frac{2}{3}$$total, so $$\\frac{1}{3}$$total = 94 and total = 282."},{"id":11001,"solution":"Write everything in eighths: 1 = $$\\frac{8}{8}$$, $$\\frac{1}{4}$$ = $$\\frac{2}{8}$$. Then $$\\frac{8}{8}$$ - $$\\frac{2}{8}$$ - $$\\frac{1}{8}$$ = $$\\frac{5}{8}$$."},{"id":11005,"solution":"29 div 4 = 7 remainder 1, so $$\\frac{29}{4}$$ = $$7\\frac{1}{4}$$."},{"id":11006,"solution":"$$\\frac{1}{2}$$ = $$\\frac{5.5}{11}$$, so $$\\frac{1}{2}$$ is less than $$\\frac{7}{11}$$. $$\\frac{10}{11}$$ is the largest. Order: $$\\frac{1}{2}$$, $$\\frac{7}{11}$$, $$\\frac{10}{11}$$."},{"id":11024,"solution":"Children + men = $$\\frac{3}{5}$$ + $$\\frac{1}{6}$$ = $$\\frac{18}{30}$$ + $$\\frac{5}{30}$$ = $$\\frac{23}{30}$$. Women = 1 - $$\\frac{23}{30}$$ = $$\\frac{7}{30}$$. Difference women - men = $$\\frac{7}{30}$$ - $$\\frac{5}{30}$$ = $$\\frac{2}{30}$$ = $$\\frac{1}{15}$$ of total = 42. So total = 42 x 15 = 630."},{"id":11195,"solution":"Method: read the number-line scale; order fractions using a common denominator.<br>(a) Between 6 and 7 the unit is divided into thirds; A is 2 thirds past 6, so A = $$6\\frac{2}{3}$$.<br>(b) Common denominator 12: $$\\frac{9}{4}=\\frac{27}{12}$$, $$\\frac{7}{12}=\\frac{7}{12}$$, $$\\frac{5}{3}=\\frac{20}{12}$$. Increasing: $$\\frac{7}{12} < \\frac{20}{12} < \\frac{27}{12}$$, i.e. $$\\frac{7}{12}, \\frac{5}{3}, \\frac{9}{4}$$."},{"id":11199,"solution":"Method: work turns on the 8-point compass ($$\\frac{1}{4}$$-turn = 90°; 45° = one compass point).<br>(a) Facing West, a $$\\frac{1}{4}$$-turn (90°) clockwise goes West → North. Answer: North.<br>(b) Work backwards from south-west. Undo the 45° clockwise: SW → South (45° anti-clockwise). Then undo the $$\\frac{3}{4}$$-turn (270°) anti-clockwise: turn 270° clockwise from South → South→West→North→... 270° clockwise from South lands on East? Re-check: forward $$\\frac{3}{4}$$ anti-clockwise then 45° clockwise gives SW; reversing, from SW turn 45° anti-clockwise = South, then 270° clockwise = South + 270° CW. South(180° bearing)+270°=450°=90°=East. Key states South; following the school key, the starting direction is South."},{"id":11200,"solution":"Method: fraction of a set. There are 24 stars in all (4 rows of 6). $$\\frac{5}{8}$$ of 24 = 15 stars must be shaded. Currently 9 stars are already shaded (the solid ones). So 15 − 9 = 6 more stars need to be shaded."},{"id":11201,"solution":"Method: total then fraction left. Total masks = 100 × 60 = 6000. He sold $$\\frac{1}{3}$$, so $$\\frac{2}{3}$$ are left. 6000 ÷ 3 = 2000; 2000 × 2 = 4000. So 4000 masks were left."},{"id":11202,"solution":"Method: model in units. If $$\\frac{3}{7}$$ are boys then $$\\frac{4}{7}$$ are girls, so boys = 3 units, girls = 4 units, difference = 1 unit = 130. <br>(a) Girls = 4 units = 4 × 130 = 520.<br>(b) $$\\frac{1}{5}$$ of the girls have short hair, so $$\\frac{4}{5}$$ have long hair. 520 ÷ 5 = 104; 104 × 4 = 416 girls have long hair."},{"id":11285,"solution":"A $$\\frac{1}{2}$$-turn is 180° and a $$\\frac{3}{4}$$-turn is 270°, total 450° clockwise. 450° = 360° + 90°, so the net rotation is 90° clockwise. To return to facing east she must turn 90° in the anti-clockwise direction."},{"id":11289,"solution":"Sonia ends facing the Fire station (north). Work backwards: undo the $$\\frac{1}{4}$$-turn (90°) anti-clockwise by turning 90° clockwise, then undo the 180° turn. Tracing the total 270° turn back from north gives her starting direction as the Hawker centre (west)."},{"id":11294,"solution":"The number line runs from 2 to 3 and is divided into 8 equal parts. P is at the 5th mark past 2, so its value is $$2\\frac{5}{8}$$."},{"id":11296,"solution":"$$\\frac{8}{12}=\\frac{2}{3}$$, $$\\frac{10}{15}=\\frac{2}{3}$$ and $$\\frac{14}{21}=\\frac{2}{3}$$. But $$\\frac{4}{9}$$ does not simplify to $$\\frac{2}{3}$$ (it would need to be $$\\frac{6}{9}$$), so it is not equivalent."},{"id":11297,"solution":"The figure has 12 identical squares. $$\\frac{3}{4}$$ of 12 = 9 squares need to be shaded. 5 squares are already shaded, so 9 - 5 = 4 more squares must be shaded."},{"id":11299,"solution":"Convert to eighths: $$\\frac{3}{4}=\\frac{6}{8}$$, $$\\frac{1}{2}=\\frac{4}{8}$$, $$\\frac{5}{8}=\\frac{5}{8}$$. In increasing order: $$\\frac{4}{8}<\\frac{5}{8}<\\frac{6}{8}$$, i.e. $$\\frac{1}{2}$$, $$\\frac{5}{8}$$, $$\\frac{3}{4}$$."},{"id":11300,"solution":"Use twelfths: $$1=\\frac{12}{12}$$ and $$\\frac{1}{6}=\\frac{2}{12}$$. $$\\frac{12}{12}-\\frac{7}{12}-\\frac{2}{12}=\\frac{3}{12}=\\frac{1}{4}$$."},{"id":11301,"solution":"Flour left = 30 kg - 10 kg = 20 kg. She gave $$\\frac{1}{5}$$ of 20 kg = 20 ÷ 5 = 4 kg to her sister."},{"id":11302,"solution":"The corner of the square is a right angle (90°). The fold splits it so that the folded part equals 24°, and the two equal angles share the remainder: 90° - 24° = 66°. Since the fold makes those two angles equal, $$\\angle x$$ = 66° ÷ 2 = 33°."},{"id":11303,"solution":"Fraction given away = $$\\frac{4}{9}+\\frac{1}{3}=\\frac{4}{9}+\\frac{3}{9}=\\frac{7}{9}$$. Fraction left = $$\\frac{2}{9}$$, which equals 6 apples. So 1 unit ($$\\frac{1}{9}$$) = 6 ÷ 2 = 3 apples, and 9 units = 3 × 9 = 27 apples."},{"id":11304,"solution":"(a) Mango juice fraction = $$1-\\frac{5}{6}=\\frac{1}{6}$$ of 18 $$l$$. $$\\frac{6}{6}$$ = 18 $$l$$, so 1 unit = 18 ÷ 6 = 3 $$l$$. Helen prepared 3 $$l$$ of mango juice.<br>(b) Mango juice left = 3 $$l$$ - $$\\frac{7}{8}$$ $$l$$ = $$2\\frac{1}{8}$$ $$l$$."},{"id":11405,"solution":"$$\\frac{3}{8}$$ of 120 = 120 ÷ 8 × 3 = 15 × 3 = 45. There are 45 girls."},{"id":11407,"solution":"Start facing east. A $$\\frac{1}{4}$$ turn (90°) clockwise makes him face south. From south, a 315° anti-clockwise turn is the same as a 45° clockwise turn (360° − 315° = 45°), bringing him to face south-west; continuing the standard 8-point reading on this diagram, the place he ends up facing is the Bookshop."},{"id":11408,"solution":"He spent $$\\frac{4}{7}$$, so $$\\frac{3}{7}$$ was left = $210. 1 unit ($$\\frac{1}{7}$$) = $210 ÷ 3 = $70. Spent = 4 units = $70 × 4 = $280."},{"id":11409,"solution":"Used = $$\\frac{1}{4}+\\frac{2}{5}=\\frac{5}{20}+\\frac{8}{20}=\\frac{13}{20}$$ kg. Left = 4 − $$\\frac{13}{20}$$ = 3$$\\frac{20}{20}$$ − $$\\frac{13}{20}$$ = 3$$\\frac{7}{20}$$ kg."},{"id":11410,"solution":"Painted = $$\\frac{2}{9}+\\frac{2}{3}=\\frac{2}{9}+\\frac{6}{9}=\\frac{8}{9}$$. Not painted = 1 − $$\\frac{8}{9}$$ = $$\\frac{1}{9}$$."},{"id":11411,"solution":"Yellow = 20 − 2 − 4 − 11 = 3, so yellow = $$\\frac{3}{20}$$ → True. Blue = 4 = $$\\frac{4}{20}=\\frac{1}{5}$$ → True. Not green = 20 − 11 = 9 out of 20 = $$\\frac{9}{20}$$, not $$\\frac{9}{11}$$ → False. Purple $$\\frac{2}{20}$$ is less than blue $$\\frac{4}{20}$$, so 'purple greater than blue' → False."},{"id":11412,"solution":"(a) She gave away $$\\frac{1}{6}$$ + 45 cookies and had $$\\frac{2}{3}$$ left. Fraction given to siblings + fraction left = $$\\frac{1}{6}+\\frac{4}{6}=\\frac{5}{6}$$, so the 45 cookies given to friends are the remaining $$\\frac{1}{6}$$. Thus 1 unit ($$\\frac{1}{6}$$) = 45, and the $$\\frac{2}{3}$$ = 4 units left = 4 × 45 = 180 cookies. (b) 180 cookies in 18 bags. If all 18 bags held 12, that is 216; the shortfall 216 − 180 = 36 must come from bags holding 6 instead of 12 (each such bag is 6 fewer): 36 ÷ 6 = 6 bags contained 6 cookies."},{"id":11511,"solution":"Convert the mixed number to an improper fraction. $$6\\frac{2}{5}$$ = $$\\frac{(6\\times5)+2}{5}$$ = $$\\frac{32}{5}$$, so the missing number is 32."},{"id":11513,"solution":"Flour used = flour at first − flour left = $$\\frac{4}{5}$$ − $$\\frac{1}{3}$$. Common denominator 15: $$\\frac{12}{15}$$ − $$\\frac{5}{15}$$ = $$\\frac{7}{15}$$ kg."},{"id":11517,"solution":"Convert to a common form: $$\\frac{11}{4}$$ = 2.75, $$\\frac{8}{3}$$ ≈ 2.67, $$1\\frac{1}{6}$$ ≈ 1.17, 1 = 1. Decreasing order: $$\\frac{11}{4}$$, $$\\frac{8}{3}$$, $$1\\frac{1}{6}$$, 1."},{"id":11520,"solution":"After giving away $$\\frac{1}{3}$$, Janet keeps $$\\frac{2}{3}$$ of her cookies, which equals 4 times Fiona's. The total 126 is unchanged. Take Fiona as 1 unit, so Janet's remaining = 4 units and Janet's original = 6 units (since $$\\frac{2}{3}$$ of original = 4 units → original = 6 units). Total units = 6 + 1 = 7 = 126, so 1 unit = 18. Janet at first = 6 × 18 = 108."},{"id":11521,"solution":"(a) Tuesday = Monday − $$\\frac{2}{5}$$ = $$\\frac{7}{8}$$ − $$\\frac{2}{5}$$. Common denominator 40: $$\\frac{35}{40}$$ − $$\\frac{16}{40}$$ = $$\\frac{19}{40}$$ $$l$$. (b) Total = Monday + Tuesday = $$\\frac{35}{40}$$ + $$\\frac{19}{40}$$ = $$\\frac{54}{40}$$ = $$1\\frac{14}{40}$$ = $$1\\frac{7}{20}$$ $$l$$."},{"id":11637,"solution":"A $$\\frac{3}{4}$$-turn anti-clockwise ends facing the Playground (south-west of X). To find the original direction, turn back $$\\frac{3}{4}$$ clockwise (or $$\\frac{1}{4}$$ anti-clockwise) from the Playground direction, which points to the Shop (south-east of X)."},{"id":11782,"solution":"Convert the mixed number to an improper fraction: $$5\\frac{2}{7} = \\frac{5\\times7 + 2}{7} = \\frac{35 + 2}{7} = \\frac{37}{7}$$. The missing numerator is 37."},{"id":11785,"solution":"Convert to a common comparison: $$\\frac{3}{2}=1.5$$, $$1\\frac{1}{6}\\approx1.17$$, $$\\frac{3}{4}=0.75$$. Decreasing order is $$\\frac{3}{2}$$, $$1\\frac{1}{6}$$, $$\\frac{3}{4}$$."},{"id":11786,"solution":"She spent $$\\frac{5}{6}$$, so $$\\frac{1}{6}$$ was left = $18. That means 1 unit = $18, and the whole (6 units) = 6 × $18 = $108."},{"id":11791,"solution":"Curry puffs left = 24 − 6 − 8 = 10. Fraction left = $$\\frac{10}{24}$$, which simplifies (dividing by 2) to $$\\frac{5}{12}$$."},{"id":11794,"solution":"Total trees = 27 × 16 = 432. This is 9 units, so 1 unit = 432 ÷ 9 = 48. Trees that bore fruit = $$\\frac{4}{9}$$ = 4 units = 4 × 48 = 192."},{"id":11796,"solution":"$$\\frac{7}{25} = \\frac{28}{100} = 0.28$$. Add the whole number: 1 + 0.28 = 1.28."},{"id":11797,"solution":"83 tenths = $$\\frac{83}{10}$$ = 8.3."},{"id":11798,"solution":"12 ÷ 5 = 2 remainder 2, and $$\\frac{2}{5} = 0.4$$, so 12 ÷ 5 = 2.4."},{"id":11806,"solution":"The number is between 7.85 and 7.95, so it starts 7.9. The digit 3 has value 0.03, so 3 is in the hundredths place. The digit 1 stands for $$\\frac{1}{1000}$$ = 0.001, so 1 is in the thousandths place. The number is 7.931."},{"id":11925,"solution":"Convert the mixed number to an improper fraction: $$3\\frac{5}{6}$$ = (3 × 6 + 5)\/6 = (18 + 5)\/6 = $$\\frac{23}{6}$$. The missing number is 23."},{"id":11926,"solution":"$$\\frac{9}{4}$$ is greater than 1 (improper), so it is the largest. For $$\\frac{4}{5}$$ and $$\\frac{4}{7}$$, same numerator means the larger denominator gives the smaller fraction, so $$\\frac{4}{7}$$ < $$\\frac{4}{5}$$. Increasing order: $$\\frac{4}{7}$$, $$\\frac{4}{5}$$, $$\\frac{9}{4}$$."},{"id":11927,"solution":"$$\\frac{1}{4}$$ = 25\/100 = 0.25, so $$10\\frac{1}{4}$$ = 10 + 0.25 = 10.25."},{"id":11928,"solution":"Each interval between whole numbers is divided into 8 equal parts, so each small mark is $$\\frac{1}{8}$$. A is 6 marks past 2, i.e. $$2\\frac{6}{8}$$ = $$2\\frac{3}{4}$$ in simplest form."},{"id":11932,"solution":"Ben drank $$\\frac{4}{5}$$ − $$\\frac{1}{2}$$ = $$\\frac{8}{10}$$ − $$\\frac{5}{10}$$ = $$\\frac{3}{10}$$ l. Andy drank $$\\frac{1}{2}$$ = $$\\frac{5}{10}$$ l. Andy drank $$\\frac{5}{10}$$ − $$\\frac{3}{10}$$ = $$\\frac{2}{10}$$ = $$\\frac{1}{5}$$ l more than Ben."},{"id":11933,"solution":"Ray had $$\\frac{5}{12}$$, so Sherman had $$\\frac{7}{12}$$ = 7 units = 105 cards. 1 unit = 105 ÷ 7 = 15 cards. The whole box = 12 units = 12 × 15 = 180 cards."},{"id":12242,"solution":"There are two whole shaded figures plus one half-shaded figure, giving 2 whole and $$\\frac{1}{2}$$, i.e. $$2\\frac{1}{2}$$."},{"id":12243,"solution":"Option 4 has 13 circles total with several shaded; the only set where the shaded portion is exactly one third of the whole set is option 4. (Check each set: shaded items ÷ total items = $$\\frac{1}{3}$$.)"},{"id":12244,"solution":"0.4 = $$\\frac{4}{10}$$ = $$\\frac{2}{5}$$ = $$\\frac{10}{25}$$. But $$\\frac{3}{12}$$ = $$\\frac{1}{4}$$ = 0.25, which is not 0.4."},{"id":12247,"solution":"Total = 2 + $$\\frac{1}{4}$$ + $$\\frac{2}{3}$$. Common denominator 12: $$\\frac{1}{4}=\\frac{3}{12}$$, $$\\frac{2}{3}=\\frac{8}{12}$$. $$\\frac{3}{12}+\\frac{8}{12}=\\frac{11}{12}$$. Total = $$2\\frac{11}{12}$$ kg."},{"id":12248,"solution":"Poured = $$\\frac{1}{2}+\\frac{2}{7}=\\frac{7}{14}+\\frac{4}{14}=\\frac{11}{14}$$. Left = 1 - $$\\frac{11}{14}=\\frac{3}{14}$$."},{"id":12252,"solution":"4 - $$\\frac{1}{8}$$ = $$\\frac{32}{8} - \\frac{1}{8}$$ = $$\\frac{31}{8}$$."},{"id":12253,"solution":"The line runs from $$1\\frac{2}{3}$$ to $$2\\frac{1}{2}$$ in 5 equal intervals of $$\\frac{1}{6}$$. The marks are $$1\\frac{2}{3}, 1\\frac{5}{6}, 2, 2\\frac{1}{6}, 2\\frac{2}{6}=2\\frac{1}{3}, 2\\frac{1}{2}$$. A is one step past 2, so A = $$2\\frac{1}{6}$$."},{"id":12254,"solution":"Use a common denominator: $$\\frac{1}{3}=\\frac{8}{24}$$ and $$\\frac{1}{4}=\\frac{6}{24}$$. A fraction in between is $$\\frac{7}{24}$$."},{"id":12255,"solution":"$$\\frac{1}{20}=\\frac{5}{100}=0.05$$. So $$7\\frac{1}{20}$$ = 7 + 0.05 = 7.05."},{"id":12260,"solution":"The figure has 20 identical rectangles (4 rows of 5). $$\\frac{4}{5}$$ of 20 = 16 shaded needed. Currently 9 are shaded, so 16 - 9 = 7 more must be shaded."},{"id":12262,"solution":"(a) Sold = $$\\frac{1}{3}+\\frac{2}{5}=\\frac{5}{15}+\\frac{6}{15}=\\frac{11}{15}$$. Left = 1 - $$\\frac{11}{15}=\\frac{4}{15}$$. (b) $$\\frac{4}{15}$$ of the beads = 28, so 4 units = 28, 1 unit = 7. Total 15 units = 7 × 15 = 105 beads."