Arrange the following fractions from the greatest to the smallest.<br>\(\dfrac{2}{3}\), \(\dfrac{5}{6}\), \(\dfrac{7}{12}\)


To round 8.745 to 2 decimal places, look at the 3rd decimal digit (5). Since it is 5, round up the 2nd decimal digit: 8.74 becomes 8.75.


Linda baked 200 cookies. She sold 60 of them. What percentage of the cookies Linda baked was sold?


A machine takes 3 min to print 4 posters. At the same rate, how long will it take to print 24 posters?

24 posters is 24 ÷ 4 = 6 times as many as 4 posters. Time = 6 × 3 min = 18 min.


1 hour = 60 minutes. 4 hours = 4 × 60 = 240 minutes.


There are 78 canned drinks in a carton. 24 of them are mango flavoured and the rest are lychee flavoured. What is the ratio of the number of mango flavoured drinks to the number of lychee flavoured drinks?

Lychee drinks = 78 − 24 = 54. Ratio mango : lychee = 24 : 54 = 4 : 9... check: 24:54 divide by 6 = 4:9. The printed key gives option 4 (9:13). Note: 24:54 simplifies to 4:9 (option 1).


Evan was given $4 pocket money every day from Monday to Friday. He spent $3.20 each day from Monday to Thursday and saved the rest. He spent 30 cents more on Friday. How much did Evan save altogether?


The average number of students in 3 classes is 32. Class A has 37 students while Class B has 3 more students than Class A. What is the number of students in Class C?

Total students = 3 × 32 = 96. Class B = 37 + 3 = 40. Class C = 96 − 37 − 40 = 19.


Devi had some 10-cents, 20-cents and 50-cents coins in a box. There were twice as many 20-cent coins as 50-cent coins in the box. The number of 10-cent coins was the same as the number of 50-cent coins. The total value of all her coins was $23. How many coins were there in the box altogether?

Let 50c coins = u, then 10c coins = u and 20c coins = 2u. Value = 50u + 10u + 20(2u) = 50u + 10u + 40u = 100u cents per unit-group = $1 per u. Total $23 means u = 23. Coins = u + u + 2u = 4u = 4 × 23 = 92.


Ali listed the factors of 18.
1, 3, 9, 18
He missed out two factors. What are the two missing factors?
<input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> and <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

The factors of 18 are 1, 2, 3, 6, 9, 18. The listed ones are 1, 3, 9, 18. The missing factors are 2 and 6.


What is the value of \(64 - (9 + 11) \div 5 \times 2\)?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Brackets first: 9 + 11 = 20. Then \(20 \div 5 = 4\), then \(4 \times 2 = 8\). Finally 64 − 8 = 56.


Find the value of \(\dfrac{4}{9} \times 5\). Leave your answer in mixed number.


Gabriel had 340 Pokemon cards for sale. He sold 35% of them yesterday. How many Pokemon cards did he sell yesterday?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Keming, Ray and Hassan shared a box of stickers in the ratio 5 : 9 : 3. Ray had 32 stickers more than Keming. How many stickers did the three boys have altogether?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Sam kept some red and blue pens in bags A and B. Bag A contained twice as many pens as bag B. In bag A, \(\dfrac{1}{5}\) of the pens were red pens and the rest were blue pens. In bag B, \(\dfrac{1}{3}\) of the pens were red pens and the rest were blue pens. What fraction of Sam's pens were red?

Let bag B = 15 units, bag A = 30 units (twice). Red in A = \(\dfrac{1}{5}\times 30 = 6\); red in B = \(\dfrac{1}{3}\times 15 = 5\). Total red = 6 + 5 = 11; total pens = 30 + 15 = 45. Fraction red = \(\dfrac{11}{45}\).


In 38 564,
(a) which digit is in the hundreds place? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) what is the value of the digit 3? <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) In 38 564 the hundreds place digit is 5. (b) The digit 3 is in the ten-thousands place, so its value is 30 000.


Two numbers add up to 674. Both numbers are 3-digit numbers. What is the greatest possible difference between the two numbers?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

To make the difference greatest, make one number as small as possible (smallest 3-digit number, 100). The other = 674 − 100 = 574. Difference = 574 − 100 = 474.


The rate to send letters to a country is: first 20 g costs $0.85, every additional 10 g or part thereof costs $0.25. Ben sent a letter that weighed 43 g. How much did Ben pay for sending the letter? Give your answer in dollars.<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

43 g = first 20 g + 23 g extra. The extra 23 g is 3 lots of 10 g (10 + 10 + 3, part thereof counts as a full block). Cost = $0.85 + 3 × $0.25 = $0.85 + $0.75 = $1.60.


In a library, books were placed on 40 shelves with equal number of books on each shelf. 6 shelves were removed and the books on these shelves were placed on the remaining 34 shelves. Because of this, the number of books on each shelf increased by 9. What was the number of books on each shelf at first?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The 6 removed shelves' books are spread over 34 shelves, raising each by 9, so total moved = 34 × 9 = 306. These came from 6 shelves, so books per shelf at first = 306 ÷ 6 = 51.


The mass of a box with 50 identical beads weighs 0.9 kg. When 30 beads are removed, the mass of the box with the remaining beads is 0.54 kg. What is the mass of each bead in kilogram?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Removing 30 beads reduced the mass by 0.9 − 0.54 = 0.36 kg. So 30 beads weigh 0.36 kg, and each bead = 0.36 ÷ 30 = 0.012 kg.


A baker baked some cupcakes for sale on Saturday morning. He sold 65% of the cupcakes on Saturday and found that he had 203 cupcakes left.
(a) How many cupcakes did the baker bake on Saturday morning? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) He sold the remaining cupcakes on Sunday at a discount of 20% off the price of $3.50 each. How much were all the cupcakes sold for on Sunday? Give your answer in dollars. $<input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) 203 left = 35% of total, so total = \(\dfrac{203}{35}\times 100 = 580\) cupcakes. (b) Discounted price = $3.50 × 80% = $2.80. The 203 remaining cupcakes sold for 2.80 × 203 = $568.40.


Mrs Chan bought an equal number of pots and pans. The ratio of the cost of a pot to the cost of a pan was 3 : 2. The average price of a pot and a pan was $11.25. Mrs Chan paid $45 less for the pans than the pots. How many pans did Mrs Chan buy?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Total of one pot + one pan = 11.25 × 2 = $22.50. The cost difference per pot-pan pair (3 − 2 = 1 unit) is 22.50 ÷ 5 = $4.50. Total difference is $45, so number of pairs = 45 ÷ 4.50 = 10. Hence 10 pans.


Two children used the same number of ice cream sticks to make toy houses. Don used \(\dfrac{6}{7}\) of his ice cream sticks while Lea used \(\dfrac{3}{4}\) of her ice cream sticks. They had a total of 4350 ice cream sticks at first.
(a) How many ice cream sticks did each of them use? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) How many ice cream sticks must Lea give Don so that they will have an equal number of ice cream sticks left? <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) \(\dfrac{6}{7}\) of Don = \(\dfrac{3}{4}\) of Lea (used equal numbers). Common multiple: Don = 7 units of his (6 used), Lea = 8 units of hers (6 used), used = 6 of each unit-set. Don:Lea total units = 7:8 = 15 parts = 4350, so 1 part = 290. Used = 6 × 290 = 1740 each. (b) Don left = 1×290 = 290; Lea left = 2×290 = 580. To equalise: (580+290)÷2 = 435, Lea gives 580 − 435 = 145.


What is three hundred and sixty thousand and six hundred in numerals?

Three hundred and sixty thousand = 360 000; plus six hundred = 600. So 360 000 + 600 = 360 600. Answer: 360 600.


What is the value of \(50 - (5 + 25 \div 5)\)?

Work inside brackets first using order of operations: 25 ÷ 5 = 5, then 5 + 5 = 10. Finally 50 − 10 = 40. Answer: 40.


4.04 has 4 ones and 4 hundredths (the 0 is in the tenths place, 4 in the hundredths place). So 4.04 = \(4\dfrac{4}{100}\).


What is the missing number in the box?<br>\(12 : 36 = [\,?\,] : 42\)

12 : 36 simplifies to 1 : 3. So the first term is one third of the second. Missing : 42 = 1 : 3, so missing = 42 ÷ 3 = 14. Answer: 14.


Which percentage is greater than 0.1 but less than 0.15?

Convert each percentage to a decimal: 0.14% = 0.0014, 1.2% = 0.012, 12% = 0.12, 15% = 0.15. Only 0.12 is greater than 0.1 and less than 0.15. Answer: 12%.


Jamie read an average of 72 pages of a book in a day. She finished reading the book in 9 days. How many pages are there in the book?

Total = average × number of items = 72 × 9 = 648 pages. Answer: 648.


Which of the following is not an example of rate?

A rate compares a quantity per unit of another quantity (per minute, per hour, per day). '75 marks in a test' has no 'per unit' comparison, so it is not a rate. Answer: option 1.


A number when divided by 8 gives a remainder of 3. Which of the following can be added to the number to make it a multiple of 4?

A number with remainder 3 when divided by 8 is of the form 8n + 3. Take 11 (8×1+3). 11 + 1 = 12, which is a multiple of 4. So adding 1 works. Answer: 1.


There are 12 lamp-posts on a straight path. The lamp-posts are installed 0.14 km apart from one another. What is the distance between the second lamp-post and the tenth lamp-post?

From the 2nd to the 10th lamp-post there are 10 − 2 = 8 gaps. Distance = 8 × 0.14 = 1.12 km. Answer: 1.12 km.


The average daily allowance of 5 students is $4. Another student joined in and their average daily allowance increased to $4.30. How much daily allowance did the last student receive?

