[{"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":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":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":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":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":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":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":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":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":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":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":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."}]