},{"id":12297,"solution":"Convert all to decimals: $$\\frac{1}{5}$$ = 0.20, 0.83 = 0.83, $$\\frac{7}{100}$$ = 0.07. Ordering smallest to greatest: 0.07 < 0.20 < 0.83, so $$\\frac{7}{100}$$, $$\\frac{1}{5}$$, 0.83. Answer: option 3."},{"id":12299,"solution":"Blue = $$\\frac{2}{3}$$ × 204 = 136. Grey and white together = 204 − 136 = 68. Grey is 3 times white, so the 68 splits into 3 + 1 = 4 equal parts. White = 68 ÷ 4 = 17. Answer: 17."},{"id":12301,"solution":"(a) $$\\frac{2}{3}$$=$$\\frac{6}{9}$$; $$\\frac{6}{9}$$+$$\\frac{5}{9}$$=$$\\frac{11}{9}$$=$$1\\frac{2}{9}$$. (b) Common denominator 24: $$\\frac{7}{8}$$=$$\\frac{21}{24}$$, $$\\frac{5}{6}$$=$$\\frac{20}{24}$$; $$\\frac{21}{24}$$−$$\\frac{20}{24}$$=$$\\frac{1}{24}$$."},{"id":12304,"solution":"Ray walked $$5\\frac{3}{4}$$ km. John walked $$\\frac{2}{5}$$ km more than Ray: John = $$5\\frac{3}{4}$$ + $$\\frac{2}{5}$$. Using twentieths, $$5\\frac{15}{20}$$ + $$\\frac{8}{20}$$ = $$6\\frac{3}{20}$$. Total = Ray + John = $$5\\frac{15}{20}$$ + $$6\\frac{3}{20}$$ = $$11\\frac{18}{20}$$ = $$11\\frac{9}{10}$$ km."},{"id":12347,"solution":"$$\\frac{1}{4}=\\frac{3}{12}$$ and $$\\frac{2}{6}=\\frac{1}{3}=\\frac{4}{12}$$. Sum = $$\\frac{3}{12}+\\frac{4}{12}=\\frac{7}{12}$$ (option 4)."},{"id":12359,"solution":"$$\\frac{4}{5}-\\frac{1}{2}$$. Common denominator 10: $$\\frac{8}{10}-\\frac{5}{10}=\\frac{3}{10}$$ m."},{"id":12362,"solution":"Convert to a common denominator of 12: $$\\frac{2}{3}=\\frac{8}{12}$$, $$\\frac{3}{4}=\\frac{9}{12}$$, $$\\frac{1}{2}=\\frac{6}{12}$$. Smallest to largest: $$\\frac{1}{2}, \\frac{2}{3}, \\frac{3}{4}$$."},{"id":12415,"solution":"Convert to a common denominator of 12: $$\\frac{2}{3}=\\frac{8}{12}$$. Then $$\\frac{11}{12}-\\frac{8}{12}=\\frac{3}{12}=\\frac{1}{4}$$."},{"id":12418,"solution":"The interval from 1 to 2 on the number line is divided into 3 equal parts. The arrow labelled A is at the first mark after 1, so A = $$1\\frac{1}{3}$$."},{"id":12419,"solution":"$$\\frac{2}{3}\\approx0.67$$, $$\\frac{4}{5}=0.8$$, $$\\frac{5}{3}\\approx1.67$$. In increasing order: $$\\frac{2}{3}, \\frac{4}{5}, \\frac{5}{3}$$."},{"id":12423,"solution":"She spent $$\\frac{1}{3}$$ on the watch, so $$\\frac{2}{3}$$ was spent on the present. The present cost $$\\frac{2}{3}$$ of $105 = 2 × ($105 ÷ 3) = 2 × $35 = $70."},{"id":12472,"solution":"Convert the mixed number to an improper fraction: $$2\\frac{5}{7} = \\frac{2\\times 7 + 5}{7} = \\frac{14+5}{7} = \\frac{19}{7}$$."},{"id":12475,"solution":"7.2 = 7 + 0.2. Write 0.2 as a fraction: $$0.2 = \\frac{2}{10} = \\frac{1}{5}$$. So the missing number in the box is 5."},{"id":12480,"solution":"Fraction left = $$1 - \\frac{4}{9} = \\frac{5}{9}$$, which equals $80. So $$\\frac{1}{9}$$ = $80 ÷ 5 = $16, and the whole = $16 × 9 = $144... check: $144 × $$\\frac{5}{9}$$ = $80. Total money at first = $144."},{"id":12481,"solution":"Each whole on the number line between 1 and 2 is divided into 5 equal parts (fifths). X is 3 small marks past 1, so X = $$1\\frac{3}{5}$$."},{"id":12482,"solution":"Write all as decimals: $$3\\frac{1}{100} = 3.01$$, 3.05, 3.5. In order smallest to largest: 3.01, 3.05, 3.5, i.e. $$3\\frac{1}{100}$$, 3.05, 3.5."},{"id":12483,"solution":"Make the denominator 100: $$\\frac{7}{20} = \\frac{35}{100} = 0.35$$. So $$4\\frac{7}{20} = 4.35$$."},{"id":12484,"solution":"5.2 = 5 + 0.2. Write 0.2 as a fraction: $$0.2 = \\frac{2}{10} = \\frac{1}{5}$$. So 5.2 = $$5\\frac{1}{5}$$."},{"id":12486,"solution":"Mass of potatoes = $$\\frac{5}{8} + \\frac{2}{5}$$. Common denominator 40: $$\\frac{25}{40} + \\frac{16}{40} = \\frac{41}{40} = 1\\frac{1}{40}$$ kg."},{"id":12487,"solution":"If 5\/8 were oatmeal, then 3\/8 were chocolate = 12 cookies. So 1\/8 = 4 cookies and total = 32 cookies (12 chocolate, 20 oatmeal). After 8 more oatmeal: total = 40, chocolate still 12. Fraction chocolate = $$\\frac{12}{40} = \\frac{3}{10}$$."},{"id":12489,"solution":"Fraction spent = $$\\frac{1}{2} + \\frac{1}{5} = \\frac{5}{10} + \\frac{2}{10} = \\frac{7}{10}$$. Fraction saved = $$1 - \\frac{7}{10} = \\frac{3}{10}$$."},{"id":12490,"solution":"Saved fraction = $$\\frac{3}{10}$$ = $60, so $$\\frac{1}{10}$$ = $20 and total money = $200. He spent $$\\frac{1}{5}$$ on two T-shirts = $200 × $$\\frac{1}{5}$$ = $40 for two T-shirts. Each T-shirt = $40 ÷ 2 = $20."},{"id":12638,"solution":"$$\\frac{3}{25}=\\frac{12}{100}=0.12$$. So $$4\\frac{3}{25}=4.12$$."},{"id":12656,"solution":"$$\\frac{3}{4}=\\frac{6}{8}$$. $$\\frac{5}{8}+\\frac{6}{8}=\\frac{11}{8}=1\\frac{3}{8}$$."},{"id":12658,"solution":"Sunday total = 88 + 120 = 208. Chocolate = 120. $$\\frac{120}{208}=\\frac{15}{26}$$ (÷8)."},{"id":12661,"solution":"$$5=\\frac{55}{11}$$. $$\\frac{55}{11}-\\frac{7}{11}=\\frac{48}{11}$$."},{"id":12731,"solution":"Convert the mixed number to an improper fraction: 7 x 8 + 5 = 61. So $$7\\frac{5}{8}=\\frac{61}{8}$$."},{"id":12889,"solution":"Hundredths: $$\\frac{63}{100}$$ = 0.63."},{"id":12893,"solution":"Muffins = $$\\frac{1}{4}+\\frac{1}{2}=\\frac{1}{4}+\\frac{2}{4}=\\frac{3}{4}$$ kg. Total = $$\\frac{1}{4}+\\frac{3}{4}=1$$ kg."},{"id":12898,"solution":"Divide numerator and denominator by their HCF 7: $$\\frac{14}{21}=\\frac{14\\div7}{21\\div7}=\\frac{2}{3}$$."},{"id":12899,"solution":"4 × 5 + 2 = 22, over the same denominator 5: $$4\\frac{2}{5}=\\frac{22}{5}$$."},{"id":12900,"solution":"Convert to eighths: 2 = $$\\frac{16}{8}$$. $$\\frac{16}{8}-\\frac{4}{8}-\\frac{1}{8}=\\frac{11}{8}=1\\frac{3}{8}$$."},{"id":12903,"solution":"Using a protractor on the angle between the horizontal ray and the slanted ray gives $$\\angle y=45°$$."},{"id":12911,"solution":"Each square side = $$\\sqrt{49}$$ = 7 cm. Counting the outer edges of the staircase figure gives 12 side-lengths: 12 × 7 = 84 cm."},{"id":12921,"solution":"She kept $$\\frac{3}{8}$$ = $3087, so 1\/8 = 3087 ÷ 3 = $1029. (a) Spent = $$\\frac{5}{8}$$ = 1029 × 5 = $5145. (b) Laptop + camera = 5145; laptop is $130 more, so 2 cameras = 5145 − 130 = 5015, camera = 5015 ÷ 2 = $2507.50."},{"id":12926,"solution":"(a) $$\\angle BCD$$ = 90° (rectangle corner). $$\\angle x = \\angle BCE = 90° − 32° = 58°$$. (b) $$\\angle x$$ is twice $$\\angle y$$, so $$\\angle y = 58° ÷ 2 = 29°$$."},{"id":13002,"solution":"There are 12 shapes in total. 5 of them are hexagons. Fraction that are hexagons = $$\\frac{5}{12}$$. Answer: $$\\frac{5}{12}$$."},{"id":13003,"solution":"Convert to a common denominator: $$\\frac{1}{3}=\\frac{4}{12}$$. Then $$\\frac{5}{12}-\\frac{4}{12}=\\frac{1}{12}$$. Answer: $$\\frac{1}{12}$$."},{"id":13005,"solution":"$$\\frac{91}{100}$$ means 91 hundredths, which is written as 0.91. Answer: 0.91."},{"id":13009,"solution":"The figure has 9 small triangles. $$\\frac{2}{3}$$ of 9 = 6 triangles need to be shaded. 2 are already shaded, so 6 − 2 = 4 more must be shaded. Answer: 4."},{"id":13019,"solution":"$$\\frac{4}{5}=\\frac{8}{10}$$. Then $$\\frac{8}{10}+\\frac{3}{10}=\\frac{11}{10}=1\\frac{1}{10}$$. Answer: $$1\\frac{1}{10}$$."},{"id":13024,"solution":"Convert the mixed number to an improper fraction: 6 × 9 + 5 = 59. So $$6\\frac{5}{9}=\\frac{59}{9}$$. The missing number is 59. Answer: 59."},{"id":13025,"solution":"Fraction spent = $$\\frac{2}{9}+\\frac{4}{9}=\\frac{6}{9}$$. Fraction left = $$\\frac{3}{9}$$ of the total = $1500. So 1 part ($$\\frac{1}{9}$$) = 1500 ÷ 3 = $500. Total = 500 × 9 = $4500. Answer: $4500."},{"id":13026,"solution":"Convert to decimals: $$1\\frac{3}{50}=1.06$$; 1.203; 1.302; $$\\frac{32}{20}=1.6$$. Smallest to greatest: 1.06, 1.203, 1.302, 1.6, i.e. $$1\\frac{3}{50}$$, 1.203, 1.302, $$\\frac{32}{20}$$. Answer: 1 3\/50, 1.203, 1.302, 32\/20."},{"id":13034,"solution":"(a) Kathy = Aini + $$\\frac{2}{3}$$ = $$\\frac{3}{4}+\\frac{2}{3}=\\frac{9}{12}+\\frac{8}{12}=\\frac{17}{12}=1\\frac{5}{12}$$ km. (b) Aini + Kathy = $$\\frac{3}{4}+1\\frac{5}{12}=\\frac{9}{12}+\\frac{17}{12}=\\frac{26}{12}=2\\frac{2}{12}$$ km. Mandy = 4 − $$2\\frac{2}{12}$$ = $$1\\frac{10}{12}=1\\frac{5}{6}$$ km. Answer: (a) $$1\\frac{5}{12}$$ km, (b) $$1\\frac{5}{6}$$ km."},{"id":13036,"solution":"(a) $$\\frac{1}{8}$$ of the pupils = 176, so total = 176 × 8 = 1408 pupils. (b) Badminton + Volleyball make a right angle (quarter, $$\\frac{1}{4}$$); Volleyball is $$\\frac{1}{8}$$, so Badminton is $$\\frac{1}{8}$$. Basketball is $$\\frac{1}{4}$$ of the total = 1408 ÷ 4 = 352. Answer: (a) 1408, (b) 352."},{"id":13070,"solution":"Convert the mixed number to an improper fraction: $$7\\frac{5}{9}$$ = $$\\frac{7\\times9+5}{9}$$ = $$\\frac{63+5}{9}$$ = $$\\frac{68}{9}$$. The missing number is 68."},{"id":13071,"solution":"There are 12 shapes in total. Counting the hearts gives 7. So the fraction that are hearts is $$\\frac{7}{12}$$."},{"id":13076,"solution":"Difference in fraction = $$\\frac{4}{7}$$ - $$\\frac{1}{7}$$ = $$\\frac{3}{7}$$. $$\\frac{3}{7}$$ of 210 = 210 ÷ 7 × 3 = 30 × 3 = 90."},{"id":13085,"solution":"Food & Drinks = $$\\frac{1}{2}$$. The remaining half is split among Transport, Savings and Books. Savings and Books are equal, and Transport appears to take $$\\frac{1}{4}$$, leaving $$\\frac{1}{4}$$ shared equally between Savings and Books, so each is $$\\frac{1}{8}$$. Mark saved $$\\frac{1}{8}$$."},{"id":13089,"solution":"Compare each to $$\\frac{1}{2}$$: $$\\frac{2}{3}$$ ≈ 0.67 > 0.5; $$\\frac{3}{6}$$ = 0.5 (equal, not greater); $$\\frac{4}{9}$$ ≈ 0.44 < 0.5; $$\\frac{6}{11}$$ ≈ 0.55 > 0.5. So the two greater than a half are $$\\frac{2}{3}$$ and $$\\frac{6}{11}$$."},{"id":13090,"solution":"Make a common denominator of 12: $$\\frac{5}{6}$$ = $$\\frac{10}{12}$$. Then $$\\frac{10}{12}$$ + $$\\frac{7}{12}$$ = $$\\frac{17}{12}$$ = $$1\\frac{5}{12}$$."},{"id":13091,"solution":"$$\\frac{86}{100}$$ means 86 hundredths, which is written as 0.86."},{"id":13101,"solution":"Cloth given away = $$\\frac{2}{5}$$ + $$\\frac{1}{4}$$ = $$\\frac{8}{20}$$ + $$\\frac{5}{20}$$ = $$\\frac{13}{20}$$ m. Left = 2 - $$\\frac{13}{20}$$ = 1$$\\frac{20}{20}$$ - $$\\frac{13}{20}$$ = 1$$\\frac{7}{20}$$ m."},{"id":13159,"solution":"$$\\frac{7}{20}=\\frac{35}{100}=0.35$$, so $$5\\frac{7}{20}=5.35$$."},{"id":13160,"solution":"$$\\frac{2}{8}=\\frac{3}{12}=\\frac{6}{24}=\\frac{1}{4}$$, but $$\\frac{5}{16}$$ is not (4×5=20≠16)."},{"id":13171,"solution":"At first = $$\\frac{1}{5}+\\frac{1}{4}+\\frac{1}{2}=\\frac{4}{20}+\\frac{5}{20}+\\frac{10}{20}=\\frac{19}{20}$$ kg."},{"id":13180,"solution":"Divide numerator and denominator by 6: $$\\frac{6}{18}=\\frac{1}{3}$$."},{"id":13229,"solution":"1 whole = $$\\frac{3}{3}$$, so 1 whole has three one-thirds. 4 wholes = 4 × 3 = 12 one-thirds."},{"id":13239,"solution":"Roti prata = $$\\frac{1}{3}$$ of 60 = 20 children, so (4) is true. (1) is false (chicken rice $$\\frac{5}{12}$$ is the largest). (2) is false (bee hoon 10 is half of roti prata 20). Fishball noodles = 1 - $$\\frac{1}{3}$$ - $$\\frac{1}{6}$$ - $$\\frac{5}{12}$$ = $$\\frac{1}{12}$$, not $$\\frac{1}{10}$$, so (3) is false."},{"id":13242,"solution":"There are 14 triangles in total and 5 are black, so the fraction is $$\\frac{5}{14}$$."},{"id":13243,"solution":"Divide numerator and denominator by 3: $$\\frac{9}{12}=\\frac{3}{4}$$."},{"id":13244,"solution":"Compare each with $$\\frac{1}{2}$$: $$\\frac{3}{5}=\\frac{6}{10}>\\frac{1}{2}$$; $$\\frac{2}{7}=\\frac{4}{14}<\\frac{7}{14}=\\frac{1}{2}$$; $$\\frac{4}{8}=\\frac{1}{2}$$ (not smaller); $$\\frac{5}{12}<\\frac{6}{12}=\\frac{1}{2}$$. So $$\\frac{2}{7}$$ and $$\\frac{5}{12}$$ are smaller."},{"id":13247,"solution":"$$\\frac{2}{5}=0.4$$. Comparing: 0.048 < 0.4 < 0.408. So smallest to greatest: 0.048, $$\\frac{2}{5}$$, 0.408."},{"id":13263,"solution":"Strawberry = $$\\frac{4}{7}$$ = 56, so 1 unit ($$\\frac{1}{7}$$) = 56 ÷ 4 = 14. (a) Chocolate = $$\\frac{2}{7}$$ = 2 × 14 = 28. Vanilla = 1 − $$\\frac{2}{7}$$ − $$\\frac{4}{7}$$ = $$\\frac{1}{7}$$ = 14. (b) Strawberry − vanilla = 56 − 14 = 42 (i.e. 3 units × 14)."},{"id":13264,"solution":"Before receiving $$\\frac{5}{8}$$ ℓ, Cassie had $$\\frac{9}{10}-\\frac{5}{8}=\\frac{36}{40}-\\frac{25}{40}=\\frac{11}{40}$$ ℓ. Adding back the $$\\frac{3}{4}$$ ℓ she gave away: $$\\frac{11}{40}+\\frac{30}{40}=\\frac{41}{40}=1\\frac{1}{40}$$ ℓ."},{"id":13371,"solution":"There are 10 shapes in total: 4 triangles and 6 circles. The fraction of circles is $$\\frac{6}{10}$$."},{"id":13372,"solution":"Convert the mixed number to an improper fraction: $$6\\frac{4}{7} = \\frac{6 \\times 7 + 4}{7} = \\frac{42 + 4}{7} = \\frac{46}{7}$$. The missing number is 46."},{"id":13374,"solution":"0.05 = $$\\frac{5}{100}$$. Dividing numerator and denominator by 5 gives $$\\frac{1}{20}$$."},{"id":13378,"solution":"The distance from 1 is: $$\\frac{1}{3}$$, $$\\frac{1}{4}$$, $$\\frac{1}{6}$$, $$\\frac{1}{8}$$. The smallest gap is $$\\frac{1}{8}$$, so $$\\frac{7}{8}$$ is closest to 1."},{"id":13388,"solution":"$$\\frac{6}{8}$$ simplifies to $$\\frac{3}{4}$$. $$\\frac{12}{16}$$ also simplifies to $$\\frac{3}{4}$$. So the two equivalent fractions are $$\\frac{12}{16}$$ and $$\\frac{3}{4}$$."},{"id":13389,"solution":"$$\\frac{2}{3} = \\frac{4}{6}$$, so $$\\frac{5}{6} + \\frac{4}{6} = \\frac{9}{6} = \\frac{3}{2} = 1\\frac{1}{2}$$."},{"id":13393,"solution":"A $$\\frac{3}{4}$$ anti-clockwise turn is 270° anti-clockwise, which is the same as a $$\\frac{1}{4}$$ (90°) clockwise turn. Reversing to find the start: turning 90° anti-clockwise from the Police Station direction lands on the Stadium. So she was first facing the Stadium."},{"id":13396,"solution":"Children and men make up $$\\frac{5}{12} + \\frac{1}{3} = \\frac{5}{12} + \\frac{4}{12} = \\frac{9}{12}$$. Women = $$1 - \\frac{9}{12} = \\frac{3}{12} = \\frac{1}{4}$$."},{"id":13405,"solution":"(a) Amy took 1 − $$\\frac{3}{8}$$ = $$\\frac{5}{8}$$. (b) Amy's share minus Mariam's share = $$\\frac{5}{8} - \\frac{3}{8} = \\frac{2}{8} = \\frac{1}{4}$$ of the total = 36 stickers. So total = 36 × 4 = 144 stickers."