Total of 5 students = 5 × $4 = $20. After 1 more student, total of 6 = 6 × $4.30 = $25.80. The new student = $25.80 − $20 = $5.80. Answer: $5.80.


Calculate \(\dfrac{3}{8} \times \dfrac{4}{7}\).

Multiply numerators and denominators: \(\dfrac{3}{8} \times \dfrac{4}{7} = \dfrac{3 \times 4}{8 \times 7} = \dfrac{12}{56}\), which simplifies to \(\dfrac{3}{14}\). Answer: \(\dfrac{3}{14}\).


Danielle has 5 times as many stamps as Jackson. Danielle has 40 stamps more than Jackson. How many stamps does Jackson have? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Danielle = 5 units, Jackson = 1 unit. The difference is 5 − 1 = 4 units = 40 stamps, so 1 unit = 40 ÷ 4 = 10. Jackson has 10 stamps. Answer: 10.


Write down a decimal between 4.6 and 4.7. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Any decimal strictly between 4.6 and 4.7 works, e.g. 4.65 (the printed key answer). Answer: 4.65.


Peter ran 3.8 km and Evelyn ran 400 m more than Peter. Calculate the total distance Evelyn and Peter ran in metres. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Karen has some red and blue marbles. For every 4 red marbles, there are 9 blue marbles. There are 36 red marbles. How many blue marbles are there? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Red : blue = 4 : 9. With 36 red, the multiplier is 36 ÷ 4 = 9. Blue = 9 × 9 = 81. Answer: 81.


Gerald deposited $3000 in his savings account which paid him an interest of 3% per year. How much interest did he receive at the end of the year? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mrs Choo baked some tarts and muffins. After giving away \(\dfrac{1}{4}\) of her tarts and \(\dfrac{1}{7}\) of her muffins, she has the same number of tarts and muffins left. What is the ratio of the number of tarts to the number of muffins Mrs Choo baked?

Tarts left = \(\dfrac{3}{4}\) of tarts; muffins left = \(\dfrac{6}{7}\) of muffins. These are equal: \(\dfrac{3}{4} \times T = \dfrac{6}{7} \times M\). So tarts left : muffins left = 12 : 12, and baked ratio T : M = \(\dfrac{12}{3/4}\) : \(\dfrac{12}{6/7}\) = 16 : 14 = 8 : 7. Answer: 8 : 7.


A printer prints 684 pages in 38 minutes. How many pages can the printer print in an hour? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Pages per minute = 684 ÷ 38 = 18. In 1 hour (60 min) = 18 × 60 = 1080 pages. Answer: 1080.


Lily paid $114 for 6 similar tops and 3 similar pairs of shorts. Hazel paid $92 for 3 similar tops and 4 similar pairs of shorts. How much did each top cost? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Double Hazel: 6 tops + 8 shorts = $184. Subtract Lily (6 tops + 3 shorts = $114): 5 shorts = 184 − 114 = $70, so 1 short = $14. Then 3 shorts = $42; 6 tops = 114 − 42 = $72; 1 top = 72 ÷ 6 = $12. Answer: $12.


Mr Leo had 12.6 kg of flour. He kept \(\dfrac{1}{3}\) of the flour and divided the remaining flour equally into 7 bags. (a) What was the mass of the flour that Mr Leo kept for himself? (b) How much flour was there in each bag? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) Kept = \(\dfrac{1}{3} \times 12.6 = 4.2\) kg. (b) Remaining = 12.6 − 4.2 = 8.4 kg, divided into 7 bags = 8.4 ÷ 7 = 1.2 kg per bag. Answers: (a) 4.2 kg, (b) 1.2 kg.


In June, Mr Chee saved 12% of his salary. In July, his monthly salary increased by $500 and he saved 10% of his new monthly salary. Given that Mr Chee saves the same amount of money every month, find Mr Chee's salary in July. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mrs Nashwa had a bag of rice. Her family ate an equal amount of rice each week. After 6 weeks, she had \(\dfrac{4}{7}\) of the rice left. After another 5 weeks, she had 2.1 kg of rice left. How much rice was in the bag at first? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

After 6 weeks, \(\dfrac{4}{7}\) left means \(\dfrac{3}{7}\) eaten in 6 weeks. As fourteenths: \(\dfrac{3}{7} = \dfrac{6}{14}\) eaten in 6 weeks. In the next 5 weeks the family ate \(\dfrac{5}{14}\) (same weekly rate, 1/14 per week), leaving \(1 - \dfrac{6}{14} - \dfrac{5}{14} = \dfrac{3}{14}\) = 2.1 kg. So 1 fourteenth = 2.1 ÷ 3 = 0.7 kg; whole bag = 14 × 0.7 = 9.8 kg. Answer: 9.8 kg.


A group of children took part in a survey on their favourite colour. \(\dfrac{3}{5}\) of them like green, \(\dfrac{1}{3}\) of the remainder and an additional 16 children like blue, and the remaining 28 children like red. How many more children like green than blue? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Green = 3/5, remainder = 2/5. Of the remainder, 1/3 likes blue (plus 16), leaving 2/3 of the remainder. Red = 28 = 2/3 of remainder minus the 16 shifted to blue: the printed working gives remainder ÷ parts so that 16 + 28 = 44 = (2/3 of remainder) → remainder ÷... Solving: total = 11 units of green/remainder. 44 ÷ 4 = 11 (one unit). Green = 11 × 9 = 99; blue = 11 × 2 + 16 = 22 + 16 = 38. More green than blue = 99 − 38 = 61. Answer: 61.


A container has a square base with sides 24 cm each. Rajan fills \(\dfrac{5}{9}\) of the container with water. Another 6.4 litres of water are needed to fill the container to the brim. What is the height of the container? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The empty part = 1 − 5/9 = 4/9 of the container = 6.4 litres. So 1/9 = 6.4 ÷ 4 = 1.6 litres, full container = 9 × 1.6 = 14.4 litres = 14 400 cm³. Volume = base area × height: 14 400 = 24 × 24 × height → height = 14 400 ÷ 24 ÷ 24 = 25 cm. Answer: 25 cm.


Chloe and Arthur have the same number of coins. Chloe has a number of fifty-cent coins and 27 twenty-cent coins. Arthur has twice as many twenty-cent coins as Chloe. (a) Who has more money and how much more? (b) Given that Chloe has $33.40, how many fifty-cent coins does Arthur have? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

(a) Both have the same total number of coins. Arthur has 2×27 = 54 twenty-cent coins where Chloe has 27, so Chloe must have 27 more fifty-cent coins than Arthur. Each swap of a fifty-cent for a twenty-cent loses 0.50 − 0.20 = $0.30, over 27 coins = 27 × 0.30 = $8.10. So Chloe has $8.10 more. (b) Chloe = $33.40. Chloe's fifty-cent value = 33.40 − 8.10 = $25.30. Arthur's twenty-cent value = 54 × 0.20 = $10.80, so Arthur's fifty-cent value = 25.30 − 10.80 = $14.50, giving 14.50 ÷ 0.50 = 29 fifty-cent coins. Answer (b): 29.


Which of the following is six million, thirty-five thousand and twenty in numerals?


Betty used \(\dfrac{3}{8}\) of her flour to bake a cake and \(\dfrac{1}{5}\) of the remaining flour to bake cookies. What fraction of her flour did she use to bake cookies?

Remaining flour after the cake \(= 1 - \dfrac{3}{8} = \dfrac{5}{8}\). Cookies use \(\dfrac{1}{5}\) of this: \(\dfrac{1}{5}\times \dfrac{5}{8} = \dfrac{1}{8}\).


Ahmad had $200. He spent $60 on a bag and saved the rest of the money. What percentage of his money did Ahmad save?


Which decimal is greater than 1.07 but smaller than 1.15?

A value between 1.07 and 1.15 is needed. 1.1 lies between them (1.07 < 1.1 < 1.15). 1.05 is too small, 1.17 and 1.5 are too large.


Aminah, Ban Leong and Colin share some stickers in the ratio 1 : 6 : 2. The biggest share is 54 stickers. What is Colin's share?

The biggest share belongs to Ban Leong: 6 units = 54, so 1 unit = 9. Colin has 2 units = 2 × 9 = 18.


Find the value of \(23 \div 9\). Express your answer as a mixed number in the simplest form.

\(23 \div 9 = \dfrac{23}{9} = 2\dfrac{5}{9}\) (since 9 goes into 23 twice with remainder 5).


How many minutes are there in \(1\dfrac{3}{5}\) h? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> min


6000 ml of paint was poured into 5 containers equally. How many litres of paint were there in one container? Express your answer as a decimal. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> ℓ

6000 ml ÷ 5 = 1200 ml per container. 1200 ml = 1.2 ℓ.


Dora folds 12 paper cranes in 10 minutes. At this rate, how many paper cranes can Dora fold in 15 minutes? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

In 15 minutes she folds \(12\times \dfrac{15}{10} = \dfrac{180}{10} = 18\) cranes. (Or 12 cranes per 10 min = 1.2 cranes/min; 1.2 × 15 = 18.)


There are 140 students in the hall. 75% of the students are boys. How many girls are there? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Jane had some water in her bottle. After she drank half the amount, she added in another \(\dfrac{4}{5}\) ℓ. There were \(1\dfrac{1}{3}\) ℓ of water left in the bottle. How many litres of water were there in the bottle at first? Express your answer as a mixed number in the simplest form.

Half the original amount remained after drinking, then \(\dfrac{4}{5}\) ℓ was added to give \(1\dfrac{1}{3}\) ℓ. So half the original \(= 1\dfrac{1}{3} - \dfrac{4}{5} = \dfrac{20}{15} - \dfrac{12}{15} = \dfrac{8}{15}\) ℓ. The original amount \(= \dfrac{8}{15}\times 2 = \dfrac{16}{15} = 1\dfrac{1}{15}\) ℓ.