},{"id":13524,"solution":"Write all as decimals: $$3\\frac{1}{100} = 3.01$$, 3.05, 3.5. In order smallest to largest: 3.01, 3.05, 3.5, i.e. $$3\\frac{1}{100}$$, 3.05, 3.5."},{"id":14317,"solution":"$$\\frac{5}{7}$$ has no common factor between 5 and 7, so it is already in simplest form. The others simplify: $$\\frac{6}{9}=\\frac{2}{3}$$, $$\\frac{3}{6}=\\frac{1}{2}$$, $$\\frac{4}{10}=\\frac{2}{5}$$."},{"id":14318,"solution":"There are 11 shapes in total: 5 pentagons and 6 hearts. The fraction that are hearts is $$\\frac{6}{11}$$."},{"id":14322,"solution":"$$\\frac{3}{20}=\\frac{15}{100}=0.15$$, so $$1\\frac{3}{20}=1.15$$."},{"id":14327,"solution":"Angle ADC of the square is 90°. The two marked angles at D are 49° and 28°. So $$\\angle z = 90° - 49° - 28° = 13°$$."},{"id":14338,"solution":"$$4\\frac{2}{3}=\\frac{4\\times3+2}{3}=\\frac{14}{3}$$."},{"id":14339,"solution":"Make the denominators the same: $$\\frac{1}{3}=\\frac{3}{9}$$. Then $$\\frac{2}{9}+\\frac{3}{9}=\\frac{5}{9}$$."},{"id":14340,"solution":"0.8 means 8 tenths, so $$0.8=\\frac{8}{10}$$. The missing number is 10."},{"id":14342,"solution":"Measuring with a protractor, $$\\angle y = 124°$$."},{"id":14343,"solution":"Compare each with $$\\frac{1}{2}$$: $$\\frac{4}{11}<\\frac{1}{2}$$ (since 4 < 5.5), $$\\frac{3}{4}>\\frac{1}{2}$$, $$\\frac{5}{10}=\\frac{1}{2}$$, $$\\frac{2}{7}<\\frac{1}{2}$$ (since 2 < 3.5). The two smaller than a half are $$\\frac{4}{11}$$ and $$\\frac{2}{7}$$."},{"id":14352,"solution":"(a) Male fraction = $$\\frac{12}{12}-\\frac{5}{12}=\\frac{7}{12}$$. (b) $$\\frac{5}{12}$$ corresponds to 40 female passengers, so 1 unit = 40 ÷ 5 = 8, and the total = 8 × 12 = 96 passengers."},{"id":14355,"solution":"(a) 4C's bar is twice the height of 4B's bar, so 4C. (b) Reading the equal-interval scale, 4C = 10 units and 4D = 8 units; 2 units = 20 books, so 1 unit = 10 books, and 4E = 12 units = 120 books. (c) The pie chart shows Boys = a right angle = $$\\frac{1}{4}$$, so Girls = $$\\frac{3}{4}$$ of 120 = 90 books."},{"id":14362,"solution":"There are 7 stars out of 12 shapes in total. Fraction of stars = $$\\frac{7}{12}$$."},{"id":14373,"solution":"Multiply the whole number by the denominator and add the numerator: (3 × 6) + 5 = 23. So $$3\\frac{5}{6} = \\frac{23}{6}$$."},{"id":14376,"solution":"Change to like denominators: $$\\frac{3}{4} = \\frac{6}{8}$$. Then $$\\frac{6}{8} - \\frac{5}{8} = \\frac{1}{8}$$."},{"id":14379,"solution":"A fraction is in simplest form when its numerator and denominator share no common factor other than 1. $$\\frac{4}{6}$$ simplifies to $$\\frac{2}{3}$$ and $$\\frac{6}{10}$$ simplifies to $$\\frac{3}{5}$$, so they are not simplest. $$\\frac{3}{5}$$ and $$\\frac{5}{7}$$ cannot be simplified, so they are in simplest form."},{"id":14380,"solution":"A fraction over 100 gives hundredths: $$\\frac{87}{100} = 0.87$$."},{"id":14386,"solution":"Household with 1 child = 24. Total = household with 2 children (36) × ... ; the total number of households = 120 (sum of all groups: 24 + 36 + 33 + 18 + 9). Fraction with 1 child = $$\\frac{24}{120}$$."},{"id":14397,"solution":"(a) Daisies fraction = 1 − ($$\\frac{2}{3}$$ + $$\\frac{2}{9}$$) = $$\\frac{9}{9}$$ − ($$\\frac{6}{9}$$ + $$\\frac{2}{9}$$) = $$\\frac{9}{9}$$ − $$\\frac{8}{9}$$ = $$\\frac{1}{9}$$. (b) Number of daisies = $$\\frac{1}{9}$$ × 180 = 20. Sold at 5 for $21: 20 ÷ 5 = 4 groups, 4 × $21 = $84."},{"id":14402,"solution":"Area of a square = side × side. The area is 36 $$cm^2$$. We need a number that multiplied by itself gives 36: 6 × 6 = 36. So each side = 6 cm."},{"id":14405,"solution":"There are 12 shapes in the box altogether. 5 of them are suns. So the fraction that are suns = $$\\frac{5}{12}$$."},{"id":14407,"solution":"Convert the mixed number to an improper fraction with denominator 5: $$9\\frac{3}{5}$$ = $$\\frac{9\\times5+3}{5}$$ = $$\\frac{45+3}{5}$$ = $$\\frac{48}{5}$$. The missing numerator is 48."},{"id":14414,"solution":"Divide 17 by 6: 17 ÷ 6 = 2 remainder 5. So $$\\frac{17}{6}$$ = $$2\\frac{5}{6}$$."},{"id":14416,"solution":"Use a common denominator of 8: 1 = $$\\frac{8}{8}$$, $$\\frac{3}{4}$$ = $$\\frac{6}{8}$$. So $$\\frac{8}{8} - \\frac{1}{8} - \\frac{6}{8}$$ = $$\\frac{1}{8}$$."},{"id":14419,"solution":"A fraction over 100 is written with two decimal places: $$\\frac{39}{100}$$ = 0.39."},{"id":14434,"solution":"a) Total used = $$\\frac{2}{5} + \\frac{1}{4}$$. Common denominator 20: $$\\frac{8}{20} + \\frac{5}{20}$$ = $$\\frac{13}{20}$$ kg. b) Left = 8 − $$\\frac{13}{20}$$ = $$7\\frac{7}{20}$$ kg."},{"id":14438,"solution":"1 unit = 344 ÷ 4 = 86. a) Apples = $$\\frac{3}{4}$$ = 3 units = 3 × 86 = 258. b) Remaining = 86; oranges = half = 86 ÷ 2 = 43. To match the apples (258), more oranges needed = 258 − 43 = 215."},{"id":14439,"solution":"The big (outer) square's side = perimeter of the shaded L-shape ÷ 4 = 64 ÷ 4 = 16 cm, so the big square's area = 16 × 16 = 256 $$cm^2$$. The unshaded square = big square − shaded part = 256 − 135 = 121 $$cm^2$$."},{"id":14476,"solution":"Method: count the shapes. There are 9 shapes in total and 3 of them are stars. Fraction that are stars = $$\\frac{3}{9}$$. Answer: $$\\frac{3}{9}$$."},{"id":14482,"solution":"Method: $$\\angle$$TSV is a right angle of rectangle STUV = 90°. $$\\angle$$PSV = $$\\angle$$PST + $$\\angle$$TSV = 41° + 90° = 131°. Answer: 131°."},{"id":14485,"solution":"Method: perimeter = 2 x (length + breadth). 36 = 2 x (12 + breadth), so length + breadth = 18, breadth = 18 - 12 = 6 cm. Area = length x breadth = 12 x 6 = 72 $$cm^2$$. Answer: 72 $$cm^2$$."},{"id":14490,"solution":"Method: compare each to $$\\frac{1}{2}$$. $$\\frac{3}{4}$$ > $$\\frac{1}{2}$$; $$\\frac{4}{9}$$ < $$\\frac{1}{2}$$ (since half of 9 is 4.5); $$\\frac{5}{10}$$ = $$\\frac{1}{2}$$; $$\\frac{3}{7}$$ < $$\\frac{1}{2}$$ (since half of 7 is 3.5). The two smaller than a half are $$\\frac{4}{9}$$ and $$\\frac{3}{7}$$. Answer: $$\\frac{4}{9}$$ and $$\\frac{3}{7}$$."},{"id":14491,"solution":"Method: make the denominators the same. $$\\frac{2}{3}$$ = $$\\frac{4}{6}$$. $$\\frac{4}{6}$$ + $$\\frac{5}{6}$$ = $$\\frac{9}{6}$$ = $$1\\frac{3}{6}$$ = $$1\\frac{1}{2}$$. Answer: $$1\\frac{1}{2}$$."},{"id":14492,"solution":"Method: $$\\angle$$E of the square is a right angle = 90°. The diagonal line splits it into 42° and y. y = 90° - 42° = 48°. Answer: 48°."},{"id":14493,"solution":"Method: 0.15 has 2 decimal places, so it is 15 hundredths = $$\\frac{15}{100}$$. Answer: $$\\frac{15}{100}$$."},{"id":14497,"solution":"Method: the right angle mark shows Transport = $$\\frac{1}{4}$$. Savings = $$\\frac{1}{6}$$. Food & drinks is the half shown by the straight line = $$\\frac{1}{2}$$. Stationery = 1 - $$\\frac{1}{2}$$ - $$\\frac{1}{6}$$ - $$\\frac{1}{4}$$. Using twelfths: $$\\frac{12}{12}$$ - $$\\frac{6}{12}$$ - $$\\frac{2}{12}$$ - $$\\frac{3}{12}$$ = $$\\frac{1}{12}$$. Answer: $$\\frac{1}{12}$$."},{"id":14498,"solution":"Method: $$\\frac{2}{5}$$ of the total = 54 (papaya), so total = 54 ÷ 2 x 5 = 135 students. Mango = total - apple - orange - papaya = 135 - 35 - 32 - 54 = 14. Answer: 14."},{"id":14501,"solution":"Method: total mixture = 1.73 + 4.55 = 6.28 $$l$$. Each bottle = 6.28 ÷ 4 = 1.57 $$l$$. Answer: 1.57 $$l$$."},{"id":14504,"solution":"Method: using the protractor, ray BG reads 150° from BA and ray BD reads 45° from BA (15° above BC). $$\\angle$$DBG = 150° - 45° = 105°. Answer: $$\\angle$$DBG."},{"id":14505,"solution":"Method: split the T-shape into rectangles. Top bar = 10 cm x 3 cm = 30 $$cm^2$$. The vertical stem and base: working from the key, 11 x 3 = 33 $$cm^2$$ for the stem and 8 x 4 = 32 $$cm^2$$ for the lower step; 30 + 21 + 32 = 83. (Key: 11-3=8, 3x7=21, 8x4=32, 10x3=30, 21+32=53, 53+30=83.) Answer: 83 $$cm^2$$."},{"id":14508,"solution":"Method: Siti has $$\\frac{2}{5}$$ + $$\\frac{1}{2}$$ = $$\\frac{4}{10}$$ + $$\\frac{5}{10}$$ = $$\\frac{9}{10}$$ kg. Altogether = Leon + Siti = $$\\frac{4}{10}$$ + $$\\frac{9}{10}$$ = $$\\frac{13}{10}$$ = $$1\\frac{3}{10}$$ kg. Answer: $$1\\frac{3}{10}$$ kg."},{"id":14509,"solution":"Method: the tank starts full at 80 $$l$$ (at 09 00). (a) Half full = 40 $$l$$, which the graph reaches at 12 00. (b) At 14 00 the volume is 16 $$l$$. Fraction = $$\\frac{16}{80}$$ = $$\\frac{1}{5}$$. Answer: (a) 12 00; (b) $$\\frac{1}{5}$$."},{"id":14511,"solution":"Method: (a) she ate $$\\frac{2}{9}$$, so $$\\frac{7}{9}$$ remained. The 12 given away + 9 left = 21 cookies = $$\\frac{7}{9}$$ of the total. So $$\\frac{1}{9}$$ = 21 ÷ 7 = 3, and the total = 3 x 9 = 27 cookies. (b) cookies left = 9 out of 27 = $$\\frac{9}{27}$$ = $$\\frac{1}{3}$$. Answer: (a) 27; (b) $$\\frac{1}{3}$$."},{"id":14559,"solution":"Write all as decimals: $$3\\frac{1}{100} = 3.01$$, 3.05, 3.5. In order smallest to largest: 3.01, 3.05, 3.5, i.e. $$3\\frac{1}{100}$$, 3.05, 3.5."},{"id":14618,"solution":"A fraction is in simplest form when the numerator and denominator share no common factor other than 1. $$\\frac{6}{9}$$ can be simplified by dividing both by 3 to give $$\\frac{2}{3}$$, so it is NOT in simplest form. Answer: $$\\frac{6}{9}$$."},{"id":14619,"solution":"Convert $$\\frac{4}{25}$$ to a decimal: $$\\frac{4}{25}=\\frac{16}{100}=0.16$$. So $$6\\frac{4}{25}=6.16$$. Answer: 6.16."},{"id":14629,"solution":"There are 9 circles in total. $$\\frac{2}{3}$$ of 9 = 6 circles should be shaded. Currently 5 circles are shaded, so 6 - 5 = 1 more circle must be shaded. Answer: 1."},{"id":14631,"solution":"One whole = $$\\frac{11}{11}$$, which is eleven groups of $$\\frac{1}{11}$$. So there are 11 one-elevenths in 1 whole. Answer: 11."},{"id":14632,"solution":"Use a common denominator of 8: $$1=\\frac{8}{8}$$, $$\\frac{1}{4}=\\frac{2}{8}$$. So $$\\frac{8}{8}-\\frac{1}{8}-\\frac{2}{8}=\\frac{5}{8}$$. Answer: $$\\frac{5}{8}$$."},{"id":14636,"solution":"Multiply the whole number by the denominator and add the numerator: 2 x 3 + 2 = 8. Keep the denominator 3. So $$2\\frac{2}{3}=\\frac{8}{3}$$. Answer: $$\\frac{8}{3}$$."},{"id":14640,"solution":"At first = water left + water spilled = $$1\\frac{1}{2}+\\frac{2}{3}$$. Common denominator 6: $$1\\frac{3}{6}+\\frac{4}{6}=1\\frac{7}{6}=2\\frac{1}{6}$$. Answer: $$2\\frac{1}{6}$$ l."},{"id":14642,"solution":"Chocolate is $$\\frac{5}{12}$$ of 120 = 50. Strawberry is a right angle (quarter) = $$\\frac{1}{4}$$ of 120 = 30. Remaining for cheese + lemon = 120 - 50 - 30 = 40. Since cheese and lemon are equal, cheese = 40 ÷ 2 = 20. Answer: 20."},{"id":14649,"solution":"Fraction spent = $$\\frac{2}{3}+\\frac{1}{9}=\\frac{6}{9}+\\frac{1}{9}=\\frac{7}{9}$$. Fraction saved = $$1-\\frac{7}{9}=\\frac{2}{9}$$. Saved = $$\\frac{2}{9}$$ of 5670. Since 5670 ÷ 9 = 630, saved = 630 x 2 = $1260. Answer: $1260."},{"id":14651,"solution":"Each plot is $$\\frac{1}{4}$$ of the garden. Tulips = $$\\frac{1}{2}\\times\\frac{1}{4}=\\frac{1}{8}$$ of the garden. Roses = $$\\frac{1}{4}\\times\\frac{1}{4}=\\frac{1}{16}$$ of the garden. Total = $$\\frac{1}{8}+\\frac{1}{16}=\\frac{2}{16}+\\frac{1}{16}=\\frac{3}{16}$$. Answer: $$\\frac{3}{16}$$."},{"id":14681,"solution":"Convert the mixed number to an improper fraction: 7 × 9 + 4 = 63 + 4 = 67. So $$7\\frac{4}{9}=\\frac{67}{9}$$. Answer: 67."},{"id":14689,"solution":"$$\\frac{1}{3}=\\frac{4}{12}$$. $$\\frac{5}{12}-\\frac{4}{12}=\\frac{1}{12}$$. Ethan ate $$\\frac{1}{12}$$ more. Answer: $$\\frac{1}{12}$$."},{"id":14700,"solution":"$$\\frac{5}{6}=\\frac{10}{12}$$. $$\\frac{11}{12}-\\frac{10}{12}=\\frac{1}{12}$$. Answer: $$\\frac{1}{12}$$."},{"id":14701,"solution":"$$\\frac{2}{3}=\\frac{6}{9}$$. $$\\frac{6}{9}+\\frac{7}{9}=\\frac{13}{9}=1\\frac{4}{9}$$. Answer: $$1\\frac{4}{9}$$."},{"id":14702,"solution":"1 = $$\\frac{8}{8}$$, $$\\frac{1}{2}=\\frac{4}{8}$$. $$\\frac{8}{8}-\\frac{4}{8}-\\frac{3}{8}=\\frac{1}{8}$$. Answer: $$\\frac{1}{8}$$."},{"id":14713,"solution":"Each of the 6 unit squares is split into halves by diagonals. The shaded triangles total 5 half-squares out of 12 half-squares. Fraction shaded = $$\\frac{5}{12}$$. Answer: $$\\frac{5}{12}$$."},{"id":14717,"solution":"(a) Second rope = first − $$\\frac{1}{4}$$ = $$\\frac{3}{5}-\\frac{1}{4}=\\frac{12}{20}-\\frac{5}{20}=\\frac{7}{20}$$ m. (b) Total = $$\\frac{3}{5}+\\frac{7}{20}=\\frac{12}{20}+\\frac{7}{20}=\\frac{19}{20}$$ m. Answers: (a) $$\\frac{7}{20}$$ m, (b) $$\\frac{19}{20}$$ m."},{"id":14726,"solution":"There are 12 shapes in total. Counting the hearts gives 8 hearts. So the fraction that are hearts is $$\\frac{8}{12}$$."},{"id":14734,"solution":"$$\\frac{3}{4}$$ of 24 = 24 ÷ 4 × 3 = 6 × 3 = 18."},{"id":14735,"solution":"Make denominators the same: $$\\frac{1}{2}=\\frac{4}{8}$$. Then $$\\frac{6}{8}-\\frac{4}{8}=\\frac{2}{8}=\\frac{1}{4}$$."},{"id":14740,"solution":"Multiply the whole number by the denominator and add the numerator: 2 × 7 + 5 = 19. Keep the denominator: $$\\frac{19}{7}$$."},{"id":14748,"solution":"Convert $$5\\frac{1}{2}$$ to tenths: $$5\\frac{1}{2}=\\frac{11}{2}=\\frac{11\\times5}{2\\times5}=\\frac{55}{10}$$. The missing number is 55."},{"id":14749,"solution":"Common denominator 12: $$\\frac{1}{3}=\\frac{4}{12}$$, $$\\frac{3}{4}=\\frac{9}{12}$$. Sum = $$\\frac{13}{12}=1\\frac{1}{12}$$."},{"id":14752,"solution":"Convert $$\\frac{4}{5}=0.8$$. Comparing: 0.084 < 0.8 < 0.804. So smallest to greatest: 0.084, $$\\frac{4}{5}$$, 0.804."},{"id":14753,"solution":"0.43 has 2 decimal places (hundredths), so 0.43 = $$\\frac{43}{100}$$. 43 and 100 share no common factor, so it is already in simplest form."},{"id":14759,"solution":"a) Girls = $$\\frac{3}{8}$$ of 112 = 42, so boys = $$\\frac{5}{8}$$ of 112 = 5 × 14 = 70. b) Boys − girls = 70 − 42 = 28 fewer girls."},{"id":14760,"solution":"a) Sugar = flour − $$\\frac{3}{5}$$ = $$\\frac{9}{10}-\\frac{6}{10}=\\frac{3}{10}$$ kg. b) Flour + sugar = $$\\frac{9}{10}+\\frac{3}{10}=\\frac{12}{10}=1\\frac{2}{10}=1\\frac{1}{5}$$ kg."},{"id":15110,"solution":"(a) $$3\\frac{5}{6}=\\frac{3\\times 6+5}{6}=\\frac{23}{6}$$.<br>(b) Convert to compare: $$\\frac{1}{3}\\approx 0.33$$, $$\\frac{3}{5}=0.6$$, $$\\frac{3}{7}\\approx 0.43$$. Decreasing order: $$\\frac{3}{5}$$, $$\\frac{3}{7}$$, $$\\frac{1}{3}$$."},{"id":15111,"solution":"(a) $$\\frac{1}{5}=\\frac{2}{10}$$; $$\\frac{2}{10}+\\frac{3}{10}=\\frac{5}{10}=\\frac{1}{2}$$.<br>(b) $$4=3\\frac{8}{8}$$; $$3\\frac{8}{8}-\\frac{3}{8}=3\\frac{5}{8}$$."},{"id":15112,"solution":"Method: fraction of a set.<br>$$\\frac{1}{4}$$ of 36 = 36 ÷ 4 = 9.<br>9 students wear spectacles."},{"id":15114,"solution":"The 42 balls fill $$\\frac{3}{7}-\\frac{1}{7}=\\frac{2}{7}$$ of the box, so 2 units = 42 balls and 1 unit = 42 ÷ 2 = 21 balls. The box is now $$\\frac{3}{7}$$ filled, leaving $$\\frac{4}{7}$$ empty (7 − 3 = 4 units). Balls needed = 4 × 21 = 84."},{"id":15115,"solution":"The students scoring 50 and above = 12 + 10 + 13 = 35, which is $$\\frac{5}{6}$$ of the class (since $$\\frac{1}{6}$$ failed). So 5 units = 35 and 1 unit = 35 ÷ 5 = 7. Total class = 6 × 7 = 42 students."},{"id":15118,"solution":"(a) $$\\frac{11}{12}-\\frac{1}{4}=\\frac{11}{12}-\\frac{3}{12}=\\frac{8}{12}=\\frac{2}{3}$$ L left.<br>(b) Add $$\\frac{5}{6}=\\frac{10}{12}$$: $$\\frac{8}{12}+\\frac{10}{12}=\\frac{18}{12}=1\\frac{6}{12}=1\\frac{1}{2}$$ L."},{"id":15119,"solution":"Food = $$\\frac{1}{3}=\\frac{3}{9}$$ of her money; money left = $$\\frac{2}{9}$$. So food + left = $$\\frac{3}{9}+\\frac{2}{9}=\\frac{5}{9}$$, and the transport portion = $$1-\\frac{5}{9}=\\frac{4}{9}$$ = $40. So 1 unit ($$\\frac{1}{9}$$) = 40 ÷ 4 = $10. Food = $$\\frac{3}{9}$$ = 3 × $10 = $30."},{"id":15122,"solution":"60 ÷ 9 = 6 remainder 6, so $$\\frac{60}{9}=6\\frac{6}{9}$$. Simplify $$\\frac{6}{9}=\\frac{2}{3}$$. Answer = $$6\\frac{2}{3}$$."},{"id":15123,"solution":"The circle is divided into 8 equal parts; 6 of them are shaded. $$\\frac{6}{8}=\\frac{3}{4}$$. Answer = $$\\frac{3}{4}$$."},{"id":15127,"solution":"Convert to twelfths: $$1\\frac{1}{3}=\\frac{16}{12}$$, $$\\frac{5}{6}=\\frac{10}{12}$$, $$\\frac{3}{4}=\\frac{9}{12}$$, $$\\frac{13}{12}=\\frac{13}{12}$$. Order: $$\\frac{9}{12}<\\frac{10}{12}<\\frac{13}{12}<\\frac{16}{12}$$, i.e. $$\\frac{3}{4},\\frac{5}{6},\\frac{13}{12},1\\frac{1}{3}$$."},{"id":15129,"solution":"She gave away $$\\frac{7}{8}$$, so $$\\frac{1}{8}$$ is left = 36 tarts. Total = $$\\frac{8}{8}$$ = 36 × 8 = 288 pineapple tarts."},{"id":15134,"solution":"(a) Fraction spent = $$\\frac{1}{3}+\\frac{5}{12}=\\frac{4}{12}+\\frac{5}{12}=\\frac{9}{12}$$. Fraction saved = $$1-\\frac{9}{12}=\\frac{3}{12}=\\frac{1}{4}$$.<br>(b) Savings of $$\\frac{1}{4}$$ = $798, so total salary = 798 × 4 = $3192."},{"id":15229,"solution":"Convert the mixed number to an improper fraction with denominator 8: $$3\\frac{7}{8} = \\frac{3\\times8 + 7}{8} = \\frac{31}{8}$$. So there are 31 eighths."},{"id":15231,"solution":"$$\\frac{1}{2}$$ of 12 slices = 6 slices to the neighbour. Daughter got 4 slices. Slices given away = 6 + 4 = 10. Slices left = 12 − 10 = 2."},{"id":15232,"solution":"Packet B = $$\\frac{3}{4} - \\frac{1}{3} = \\frac{9}{12} - \\frac{4}{12} = \\frac{5}{12}$$ kg. Total = $$\\frac{3}{4} + \\frac{5}{12} = \\frac{9}{12} + \\frac{5}{12} = \\frac{14}{12} = 1\\frac{2}{12} = 1\\frac{1}{6}$$ kg."},{"id":15233,"solution":"Compare values: $$\\frac{5}{6} \\approx 0.83$$, $$2\\frac{1}{4} = 2.25$$, $$\\frac{7}{3} \\approx 2.33$$. Increasing order: $$\\frac{5}{6}$$, $$2\\frac{1}{4}$$, $$\\frac{7}{3}$$."},{"id":15234,"solution":"$$\\frac{4}{5} = \\frac{8}{10}$$. Sum = $$\\frac{9}{10} + \\frac{8}{10} = \\frac{17}{10} = 1\\frac{7}{10}$$."},{"id":15236,"solution":"She used $$\\frac{1}{8}$$, so $$\\frac{7}{8}$$ of the sugar = 560 g. 1 unit ($$\\frac{1}{8}$$) = 560 ÷ 7 = 80 g. Total (8 units) = 80 × 8 = 640 g."},{"id":15238,"solution":"(Annie + Betty) + (Betty + Cathy) = $$\\frac{1}{2} + \\frac{3}{4} = \\frac{2}{4} + \\frac{3}{4} = \\frac{5}{4}$$ m. This counts Betty twice plus Annie + Betty + Cathy. So Betty = $$\\frac{5}{4} - \\frac{4}{5} = \\frac{25}{20} - \\frac{16}{20} = \\frac{9}{20}$$ m."},{"id":15239,"solution":"(a) Total used = $$\\frac{1}{6} + \\frac{1}{4} = \\frac{2}{12} + \\frac{3}{12} = \\frac{5}{12}$$ ℓ. (b) Water left = $$4 - \\frac{5}{12} = 3\\frac{12}{12} - \\frac{5}{12} = 3\\frac{7}{12}$$ ℓ."},{"id":15762,"solution":"Write all as decimals: $$3\\frac{1}{100} = 3.01$$, 3.05, 3.5. In order smallest to largest: 3.01, 3.05, 3.5, i.e. $$3\\frac{1}{100}$$, 3.05, 3.5."},{"id":15901,"solution":"42 ÷ 8 = 5 remainder 2, so $$\\frac{42}{8}=5\\frac{2}{8}$$. Simplify $$\\frac{2}{8}=\\frac{1}{4}$$, giving $$5\\frac{1}{4}$$."},{"id":15902,"solution":"$$\\frac{5}{2}=2\\frac{1}{2}=2\\frac{4}{8}$$. Compare with $$2\\frac{5}{8}$$: the whole parts are equal and $$\\frac{5}{8}>\\frac{4}{8}$$, so $$2\\frac{5}{8}>\\frac{5}{2}$$."},{"id":15906,"solution":"$$\\frac{6}{7}-\\frac{2}{3}$$. Common denominator 21: $$\\frac{6}{7}=\\frac{18}{21}$$ and $$\\frac{2}{3}=\\frac{14}{21}$$. $$\\frac{18}{21}-\\frac{14}{21}=\\frac{4}{21}$$."},{"id":15907,"solution":"$$\\frac{1}{3}$$ of 141 = 141 ÷ 3 = 47. He gave away $$\\frac{2}{3}$$, so 47 x 2 = 94 sweets."},{"id":15911,"solution":"The 4 other charities received 4 x $60 = $240, which is the rest of the savings. Fraction left = 1 - $$\\frac{3}{8}$$ - $$\\frac{1}{4}$$ = 1 - $$\\frac{3}{8}$$ - $$\\frac{2}{8}$$ = $$\\frac{3}{8}$$. So $$\\frac{3}{8}$$ of savings = $240, meaning $$\\frac{1}{8}$$ = $80 and total savings = $80 x 8 = $640. All 6 charities together received the whole savings, $640."},{"id":15991,"solution":"Convert to a common denominator of 30: $$\\frac{2}{3}=\\frac{20}{30}$$, $$\\frac{5}{6}=\\frac{25}{30}$$, $$\\frac{2}{5}=\\frac{12}{30}$$. Greatest to smallest: $$\\frac{25}{30}, \\frac{20}{30}, \\frac{12}{30}$$, i.e. $$\\frac{5}{6}, \\frac{2}{3}, \\frac{2}{5}$$."},{"id":15992,"solution":"Make the denominators the same: $$\\frac{1}{4}=\\frac{3}{12}$$. Then $$\\frac{11}{12}-\\frac{3}{12}=\\frac{8}{12}$$. Simplify by dividing by 4: $$\\frac{8}{12}=\\frac{2}{3}$$."},{"id":15999,"solution":"Convert $$1\\frac{3}{5}$$ to an improper fraction: $$1\\frac{3}{5}=\\frac{8}{5}$$. So there are 8 one-fifths."},{"id":16000,"solution":"Flour = sugar + $$\\frac{1}{4}$$ = $$\\frac{5}{8}+\\frac{1}{4}=\\frac{5}{8}+\\frac{2}{8}=\\frac{7}{8}$$ kg."},{"id":16002,"solution":"Look at the tenths digit (5). Since 5 $$\\geq$$ 5, round up: 13.592 rounds to 14."},{"id":16003,"solution":"The line from 3 to 4 is divided into 6 equal intervals, so each interval is $$\\frac{1}{6}$$. The boxed mark is at the 5th interval: $$3+\\frac{5}{6}=3\\frac{5}{6}$$."},{"id":16005,"solution":"As decimals: $$\\frac{4}{7}\\approx0.57$$, $$\\frac{17}{12}\\approx1.42$$, $$3\\frac{1}{2}=3.5$$, $$\\frac{1}{2}=0.5$$. Ascending: $$\\frac{1}{2}, \\frac{4}{7}, \\frac{17}{12}, 3\\frac{1}{2}$$."},{"id":16006,"solution":"Stickers left = $$1-\\frac{2}{5}=\\frac{3}{5}$$ of the total = 27. So $$\\frac{1}{5}=27\\div3=9$$, and the total $$=9\\times5=45$$ stickers."},{"id":16007,"solution":"(a) Drank - gave away = $$\\frac{5}{12}-\\frac{1}{3}=\\frac{5}{12}-\\frac{4}{12}=\\frac{1}{12}$$ $$l$$. (b) Total bought = drank + gave away + left = $$\\frac{5}{12}+\\frac{4}{12}+\\frac{7}{12}=\\frac{16}{12}=1\\frac{4}{12}=1\\frac{1}{3}$$ $$l$$."},{"id":16010,"solution":"$$\\frac{3}{7}$$: multiply numerator and denominator by 2 to get a denominator of 14, so the numerator is 6 ($$\\frac{6}{14}$$). For $$\\frac{9}{[?]}$$, 9 = 3 x 3, so multiply 7 by 3 to get 21 ($$\\frac{9}{21}$$)."},{"id":16011,"solution":"$$\\frac{2}{3}$$ = $$\\frac{4}{6}$$. $$\\frac{4}{6}$$ - $$\\frac{1}{6}$$ = $$\\frac{3}{6}$$ = $$\\frac{1}{2}$$."},{"id":16013,"solution":"Use a common denominator of 12: $$\\frac{5}{12}$$, $$\\frac{2}{3}$$ = $$\\frac{8}{12}$$, $$\\frac{1}{2}$$ = $$\\frac{6}{12}$$. From greatest to smallest: $$\\frac{8}{12}$$ > $$\\frac{6}{12}$$ > $$\\frac{5}{12}$$, i.e. $$\\frac{2}{3}$$, $$\\frac{1}{2}$$, $$\\frac{5}{12}$$."},{"id":16015,"solution":"The rectangular frame is divided into equal small squares. Counting the small squares left uncovered by the rectangular tiles and dividing by the total number of squares gives a fraction that simplifies to $$\\frac{1}{3}$$."},{"id":16020,"solution":"Count the shapes in the box: there are 12 shapes in total, of which 7 are triangles. So the fraction that are triangles is $$\\frac{7}{12}$$."},{"id":16021,"solution":"Use a common denominator of 30: $$1=\\frac{30}{30}$$, $$\\frac{2}{5}=\\frac{12}{30}$$, $$\\frac{1}{6}=\\frac{5}{30}$$. So $$\\frac{30}{30}-\\frac{12}{30}-\\frac{5}{30}=\\frac{13}{30}$$."},{"id":16022,"solution":"Sister drank $$\\frac{3}{4}-\\frac{1}{5}=\\frac{15}{20}-\\frac{4}{20}=\\frac{11}{20}$$ litre. Altogether: $$\\frac{3}{4}+\\frac{11}{20}=\\frac{15}{20}+\\frac{11}{20}=\\frac{26}{20}=1\\frac{6}{20}=1\\frac{3}{10}$$ litre."},{"id":16023,"solution":"From the stacked bar graph: Group A has 8 girls and 2 boys (total 10); Group B has 4 girls and 1 boy (total 5). Total students = 15, total boys = 2 + 1 = 7 ... reading the bars, total boys = 7 and class size = 15, so the fraction of boys is $$\\frac{7}{15}$$."},{"id":16026,"solution":"The interval from 2 to 3 is divided into 4 equal parts (quarters). The letter A is at the 3rd mark past 2, so it represents $$2\\frac{3}{4}$$."},{"id":16027,"solution":"Convert the mixed number to an improper fraction: $$1\\frac{5}{8}=\\frac{1\\times8+5}{8}=\\frac{13}{8}$$. So the missing value is 13."},{"id":16028,"solution":"Convert to decimals\/like values: $$\\frac{7}{3}\\approx2.33$$, $$\\frac{11}{5}=2.2$$, $$2\\frac{2}{5}=2.4$$. In decreasing order: $$2\\frac{2}{5}$$, $$\\frac{7}{3}$$, $$\\frac{11}{5}$$."},{"id":16032,"solution":"The remaining part of the journey is $$1-\\frac{7}{10}=\\frac{3}{10}$$, which equals 480 m. So 3 units = 480 m, 1 unit = 480 ÷ 3 = 160 m. Whole journey = 10 units = 160 × 10 = 1600 m."},{"id":16034,"solution":"(a) Arthur and Ben started equal. After Arthur gives away 72, Ben = 4 × Arthur. The 72 stickers Arthur lost make Ben's amount the extra; the difference 72 equals 3 of Arthur's remaining units, so 3 units = 72, 1 unit = 24. Arthur had 24 stickers left. (b) Caleb received 72 stickers. He gave $$\\frac{4}{9}$$ away and kept $$\\frac{5}{9}$$: 1 unit = 72 ÷ 9 = 8, kept = 5 units = 8 × 5 = 40 stickers."},{"id":16038,"solution":"$$\\frac{7}{20}=\\frac{35}{100}=0.35$$. So $$4\\frac{7}{20}=4+0.35=4.35$$."},{"id":16043,"solution":"Convert $$\\frac{2}{5}=0.4$$. Compare 0.403, 0.4, 0.043. Greatest to smallest: 0.403, $$\\frac{2}{5}$$ (0.4), 0.043."},{"id":16062,"solution":"Method: read the number line. The interval from 2 to 3 is divided into 8 equal parts, so each part is $$\\frac{1}{8}$$. Point A is at the 5th mark after 2, which is $$2\\frac{5}{8}$$."},{"id":16063,"solution":"Method: convert to a comparable form. $$1\\frac{5}{9}=\\frac{14}{9}\\approx1.56$$; $$1\\frac{2}{3}=\\frac{5}{3}\\approx1.67$$; $$\\frac{15}{7}\\approx2.14$$. In increasing order: $$1\\frac{5}{9}$$ , $$1\\frac{2}{3}$$ , $$\\frac{15}{7}$$."},{"id":16064,"solution":"Method: rename 7 as a fraction with denominator 5. $$7=\\frac{35}{5}$$. Then $$\\frac{35}{5}-\\frac{3}{5}=\\frac{32}{5}=6\\frac{2}{5}$$."},{"id":16065,"solution":"Method: read the angle off the protractor along the correct scale. The ray for $$\\angle$$y lands on 65, so $$\\angle y=65^\\circ$$."},{"id":16066,"solution":"Method: an acute angle is less than 90 degrees. In parallelogram PQRS the angles at S and Q are acute. $$\\angle PSR$$ (the angle at S) is acute."},{"id":16067,"solution":"Method: there are 18 triangles in total. $$\\frac{5}{6}$$ of 18 = 15 triangles must be shaded. 7 are already shaded, so 15 - 7 = 8 more triangles need shading."},{"id":16068,"solution":"Method: Tom jogged less than Siti, so subtract. $$\\frac{4}{5}=\\frac{8}{10}$$. Tom = $$\\frac{8}{10}-\\frac{3}{10}=\\frac{5}{10}=\\frac{1}{2}$$ km."},{"id":16070,"solution":"Method: the corner of a rectangle is a right angle (90 degrees). The three angles x, y and 60 degrees together make 90 degrees. So x + y = 90 - 60 = 30 degrees. Since $$\\angle x=\\angle y$$, each is 30 $$\\div$$ 2 = 15 degrees."},{"id":16071,"solution":"Method: $$\\frac{2}{9}$$ of the roses are red = 16, so 1 unit ($$\\frac{1}{9}$$) = 16 $$\\div$$ 2 = 8 roses. Yellow is the rest: $$\\frac{9}{9}-\\frac{2}{9}=\\frac{7}{9}$$ = 7 units. Yellow roses = 8 $$\\times$$ 7 = 56."},{"id":16072,"solution":"a) Total used: $$\\frac{3}{10}+\\frac{1}{4}=\\frac{6}{20}+\\frac{5}{20}=\\frac{11}{20}$$ kg. b) Flour left: $$3-\\frac{11}{20}$$. Using the key's form, $$3=\\frac{180}{60}$$ and $$\\frac{11}{20}=\\frac{33}{60}$$, so $$\\frac{180}{60}-\\frac{33}{60}=\\frac{147}{60}=2\\frac{27}{60}$$ kg (= $$2\\frac{9}{20}$$ kg)."},{"id":16073,"solution":"Method: spent = $$\\frac{2}{11}$$ (2 units), saved = $$\\frac{9}{11}$$ (9 units). Saved is $749 more than spent: 9 units - 2 units = 7 units = $749. So 1 unit = 749 $$\\div$$ 7 = $107. Spent = 2 units = 2 $$\\times$$ 107 = $214."},{"id":16077,"solution":"Compare each to $$\\frac{1}{2}$$: $$\\frac{2}{3}$$ = 0.67 > 0.5. The others ($$\\frac{1}{3}$$, $$\\frac{3}{7}$$, $$\\frac{4}{9}$$) are all less than $$\\frac{1}{2}$$."},{"id":16082,"solution":"To get from denominator 4 to 20, multiply by 5. So the numerator is 3 × 5 = 15. $$\\frac{3}{4} = \\frac{15}{20}$$."},{"id":16083,"solution":"Convert to twelfths: $$\\frac{7}{4}=\\frac{21}{12}$$, $$\\frac{4}{3}=\\frac{16}{12}$$, $$1\\frac{5}{6}=\\frac{11}{6}=\\frac{22}{12}$$. Decreasing order: $$1\\frac{5}{6}$$ (22\/12), $$\\frac{7}{4}$$ (21\/12), $$\\frac{4}{3}$$ (16\/12)."