There were 30 lampposts along a road. The lampposts were at the same distance apart. The distance between the first and third lamppost was 90 m. What was the distance between the first and the last lamppost? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> m

First to third lamppost spans 2 gaps = 90 m, so 1 gap = 45 m. From the 1st to the 30th lamppost there are 30 − 1 = 29 gaps. Distance = 45 × 29 = 1305 m.


Emerlyn had 3.45 m of string. She used 102 cm of it to tie a parcel and cut another 4 pieces of string, each 33 cm long, to wrap some presents. How much string did she have left? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> m


A pastry chef takes 27 minutes to decorate a cake. At this rate, how much time is needed to decorate 8 cakes? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> h <input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> min

8 cakes × 27 min = 216 min. 216 ÷ 60 = 3 remainder 36, so 216 min = 3 h 36 min.


Jolie decorated the room with green and blue lights. The green light blinked every 6 minutes and the blue light blinked every 8 minutes. Both lights were turned on at the same time. After how many minutes did all the lights blink together for the 7th time? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> min

Both lights blink together every LCM(6, 8) = 24 minutes. The 7th time together is at 24 × 7 = 168 minutes.


Mr Low took a taxi from home to his office. His taxi fare was based on the charges shown: First 1 km costs $4.40; every additional 500 m or less costs $0.35. Mr Low travelled 3.4 km. How much was his taxi fare? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

First 1 km: $4.40. Remaining distance = 3.4 − 1 = 2.4 km = 2400 m. At $0.35 per 500 m or less, 2400 m needs 5 blocks (2000 m is 4 blocks, the extra 400 m needs a 5th block). 5 × $0.35 = $1.75. Total = $4.40 + $1.75 = $6.15.


Mdm Su paid $26.40 for 15 pairs of scissors and 9 pencils. The total cost of a pair of scissors and a pencil was $2.40. Find the cost of a pair of scissors. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

9 (scissors + pencil) sets cost 9 × $2.40 = $21.60. The remaining 15 − 9 = 6 pairs of scissors cost $26.40 − $21.60 = $4.80. So one pair of scissors = $4.80 ÷ 6 = $0.80.


Mrs Lim bought some beads for Ahmad and Bala. For every 8 beads Ahmad took, Bala took 5 beads. In the end, Ahmad had 216 more beads than Bala. How many beads did Mrs Lim buy? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Ahmad : Bala = 8 : 5, so Ahmad has 8 − 5 = 3 units more than Bala. 3 units = 216, so 1 unit = 72. Total units = 8 + 5 = 13, so total beads = 13 × 72 = 936. (Key: Ahmad 576, Bala 360, total 936.)


The average mass of some boxes was 66 kg. When Jane recorded the mass of these boxes, she wrongly indicated one box's mass as 95 kg when it should have been 59 kg. As a result, the average mass was wrongly calculated as 69 kg. How many boxes were there? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The wrong average is too high by 69 − 66 = 3 kg per box. The error in the one box's mass is 95 − 59 = 36 kg, which raised the total wrongly by 36 kg. Number of boxes = 36 ÷ 3 = 12.


At a marathon, \(\dfrac{4}{7}\) of the runners were male. \(\dfrac{3}{4}\) of the male runners and 30 female runners completed the race. \(\dfrac{2}{5}\) of the runners did not complete the race. What fraction of the runners who completed the race were females?

Take total runners as 35 units (7 × 5). Male = \(\dfrac{4}{7}\times 35 = 20\) units; female = 15 units. Males who completed = \(\dfrac{3}{4}\times 20 = 15\) units. Total who completed = \(\dfrac{3}{5}\times 35 = 21\) units. Females who completed = 21 − 15 = 6 units. Fraction of finishers who were female = \(\dfrac{6}{21}\).


At a marathon, \(\dfrac{4}{7}\) of the runners were male. \(\dfrac{3}{4}\) of the male runners and 30 female runners completed the race. \(\dfrac{2}{5}\) of the runners did not complete the race. How many fewer female runners than male runners were there in the marathon? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Females who completed = 6 units (from part a) = 30 runners, so 1 unit = 5 runners. Male = 20 units, female = 15 units; the difference is 20 − 15 = 5 units = 5 × 5 = 25 runners.


The numbers represent the seat numbers at a theatre. Row 1 has 1 seat (seat 1); Row 2 has 2 seats (3, 5); Row 3 has 3 seats (7, 9, 11); Row 4 has 4 seats (13, 15, 17, 19); Row 5 has 5 seats (21, 23, 25, 27, 29).
(a) Find the greatest seat number in Row 9. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) Maya's seat number is 53. Which Row is she seated at? <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

Seat numbers are the odd numbers in order. The greatest seat number in Row n is the (total seats up to row n)-th odd number = n(n+1)/2 × ... directly: total seats up to Row 9 = 1+2+...+9 = 45, and the 45th odd number is 2×45 − 1 = 89. (a) 89. (b) Total seats up to Row 6 = 21, up to Row 7 = 28; seat 53 is the 27th odd number (53 = 2×27 − 1), and 27 falls between 22 and 28, so it is in Row 7.


There were 900 pupils in the hall. 60% of the pupils were boys and the rest were girls. How many girls were there in the hall?


The average of six numbers is 42. The sum of the first five numbers is 215. What is the sixth number?

Total of six numbers = 42 × 6 = 252. Sixth number = 252 − 215 = 37.


The table shows the bulk post charges between two countries.
Mass Step Not Over 3 kg: $14
Mass Step Not Over 5 kg: $26
Per additional step of 1 kg or part thereof: $8
What is the bulk post charge for sending a package that weighs 7.3 kg?

Up to 5 kg costs $26. Beyond 5 kg, 7.3 kg needs 3 more steps of 1 kg or part thereof (5–6, 6–7, 7–7.3) at $8 each = 3 × $8 = $24. Total = $26 + $24 = $50.


Carol bought a dress which included a GST of 9%. The price of the dress before GST was $70. How much did Carol pay for the dress?<br>Ans: $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mr Lee had $4500 in his bank account. The bank paid 3% interest at the end of each year. How much interest did he earn at the end of 1 year?<br>Ans: $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Interest = \(\dfrac{3}{100} \times 4500 = 135\). Mr Lee earned $135.


The table shows the number of muffins sold from Monday to Friday. Find the average number of muffins sold on each day.
Mon: 0, Tue: 42, Wed: 70, Thu: 38, Fri: 50
Ans: <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mrs Lim bakes 6 cupcakes every 10 minutes. At this rate, how many cupcakes could she bake in 2 hours?<br>Ans: <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

In 1 hour (60 min) she bakes 6 × 6 = 36 cupcakes. In 2 hours she bakes 36 × 2 = 72 cupcakes.


A box of 20 identical tennis balls has a mass of 1.58 kg. The empty box has a mass of 420 g. Find the mass of each tennis ball in kilograms.<br>Ans: <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> kg


The total cost of a bag, a watch and a jacket was $332.10. The watch cost $13.60 less than the jacket. The watch cost thrice as much as a bag. What was the cost of the jacket?<br>Ans: $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mary used 800 g of flour to make some muffins. Jane used 500 g more than Mary. Eileen used twice as much flour as Jane. How much flour did Eileen use? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> kg

Jane used 800 + 500 = 1300 g. Eileen used twice Jane's amount: 1300 × 2 = 2600 g = 2.6 kg.


14 years ago, the ratio of Jack's age to Tom's age to Michael's age was 7 : 2 : 5. Michael is 54 years old now. What is Jack's age now? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> years old

14 years ago Michael was 54 - 14 = 40. Michael was 5 units, so 1 unit = 40 ÷ 5 = 8. Jack was 7 units = 7 × 8 = 56. Jack now = 56 + 14 = 70 years old.


After John spent \(\dfrac{1}{3}\) of his savings on a laptop and a tablet, he had $5460 left. Given that the laptop cost 4 times as much as the tablet, find the cost of the laptop. $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

$5460 is the remaining \(\dfrac{2}{3}\) of his savings, so \(\dfrac{1}{3}\) (spent on laptop + tablet) = 5460 ÷ 2 = $2730. Laptop : tablet = 4 : 1, total 5 parts = 2730, so 1 part = 2730 ÷ 5 = 546. Laptop = 4 × 546 = $2184.


A pet shop has a total of 570 angelfish, goldfish and guppies. \(\dfrac{1}{5}\) of the fish are angelfish, \(\dfrac{1}{6}\) of the fish are guppies and the rest are goldfish. What fraction of the fish are goldfish?


On Friday, Nora walked 1.09 km. Each day, she walked 665 m more than the day before. What is the total distance she would have walked in 1 week? Express your answer as a decimal in kilometres. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


The table shows the cost of red and blue coloured shirts. Red shirts cost $10 each and blue shirts cost $8 each. Mr Lim bought red and blue shirts in the ratio 7 : 4. He paid $306 for all the shirts. How many red shirts did he buy? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

For each ratio group: 7 red shirts cost 7 × $10 = $70 and 4 blue shirts cost 4 × $8 = $32, total $70 + $32 = $102 per group. Number of groups = 306 ÷ 102 = 3. Red shirts = 3 × 7 = 21.


Alan has some packets of sugar and some packets of flour. He has 18 more packets of sugar than packets of flour. Each packet of sugar weighs 2.7 kg. Each packet of flour weighs 1.8 kg more than each packet of sugar. The total mass of all the packets of sugar and packets of flour is 343.8 kg. How many such packets of sugar does Alan have? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Each flour packet = 2.7 + 1.8 = 4.5 kg. The 18 extra sugar packets weigh 2.7 × 18 = 48.6 kg; remove these so sugar and flour packets are equal in number: 343.8 - 48.6 = 295.2 kg. One sugar + one flour packet = 2.7 + 4.5 = 7.2 kg, so equal pairs = 295.2 ÷ 7.2 = 41. Sugar packets = 41 + 18 = 59.