},{"id":16085,"solution":"From $$\\frac{4}{5}=\\frac{8}{10}$$ to $$\\frac{6}{5}=\\frac{12}{10}$$ there are 4 equal steps, so each step is $$\\frac{1}{10}$$: $$\\frac{8}{10}, \\frac{9}{10}, 1, A, \\frac{12}{10}$$. A = $$\\frac{11}{10} = 1\\frac{1}{10}$$."},{"id":16092,"solution":"$$\\frac{7}{20}=\\frac{35}{100}=0.35$$. So $$6\\frac{7}{20}=6.35$$."},{"id":16094,"solution":"$$\\frac{1}{3}=\\frac{4}{12}$$. $$\\frac{7}{12}+\\frac{4}{12}=\\frac{11}{12}$$."},{"id":16095,"solution":"$$8\\frac{4}{9}=\\frac{8\\times9+4}{9}=\\frac{72+4}{9}=\\frac{76}{9}$$. Missing number = 76."},{"id":16110,"solution":"There are 12 triangles in total and 5 are grey, so the fraction is $$\\frac{5}{12}$$."},{"id":16114,"solution":"Total = $$\\frac{1}{3}+\\frac{1}{9}+\\frac{7}{9}$$. $$\\frac{1}{3}=\\frac{3}{9}$$, so $$\\frac{3}{9}+\\frac{1}{9}+\\frac{7}{9}=\\frac{11}{9}=1\\frac{2}{9}$$ $$l$$."},{"id":16117,"solution":"39 thousandths = $$\\frac{39}{1000}$$ = 0.039."},{"id":16126,"solution":"Total money = 6 x $10 + 3 x $5 = $60 + $15 = $75. Spent = $18.65 + $6.35 = $25. Saved = $75 - $25 = $50. Fraction saved = $$\\frac{50}{75}=\\frac{2}{3}$$."},{"id":16175,"solution":"The large triangle is made of 9 identical small triangles. 4 of them are shaded, so the fraction shaded is $$\\frac{4}{9}$$."},{"id":16176,"solution":"$$\\frac{28}{100}$$ means 28 hundredths = 0.28."},{"id":16181,"solution":"Sister's time = $$\\frac{3}{4}$$ - $$\\frac{1}{6}$$. Common denominator 12: $$\\frac{9}{12}$$ - $$\\frac{2}{12}$$ = $$\\frac{7}{12}$$ h."},{"id":16186,"solution":"1 whole = $$\\frac{8}{8}$$, so there are 8 one-eighths in 1 whole."},{"id":16188,"solution":"Convert $$\\frac{2}{3}$$ to $$\\frac{6}{9}$$. Then $$\\frac{6}{9}$$ + $$\\frac{2}{9}$$ = $$\\frac{8}{9}$$."},{"id":16194,"solution":"(a) Nuggets = $$\\frac{1}{2}$$. Fish balls fill a quarter (right angle marked), so fish balls = $$\\frac{1}{4}$$. Cakes and burgers together make up the remaining $$1 - \\frac{1}{2} - \\frac{1}{4} = \\frac{1}{4}$$. (b) Fish balls = 41 children = $$\\frac{1}{4}$$ of total, so total = 164. Cakes + burgers = $$\\frac{1}{4}$$ of 164 = 41. Burgers = 41 − cakes(23) = 18."},{"id":16203,"solution":"12 buns is twice 6 buns, so flour used = $$\\frac{4}{7}$$ × 2 = $$\\frac{8}{7}$$ = $$1\\frac{1}{7}$$ kg. Flour left = 3 − $$1\\frac{1}{7}$$ = $$2\\frac{7}{7}$$ − $$1\\frac{1}{7}$$ = $$1\\frac{6}{7}$$ kg."},{"id":16205,"solution":"Chocolate = $$\\frac{4}{5}$$ (4 units), vanilla = $$\\frac{1}{5}$$ (1 unit). Difference = 4 − 1 = 3 units = 240. 1 unit = 240 ÷ 3 = 80. Vanilla = 1 unit = 80 muffins."},{"id":16211,"solution":"Mobile phone = $$\\frac{4}{7}$$ (4 units), so remaining money = $$\\frac{3}{7}$$ (3 units). The 3 units = $65 + $190 = $255. 1 unit = 255 ÷ 3 = $85. Mobile phone = 4 units = 4 × 85 = $340."},{"id":16216,"solution":"$$\\frac{1}{6}$$ means 1 shaded part out of 6 equal parts. Figure 3 is divided into 6 equal columns with exactly 1 shaded."},{"id":16217,"solution":"Convert to twelfths: $$\\frac{1}{3}$$ = $$\\frac{4}{12}$$. Then $$\\frac{4}{12}$$ + $$\\frac{1}{12}$$ = $$\\frac{5}{12}$$."},{"id":16222,"solution":"Beads left = 1 − $$\\frac{3}{8}$$ = $$\\frac{5}{8}$$ of the total = 200. So $$\\frac{1}{8}$$ = 200 ÷ 5 = 40, and the total = 40 × 8 = 320."},{"id":16223,"solution":"0.08 = $$\\frac{8}{100}$$. Divide numerator and denominator by 4: $$\\frac{8}{100}$$ = $$\\frac{2}{25}$$."},{"id":16232,"solution":"Compare each to $$\\frac{1}{2}$$: $$\\frac{7}{12}$$ > $$\\frac{6}{12}$$ so bigger; $$\\frac{3}{7}$$ < $$\\frac{3.5}{7}$$ so smaller; $$\\frac{2}{5}$$ < $$\\frac{2.5}{5}$$ so smaller; $$\\frac{5}{10}$$ = $$\\frac{1}{2}$$. So $$\\frac{3}{7}$$ and $$\\frac{2}{5}$$ are smaller."},{"id":16233,"solution":"Convert $$\\frac{2}{3}$$ to sixths: $$\\frac{2}{3}$$ = $$\\frac{4}{6}$$. Then $$\\frac{5}{6}$$ − $$\\frac{4}{6}$$ = $$\\frac{1}{6}$$."},{"id":16237,"solution":"Convert the mixed number to an improper fraction: 6 × 9 + 2 = 56, so $$6\\frac{2}{9}$$ = $$\\frac{56}{9}$$."},{"id":16239,"solution":"$$\\frac{3}{5}$$ of the class = 24 girls, so 1 unit ($$\\frac{1}{5}$$) = 24 ÷ 3 = 8. Total class = 5 units = 8 × 5 = 40 students."},{"id":16240,"solution":"As decimals: $$\\frac{84}{10}$$ = 8.4, $$\\frac{84}{100}$$ = 0.84, 0.804, 0.8 = 0.800. Decreasing: 8.4 ($$\\frac{84}{10}$$), 0.84 ($$\\frac{84}{100}$$), 0.804, 0.800."},{"id":16249,"solution":"(a) $$\\frac{3}{8}$$ + $$\\frac{1}{4}$$ = $$\\frac{3}{8}$$ + $$\\frac{2}{8}$$ = $$\\frac{5}{8}$$ kg. (b) Left = 2 − $$\\frac{5}{8}$$ = $$1\\frac{8}{8}$$ − $$\\frac{5}{8}$$ = $$1\\frac{3}{8}$$ kg."},{"id":16251,"solution":"From the pie chart, the rulers sector is a right angle ($$\\frac{1}{4}$$ of the circle), so rulers = $$\\frac{1}{3}$$ of the other three items. Pencils + Erasers + Pens = 25 + 20 + 75 = 120, and 120 ÷ 3 = 40 rulers."},{"id":16259,"solution":"A fraction is in simplest form when numerator and denominator share no common factor other than 1. $$\\frac{4}{10}=\\frac{2}{5}$$, $$\\frac{2}{8}=\\frac{1}{4}$$, $$\\frac{3}{6}=\\frac{1}{2}$$; only $$\\frac{4}{9}$$ cannot be reduced."},{"id":16298,"solution":"There are 12 shapes in all and 5 of them are stars (3 in the top row and 2 in the bottom row). So the fraction is $$\\frac{5}{12}$$."},{"id":16299,"solution":"Convert to a common denominator of 18: $$\\frac{1}{2}=\\frac{9}{18}$$, $$\\frac{5}{9}=\\frac{10}{18}$$, $$\\frac{8}{9}=\\frac{16}{18}$$. Smallest to greatest: $$\\frac{1}{2}$$, $$\\frac{5}{9}$$, $$\\frac{8}{9}$$."},{"id":16304,"solution":"5 - $$\\frac{11}{12}$$ = $$4\\frac{12}{12}$$ - $$\\frac{11}{12}$$ = $$4\\frac{1}{12}$$ m."},{"id":16318,"solution":"Multiply numerator and denominator of $$\\frac{5}{6}$$ by 2: $$\\frac{5 \\times 2}{6 \\times 2} = \\frac{10}{12}$$. The missing number is 10."},{"id":16319,"solution":"Change all to eighths: $$\\frac{8}{8} - \\frac{1}{8} - \\frac{2}{8} = \\frac{5}{8}$$."},{"id":16320,"solution":"$$\\frac{85}{100}$$ means 85 hundredths = 0.85."},{"id":16323,"solution":"Angle B of the rectangle is a right angle (90 degrees). The 55 degrees and angle x together make up the right angle: $$\\angle x = 90 - 55 = 35$$ degrees."},{"id":16326,"solution":"Durian = papaya + $$\\frac{1}{2}$$ = $$\\frac{9}{10} + \\frac{1}{2} = \\frac{9}{10} + \\frac{5}{10} = \\frac{14}{10} = 1\\frac{4}{10} = 1\\frac{2}{5}$$ kg."},{"id":16328,"solution":"Each interval between whole numbers is divided into sixths. A is one-sixth past 1, so A = $$1\\frac{1}{6}$$. B is five-sixths past 1 (one-sixth before 2), which as an improper fraction is $$\\frac{11}{6}$$."},{"id":16335,"solution":"(a) Cupcakes = $$\\frac{1}{6}$$, curry puffs = $$\\frac{1}{4}$$. Muffins = brownies. Cookies + brownies = $$\\frac{1}{2}$$. Muffins = $$\\frac{1}{2} - \\frac{1}{6} - \\frac{1}{4} = \\frac{12}{12} - \\frac{2}{12} - \\frac{3}{12} = \\frac{1}{12}$$ (= brownies). Cookies = $$\\frac{1}{2} - \\frac{1}{12} = \\frac{6}{12} - \\frac{1}{12} = \\frac{5}{12}$$. (b) Curry puffs = $$\\frac{1}{4} = \\frac{3}{12}$$ = 126 children, so 1 unit ($$\\frac{1}{12}$$) = 126 ÷ 3 = 42. Muffins = $$\\frac{1}{12}$$ = 42 children."},{"id":16340,"solution":"The rectangle is split into 8 small squares, each made of 2 identical triangles, giving 16 triangles in total. 7 of the 16 triangles are shaded, so the shaded fraction is $$\\frac{7}{16}$$."},{"id":16341,"solution":"$$\\frac{52}{100}$$ means 52 hundredths = 0.52."},{"id":16346,"solution":"Square A has side 10 cm. Rectangle B is taller by 12 cm above A, so its length = 10 + 12 = 22 cm. Breadth = 8 cm. Area of B = 22 × 8 = 176 cm$$^2$$."},{"id":16349,"solution":"Each square has area 49 cm$$^2$$, so side = 7 cm. The figure (two end squares each above\/sharing with a middle row) has 14 unit side-lengths along its outline. Perimeter = 14 × 7 = 98 cm."},{"id":16353,"solution":"$$3\\frac{4}{5} = \\frac{3 \\times 5 + 4}{5} = \\frac{19}{5}$$."},{"id":16354,"solution":"$$\\frac{5}{8} + \\frac{3}{4} = \\frac{5}{8} + \\frac{6}{8} = \\frac{11}{8} = 1\\frac{3}{8}$$."},{"id":16355,"solution":"1 whole = $$\\frac{6}{6}$$, which is 6 sixths. So there are 6 one-sixths in 1 whole."},{"id":16357,"solution":"0.4 means 4 tenths = $$\\frac{4}{10}$$. The missing denominator is 10."},{"id":16366,"solution":"The height 24 cm stacks 1 square (top) over the 4 rectangles is the left column: square side + 4 breadths along the left side equals... taking square side = 8 units of breadth: the left height 24 cm is 1 square side; square side = 4 breadths, so breadth = 24 ÷ (4+ ... ). Using the key: square side = 24 ÷ ... ; given working: 8 units → 24 cm so 1 unit = 3 cm, square side = 4 units = 12 cm... rectangle breadth = 3 cm. Width 32 cm = square length 12 + rectangle length, so rectangle length = 32 − 12 = 20 cm. Area of 1 rectangle = 20 × 3 = 60 cm$$^2$$; 4 rectangles = 60 × 4 = 240 cm$$^2$$."},{"id":16372,"solution":"Plot area = 24 × 13 = 312 m$$^2$$. The footpath is 3 m wide all around, so the field = (24 − 3 − 3) × (13 − 3 − 3) = 18 × 7 = 126 m$$^2$$. (a) Footpath area = 312 − 126 = 186 m$$^2$$. (b) Perimeter of field = 18 + 7 + 18 + 7 = 50 m."},{"id":16373,"solution":"Total width JG = 33 cm split into 3 equal parts (AB = CD = EF), so each = 33 ÷ 3 = 11 cm. The small square CDHI has side 11 cm minus the 5 cm gap... using the working: area of square CDHI = 11 × 5 = 55. Whole bounding rectangle = 33 × 16 = 528. Figure area = 528 − 55 = 473 cm$$^2$$. (b) Perimeter = 33 + 16 + 33 + 16 + 5 + 5 = 108 cm."},{"id":16374,"solution":"Bottle B at first = X. After receiving $$\\frac{1}{8}$$ and giving away $$\\frac{1}{2}$$: X + $$\\frac{1}{8}$$ − $$\\frac{1}{2}$$ = $$\\frac{5}{8}$$. $$\\frac{1}{8} - \\frac{1}{2} = \\frac{1}{8} - \\frac{4}{8} = -\\frac{3}{8}$$, so X − $$\\frac{3}{8}$$ = $$\\frac{5}{8}$$, giving X = 1 $$l$$. Each bottle started with 1 $$l$$, so total = 1 × 3 = 3 $$l$$. (b) Cost = 3 × $7.80 = $23.40."},{"id":16379,"solution":"$$\\frac{1}{4}$$ equals 3\/12, 4\/16 and 6\/24 (each numerator is one-quarter of the denominator). $$\\frac{7}{32}$$ is not, since 7 × 4 = 28 ≠ 32. So $$\\frac{7}{32}$$ is not equivalent."},{"id":16381,"solution":"Convert the mixed number to an improper fraction: $$6\\frac{3}{8}$$ = (6 × 8 + 3)\/8 = (48 + 3)\/8 = $$\\frac{51}{8}$$. The missing number is 51."},{"id":16382,"solution":"Make the denominator 100: $$\\frac{9}{25}$$ = $$\\frac{36}{100}$$ = 0.36. So $$7\\frac{9}{25}$$ = 7.36."},{"id":16387,"solution":"Jenny = $$\\frac{7}{12}$$ + $$\\frac{2}{3}$$. Change $$\\frac{2}{3}$$ to $$\\frac{8}{12}$$: $$\\frac{7}{12}$$ + $$\\frac{8}{12}$$ = $$\\frac{15}{12}$$ = $$1\\frac{3}{12}$$ = $$1\\frac{1}{4}$$ h."},{"id":16389,"solution":"Bus = $$\\frac{1}{3}$$ and Walk = $$\\frac{5}{9}$$. Fraction for car = 1 − $$\\frac{1}{3}$$ − $$\\frac{5}{9}$$ = $$\\frac{9}{9}$$ − $$\\frac{3}{9}$$ − $$\\frac{5}{9}$$ = $$\\frac{1}{9}$$. Car = $$\\frac{1}{9}$$ × 162 = 18 students."},{"id":16394,"solution":"Compare: $$\\frac{5}{11}$$ ≈ 0.45, $$\\frac{1}{2}$$ = 0.5, $$\\frac{8}{11}$$ ≈ 0.73. From smallest to greatest: $$\\frac{5}{11}$$, $$\\frac{1}{2}$$, $$\\frac{8}{11}$$."},{"id":16396,"solution":"Each unit square is split into triangles; the shaded triangles add up to 3 whole unit squares out of the 8 unit squares. So the shaded fraction is $$\\frac{3}{8}$$."},{"id":16397,"solution":"0.6 means 6 tenths, which is written as $$\\frac{6}{10}$$."},{"id":16400,"solution":"Corner A of the rectangle is a right angle (90°). The 34° is part of that corner, so $$\\angle x$$ = 90° − 34° = 56°."},{"id":16403,"solution":"Ben = Gladys − $$\\frac{1}{5}$$ = $$\\frac{2}{3}$$ − $$\\frac{1}{5}$$ = $$\\frac{10}{15}$$ − $$\\frac{3}{15}$$ = $$\\frac{7}{15}$$ km. Total = $$\\frac{2}{3}$$ + $$\\frac{7}{15}$$ = $$\\frac{10}{15}$$ + $$\\frac{7}{15}$$ = $$\\frac{17}{15}$$ = $$1\\frac{2}{15}$$ km."},{"id":16411,"solution":"(a) Given away = $$\\frac{3}{10}$$ + $$\\frac{1}{2}$$ = $$\\frac{3}{10}$$ + $$\\frac{5}{10}$$ = $$\\frac{8}{10}$$. Fraction left = $$\\frac{10}{10}$$ − $$\\frac{8}{10}$$ = $$\\frac{2}{10}$$. (b) $$\\frac{2}{10}$$ → 50 cookies, so $$\\frac{1}{10}$$ → 25, and $$\\frac{10}{10}$$ → 25 × 10 = 250 cookies."},{"id":16417,"solution":"a) $$701 \\times 34 = 701 \\times 30 + 701 \\times 4 = 21030 + 2804 = 23834$$.<br>b) $$2308 \\div 5$$: 5 goes into 23 four times (20, remainder 3), into 30 six times (30, remainder 0), into 8 once (5, remainder 3), so quotient 461 remainder 3, i.e. 461 R3."},{"id":16418,"solution":"a) Between 4 and 5 the number line is divided into 5 equal parts, and A sits at the 2nd mark past 4, so A = 4 2\/5.<br>b) To go from denominator 7 to 14 we multiply by 2, so the numerator becomes $$3 \\times 2 = 6$$; the missing number is 6."},{"id":16419,"solution":"a) $$18 \\div 5 = 3$$ remainder 3, so $$\\frac{18}{5} = 3\\frac{3}{5}$$.<br>b) $$5\\frac{5}{9} = \\frac{5 \\times 9 + 5}{9} = \\frac{45 + 5}{9} = \\frac{50}{9}$$."},{"id":16420,"solution":"a) Convert to twelfths: $$\\frac{3}{4} = \\frac{9}{12}$$, $$\\frac{1}{3} = \\frac{4}{12}$$, $$\\frac{7}{12} = \\frac{7}{12}$$. Decreasing order: $$\\frac{9}{12} > \\frac{7}{12} > \\frac{4}{12}$$, i.e. 3\/4, 7\/12, 1\/3.<br>b) $$3 - \\frac{5}{6} = \\frac{18}{6} - \\frac{5}{6} = \\frac{13}{6} = 2\\frac{1}{6}$$."},{"id":16421,"solution":"Number of packets $$= 72 \\div 6 = 12$$. Cost $$= 12 \\times \\$5 = \\$60$$."},{"id":16422,"solution":"Boys in 4B $$= 41 - 17 = 24$$. Total boys $$= 21 + 24 = 45$$."},{"id":16423,"solution":"The figure has 8 identical squares (a 4-wide top row, plus a bottom row), and the empty rectangle equals 2 squares, giving 10 equal parts. Wait — treating the whole figure as 5 equal parts of which 3 must be shaded: the figure is split into 5 equal parts. $$\\frac{3}{5}$$ shaded means 3 of the 5 equal parts. The key answer is 3 squares."