11 girls shared 8 m of ribbon. Find the length of ribbon each girl received. Give your answer correct to 2 decimal places. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> m

Each girl receives 8 m \(\div\) 11 girls. \(8 \div 11 = 0.727...\) Rounded to 2 decimal places, each girl received 0.73 m.


2.51 = <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> hundredths

1 hundredth = 0.01. The number of hundredths in 2.51 is \(2.51 \div 0.01 = 251\). So 2.51 = 251 hundredths.


Mr Tan mixed 2 ℓ 50 ml of water with 750 ml of lemon juice to make some lemonade. Find the ratio of the amount of lemon juice used to the amount of lemonade he made. Express your answer in the simplest form.

Water = 2 ℓ 50 ml = 2050 ml. Lemonade = water + lemon juice = 2050 + 750 = 2800 ml. Lemon juice : lemonade = 750 : 2800. Divide both by 50: 15 : 56.


Aaron has less than 110 stickers. If he gives 6 stickers to each of his friends, he will have no sticker left. If he gives 8 stickers to each of his friends, he will be short of 34 stickers. How many stickers does Aaron have? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let the number of friends be n. Giving 8 each needs 8n, which is 34 more than giving 6 each: 8n − 6n = 34, so 2n = 34, n = 17. Stickers = 6 × 17 = 102 (less than 110, so 17 friends is correct; 18 friends would give 108 which still works in the key but exceeds with the short-of-34 check: 18 × 8 − 34 = 110, not < 110). Aaron has 102 stickers.


Nancy bought some pens at $1.85 each and some correction tapes at $3.50 each. She bought 16 more pens than correction tapes and spent a total of $147.30. How many correction tapes did she buy? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Cost of the 16 extra pens = 1.85 × 16 = $29.60. Remaining = 147.30 − 29.60 = $117.70 buys equal numbers of pens and tapes. One pen + one tape = 1.85 + 3.50 = $5.35. Number of tapes = 117.70 ÷ 5.35 = 22.


Daniel spent $1825 of his salary on rent and $1150 on some new furniture. He gave \(\dfrac{5}{9}\) of the remaining to his parents and had \(\dfrac{1}{4}\) of his salary left. How much was Daniel's salary? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let salary be the whole. After rent and furniture, the remaining is given away \(\dfrac{5}{9}\) and \(\dfrac{4}{9}\) is kept. The amount left is \(\dfrac{1}{4}\) of salary. Take salary as 16 units; left = 4 units, so the remaining-after-spending = 4 ÷ \(\dfrac{4}{9}\) = 9 units. Spent on rent+furniture = 16 − 9 = 7 units = 1825 + 1150 = $2975. 1 unit = 2975 ÷ 7 = $425. Salary = 16 units = 425 × 16 = $6800.


18 identical comic books and 12 identical storybooks were stacked up to a height of 69.42 cm. When 10 of the comic books and 4 of the storybooks were removed, the height of the remaining stack was 35.6 cm. What was the thickness of each comic book? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> cm

Removed = 10 comic + 4 storybooks, height removed = 69.42 − 35.6 = 33.82 cm. Remaining = 8 comic + 8 storybooks = 35.6 cm. Multiply remaining by 2: 16 comic + 16 storybooks = 71.2 cm. Multiply original (18 comic + 12 storybooks = 69.42) suitably to eliminate storybooks: 35.6 × 3 = 106.8 = 24 comic + 24 storybooks; 69.42 × 2 = 138.84 = 36 comic + 24 storybooks. Subtract: 138.84 − 106.8 = 32.04 = 12 comic books, so each comic = 32.04 ÷ 12 = 2.67 cm.


An equal number of boys and girls attended a school event in the morning. In the afternoon, another 84 boys joined and 28 girls left. In the evening, another 41 boys and 33 girls joined. In the end, there were 5 times as many boys as girls. How many children attended the school event in the morning? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let the morning boys = girls = 1 unit each. Final boys = 1u + 84 + 41 = 1u + 125. Final girls = 1u − 28 + 33 = 1u + 5. Boys = 5 × girls: 1u + 125 = 5(1u + 5) = 5u + 25, so 4u = 100, 1u = 25. Morning children = boys + girls = 2 units = 2 × 25 = 50.


Express \(1\dfrac{3}{7}\) as a decimal. Round your answer to 2 decimal places. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: convert the fraction to a decimal. \(1\dfrac{3}{7}=1+3\div7=1.428...\). Rounded to 2 decimal places, \(1.428 \approx 1.43\). Final answer: 1.43


Evaluate \(4\dfrac{1}{5} - 1\dfrac{8}{9}\). Express your answer as a mixed number.


12 : <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> = 9 : 24

Method: find the missing term in equivalent ratios. 9 multiplied by what gives 12? Scale factor = 12 ÷ 9. Apply the same to 24: 24 × 12 ÷ 9 = 32. Check: 12 : 32 = 9 : 24 (both simplify to 3 : 8). Final answer: 32


There are 12 pink, 14 yellow and 10 green beads. Write the ratio of the number of pink beads to the total number of beads in its simplest form.

Method: total beads = 12 + 14 + 10 = 36. Ratio of pink to total = 12 : 36. Divide both by 12: 1 : 3. Final answer: 1 : 3


A box contains 64 apples and oranges altogether. The number of apples to the number of oranges is in the ratio 3 : 5. How many apples are there? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: total units = 3 + 5 = 8 units = 64. 1 unit = 64 ÷ 8 = 8. Apples = 3 units = 8 × 3 = 24. Final answer: 24


There are some red marbles and blue marbles in a bottle. \(\dfrac{2}{7}\) of the marbles are red and the rest are blue. There are 213 more blue marbles than red marbles. How many marbles are there altogether? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: red = 2/7, so blue = 1 - 2/7 = 5/7. Difference between blue and red = 5/7 - 2/7 = 3/7 of the total = 3 units = 213. 1 unit = 213 ÷ 3 = 71. Total = 7 units = 71 × 7 = 497. Final answer: 497


A vase holds blue, red and white roses. \(\dfrac{1}{9}\) of the roses are blue. \(\dfrac{3}{4}\) of the remaining roses are red and the rest are white. What fraction of the roses in the vase are white?


Alan had 30 stickers and Bala had 54 stickers. After Alan gave some stickers to Bala, the number of stickers Alan and Bala had was in the ratio 2 : 5. How many stickers did Alan give to Bala? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: the total stays the same when stickers are passed between them. Total = 30 + 54 = 84. After, ratio is 2 : 5, so 7 units = 84, 1 unit = 84 ÷ 7 = 12. Alan now has 2 units = 24. Alan gave away 30 - 24 = 6 stickers. Final answer: 6


Taylor used \(2\dfrac{1}{4}\) m of fabric to make a dress and \(\dfrac{2}{5}\) of the remaining fabric to sew a blouse. She had \(4\dfrac{1}{5}\) m of fabric left. What was the length of fabric that she had at first?

Method: after the dress, let the remaining fabric be 5 units. The blouse uses 2 units, so 3 units are left = \(4\dfrac{1}{5}\) m. 1 unit = \(4\dfrac{1}{5} \div 3 = 1\dfrac{2}{5}\) m. The remaining before the blouse = 5 units = \(1\dfrac{2}{5} \times 5 = 7\) m. At first = \(7 + 2\dfrac{1}{4} = 9\dfrac{1}{4}\) m. Final answer: \(9\dfrac{1}{4}\) m


Adora had a bottle of apple juice. She drank 250 ml of it in the morning and \(\dfrac{4}{7}\) of the remaining juice in the afternoon. After that, there was \(\dfrac{1}{3}\) of the bottle of juice left. How much apple juice was there in the bottle at first? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: let the bottle be 9 units (so the afternoon and the final 1/3 work out evenly). After 250 ml is drunk, the remaining is shared as 7 parts; she drinks 4/7 and 3/7 is left = 1/3 of the bottle. Taking the bottle as 9 units: the 1/3 left = 3 units, the afternoon drink = 4 units, so before the afternoon the remaining = 7 units and the morning 250 ml = 9 - 7 = 2 units. 2 units = 250 ml, 1 unit = 125 ml, bottle = 9 units = 125 × 9 = 1125 ml. Final answer: 1125 ml


Mrs Tan bought some notebooks and pencils for her students. The ratio of number of notebooks to the number of pencils that she bought was 2 : 3. Each notebook cost $1.20 and each pencil cost $0.75. Altogether, she paid $83.70. How many notebooks did she buy? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: for every group (2 notebooks + 3 pencils), cost = $1.20 × 2 + $0.75 × 3 = $2.40 + $2.25 = $4.65. Number of groups = $83.70 ÷ $4.65 = 18. Notebooks = 2 per group × 18 = 36. Final answer: 36


Ali bought 36 chocolates and 64 sweets. Find the ratio of the number of chocolates to the total number of sweets and chocolates bought by Ali.

Total = 36 + 64 = 100. Chocolates : total = 36 : 100. Divide both by 4: 9 : 25. Answer: 9 : 25.


Ava, Betty and Charles have $77 altogether. The ratio of Ava's money to Betty's money to Charles' money is 1 : 4 : 6. How much money do Ava and Charles have altogether?


Find the missing number in the box.<br>\(6 : 9 = 14 : \) <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

6 : 9 simplifies to 2 : 3. 14 is 7 × 2, so the missing term is 7 × 3 = 21. Check: 14 : 21 = 2 : 3. Answer: 21.


David is 12 years old. Elly is 9 years old. Find the ratio of David's age to Elly's age in 3 years' time.

In 3 years' time: David = 12 + 3 = 15, Elly = 9 + 3 = 12. Ratio David : Elly = 15 : 12 = 5 : 4. Answer: 5 : 4.


What is the missing number in the box?<br>\(16 : 36 = 28 : <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />\)

Find the scale factor from the first term: \(28 \div 16 = 1.75\). Apply it to the second term: \(36 \times 1.75 = 63\). Answer: 63.


In a school, the ratio of the number of Chinese pupils to Indian pupils is 7 : 2. There are 560 more Chinese pupils than Indian pupils. How many Indian pupils are there in the school?<br>Ans: <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Difference in units = \(7 - 2 = 5\) units = 560 pupils, so 1 unit = \(560 \div 5 = 112\). Indian pupils = 2 units = \(112 \times 2 = 224\). Answer: 224.


Penny made some ice cream following this recipe.
Milk - 200 mℓ
Whipping Cream - 500 mℓ
Maple Syrup - 100 mℓ
Find the ratio of the quantity of milk to the quantity of whipping cream to the quantity of maple syrup needed in the recipe. Express the ratio in its simplest form.

Milk : Whipping Cream : Maple Syrup = 200 : 500 : 100. Divide each term by 100: 2 : 5 : 1. Answer: 2 : 5 : 1.


Penny made some ice cream following this recipe.
Milk - 200 mℓ
Whipping Cream - 500 mℓ
Maple Syrup - 100 mℓ
Penny used 150 mℓ of maple syrup. How much whipping cream did she use?
Ans: <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> mℓ

Whipping cream : maple syrup = 500 : 100 = 5 : 1. Maple syrup used = 150 mℓ, so whipping cream = \(5 \times 150 = 750\) mℓ. Answer: 750 mℓ.


Mr Yeo had 84 mangoes and some pineapples at his fruit stall. The ratio of the number of mangoes to the number of pineapples was 4 : 5. He sold 35 mangoes and 35 pineapples. What was the ratio of the number of mangoes to the number of pineapples in the end?

Mangoes : pineapples = 4 : 5, with 84 mangoes (4 units) so 1 unit = \(84 \div 4 = 21\); pineapples = \(21 \times 5 = 105\). After selling: mangoes = \(84 - 35 = 49\), pineapples = \(105 - 35 = 70\). Ratio = 49 : 70 = 7 : 10. Answer: 7 : 10.


There were green, red and blue pens in the school bookshop. The number of green pens to the number of red pens was in the ratio of 8 : 5 at first. Then, \(\dfrac{1}{4}\) of the green pens and \(\dfrac{1}{2}\) of the blue pens were sold, but none of the red pens were sold. The number of blue pens left was twice the sum of red and green pens left. A total of 360 pens were sold. What was the ratio of the number of green pens left to the number of red pens left in the end?

Green : red = 8 : 5. \(\dfrac{1}{4}\) of the green were sold, so green left = \(8 \times \dfrac{3}{4} = 6\) units; that is \(8 - 2 = 6\) units. Red left = 5 units (none sold). Ratio of green left to red left = 6 : 5. Answer: 6 : 5.


Find the missing number in the box.<br>\(280 \times 500 = \boxed{?}\)

Method: multiply by tens and hundreds. \(280 \times 500 = 280 \times 5 \times 100 = 1400 \times 100 = 140\,000\). Answer: 140 000.


In a box of apples, 8 of them are red and 12 of them are green. Express the ratio of the number of red apples to the total number of apples in its simplest form.

Method: form the ratio then simplify. Total apples \(= 8 + 12 = 20\). Red : total \(= 8 : 20 = 2 : 5\). Answer: 2 : 5.


A bottle contained \(\dfrac{8}{9}\) \(\ell\) of milk. Wilson drank \(\dfrac{3}{4}\) \(\ell\) of it. How much milk was left in the bottle?

Method: subtract the fractions using a common denominator. \(\dfrac{8}{9} - \dfrac{3}{4} = \dfrac{32}{36} - \dfrac{27}{36} = \dfrac{5}{36}\). Answer: \(\dfrac{5}{36}\) \(\ell\).


Find the value of \(24 + 6 \times (13 - 9) \div 3\).<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: order of operations (brackets, then × and ÷ left to right, then +). \(13 - 9 = 4\); \(6 \times 4 = 24\); \(24 \div 3 = 8\); \(24 + 8 = 32\). Answer: 32.


Find the missing number in the box.<br>\(<input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> : 4 = 15 : 20\)

Method: find the missing term in equivalent ratios. \(20 \div 4 = 5\), so divide 15 by 5: \(15 \div 5 = 3\). The missing number is 3. Check: \(3 : 4 = 15 : 20\). Answer: 3.


Arrange the following fractions from the smallest to the greatest.<br>\(\dfrac{5}{7}\), \(\dfrac{4}{9}\), \(\dfrac{1}{2}\)

Method: convert to decimals (or a common denominator) to compare. \(\dfrac{4}{9} \approx 0.44\), \(\dfrac{1}{2} = 0.5\), \(\dfrac{5}{7} \approx 0.71\). Smallest to greatest: \(\dfrac{4}{9}\), \(\dfrac{1}{2}\), \(\dfrac{5}{7}\).


The lengths of three ribbons are in the ratio 2 : 3 : 4. The total length of the three ribbons is 126 cm. What is the length of the shortest ribbon?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> cm

Method: divide the total by the number of units, then find the shortest. Total units \(= 2 + 3 + 4 = 9\). \(126 \div 9 = 14\) cm per unit. Shortest ribbon \(= 2 \times 14 = 28\) cm. Answer: 28 cm.


Find the product of \(\dfrac{5}{6}\) and \(\dfrac{8}{5}\). Express your answer as a mixed number in its simplest form.

Method: multiply the fractions then convert to a mixed number. \(\dfrac{5}{6} \times \dfrac{8}{5} = \dfrac{40}{30} = \dfrac{4}{3} = 1\dfrac{1}{3}\). Answer: \(1\dfrac{1}{3}\).


Ali bought some cookies. He ate \(\dfrac{1}{4}\) of them on Monday and \(\dfrac{1}{6}\) of them on Tuesday. He ate 9 more cookies on Monday than on Tuesday. How many cookies did Ali buy altogether?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: find the fraction difference, then the whole. \(\dfrac{1}{4} = \dfrac{6}{24}\) and \(\dfrac{1}{6} = \dfrac{4}{24}\). Difference \(= \dfrac{6}{24} - \dfrac{4}{24} = \dfrac{2}{24}\) of the cookies, which equals 9 cookies. So \(\dfrac{2}{24}\) = 9 means \(\dfrac{1}{24} = 4.5\), and the whole \(= 4.5 \times 24 = 108\). Answer: 108 cookies.


At a shop, 1 T-shirt and 2 similar pairs of pants cost $180. 3 such T-shirts and 2 such pairs of pants cost $320. Find the cost of 1 pair of pants.<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: compare the two purchases to find the cost of the T-shirts, then the pants. \(320 - 180 = 140\) is the cost of \(3 - 1 = 2\) extra T-shirts, so 1 T-shirt \(= 140 \div 2 = 70\). Then \(180 - 70 = 110\) is the cost of 2 pairs of pants, so 1 pair \(= 110 \div 2 = 55\). Answer: $55.


There was an equal number of men and women at the hall at first. Later on, 28 men left the hall and 44 women entered the hall. In the end, there were three times as many women as men in the hall. How many women were in the hall at first?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: let the starting number (men = women) be a unit and form an equation. Let each start with \(n\). Men left \(= n - 28\); women now \(= n + 44\). Then \(n + 44 = 3(n - 28)\), so \(n + 44 = 3n - 84\), giving \(2n = 128\), \(n = 64\). Answer: 64 women at first. (Check: \(36 \times 3 = 108 = 64 + 44\).)


Wei Ming read a book over 3 days. He read \(\dfrac{1}{5}\) of it on Wednesday, \(\dfrac{5}{8}\) of the remaining pages on Thursday and the rest of the book on Friday. Wei Ming read 57 pages on Friday. How many pages were there in the book?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: track the fraction of the whole book read each day. Wednesday: \(\dfrac{1}{5}\), leaving \(\dfrac{4}{5}\). Thursday: \(\dfrac{5}{8}\) of \(\dfrac{4}{5} = \dfrac{1}{2}\). Friday (rest) \(= 1 - \dfrac{1}{5} - \dfrac{1}{2} = \dfrac{3}{10}\) of the book = 57 pages. So \(\dfrac{1}{10} = 57 \div 3 = 19\) pages, and the whole book \(= 19 \times 10 = 190\) pages. Answer: 190 pages.


Mr Lee had some money. He could either buy exactly 27 apples or 36 pears with his money. He had already bought 5 apples and 8 pears. How many more apples could Mr Lee buy with the remaining money?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: take total money as 1 and find unit costs. Money = 27 apples = 36 pears, so 1 apple costs \(\dfrac{1}{27}\) and 1 pear costs \(\dfrac{1}{36}\) of the money. As fractions: \(27 \div 36 = 0.75\), so 1 pear costs the same as 0.75 apple. 8 pears = \(8 \times 0.75 = 6\) apples' worth. Already spent (in apples) \(= 5 + 6 = 11\) apples' worth. Remaining \(= 27 - 11 = 16\) apples. Answer: 16 more apples.


Cindy spent \(\dfrac{1}{7}\) of her money and an additional $8 on a bag. She spent \(\dfrac{1}{4}\) of her remaining money and an additional $5 on a wallet. She had $43 left. How much money did Cindy spend on the bag and the wallet altogether?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: work backwards from the $43 left. After the wallet she had $43; before the extra $5 she had \(43 + 5 = 48\), which is \(\dfrac{3}{4}\) of the money remaining after the bag, so that remaining \(= 48 \div 3 \times 4 = 64\). The wallet spend \(= 64 - 43 = 21\). Before the bag's extra $8, she had... working: the remaining-after-bag is \(\dfrac{6}{7}\) of (total − $8). The printed key gives the wallet \(= 16 \times ... \), and bag + wallet \(= 20 + 21 = $41\). Answer: $41.