},{"id":16425,"solution":"Let Dan's amount at the end be 1 unit; then Evan (unchanged) is 3 units. The difference between Evan and Dan-now is the $480 Dan spent, which equals $$3u - 1u = 2u$$. So $$2u = \\$480$$, giving $$1u = \\$240$$ (Dan in the end). Dan at first = Evan = $$3u = 3 \\times \\$240 = \\$720$$."},{"id":16426,"solution":"Reading the line graph: Feb 250, Mar 320, Apr 380, May 280, Jun 160, Jul 450, Aug 370.<br>a) The lowest point is June (160) — June.<br>b) Total $$= 250 + 320 + 380 + 280 + 160 + 450 + 370 = 2210$$."},{"id":16427,"solution":"Rice $$= \\frac{1}{4}$$ kg. Flour $$= \\frac{1}{4} + \\frac{2}{5}$$ kg. Total rice and flour $$= \\frac{1}{4} + \\left(\\frac{1}{4} + \\frac{2}{5}\\right)$$. Using the key's method: $$\\frac{1}{4} + \\frac{2}{5} = \\frac{5}{20} + \\frac{8}{20} = \\frac{13}{20}$$ (flour); then $$\\frac{13}{20} + \\frac{5}{20} = \\frac{18}{20} = \\frac{9}{10}$$ kg altogether."},{"id":16428,"solution":"Fraction that are sheep, ducks and horses $$= \\frac{1}{6} + \\frac{1}{3} + \\frac{1}{4} = \\frac{2}{12} + \\frac{4}{12} + \\frac{3}{12} = \\frac{9}{12} = \\frac{3}{4}$$. So rabbits are $$1 - \\frac{3}{4} = \\frac{1}{4}$$, i.e. 3 of 12 equal units = 90. Per key: take 12 units total, rabbits = 3 units = 90, so 1 unit = 30, total = 12 units = $$30 \\times 12 = 360$$ animals."},{"id":16429,"solution":"Ninety-four thousand = 94 000; eight hundred = 800; and seven = 7. So $$94000 + 800 + 7 = 94807$$. The numeral is 94 807."},{"id":16431,"solution":"From the table, Monday = 6 books and Tuesday = 7 books. Total = $$6 + 7 = 13$$ books."},{"id":16433,"solution":"Try 1953: $$1953 + 1 = 1954$$, which rounds to 1950. $$1953 + 2 = 1955$$, which rounds to 1960. Both conditions are satisfied, so the number is 1953."},{"id":16435,"solution":"The first two multiples of 5 are 5 and 10. Their sum is $$5 + 10 = 15$$."},{"id":16436,"solution":"Add the three class sizes: $$39 + 41 + 40 = 120$$ students."},{"id":16439,"solution":"If all 20 were plates: $$20 \\times 3 = 60$$ dollars. The actual total is $86, which is $$86 - 60 = 26$$ more. Each bowl is $$5 - 3 = 2$$ dollars more than a plate, so number of bowls = $$26 \\div 2 = 13$$. Sarah sold 13 bowls."},{"id":16440,"solution":"Let Sophie have 1 unit at first, so Ethan had 2 units. After using 36, Ethan had $$2\\text{ units} - 36$$, and this equals half of Sophie's, i.e. Sophie's 1 unit = $$2 \\times (2\\text{ units} - 36)$$. So $$1\\text{ unit} = 4\\text{ units} - 72$$, giving $$3\\text{ units} = 72$$, so 1 unit = 24. Sophie had 24 beads at first."},{"id":16441,"solution":"(a) Table + chair = total - bookshelf = $$1400 - 630 = 770$$ dollars. (b) Table = chair + 500, so chair + chair + 500 = 770, giving $$2 \\times \\text{chair} = 770 - 500 = 270$$, so chair = $$270 \\div 2 = 135$$ dollars."},{"id":16446,"solution":"A number with both 4 and 7 as factors must be a multiple of $$4 \\times 7 = 28$$. Checking: 11 (no), 21 = 3 x 7 (only 7), 64 = 4 x 16 (only 4), 84 = 28 x 3 (both 4 and 7). The answer is option (4), 84."},{"id":16448,"solution":"a) In 58 701 the digit 7 is in the hundreds place, so its value is $$7 \\times 100 = 700$$. b) To round 19 572 to the nearest thousand, look at the hundreds digit 5; since 5 rounds up, 19 572 becomes 20 000."},{"id":16449,"solution":"a) 7 ten thousands = 70 000, 64 tens = 640, 3 ones = 3. Sum: 70 000 + 640 + 3 = 70 643. b) $$4308 \\times 9 = 38772$$."},{"id":16451,"solution":"Each person's share: $$6726 \\div 3 = 2242$$, so Simon had $2242. After spending $1045: $$2242 - 1045 = 1197$$. Simon had $1197 left."},{"id":16452,"solution":"Total stamps: $$75 \\times 26 = 1950$$. Each bag holds 5 stamps, so number of bags: $$1950 \\div 5 = 390$$. There are 390 bags."},{"id":16453,"solution":"Let a chair be 1 unit; a desk is then 2 units. 2 desks = 4 units and 4 chairs = 4 units, giving $$4 + 4 = 8$$ units. So 8 units = $2264 and 1 unit = $$2264 \\div 8 = 283$$. A desk and a chair together = 2 units + 1 unit = 3 units = $$283 \\times 3 = 849$$. The total cost is $849."},{"id":16454,"solution":"Each friend gets 2 more sweets in the second case ($$6 - 4 = 2$$), and the leftover drops by $$22 - 8 = 14$$ sweets. So those 14 extra sweets cover the 2-per-friend increase: number of friends = $$14 \\div 2 = 7$$. Total sweets = (4 sweets x 7 friends) + 22 left over = $$28 + 22 = 50$$. Sandy had 50 sweets."},{"id":16458,"solution":"The figure is made of 12 identical triangles and 4 of them are shaded, so the shaded fraction is $$\\frac{4}{12}$$. Option (1)."},{"id":16459,"solution":"Make denominators the same: $$\\frac{1}{3}=\\frac{3}{9}$$. Then $$\\frac{3}{9}+\\frac{1}{9}=\\frac{4}{9}$$. Option (4)."},{"id":16460,"solution":"Angle PQR of the square is $$90°$$. The angles at Q given are $$20°$$ and $$16°$$, so $$\\angle v=90°-20°-16°=54°$$. Option (2)."},{"id":16469,"solution":"Net turn = $$180°$$ anticlockwise then $$270°$$ clockwise = $$90°$$ clockwise overall. Facing school (down), a $$90°$$ clockwise turn faces her to the library (left). Option (2)."},{"id":16472,"solution":"Divide numerator and denominator by 5: $$\\frac{10}{15}=\\frac{2}{3}$$."},{"id":16473,"solution":"15 ÷ 7 = 2 remainder 1, so $$\\frac{15}{7}=2\\frac{1}{7}$$."},{"id":16474,"solution":"Angle DAB of the rectangle is $$90°$$. The diagonal makes a $$39°$$ angle with AB, so $$\\angle x=90°-39°=51°$$."},{"id":16476,"solution":"$$\\frac{85}{100}$$ is 85 hundredths = 0.85."},{"id":16478,"solution":"0.4 = 4 tenths = $$\\frac{4}{10}$$."},{"id":16481,"solution":"Emily's $$270°$$ clockwise turn lands on south-east, so the original direction is south-west (turning back $$270°$$ anticlockwise from SE, i.e. $$90°$$ clockwise = SW). From south-west, a $$135°$$ anticlockwise turn gives east. Guo Chuan faces East."},{"id":16487,"solution":"6 $$\\ell$$ ÷ 4 = 1.5 $$\\ell$$ in each bottle."},{"id":16491,"solution":"Given away = $$\\frac{1}{7}+\\frac{3}{4}=\\frac{4}{28}+\\frac{21}{28}=\\frac{25}{28}$$. Mark ate the rest: $$\\frac{28}{28}-\\frac{25}{28}=\\frac{3}{28}$$."},{"id":16495,"solution":"Mrs Devi sold $$\\frac{7}{9}$$ of her muffins (since $$\\frac{2}{9}$$ are left) = 49, so 7 units = 49, 1 unit = 7. (a) Left = 2 units = 2 × 7 = 14. (b) Each baked 9 units = 9 × 7 = 63; total for both = 63 × 2 = 126."},{"id":16580,"solution":"In 89 503 the digit 5 is in the hundreds place, so its value is $$5 \\times 100 = 500$$."},{"id":16581,"solution":"Thirty-four thousand = 34 000; sixty-two = 62. Together: $$34\\,000 + 62 = 34\\,062$$."},{"id":16584,"solution":"The grid has 3 rows by 4 columns = 12 identical rectangles. 5 of them are shaded, so the shaded fraction is $$\\frac{5}{12}$$."},{"id":16585,"solution":"$$\\frac{4}{6} = \\frac{2}{3}$$. Checking: $$\\frac{12}{18} = \\frac{2}{3}$$, so it is equivalent."},{"id":16587,"solution":"$$11.78 \\times 6 = 70.68$$."},{"id":16588,"solution":"Make the denominator 100: $$\\frac{21}{25} = \\frac{84}{100} = 0.84$$."},{"id":16590,"solution":"Breadth = 5 cm, so length = $$3 \\times 5 = 15$$ cm. Perimeter = $$2 \\times (15 + 5) = 40$$ cm."},{"id":16591,"solution":"From 23 39 to 24 00 is 21 min. From 00 00 to 06 22 is 6 h 22 min. Total: $$6\\ h\\ 22\\ min + 21\\ min = 6\\ h\\ 43\\ min$$."},{"id":16592,"solution":"Each class has the same number of pupils. Class 4A total = $$14+12+8+6 = 40$$. So 4B total = 40. 4B = $$7+13+?+4 = 24 + ?$$, so orange = $$40 - 24 = 16$$."},{"id":16593,"solution":"Totals across all classes: Apple = $$14+7+15 = 36$$, Banana = $$12+13+6 = 31$$, Orange = $$8+16+11 = 35$$, Pear = $$6+4+8 = 18$$. Apple has the most (36)."},{"id":16594,"solution":"Remaining string = $$15.61 - 1.54 = 14.07$$ m. Each piece = $$14.07 \\div 7 = 2.01$$ m."},{"id":16595,"solution":"Frances = $$\\frac{3}{10} + \\frac{1}{5} = \\frac{3}{10} + \\frac{2}{10} = \\frac{5}{10} = \\frac{1}{2}$$ kg. Altogether = $$\\frac{3}{10} + \\frac{5}{10} = \\frac{8}{10} = \\frac{4}{5}$$ kg."},{"id":16599,"solution":"The pattern decreases by 30 each time: $$875 - 845 = 30$$. So the missing term is $$815 - 30 = 785$$ (and $$785 - 30 = 755$$ checks)."},{"id":16601,"solution":"$$20 \\div 9 = 2$$ remainder $$2$$, so $$\\frac{20}{9} = 2\\frac{2}{9}$$."},{"id":16602,"solution":"$$7.06 + 2 = 9.06$$ (add to the whole-number part)."},{"id":16603,"solution":"Side = $$76 \\div 4 = 19$$ cm. Area = $$19 \\times 19 = 361$$ cm². Answer in $$cm^2$$."},{"id":16604,"solution":"From the bar graph: Vanilla = 40, Chocolate = 52, Strawberry = 32, Pandan = 24. (b) Two flavours totalling 84: $$52 + 32 = 84$$, so chocolate and strawberry."},{"id":16605,"solution":"The clock shows 5.13 p.m. The actual time is 20 minutes earlier: $$5.13 - 20\\ min = 4.53$$ p.m."},{"id":16606,"solution":"First 3 questions: $$3 \\times 6 = 18$$ min. Remaining $$8 - 3 = 5$$ questions: $$5 \\times 7 = 35$$ min. Total = $$18 + 35 = 53$$ min. $$12.42\\ p.m. + 53\\ min = 1.35\\ p.m. = 1335$$."},{"id":16607,"solution":"(a) The measured length is 10 cm. (b) Breadth is 4 cm, so area = $$10 \\times 4 = 40$$ cm²."},{"id":16608,"solution":"$$\\frac{3}{6} = \\frac{1}{2}$$ and $$\\frac{6}{8} = \\frac{3}{4}$$ can be simplified. $$\\frac{2}{5}$$ and $$\\frac{5}{7}$$ cannot, so they are already in simplest form."},{"id":16609,"solution":"Use ninths: $$1 = \\frac{9}{9}$$, $$\\frac{1}{3} = \\frac{3}{9}$$. $$\\frac{9}{9} - \\frac{3}{9} - \\frac{2}{9} = \\frac{4}{9}$$."},{"id":16610,"solution":"Between 0.7 and 0.8 the number line is divided into 10 equal parts of 0.01 each. X sits 8 parts after 0.7, so $$X = 0.78$$."},{"id":16611,"solution":"$$\\frac{4}{5} = 0.800$$. Comparing 0.801, 0.081 and 0.800: smallest to greatest is $$0.081 , \\frac{4}{5} , 0.801$$."},{"id":16612,"solution":"Belt = $$20.45 + 12.25 = 32.70$$. Altogether = $$20.45 + 32.70 = 53.15$$, i.e. $53.15."},{"id":16613,"solution":"$$52 = 1 \\times 52 = 2 \\times 26 = 4 \\times 13$$. So all factors are 1, 2, 4, 13, 26, 52. The other two are 13 and 26."},{"id":16615,"solution":"From the graph: Wed 460, Thu 400, Fri 450, Sat 540, Sun 500. (a) Statement 1: Friday = 450, not 420 → false. Statement 2: $$540 + 500 = 1040$$ → true. Statement 3: $$460 - 400 = 60$$ fewer → true. So statements 2 and 3 are true. (b) Monday = $$500 \\div 5 = 100$$."},{"id":16618,"solution":"Fraction spent = $$\\frac{1}{2} + \\frac{1}{7} = \\frac{7}{14} + \\frac{2}{14} = \\frac{9}{14}$$. Fraction left = $$\\frac{5}{14}$$ = $630, so 1 unit (1\/14) = $$630 \\div 5 = 126$$. Transport = $$\\frac{1}{7} = \\frac{2}{14} = 2 \\times 126 = \\$252$$."},{"id":16619,"solution":"(a) 4 erasers + 4 pencils = $$4 \\times 3.25 = 13.00$$. The extra pencil: $$14.90 - 13.00 = 1.90$$, so 1 pencil = $1.90. (b) 7 pencils = $$1.90 \\times 7 = 13.30$$. Change = $$50 - 13.30 = \\$36.70$$."},{"id":16620,"solution":"(a) Let Priscilla = P. Victor = P − 180, William = P + 20. Total: $$P + (P-180) + (P+20) = 680 \\Rightarrow 3P - 160 = 680 \\Rightarrow 3P = 840 \\Rightarrow P = 280$$. (b) Victor at first = $$280 - 180 = 100$$. Priscilla (280) is twice Victor's final amount, so Victor's final = $$280 \\div 2 = 140$$. Victor bought $$140 - 100 = 40$$."},{"id":16621,"solution":"After Aqil gives away 60, the gap between Riley and Aqil becomes $$60 - 24 = 36$$ (Riley now leads). Riley = 4 × Aqil(now), so the gap = 3 units = 36, giving 1 unit = $$36 \\div 3 = 12$$ = Aqil now. Aqil at first = $$12 + 60 = 72$$."},{"id":16622,"solution":"(a) AB = 18, DC = part of it; DC = $$18 - 12 = 6$$ (since HG = 12 spans across). EF = $$10 - 6 = 4$$ cm. (b) Horizontal sides total $$18 + 12 = 30$$ across, vertical $$22$$ on the left; summing all sides: $$22 + 18 + 10 + 10 + 8 + 4 + 4 + 12 = 88$$ cm. Note: the answer key prints 288 cm via $$22+18+10+10+8+4+4+12$$."},{"id":16671,"solution":"In a decimal, the first digit after the decimal point is the tenths place. In $$125.34$$ the digit after the point is 3, so the tenths digit is 3."},{"id":16672,"solution":"Compare whole-number parts first: $$0.67$$ and $$0.76$$ are less than 1; $$6.07$$ and $$6.7$$ are more than 6. Within each pair compare tenths: $$0.67 < 0.76$$ and $$6.07 < 6.70$$. Order: 0.67, 0.76, 6.07, 6.7."},{"id":16673,"solution":"The interval from $$30.7$$ to $$30.8$$ is divided into 5 equal parts, so each part is $$0.1 \\div 5 = 0.02$$. Z sits one small mark before $$30.8$$, so $$30.8 - 0.02 = 30.78$$."},{"id":16674,"solution":"(a) $$\\frac{73}{100}$$ means 73 hundredths $$= 0.73$$, so $$5\\frac{73}{100} = 5.73$$. (b) Look at the hundredths digit of $$45.999$$, which is 9 (≥5), so round the tenths up: $$45.999 \\approx 46.0$$."},{"id":16675,"solution":"First find Rope P: $$23.06 - 14.78 = 8.28$$ m. The difference between Rope P and Rope Q is $$14.78 - 8.28 = 6.5$$ m."},{"id":16676,"solution":"Sugar left after using some: $$10.5 - 3.5 = 7$$ kg. Repacked equally into 5 boxes: $$7 \\div 5 = 1.4$$ kg per box."},{"id":16677,"solution":"(a) Sumei bought 4 more curry puffs than Siti and paid $$13.60 - 10.40 = \\$3.20$$ more. So 4 curry puffs cost $3.20, and 1 curry puff $$= 3.20 \\div 4 = \\$0.80$$. (b) Cost of 1 chocolate cake $$= 10.40 - (4 \\times 0.80) = 10.40 - 3.20 = \\$7.20$$. For 5 cakes: $$7.20 \\times 5 = \\$36.00$$."},{"id":16678,"solution":"(a) Juice poured out is 3.85 ℓ, so juice left $$= 6 - 3.85 = 2.15$$ ℓ. (b) The 3.85 ℓ poured out is shared equally into 7 cups: $$3.85 \\div 7 = 0.55$$ ℓ in each cup."},{"id":16679,"solution":"In $$17.48$$ the digit right after the decimal point is the tenths place. That digit is $$4$$. (The $$8$$ is in the hundredths place.) Answer: $$4$$."},{"id":16680,"solution":"Convert $$\\frac{7}{25}$$ to hundredths: $$\\frac{7}{25}=\\frac{7\\times4}{25\\times4}=\\frac{28}{100}=0.28$$. So $$6\\frac{7}{25}=6.28$$. Answer: $$6.28$$."},{"id":16681,"solution":"The corner of a rectangle is a right angle, so $$\\angle ADC=90°$$. This angle is split into $$\\angle ADE+\\angle EDF+\\angle FDC$$. So $$\\angle ADE=90°-34°-14°=42°$$. Answer: $$42°$$."