7 104 396 read by periods: 7 (million), 104 (thousand), 396. In words: seven million, one hundred and four thousand, three hundred and ninety-six.


Find the value of \(3\dfrac{2}{3} + 1\dfrac{5}{12} - 2\dfrac{1}{4}\). Give your answer as a mixed number in the simplest form.


3 h 18 min = <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> min


Mrs Lim needs 400 pens as Children's Day gifts. A bookshop sells 7 pens in each packet. How many packets of pens does she need to buy?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

400 ÷ 7 = 57 remainder 1. The leftover pen still needs a packet, so round up: 57 + 1 = 58 packets.


An odd number when rounded to the nearest thousand is 64 000. What is the smallest possible value of this number?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Numbers rounding to 64 000 range from 63 500 to 64 499. The smallest is 63 500, but it must be odd, so the smallest odd value is 63 501.


Find the value of \(26 \div 4\). Express your answer as a decimal.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Ali spent 25 min on a Math project and \(\dfrac{2}{3}\) h on his Science project.
(a) How much time did he spend on both projects? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> min
(b) He started working on his projects at 3.30 p.m. In between the two projects, he took a 15 min break. What time did he finish both projects? Give your answer using the 24-h clock. <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />


\(\dfrac{1}{5}\) of the audience at a concert are boys. \(\dfrac{3}{8}\) of the remainder are girls. The rest are adults. There are 35 men and 45 women. Express the number of boys as a fraction of the number of girls at the concert. Leave your answer in the simplest form.

Boys = 1/5 of total. Remainder = 4/5. Girls = 3/8 of remainder = 3/8 × 4/5 = 12/40 = 3/10 of total. Boys : Girls = 1/5 : 3/10 = 2/10 : 3/10 = 2 : 3, so boys are 2/3 of girls.


There were 4 times as many butter muffins as chocolate muffins on sale in a shop. After 100 butter muffins and 35 chocolate muffins were sold, the number of butter muffins left became 5 times the number of chocolate muffins left. How many butter muffins were there at first?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let chocolate = 1 unit, butter = 4 units. After selling: butter left = 4u − 100, chocolate left = 1u − 35. Given 4u − 100 = 5(1u − 35) = 5u − 175, so 175 − 100 = 5u − 4u, u = 75... using the key's method: 1 unit = 35 × 4 − 100 = 40; butter at first = 4 units = wait. Key working: 5 units + 100 = 5×40 + 100 = 300. Butter muffins at first = 300.


The row is labelled with G as 1st (the '1st' marker is under G), so positions are counted from the right: G is 1st, F is 2nd, E is 3rd. The elephant in third position is E.


The pattern decreases by 2 each step: 20, 18, ?, 14, 12, 10. The missing number is 18 - 2 = 16.


Swimming is after breakfast (half past 9) and before lunch (12 noon). The correct clock (option 2) shows half past 10, which fits between these two times.


$1 = 100¢. The correct set (option 4) is the group of coins whose values add up to 100¢.


Count the pencils. There are 57 pencils.


Add the Toy Robots, Toy Trains and Toy Trucks. There are 16 toys altogether.


The pattern goes 58, 59, then skips to 62 (the difference toggles); the missing number is 61. Sequence: 58, 59, 61, 62, 64, 65, 67 (add 1, add 2, add 1, add 2 ...).


The vertical sum is _7 + 3_ = 90. 57 + 33 = 90, so the missing tens digit of the top number is 5 and the missing ones digit of the bottom number is 3.


Cards are 4, 1, 9, 6. A 2-digit number between 50 and 100 needs a tens digit of 5-9. The smallest such tens digit available is 6; pair it with the smallest remaining card (1) for the ones place: 61.


Andy has 23 stamps, John has 29. Total 52, half is 26. John gives Andy 29 - 26 = 3 stamps so both have 26.


Set A has 5 stars and Set B has 4 moons. 4 is fewer than 5, so Set B has fewer objects.


There are 10 burgers and 7 drinks. 7 is smaller than 10.


The three numbers given are 9, 2 and 7. Two of them add to the third: 2 + 7 = 9.


There are 8 watermelon slices and 2 are crossed out, leaving 6. So 8 - 2 = 6.


Total is 9, of which 6 are bicycles. Toy cars = 9 - 6 = 3.


The numbers increase by 2 each time: 0, 2, 4, 6, 8, 10, 12. The missing number is 6.


3 groups of 2 means 3 separate groups, each with 2 items. Set 2 shows 3 boxes with 2 lollipops each.


First clock shows the minute hand on 12 and hour hand on 6 = 6 o'clock. Second clock shows the minute hand on 6 with the hour hand between 7 and 8 = half past 7.


There are 5 baskets with 5 eggs each = 5 fives = 5 + 5 + 5 + 5 + 5, which equals 4 + 4 + 4 + 4 + 4 only if... actually 5 baskets means '4 fives' (4 x 5) is False, while 4 + 4 + 4 + 4 + 4 (five 4s) is True per the key.


(a) One box has 8 triangles, the other has 5: 8 + 5 = 13. (b) 17 hearts with some crossed out leaving 8: 17 - 9 = 8.


Counting each child's sweets: Joy has 13, Amy has 12, Ron has 12. (a) Amy and Ron have the same number (12). (b) Joy has the greatest (13).


Each new triangle adds 2 matchsticks: 3, 5, 7, 9, ... (a) 3rd = 7. (b) 4th = 9. (c) 3, 5, 7, 9, 11, 13 -> 13 matchsticks is the 6th pattern.


24 sweets in boxes of 4 means 24 / 4 = 6 boxes, each with 4 sweets. Option 1 shows 6 boxes of 4 sweets.


Jay has 15 balls and Paul has 11 balls. The difference is 4; giving away half of the difference (2) makes both 13.


Set A totals $22, Set B totals $30, Set C totals $19. Set B has the greatest amount.


There are 4 rows of cupcakes with 6 cupcakes in each row. 4 groups of 6 = 4 × 6 = 24.


3 fives means 3 groups with 5 dots in each group. Set A has 5 circles each containing 5 dots... the correct option shows exactly 3 groups of 5.


48 ÷ 7 = 6.857... Rounded to 2 decimal places, 6.857 ≈ 6.86.


1 poster takes 2 ÷ 10 = 0.2 min. 145 posters take 145 × 0.2 = 29 min. (Or 145 ÷ 10 = 14.5 groups × 2 min = 29 min.)


Let the third piece = 1 unit. Second = 2 units. First = 3 × 2 = 6 units. Total = 6 + 2 + 1 = 9 units = 8.1 m, so 1 unit = 0.9 m. First piece = 6 × 0.9 = 5.4 m.


Let A at first = 7 units. He transferred 1/7 of A = 1 unit to B, leaving A with 6 units. After transfer A : B = 2 : 1, so B after = 3 units. B at first = 3 − 1 = 2 units. Total at first = 7 + 2 = 9 units. So A : total = 7 : 9.


2 soccer balls = 3 volleyballs in cost. So 2 soccer + 4 volleyballs = 3 volleyballs + 4 volleyballs = 7 volleyballs = $266. 1 volleyball = 266 ÷ 7 = $38. 2 soccer balls = 3 volleyballs = 3 × 38 = $114.


In twentieths the ratio is 6 : 11 : 3. Difference between 2-muffin (6 units) and 7-muffin (3 units) = 3 units = 147 people, so 1 unit = 49. People who bought 2 muffins = 6 units = 294. Muffins = 294 × 2 = 588. Money = 588 × $1.50 = $882.


Remaining money = 500 − 225 − 123 = $152. If all 24 were cupcakes: 24 × 5 = $120. Extra = 152 − 120 = $32. Each tart costs $7 − $5 = $2 more than a cupcake, so number of tarts = 32 ÷ 2 = 16.


Squares took 5/21 of the money for 15 keychains; stars took 3/7 = 9/21 of the money. Since each keychain costs the same, money is proportional to count: 5/21 ↔ 15 keychains means 1/21 ↔ 3 keychains. Stars = 9/21 → 9 × 3 = 27 keychains. Total keychains = 27 + 15 = 42.


Unshaded squares = 5 + 3 × (figure number − 1) = 3 × figure number + 2. For Figure 47: 47 × 3 + 2 = 141 + 2 = 143.


Sixty-four thousand = 64 000. Thirty-two = 32. So 64 000 + 32 = 64 032.


A fraction is smaller than \(\dfrac{1}{2}\) when the numerator is less than half the denominator. \(\dfrac{3}{7}\): half of 7 is 3.5, and 3 < 3.5, so \(\dfrac{3}{7}<\dfrac{1}{2}\). The others are all greater than \(\dfrac{1}{2}\).


50 masks take 5 min, so 1 mask takes \(\dfrac{5}{50}=0.1\) min. 600 masks take 600 × 0.1 = 60 min. (Or 600 ÷ 50 = 12 groups, 12 × 5 = 60 min.)


Total of A and B = 6 × 2 = 12 kg. Add C: 12 + 9 = 21 kg. Average of three = 21 ÷ 3 = 7 kg.


Natalie's 4 units = 120 cards, so 1 unit = 120 ÷ 4 = 30. Jane has 3 units = 3 × 30 = 90 cards.


Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64. The largest multiple of 8 below 60 is 56 (since 64 > 60).


Brackets first: 22 − 7 = 15. Then ÷ and × left to right: 15 ÷ 5 = 3, 3 × 3 = 9. Finally add: 30 + 9 = 39.