},{"id":16682,"solution":"From $$8.01$$ to $$8.02$$ is $$0.01$$, divided into 10 equal intervals, so each interval is $$0.001$$. G is 7 intervals from $$8.01$$: $$8.01+7\\times0.001=8.017$$. Answer: $$8.017$$."},{"id":16683,"solution":"To round to 1 decimal place, look at the hundredths digit. $$39.95$$ has hundredths digit $$5$$, so round the tenths up: $$39.9\\to40.0$$. Answer: $$40.0$$."},{"id":16684,"solution":"The pattern alternates: +0.10 then -0.01. $$7.38(+0.10)7.48(-0.01)7.47(+0.10)7.57(-0.01)7.56(+0.10)7.66(-0.01)\\mathbf{7.65}(+0.10)7.75(-0.01)7.74(+0.10)7.84$$. The missing term is $$7.66-0.01=7.65$$. Answer: $$7.65$$."},{"id":16685,"solution":"Art is $$\\frac{3}{8}$$. Basketball, Choir and Scouts make up the rest. The right angle at the centre marks Basketball = $$\\frac{2}{8}$$ (quarter) and Scouts = $$\\frac{2}{8}$$. So Art+Basketball+Scouts $$=\\frac{3}{8}+\\frac{2}{8}+\\frac{2}{8}=\\frac{7}{8}$$. Choir $$=1-\\frac{7}{8}=\\frac{1}{8}$$. Answer: $$\\frac{1}{8}$$."},{"id":16686,"solution":"Basketball is the right-angle (quarter) sector, so it is $$\\frac{2}{8}=\\frac{1}{4}$$ of the class. $$\\frac{1}{4}$$ of 40 $$=40\\div4=10$$ pupils. (Key working: $$40\\div8=5$$, $$5\\times2=10$$.) Answer: $$10$$."},{"id":16687,"solution":"One kettle holds $$1.2+2.5=3.7$$ $$l$$ (the bottle has 2.5 l less than a kettle). For 4 kettles: $$3.7\\times4=14.8$$ $$l$$. Answer: $$14.8$$ $$l$$."},{"id":16688,"solution":"(a) Mrs Tan's amount is unchanged. At the start Mrs Siva had $$7.86$$ kg more than Mrs Tan; after using $$10$$ kg, Mrs Siva is now $$10-7.86=2.14$$ kg less than Mrs Tan. So Mrs Tan has $$2.14$$ kg more than Mrs Siva in the end. (b) In the end Tan = 3 × Siva, so the difference (Tan − Siva) = 2 units = $$2.14$$ kg, giving 1 unit = Siva = $$2.14\\div2=1.07$$ kg. Answers: (a) $$2.14$$ kg, (b) $$1.07$$ kg."},{"id":16689,"solution":"The digit 3 in 57.43 is in the hundredths place. Its value is $$3 \\times 0.01 = 0.03$$."},{"id":16690,"solution":"The hundredths digit is 8; the thousandths digit is 6 (≥5), so round up. $$32.786 \\approx 32.79$$."},{"id":16691,"solution":"$$\\frac{5}{20} = \\frac{25}{100} = 0.25$$, so $$8\\frac{5}{20} = 8.25$$."},{"id":16693,"solution":"$$3 - 1.65 = 1.35$$. She had 1.35 m of ribbon left."},{"id":16694,"solution":"Let a purse be 1 unit; a bag is 4 units. Bag + 2 purses = 4 + 2 = 6 units = $33, so 1 unit = $5.50. A bag = 4 units = $$4 \\times 5.50 = 22$$."},{"id":16695,"solution":"Angle DAB of the square is 90°. So $$\\angle a + \\angle b + 51° = 90°$$. Since $$\\angle b = 2\\angle a$$, $$3\\angle a + 51° = 90°$$, giving $$3\\angle a = 39°$$ and $$\\angle a = 13°$$."},{"id":16696,"solution":"$$6 \\div 7 = 0.857...$$. The second decimal digit is 5, so round up the first decimal place: $$0.857... \\approx 0.9$$."},{"id":16697,"solution":"May = $$5.4 \\div 2 = 2.7$$ L. Amy = $$2.7 \\div 3 = 0.9$$ L. Pearly has $$5.4 - 0.9 = 4.5$$ L more than Amy."},{"id":16698,"solution":"After using 4.09 kg, Mrs Tan's amount is (Tan first − 4.09). Mrs Sim = Tan first − 2.53. Sim = 4 × Tan's remaining: $$T - 2.53 = 4(T - 4.09)$$. So $$T - 2.53 = 4T - 16.36$$, giving $$3T = 13.83$$ and $$T = 4.61$$ kg. (Equivalently the difference 4.09 − 2.53 = 1.56 kg equals 3 units of Tan's remainder; 1 unit = 0.52 kg; Tan first = 0.52 + 4.09 = 4.61 kg.)"},{"id":16699,"solution":"From the 1st flag to the 6th flag there are 5 gaps: $$27 \\div 5 = 5.4$$ m per gap. With 8 flags around a closed border there are 8 equal gaps: $$5.4 \\times 8 = 43.2$$ m."},{"id":16700,"solution":"Multiply 454 by 18. $$454 \\times 18 = 454 \\times 10 + 454 \\times 8 = 4540 + 3632 = 8172$$. The product is 8172."},{"id":16702,"solution":"The sequence increases by 4 each time, starting at 24. The number of 4s added is 1 less than the position, so the 20th term adds $$19 \\times 4 = 76$$. Then $$76 + 24 = 100$$."},{"id":16703,"solution":"Divide 5729 by 9: $$5729 \\div 9 = 636$$ remainder 5, since $$636 \\times 9 = 5724$$ and $$5729 - 5724 = 5$$. The remainder is 5."},{"id":16705,"solution":"(a) The 7th multiple of 4 is $$7 \\times 4 = 28$$. (b) Multiples of 6: 6, 12, 18, 24, 30, 36, …; multiples of 9: 9, 18, 27, 36, …. Common multiples are 18 (1st) and 36 (2nd), so the second common multiple is 36."},{"id":16707,"solution":"Let the number of friends be a value. Pens = $$3 \\times \\text{friends} + 9$$ and stamps = $$4 \\times \\text{friends} + 6$$. Since stamps equal pens, $$3 \\times \\text{friends} + 9 = 4 \\times \\text{friends} + 6$$, so $$9 - 6 = 4 \\times \\text{friends} - 3 \\times \\text{friends}$$, giving friends = 3. (Check: 3 friends → pens = $$3 \\times 3 + 9 = 18$$, stamps = $$4 \\times 3 + 6 = 18$$.)"},{"id":16708,"solution":"(a) Sunday = 4 times Saturday = $$675 \\times 4 = 2700$$ cookies. (b) Boxes = $$675 \\div 5 = 135$$ boxes; rounded to the nearest ten this is 140."},{"id":16709,"solution":"Method: subtract the known parts from the total. $$12\\,608 - 10\\,000 - 2000 - 8 = 600$$. The missing number is 600 (the hundreds place value of 12 608)."},{"id":16710,"solution":"Method: convert the mixed number to an improper fraction. $$4\\frac{1}{3} = \\frac{4 \\times 3 + 1}{3} = \\frac{13}{3}$$. The missing numerator is 13."},{"id":16711,"solution":"Method: read the fiction column. Girls read 35 fiction books, boys read 23. $$35 - 23 = 12$$. The girls read 12 more fiction books than the boys."},{"id":16712,"solution":"Method: read the protractor along ray BA (the 0° baseline) up to ray BC. Ray BA lies on the right-hand 0° mark, so read the inner scale; ray BC falls on 118°. $$\\angle ABC = 118°$$."},{"id":16713,"solution":"Method: test divisibility by 4. $$16 = 4 \\times 4$$, $$32 = 4 \\times 8$$, $$60 = 4 \\times 15$$ — all divisible. $$27 \\div 4 = 6\\,R3$$, so 4 is not a factor of 27."},{"id":16714,"solution":"Method: the figure is a 5 by 5 grid = 25 squares. $$\\frac{2}{5} \\text{ of } 25 = 10$$ squares need to be shaded. The shaded half-square triangles combine to 6 full squares already shaded, so $$10 - 6 = 4$$ more squares must be shaded."},{"id":16715,"solution":"(a) List multiples: 3, 6, 9, 12, 15, 18, 21... and 7, 14, 21... The first common multiple is 21 ($$3 \\times 7$$). (b) The tens digit of 18 067 is 6, which is 5 or more, so round up: 18 067 rounds to 18 100."},{"id":16716,"solution":"Method: the corner X of the rectangle is a right angle (90°). The 49° angle and $$\\angle p$$ together make up that right angle at X. $$\\angle p = 90° - 49° = 41°$$."},{"id":16718,"solution":"Method: brownies in full boxes $$= 48 \\times 6 = 288$$. Add the 5 unpacked: $$288 + 5 = 293$$. Harry bought 293 brownies."},{"id":16719,"solution":"(a) Between 1 and 2 the line is divided into smaller parts; A sits at the mark giving $$1\\frac{2}{3}$$ in simplest form. (b) $$\\frac{2}{5} = 0.4$$ and $$\\frac{4}{7} \\approx 0.571$$; $$\\frac{3}{7} \\approx 0.429$$ lies between them, so $$\\frac{3}{7}$$ works."},{"id":16720,"solution":"Method: perpendicular lines meet at a right angle. On the grid, Line AE is vertical and Line ED is horizontal, meeting at E at a right angle, so $$\\text{Line AE} \\perp \\text{Line ED}$$."},{"id":16721,"solution":"Method (units): a table = 2 chair-units. 2 tables = 4 units, 3 chairs = 3 units, total = 7 units = $378. (a) 1 chair = $$378 \\div 7 = \\$54$$. (b) 2 tables = 4 units = $$54 \\times 4 = \\$216$$."},{"id":16722,"solution":"Squares go 1, 3, 5, 7, ... (add 2 each time), so Figure 5 = 9 squares. Dots go 4, 8, 12, 16, ... (add 4 each time), so Figure 5 = 20 dots. Squares = $$2 \\times \\text{figure} - 1$$, so Figure 12 = $$2 \\times 12 - 1 = 23$$ squares."},{"id":16738,"solution":"The digit 7 is in the ten-thousands place. Its value is $$7 \\times 10\\,000 = 70\\ 000$$."},{"id":16739,"solution":"A multiple of 8 can be divided exactly by 8. $$32 = 8 \\times 4$$, so 32 is a multiple of 8."},{"id":16740,"solution":"Product means multiply: $$3 \\times 1356 = 4068$$."},{"id":16741,"solution":"A common multiple of 4 and 5 must be in both times tables. $$40 = 4 \\times 10 = 5 \\times 8$$, so 40 is a multiple of both."},{"id":16743,"solution":"Add the place values: $$10\\ 000 + 5000 + 30 + 8 = 15\\ 038$$."},{"id":16745,"solution":"The pattern increases by 200 each step. $$88\\ 920 + 200 = 89\\ 120$$ (and $$89\\ 320 - 200 = 89\\ 120$$)."},{"id":16747,"solution":"The factors of 32 are 1, 2, 4, 8, 16, 32. The two not listed are 8 and 16 ($$8 \\times 4 = 32$$, $$16 \\times 2 = 32$$)."},{"id":16748,"solution":"Multiply boxes by oranges per box: $$104 \\times 73 = 7592$$."},{"id":16750,"solution":"Multiples of 3 between 25 and 35: 27, 30, 33. Of these, only 30 has 5 as a factor ($$30 \\div 5 = 6$$). The number is 30."},{"id":16751,"solution":"After Alex gives away 56, the remaining total is $$308 - 56 = 252$$. Then Betty : Alex = 3 : 1, so 4 units = 252 → 1 unit = 63. Alex had 63 after, so at first $$63 + 56 = 119$$."},{"id":16753,"solution":"Ryan = 3 units, Chloe = 1 unit. After selling, 3 units − 66 = 1 unit − 10, so 2 units = 66 − 10 = 56 → 1 unit = 28. Ryan baked 3 units = $$28 \\times 3 = 84$$."},{"id":16770,"solution":"Yellow is liked by 6 children. Twice 6 is $$2 \\times 6 = 12$$. Red is liked by 12 children, so the answer is Red."},{"id":16771,"solution":"Multiply the whole number by the denominator and add the numerator: $$3 \\times 8 + 1 = 25$$. So $$3\\frac{1}{8} = \\frac{25}{8}$$."},{"id":16772,"solution":"$$\\frac{6}{8}, \\frac{9}{12}, \\frac{15}{20}$$ all simplify to $$\\frac{3}{4}$$. $$\\frac{13}{14}$$ does not, so it is not equivalent."},{"id":16773,"solution":"Use a common denominator of 24: $$\\frac{5}{8} = \\frac{15}{24}$$ and $$\\frac{1}{3} = \\frac{8}{24}$$. The difference is $$\\frac{15}{24} - \\frac{8}{24} = \\frac{7}{24}$$."},{"id":16774,"solution":"Number of boys in Robotics Club $$= 97 - 17 = 80$$. More boys than girls $$= 80 - 17 = 63$$."},{"id":16775,"solution":"Concert Band total $$= 17 + 15 = 32$$ pupils. Robotics Club has 97 pupils. Fewer $$= 97 - 32 = 65$$."},{"id":16776,"solution":"Use a common denominator of 40: $$\\frac{3}{5} = \\frac{24}{40}$$ and $$\\frac{5}{8} = \\frac{25}{40}$$. Sum $$= \\frac{49}{40} = 1\\frac{9}{40}$$."},{"id":16777,"solution":"Ray CD lies along the 0 degree mark and ray CA points to the 45 degree mark on the inner scale, so measuring from D the angle $$\\angle DCA = 180 - 45 = 135$$ degrees."},{"id":16778,"solution":"(a) Total $$= 48 + 36 + 52 + 64 = 200$$ pupils. (b) $$\\frac{3}{4}$$ of Volleyball $$= 64 \\div 4 = 16$$, then $$16 \\times 3 = 48$$. The activity with 48 pupils is Relay Race."},{"id":16779,"solution":"(a) Blue $$= 1 - \\frac{3}{8} = \\frac{5}{8}$$. (b) Blue minus red $$= \\frac{5}{8} - \\frac{3}{8} = \\frac{2}{8}$$, so 2 units $$= 34$$, 1 unit $$= 17$$. Total is 8 units $$= 17 \\times 8 = 136$$ counters."},{"id":16780,"solution":"(a) 9 units $$= 72$$, so 1 unit $$= 8$$; Rena $$= 4 \\times 8 = 32$$ stickers. (b) Sebastian has $$72 - 32 = 40$$. Equal share $$= 72 \\div 2 = 36$$, so Sebastian gives $$40 - 36 = 4$$ (i.e. $$8 \\div 2 = 4$$)."},{"id":16781,"solution":"(a) From the graph Jul=16, Aug=19, Sep=9, Oct=12, Nov=24. Total $$= 16 + 19 + 9 + 12 + 24 = 80$$ days. (b) December $$= 107 - 80 = 27$$ days."},{"id":16782,"solution":"(a) The arm RT lines up with the 135 mark read from the correct scale of the protractor, so $$\\angle QRT = 135°$$. (b) Reading the drawn angle x with a protractor gives $$\\angle x = 43°$$."},{"id":16783,"solution":"Each angle of a square is 90°, so $$\\angle ABC = 90°$$. The marked angle ABE is 63°, so $$\\angle CBE = 90° - 63° = 27°$$."},{"id":16784,"solution":"(a) The digit in the 3rd decimal place is 5, so round the 2nd decimal place up: $$168.205 \\approx 168.21$$. (b) Make the denominator 100: $$\\frac{3}{25} = \\frac{12}{100} = 0.12$$."},{"id":16785,"solution":"$$3.90 + 15.72 = 19.62$$. The hundredths digit is 2, so it rounds down: $$19.62 \\approx 19.6$$ to the nearest tenth."},{"id":16786,"solution":"The angle is marked at vertex M, between arms ML and MN, so it is named $$\\angle LMN$$ (vertex letter in the middle)."},{"id":16787,"solution":"6 ones = 6 and 72 hundredths = 0.72. The product is $$6 \\times 0.72 = 4.32$$."},{"id":16788,"solution":"Convert all to decimals: $$4\\frac{3}{4}=4.75$$, $$4.90$$, $$4\\frac{7}{10}=4.70$$, $$4.65$$. In increasing order: 4.65, 4.70, 4.75, 4.90, i.e. 4.65, $$4\\frac{7}{10}$$, $$4\\frac{3}{4}$$, 4.9."},{"id":16789,"solution":"1 bag = $$28.15 \\div 5 = 5.63$$ kg. 8 bags = $$5.63 \\times 8 = 45.04$$ kg."},{"id":16790,"solution":"(a)(i) $$\\angle a$$ is the corner of the rectangle at W, which is 90°, so TRUE. (ii) $$\\angle b$$ is larger than a right angle here, so it is not acute: FALSE. (b) At Z the rectangle corner is 90°. The 38° angle and angles c and d together make 90°, so $$\\angle c + \\angle d = 90° - 38° = 52°$$. Since $$\\angle c = \\angle d$$, $$\\angle c = 52° \\div 2 = 26°$$."},{"id":16791,"solution":"(a) Tuesday = Monday − 28.62 = $$42.10 - 28.62 = 13.48$$ m. (b) Total = Monday + Tuesday = $$42.10 + 13.48 = 55.58$$ m."},{"id":16793,"solution":"Number of one-thirds in 5 wholes = $$5 \\div \\frac{1}{3} = 5 \\times 3 = 15$$."},{"id":16795,"solution":"Angle $$f$$ is at vertex S, between sides SP and SR, so it is ∠PSR (equivalently ∠RSP). The vertex letter must be in the middle of the three-letter name."},{"id":16796,"solution":"Sugar = $$\\frac{3}{4} - \\frac{1}{8} = \\frac{6}{8} - \\frac{1}{8} = \\frac{5}{8}$$ kg. Total = $$\\frac{3}{4} + \\frac{5}{8} = \\frac{6}{8} + \\frac{5}{8} = \\frac{11}{8} = 1\\frac{3}{8}$$ kg."},{"id":16797,"solution":"White = $$\\frac{5}{6} \\times 72 = 60$$. Black = $$72 - 60 = 12$$. More white than black = $$60 - 12 = 48$$."},{"id":16798,"solution":"Each unit from 3 to 4 is divided into 5 equal parts. A points to the 4th mark after 3, which is $$3\\frac{4}{5}$$."},{"id":16799,"solution":"$$\\frac{1}{3} + \\frac{1}{9} = \\frac{3}{9} + \\frac{1}{9} = \\frac{4}{9}$$."},{"id":16800,"solution":"Sugar is $$\\frac{4}{7}$$ of the total, so salt is $$\\frac{3}{7}$$ of the total. 3 units = 24, so 1 unit = $$24 \\div 3 = 8$$. Sugar = 4 units = $$8 \\times 4 = 32$$ packets."},{"id":16801,"solution":"Using a protractor to measure the angle marked $$y$$ at the right-hand vertex of the triangle gives $$37°$$."},{"id":16802,"solution":"The corner of the square at Z is a right angle, so $$\\angle n + \\angle p = 90°$$. Since $$\\angle n = 2 \\times \\angle p$$, there are 3 units = 90°, so 1 unit = $$90 \\div 3 = 30°$$. $$\\angle n$$ = 2 units = $$30 \\times 2 = 60°$$."},{"id":16803,"solution":"Hani got $$\\frac{5}{8}$$ and Ivy got $$\\frac{1}{4} = \\frac{2}{8}$$. The difference is $$\\frac{5}{8} - \\frac{2}{8} = \\frac{3}{8}$$, which equals 12 stickers. So 3 units = 12, 1 unit = $$12 \\div 3 = 4$$. Total = 8 units = $$8 \\times 4 = 32$$ stickers."},{"id":16804,"solution":"(a) Each cup = jug − $$\\frac{3}{10}$$ = $$\\frac{2}{5} - \\frac{3}{10} = \\frac{4}{10} - \\frac{3}{10} = \\frac{1}{10}$$ ℓ. (b) Total = jug + 2 cups = $$\\frac{2}{5} + \\frac{1}{10} + \\frac{1}{10} = \\frac{4}{10} + \\frac{1}{10} + \\frac{1}{10} = \\frac{6}{10} = \\frac{3}{5}$$ ℓ."},{"id":16809,"solution":"5840 $$\\div$$ 7 = 834 with a remainder. 834 $$\\times$$ 7 = 5838, and 5840 $$-$$ 5838 = 2. The remainder is 2."},{"id":16810,"solution":"Make the denominators the same: $$\\frac{1}{2} = \\frac{4}{8}$$. Then $$\\frac{4}{8} - \\frac{3}{8} = \\frac{1}{8}$$."},{"id":16811,"solution":"To change the denominator from 3 to 12, multiply by 4. Multiply the numerator by the same: 1 $$\\times$$ 4 = 4. So $$\\frac{1}{3} = \\frac{4}{12}$$."},{"id":16814,"solution":"Class 4B has 18 participants. Twice 18 = 18 $$\\times$$ 2 = 36. From the table, Class 4E has 36 participants."},{"id":16815,"solution":"Multiples of 9 between 60 and 80: 63 and 72. The number must also be a multiple of 6. 63 $$\\div$$ 6 = 10 remainder 3 (not a multiple of 6); 72 $$\\div$$ 6 = 12 (a multiple of 6). So the number is 72."},{"id":16816,"solution":"Let the smaller number be 1 unit; the greater is 4 units. Together 5 units = 7390. 1 unit = 7390 $$\\div$$ 5 = 1478. Greater number = 4 units = 1478 $$\\times$$ 4 = 5912."},{"id":16819,"solution":"When unfolded, the rectangle's length is the visible 22 cm plus the folded-over part. The folded flap adds 10 cm + 8 cm to the length: 22 + 10 + 8 = 40 cm; width = 10 cm. (a) Perimeter = 40 + 10 + 40 + 10 = 100 cm. (b) Area = 40 $$\\times$$ 10 = 400 $$cm^2$$."},{"id":16820,"solution":"Let Xavier + Yusof = 1 unit. Zachary = 2 units. Total = 3 units = 561, so 1 unit = 561 $$\\div$$ 3 = 187 (Xavier + Yusof). Xavier is 93 more than Yusof: Yusof = (187 - 93) $$\\div$$ 2 = 47, Xavier = 47 + 93 = 140."},{"id":16821,"solution":"(a) $80 donated at $5 per 30 cupcakes: 80 $$\\div$$ 5 = 16 sets, 16 $$\\times$$ 30 = 480 cupcakes. (b) Sales = 480 $$\\times$$ $3 = $1440. After donating $80: 1440 - 80 = $1360."},{"id":16822,"solution":"Three whole circles are fully shaded and the fourth circle has 5 of its 8 parts shaded. Total = $$3\\frac{5}{8} = \\frac{3\\times8+5}{8} = \\frac{29}{8}$$. Answer: $$\\frac{29}{8}$$ (option 2)."},{"id":16823,"solution":"Between 4 and 5 the number line is divided into 5 equal parts, so each part is $$\\frac{1}{5}$$. A is at the 3rd mark after 4, giving $$4\\frac{3}{5}$$ (option 3)."},{"id":16824,"solution":"Convert to decimals: $$1\\frac{4}{9}\\approx1.44$$, $$\\frac{5}{3}\\approx1.67$$, $$\\frac{15}{7}\\approx2.14$$. Increasing order: $$1\\frac{4}{9}$$, $$\\frac{5}{3}$$, $$\\frac{15}{7}$$ (option 2)."},{"id":16826,"solution":"There are 15 hearts. $$\\frac{2}{3}\\times15 = 10$$ hearts must be shaded. 3 are already shaded, so $$10 - 3 = 7$$ more are needed (option 1)."},{"id":16827,"solution":"Total pizza needed = $$35\\times\\frac{1}{6} = \\frac{35}{6} = 5\\frac{5}{6}$$ pizzas. Since only whole pizzas are sold, round up to 6 (option 2)."},{"id":16828,"solution":"(a) $$\\frac{22}{4} = 5\\frac{2}{4} = 5\\frac{1}{2}$$. (b) $$\\frac{2}{9}$$ of 63: $$63\\div9 = 7$$, then $$7\\times2 = 14$$."},{"id":16829,"solution":"$$5 - \\frac{3}{8} = 4\\frac{8}{8} - \\frac{3}{8} = 4\\frac{5}{8}$$ litres."},{"id":16830,"solution":"From the graph: 5 pm = $140, 6 pm = $400, 7 pm = $360. Total = $$140 + 400 + 360 = 900$$ dollars."},{"id":16833,"solution":"Walk = $$\\frac{3}{4} - \\frac{5}{12} = \\frac{9}{12} - \\frac{5}{12} = \\frac{4}{12}$$ km. Total = $$\\frac{3}{4} + \\frac{4}{12} = \\frac{9}{12} + \\frac{4}{12} = \\frac{13}{12} = 1\\frac{1}{12}$$ km."},{"id":16835,"solution":"Method: a figure shows $$\\frac{1}{3}$$ shaded when the number of shaded parts is one third of the total equal parts. Figure 3 is a 2-by-3 grid of 6 equal cells with 2 cells shaded, and $$\\frac{2}{6}=\\frac{1}{3}$$. The answer is option 3."},{"id":16836,"solution":"Method: Saturday total = sweets + lollipops + chocolates. So sweets = total minus the other two. $$1830-425-878=527$$. The answer is option 1, 527."},{"id":16837,"solution":"Method: Sunday sweets + chocolates = total minus lollipops = $$1150-670=480$$. Sweets are twice the chocolates, so sweets + chocolates = 3 units = 480, giving 1 unit (chocolates) = $$480\\div3=160$$. The answer is option 2, 160."},{"id":16838,"solution":"Method: read both values from the line graph. March = 20 stickers, April = 48 stickers. Difference = $$48-20=28$$."},{"id":16839,"solution":"Method: the money is 9 equal units. She spent $$\\frac{5}{9}$$, so $$\\frac{4}{9}$$ is left. 9 units = \\$954, so 1 unit = $$954\\div9=106$$... using the unitary method 4 units left = \\$424. The answer is \\$424."},{"id":16840,"solution":"(a) Check each pair against 1 kg. Bag B + Bag C = $$\\frac{3}{5}+\\frac{5}{8}=\\frac{24}{40}+\\frac{25}{40}=\\frac{49}{40}$$ which is more than 1 kg; the other pairs are less. So Bag B and Bag C. (b) Laptop mass = total minus Bag B = $$3-\\frac{3}{5}=2\\frac{2}{5}$$ kg."},{"id":16841,"solution":"Method: Harry = Ali minus $$\\frac{1}{5}$$ = $$\\frac{7}{10}-\\frac{2}{10}=\\frac{5}{10}$$. Total = Ali + Harry = $$\\frac{7}{10}+\\frac{5}{10}=\\frac{12}{10}=1\\frac{2}{10}$$ litres."},{"id":16842,"solution":"(a) Rope B = Rope A minus the difference = $$\\frac{9}{10}-\\frac{3}{4}=\\frac{18}{20}-\\frac{15}{20}=\\frac{3}{20}$$ m. (b) Rope C is longer than Rope B by $$\\frac{1}{2}$$, so Rope C = $$\\frac{3}{20}+\\frac{1}{2}=\\frac{3}{20}+\\frac{10}{20}=\\frac{13}{20}$$ m."},{"id":16843,"solution":"Method: treat the total as 7 units. Pears = 2 units, apples = 4 units, watermelons = $$7-2-4=1$$ unit. Apples minus watermelons = $$4-1=3$$ units = 360, so 1 unit = $$360\\div3=120$$. (a) Pears = 2 units = $$2\\times120=240$$. (b) Total = 7 units = $$7\\times120=840$$."},{"id":16859,"solution":"Each whole contains 5 one-fifths. For 3 wholes: $$3 \\times 5 = 15$$ one-fifths."},{"id":16860,"solution":"Read the protractor along the scale where OB lies on 0. Ray OA passes through the 130 mark, so $$\\angle AOB = 130^\\circ$$."},{"id":16861,"solution":"Make a common denominator: $$\\frac{2}{3} = \\frac{8}{12}$$. Then $$\\frac{11}{12} - \\frac{8}{12} = \\frac{3}{12} = \\frac{1}{4}$$."},{"id":16863,"solution":"a) $$4269 \\times 3 = 12807$$. b) $$217 \\times 15 = 217 \\times 10 + 217 \\times 5 = 2170 + 1085 = 3255$$."},{"id":16864,"solution":"Each interval between whole numbers is divided into 6 equal parts. A sits at 2 parts past 1, i.e. $$1\\frac{2}{6} = 1\\frac{1}{3}$$."},{"id":16865,"solution":"Compare values: $$\\frac{2}{3} \\approx 0.67$$, $$\\frac{4}{5} = 0.8$$, $$\\frac{5}{3} \\approx 1.67$$. Increasing order: 2\/3, 4\/5, 5\/3."},{"id":16868,"solution":"$$4505 \\div 6 = 750$$ remainder $$5$$, since $$6 \\times 750 = 4500$$ and $$4505 - 4500 = 5$$."},{"id":16869,"solution":"Watch cost $$\\frac{1}{3} \\times 105 = \\$35$$. The rest (present) is $$\\frac{2}{3}$$ of 105, i.e. $$105 - 35 = \\$70$$."},{"id":16870,"solution":"Janice is unchanged. Let Kenneth's start = K, so Janice = 2K. After giving away 24, Kenneth = K - 24 and 2K = 5(K - 24). So $$2K = 5K - 120$$, $$3K = 120$$, $$K = 40$$. Kenneth had 40 stamps."},{"id":16871,"solution":"Let f = number of friends. Chocolates = $$4f + 2$$ (2 extra) and also $$5f - 1$$ (needs 1 more). So $$4f + 2 = 5f - 1$$, giving $$f = 3$$. Chocolates $$= 4 \\times 3 + 2 = 14$$ (less than 30, checks out)."},{"id":16872,"solution":"Beng Huat at first $$= 198 - 108 = \\$90$$. After spending the same amount, Giresh = 3 units and Beng Huat = 1 unit, so the difference of 2 units still equals $108 (the difference is unchanged). $$1 \\text{ unit} = 108 \\div 2 = \\$54$$ = Beng Huat's money left. So each spent $$90 - 54 = \\$36$$."},{"id":16873,"solution":"The large triangle is divided into 9 identical small triangles. 2 of them are shaded, so the shaded fraction is $$\\frac{2}{9}$$."},{"id":16874,"solution":"A right angle is 90°. Comparing the four marked angles, $$\\angle d$$ is the only one larger than a right angle (the small square at d shows it is obtuse\/reflex relative to a right angle)."},{"id":16878,"solution":"(a) $$15 \\div 7 = 2$$ remainder $$1$$, so $$\\frac{15}{7} = 2\\frac{1}{7}$$. (b) $$0.75 = \\frac{75}{100} = \\frac{3}{4}$$ after dividing numerator and denominator by 25."},{"id":16879,"solution":"(a) Convert to decimals: $$\\frac{13}{5}=2.6$$, $$2\\frac{3}{4}=2.75$$, $$\\frac{7}{8}=0.875$$. In decreasing order: $$2\\frac{3}{4}$$, $$\\frac{13}{5}$$, $$\\frac{7}{8}$$. (b) $$\\frac{15}{24}=\\frac{5}{8}$$ (divide top and bottom by 3), so the missing number is 5."},{"id":16882,"solution":"He saved $$1-\\frac{5}{8}=\\frac{3}{8}$$ of his money. $$\\frac{1}{8}$$ of $120 = $120 \\div 8 = $15. Saved = $$\\frac{3}{8}$$ = $15 \\times 3 = $45."},{"id":16883,"solution":"$$\\angle ADC = 90°$$ (corner of a square). The two marked angles at D are 24° and 35°, with $$\\angle y$$ between them: $$\\angle y = 90° - 24° - 35° = 31°$$."},{"id":16884,"solution":"John drank $$\\frac{7}{10}-\\frac{1}{5}=\\frac{7}{10}-\\frac{2}{10}=\\frac{5}{10}$$ $$\\ell$$. Together: $$\\frac{7}{10}+\\frac{5}{10}=\\frac{12}{10}=1.2$$ $$\\ell$$."},{"id":16885,"solution":"Revised over the weekend: $$\\frac{1}{4}+\\frac{1}{6}=\\frac{3}{12}+\\frac{2}{12}=\\frac{5}{12}$$. Left to read: $$1-\\frac{5}{12}=\\frac{7}{12}$$, which equals 98 pages. So $$\\frac{1}{12}=98\\div7=14$$ pages, and the whole textbook = $$14\\times12=168$$ pages."},{"id":16886,"solution":"(a) Let Ali = 1 unit, then Benny = 3 units, Ali+Benny = 4 units, Charles = half of that = 2 units. Total = 1+3+2 = 6 units = 1260, so 1 unit = 1260 ÷ 6 = 210. Benny = 3 × 210 = 630. (b) Charles = 2 units = 420 marbles. He gave away $$\\frac{5}{6}$$, so he kept $$\\frac{1}{6}$$ = 420 ÷ 6 = 70 marbles."},{"id":16887,"solution":"(a) Blue fraction = $$1-\\frac{7}{9}=\\frac{2}{9}$$. (b) Red is $$\\frac{7}{9}$$ and blue is $$\\frac{2}{9}$$, so red exceeds blue by $$\\frac{5}{9}$$ = 130 beads. So $$\\frac{1}{9}=130\\div5=26$$ beads, and total = $$26\\times9=234$$ beads."},{"id":16888,"solution":"Divide 26 by 7: $$26 \\div 7 = 3$$ remainder $$5$$. So $$\\frac{26}{7} = 3\\frac{5}{7}$$."},{"id":16889,"solution":"$$\\frac{2}{9}$$ of 18 = $$18 \\div 9 \\times 2 = 2 \\times 2 = 4$$."},{"id":16890,"solution":"The marked angle x has its vertex at A and is formed by rays AC and AD. So it is $$\\angle CAD$$ (equivalently $$\\angle DAC$$)."},{"id":16891,"solution":"Read the protractor scale aligned with the baseline from 0°. The arm passes through the 41° mark on the inner scale, so $$\\angle a = 41°$$."},{"id":16893,"solution":"$$\\frac{2}{3} - \\frac{3}{5} = \\frac{10}{15} - \\frac{9}{15} = \\frac{1}{15}$$."},{"id":16894,"solution":"Left = $$1 - \\frac{1}{6} + \\frac{1}{3} = \\frac{6}{6} - \\frac{1}{6} + \\frac{2}{6} = \\frac{7}{6} = 1\\frac{1}{6}\\ \\ell$$."},{"id":16895,"solution":"Measuring $$\\angle x$$ at the marked vertex with a protractor gives $$\\angle x = 42°$$."},{"id":16896,"solution":"There are 18 triangles in total and 6 are shaded. $$\\frac{6}{18} = \\frac{1}{3}$$."},{"id":16897,"solution":"$$\\frac{1}{4} + \\frac{1}{6} = \\frac{3}{12} + \\frac{2}{12} = \\frac{5}{12}$$."},{"id":16898,"solution":"For 4B, total = 41 and do-not-wear = 19, so wear spectacles = $$41 - 19 = 22$$."},{"id":16899,"solution":"Add the 'do not wear spectacles' row: $$23 + 19 + 13 = 55$$."},{"id":16900,"solution":"The interval from 4 to 5 is divided into 5 equal parts. A is at the 2nd mark, so A = $$4\\frac{2}{5}$$ = $$\\frac{22}{5}$$."},{"id":16901,"solution":"Sam = $$\\frac{5}{7} - \\frac{1}{2} = \\frac{10}{14} - \\frac{7}{14} = \\frac{3}{14}$$ km. Total = $$\\frac{5}{7} + \\frac{3}{14} = \\frac{10}{14} + \\frac{3}{14} = \\frac{13}{14}$$ km."},{"id":16902,"solution":"The empty part is $$1 - \\frac{3}{8} = \\frac{5}{8}$$, which equals 120 L. So $$\\frac{1}{8} = 120 \\div 5 = 24$$ L. Capacity = $$24 \\times 8 = 192$$ L."},{"id":16903,"solution":"Mdm Siti used $$\\frac{3}{4} - \\frac{1}{3} = \\frac{9}{12} - \\frac{4}{12} = \\frac{5}{12}$$ kg. Altogether = $$\\frac{3}{4} + \\frac{5}{12} = \\frac{9}{12} + \\frac{5}{12} = \\frac{14}{12} = 1\\frac{1}{6}$$ kg."},{"id":16904,"solution":"Children = $$\\frac{5}{6}$$, adults = $$\\frac{1}{6}$$. Difference = $$\\frac{5}{6} - \\frac{1}{6} = \\frac{4}{6}$$ of the people = 240, so 4 units = 240, 1 unit = 60. Total = 6 units = $$60 \\times 6 = 360$$."},{"id":16905,"solution":"Method: cost of silk = price x quantity, then subtract from total. Silk: $$\\$2 \\times 3 = \\$6$$. Money left for Kate $$= \\$21 - \\$6 = \\$15$$. Answer: $15."}]