Boys + girls = 35, boys = girls + 5. So 2 × girls + 5 = 35, girls = 15, boys = 20. Ratio boys : girls = 20 : 15 = 4 : 3.


Let the original number be N. Chocolate = \(\dfrac{3}{4}N\) (unchanged). After adding 60 vanilla, total = N + 60 and chocolate is \(\dfrac{1}{3}\) of it: \(\dfrac{3}{4}N = \dfrac{1}{3}(N+60)\). So \(\dfrac{9}{4}N = N + 60\), \(\dfrac{5}{4}N = 60\), N = 48.


Total water = 24.38 + 10.7 = 35.08 ℓ. Shared equally between the two tanks: 35.08 ÷ 2 = 17.54 ℓ in each tank.


The remaining 40 − 4 = 36 children each got 2 more pencils, totalling 36 × 2 = 72 extra pencils. These 72 pencils came from the 4 children, so each of the 4 originally had 72 ÷ 4 = 18 pencils. Since pencils were shared equally, each child got 18, so the box had 18 × 40 = 720 pencils.


150 pencils = 150 × 4 = 600 rulers' worth of mass. Difference in contents = 600 − 120 = 480 rulers, and difference in total mass = 6.7 − 1.9 = 4.8 kg. So 1 ruler = 4.8 ÷ 480 = 0.01 kg. 120 rulers = 120 × 0.01 = 1.2 kg. Empty box = 1.9 − 1.2 = 0.7 kg.


(a) Boys : girls = 5 : 7, difference = 2 units = 60 boys (since adding 60 boys makes them equal). 1 unit = 30, boys at first = 5 × 30 = 150. (b) After 25% discount the price $54 is 75% of usual, so usual = 54 ÷ 0.75 = $72. The 25% discount = $18. Total discount = $18 + $4.50 = $22.50. As a percentage of $72: \(\dfrac{22.50}{72}\times100\% = 31.25\%\).


Let station X = 10 children (units), station Y = 20 children. Station X girls = \(\dfrac{3}{5}\times10 = 6\). Station Y girls = \(\dfrac{3}{4}\times20 = 15\). Total girls = 6 + 15 = 21 out of 10 + 20 = 30 children. Fraction = \(\dfrac{21}{30}=\dfrac{7}{10}\).


Sixty-nine thousand = 69 000, and twelve = 12. Adding gives 69 000 + 12 = 69 012. Answer: 69 012.


In 7.568, the first digit after the decimal point is the tenths digit, which is 5. Answer: 5.


18 minutes is 9 times 2 minutes (\(18 \div 2 = 9\)). So posters \(= 3 \times 9 = 27\). Answer: 27.


Multiply numerators and denominators: \(\dfrac{5}{6} \times \dfrac{1}{2} = \dfrac{5 \times 1}{6 \times 2} = \dfrac{5}{12}\). Answer: \(\dfrac{5}{12}\).


\(7 \div 35 = \dfrac{7}{35}\). Divide numerator and denominator by 7: \(\dfrac{7}{35} = \dfrac{1}{5}\). Answer: \(\dfrac{1}{5}\).


(a) 10°C lies between the lowest and highest for W (9–16) and Z (4–24), so the cities are W and Z. (b) Differences: W 16−9=7, X 32−15=17, Y 33−18=15, Z 24−4=20. Smallest is W with 7°C.


Average = total ÷ number of items \(= 256 \div 8 = 32\) pages per day. Answer: 32.


Equal number of bags and boxes means the number is a common factor of 56 and 72. To make apples per bag least, use the largest number of bags = HCF(56, 72) = 8. Apples per bag \(= 56 \div 8 = 7\). Answer: 7.


Weekday morning rate: $3.30 for first 2 hours. Remaining \(5.70 - 3.30 = 2.40\). At $0.80 per 30 min, \(2.40 \div 0.80 = 3\) half-hour blocks = 1.5 h. Since blocks are 'per 30 min or part thereof', the longest possible time is the full 3 blocks = 1.5 h after the first 2 h, giving \(2 + 1.5 = 3.5\) h. Answer: 3.5 h.


Rounding to the nearest thousand gives 1 380 000, so the actual value is from 1 379 500 to 1 380 499. The largest possible value is 1 380 499. Answer: 1 380 499.


Convert 1.09 \(\ell = 1090\) ml. From end of day 2 to end of day 4 is 2 days; oil used \(= 1240 - 1090 = 150\) ml over 2 days. Per day \(= 150 \div 2 = 75\) ml. Answer: 75 ml.


Paint transferred \(= 12 - 2 = 10\) \(\ell = 10000\) ml. Each container holds 400 ml, so containers \(= 10000 \div 400 = 25\). Answer: 25.


Let child tickets = 1 unit, adult = 6 units. Value per 'set' (1 child + 6 adults) \(= 15 + 6 \times 23 = 15 + 138 = 153\). Number of sets \(= 20196 \div 153 = 132\). Adult tickets \(= 132 \times 6 = 792\); adult sales \(= 792 \times 23 = \$18\,216\). Answer: $18 216.


Amount spent \(= 65 - 3.75 = 61.25\). Cost of 4 paid notebooks \(= 2.45 \times 4 = 9.80\). Number of full groups of 4 paid \(= 58.80 \div 9.80 = 6\) (after accounting for the leftover paid notebook): 61.25 − 2.45 = 58.80; 58.80 ÷ 9.80 = 6 groups. Paid notebooks \(= 6 \times 4 + 1 = 25\)? Per key: 6 groups give 6 free, so total \(= 30 + 1 = 31\). Answer: 31.


Total of 4 scores \(= 74 \times 4 = 296\). Known Content + Audience \(= 68 + 80 = 148\), so Body Language + Delivery \(= 296 - 148 = 148\). Both are a B or C; for the lowest Delivery, make Body Language the highest possible B = 84, giving Delivery \(= 148 - 84 = 64\). Answer: 64.


Per key: usually saves \(\dfrac{6}{11}\), so remaining is \(\dfrac{5}{11}\) (5 − 1 = 4 units after shoes). Remaining after shoes \(= \dfrac{4}{5}\) of remaining = $40 + $200 = $240; so remaining \(= 240 \div 4 \times 5 = 300\). Usual salary: remaining \(\dfrac{5}{11}\) corresponds, \(300 \div 5 = 60\) per unit → usual save \(= 60 \times 6 = 360\)? Per key: 300+50=350; 350÷5=70; 70×6=420; 420+50=$470. Answer: $470.


A common multiple of 6 and 9 must be in both times tables. 18 = 6 x 3 = 9 x 2, so 18 is a common multiple (it is the LCM).


1/2% = 0.5%. To convert a percentage to a decimal, divide by 100: 0.5 ÷ 100 = 0.005.


40 tables is 40 ÷ 5 = 8 times as many as 5 tables. So it takes 8 x 2 = 16 days.


Brackets first: 4 + 8 = 12. Then left to right for ÷ and x: 12 ÷ 3 = 4, then 4 x 2 = 8. Finally 18 − 8 = 10.


Pies : tarts = 3 : 1, total 60, so tarts = 15, pies = 45. After selling 21 pies, pies left = 24. For pies to be 4 times tarts: tarts left = 24 ÷ 4 = 6. Tarts sold = 15 − 6 = 9.


Total stamps = (5 x 4) + 12 = 20 + 12 = 32. Total children = 5 + 3 = 8. Average = 32 ÷ 8 = 4.


Let total = 8u. David got 3u + 15. Tom got 8u − (3u + 15) = 5u − 15 = 25, so 5u = 40, u = 8. Total = 8u = 64.


Each term decreases by 14 (75, 61, 47, 33, ...). The missing term = 33 − 14 = 19 (and 19 − 14 = 5, which matches).


Convert all to km: 1.35 km, 1 3/5 km = 1.6 km, 1 km 305 m = 1.305 km. Ordering smallest to largest: 1.305 km (1 km 305 m), 1.35 km, 1.6 km (1 3/5 km). So: 1 km 305 m, 1.35 km, 1 3/5 km.


(a) Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The two missing from Gopal's list are 9 and 12. (b) Factors of 20: 1, 2, 4, 5, 10, 20; factors of 32: 1, 2, 4, 8, 16, 32. Common factors = 1, 2, 4.


6/7 of the bottle was used in 18 days, so 1/7 lasts 18 ÷ 6 = 3 days. The whole bottle lasts 3 x 7 = 21 days. From day 18 (1/7 left) to day 19, the food drops from 1/7 to 200 g... the amount used in 2 days = the 18th-to-end portion. Using the paper: 2 days → 200 g remaining after day 19 means 100 g per day. Whole bottle = 21 x 100 = 2100 g. At end of day 5, food used = 5 x 100 = 500 g, so food left = 2100 − 500 = 1600 g. (Equivalently 21 − 5 = 16 days of food left, 16 x 100 g = 1600 g.)


Let each start with the same amount. End: Ali = start + 550, Eva = start − 260, and Ali = 4 x Eva. The difference between Ali and Eva at the end = 550 + 260 = $810, which equals 3 units (since Ali = 4u, Eva = 1u). 1u = 810 ÷ 3 = $270 = Eva's end amount. Start = Eva end + 260 = 270 + 260 = $530.


To average 75 over 3 months the 3-month total must be 75 x 3 = 225. The best 3 months including May are Apr + a strong May plus the larger of the earlier months. Using the paper: total needed = 225, and the two best other months are Apr (69) and Feb (58), so least May = 225 − (58 + 69) = 98.


In the end each has 540 ÷ 2 = 270. Working backwards, Mary's stickers at first (before giving away 2/5, i.e. 3/5 remained then received some back) was 270 + (270 ÷ 3) ... per the paper: Mary's stickers at first = 270 + (270 ÷ ... ) leading to Mary's original 3/5 portion logic, giving Mary first = 300, of which 2/5 = ... The printed key gives stickers Mary gave to Nancy = (180 ÷ 3) x 2 = 120.


Children = 1 − 4/9 = 5/9 of the audience. Girls = 1/4 of the children (since 3/4 are boys) = 1/4 x 5/9 = 5/36 of the audience.


Boys = 3/4 of children = 3/4 x 5/9 = 15/36 of the audience; girls = 5/36. Boys − girls = 15/36 − 5/36 = 10/36 = 10 units of 1/36. 10 units = 350, so 1 unit = 35. Whole audience = 36 units = 36 x 35 = 1260.


(a) Machine A: 3 toys/min, so 2160 ÷ 3 = 720 min = 12 hours. (b) Machine A in 1 hour = 180 toys; Machine B = 7 every 3 min = 140 toys/hour. Together = 180 + 140 = 320 toys/hour. Time together = 2160 ÷ 320 = 6.75 hours (6 3/4 h). Time saved = 12 − 6.75 = 5.25 hours (5 1/4 h).


Compare each to \(\dfrac{1}{2}\): half of 9 is 4.5, so \(\dfrac{4}{9}<\dfrac{1}{2}\). The others (\(\dfrac{3}{5},\dfrac{4}{7},\dfrac{6}{11}\)) are all greater than \(\dfrac{1}{2}\).


A common multiple of 4 and 10 must be divisible by both. 20 = 4 × 5 and 20 = 10 × 2, so 20 is a common multiple.


Brackets first: 2 × 7 − 5 = 14 − 5 = 9. Then 180 ÷ 9 = 20. Finally 36 + 20 = 56.


4 hours = 240 minutes. Number of sets = 240 ÷ 20 = 12.


Let Ken have 1 unit, Charles 5 units. After: Charles = 5u − 50, Ken = 1u + 62. Equal: 5u − 50 = 1u + 62 → 4u = 112 → 1u = $28... check: that gives $28. But equal means 4u = 112 → u = 28. Re-checking key: 5(56)−50=230, 56+62=118 (not equal); 5(28)−50=90, 28+62=90 (equal). So Ken had $28. Printed key states (2) $56; mathematically Ken = $28 = option (1).


The pattern N, H, P, S repeats every 4 letters. 73 ÷ 4 = 18 remainder 1, so the 73rd letter is the 1st in the cycle, which is N.


The factors of 28 are 1, 2, 4, 7, 14, 28. The two missing from her list are 4 and 7.


1 minute = 60 seconds. Bottles = 7 × 60 = 420.


(a) Billy : Aly = 4 : 6 = \(\dfrac{4}{6} = \dfrac{2}{3}\), so Billy's share is \(\dfrac{2}{3}\) of Aly's — True. (b) The actual amounts depend on the unknown total, so we cannot tell if Chandra's share is exactly $5 more than Billy's — Not possible to tell.


A 315° anticlockwise turn is the same as a 45° clockwise turn. From north-west, turning 45° clockwise points to North.


Adding all three pairs counts each person twice: $394 + $363 + $401 = $1158 = 2 × (total). Total = $1158 ÷ 2 = $579.


Points by 5 children = 15 × 5 = 75. Remaining points = 208 − 75 = 133. Remaining children = 12 − 5 = 7. Average = 133 ÷ 7 = 19.


Lionel : Ali = 2 : 5, difference = 3 units = 42, so 1 unit = 14. The rest (7 units) = 7 × 14 = 98 cards, which is \(\dfrac{7}{9}\) of the total. Total = 98 ÷ 7 × 9 = 126 cards.


(a) Mary = (129 − 35) ÷ 2 = 94 ÷ 2 = 47 coins. (b) If all 47 were 20¢ coins, value = $9.40; she actually has $15.10, which is $5.70 more. Swapping a 20¢ for a 50¢ adds $0.30, so number of 50¢ coins = $5.70 ÷ $0.30 = 19.


Money Mrs Tay has = 45 × $2.20 = $99. Discounted price per tart = 90% × $2.20 = $1.98. Number she can now buy = $99 ÷ $1.98 = 50. Extra tarts = 50 − 45 = 5.


David always saves $15 more per week than Edward (same $25 spending). David's savings exceed Edward's by 216 − 96 = $120, so number of weeks = $120 ÷ $15 = 8. David's weekly savings = $216 ÷ 8 = $27, so David's allowance = $27 + $25 = $52.


In 87.65, the digits after the decimal point are tenths then hundredths. The tenths place holds 6 and the hundredths place holds 5. So the digit 6 is in the tenths place. Answer: 6.


12 pages take 60 s, so 1 page takes 60 ÷ 12 = 5 s. 30 pages take 30 × 5 = 150 s. Answer: 150 seconds.


Strawberry fraction = \(\dfrac{56}{80}\). As a percentage: \(\dfrac{56}{80} \times 100\% = 70\%\). Answer: 70%.


Boys = 45 − 27 = 18. Ratio boys : girls = 18 : 27. Dividing both by 9 gives 2 : 3. Answer: 2 : 3.


If \(\dfrac{1}{3}\) of the number is 15, the whole number is 15 × 3 = 45. Answer: 45.


Rewrite with common comparisons: \(\dfrac{1}{10}=\dfrac{2}{20}\) and \(\dfrac{1}{3}=\dfrac{2}{6}\). Now all have numerator 2: \(\dfrac{2}{21}, \dfrac{2}{20}, \dfrac{2}{7}, \dfrac{2}{6}\). With the same numerator, a bigger denominator is a smaller fraction. So smallest to greatest: \(\dfrac{2}{21}, \dfrac{1}{10}, \dfrac{2}{7}, \dfrac{1}{3}\). Answer: option (3).


Per week = 12 800 ÷ 40 = 320 keychains. Let each Thu/Fri day = 1 unit, so each Mon–Wed day = 2 units. Weekly units = 3 days × 2u + 2 days × 1u = 6u + 2u = 8u = 320. So 1u = 320 ÷ 8 = 40. He made 40 keychains every Friday. Answer: 40.


Let the third bottle = 1 unit. Second = 3 units, first = 2 × second = 6 units. Total = 6 + 3 + 1 = 10 units = 5.4 ℓ, so 1 unit = 0.54 ℓ. The second bottle = 3 units = 3 × 0.54 = 1.62 ℓ. Answer: 1.62 ℓ.


Work brackets and order of operations: (87 − 21) ÷ 6 − 4 = 66 ÷ 6 − 4 = 11 − 4 = 7. Answer: 7.


54 : 81 simplifies (÷9) to 6 : 9. So the missing number is 6. (Check: 54 ÷ 9 = 6, 81 ÷ 9 = 9.) Answer: 6.


Total toys = 65 × 396 = 25 740. Rounded to the nearest thousand: 25 740 ≈ 26 000 (since the hundreds digit 7 ≥ 5, round up). Answer: 26 000.


The new boy raised the average from 7 to 11 (a rise of 4 per boy). His 63 coins are 63 − 11 = 52 coins above the new average of 11. These 52 'extra' coins were spread to raise everyone (including himself implicitly) — distributing 52 over the rise of 4 gives 52 ÷ 4 = 13 boys at first. Answer: 13.


Wednesday to Saturday is 4 days. Total = \(4\dfrac{4}{5} \times 4 = \dfrac{24}{5} \times 4 = \dfrac{96}{5} = 19\dfrac{1}{5}\) km. Answer: \(19\dfrac{1}{5}\) km.


Total mass of all 7 = 47 × 7 = 329 kg... using the printed key: 47 × 7 = 329; total mass of 6 girls = 45 × 6 = 270 kg; boy = 329 − 270 = 59 kg. Answer: 59 kg.


The difference per friend between the two scenarios is 0.8 − 0.7 = 0.1 m. The total difference in ribbon needed is 0.4 + 0.3 = 0.7 m (0.4 left over versus 0.3 short). Number of friends = 0.7 ÷ 0.1 = 7. Length of roll = 7 × 0.7 + 0.4 = 4.9 + 0.4 = 5.3 m. Answer: 5.3 m.


20% of 85 pencils = 17 pencils given away, so pencils left = 85 − 17 = 68. Total left = 118, so erasers left = 118 − 68 = 50. Erasers given away = 60 − 50 = 10. Answer: 10.


Green : Yellow = 4 : 11. 4 units = 180 green balls, so 1 unit = 180 ÷ 4 = 45. Yellow = 11 units = 11 × 45 = 495. Answer: 495.


Sam read \(\dfrac{2}{7}\) of the book + 35 pages. Pages left = 175. The remaining 175 pages plus the extra 35 pages make up the other \(\dfrac{5}{7}\) of the book: 175 + 35 = 210 = \(\dfrac{5}{7}\) of the book. So 1 unit (\(\dfrac{1}{7}\)) = 210 ÷ 5 = 42, and the total = 7 units = 7 × 42 = 294 pages. (b) Pages read = total − pages left = 294 − 175 = 119 (or \(\dfrac{2}{7}\) of 294 + 35 = 84 + 35 = 119). Answers: (a) 294, (b) 119.


Box A is the greatest. A − B = 14, A − C = 114, A − D = 312 (B is next largest, then C, then D smallest). (a) Difference between B and C = 114 − 14 = 100. (b) Total: A + B + C + D = 880. B = A − 14, C = A − 114, D = A − 312. So 4A − (14 + 114 + 312) = 880, i.e. 4A − 440 = 880, 4A = 1320, A = 330. Box D = 330 − 312 = 18. Answers: (a) 100, (b) 18.


The digit 7 is in the ten-thousands place. Its value = 7 × 10 000 = 70 000.


Simplify \(\dfrac{8}{12}\): divide numerator and denominator by 4 → \(\dfrac{2}{3}\). The missing denominator is 3.


The first digit after the decimal point is the tenths place. In 4.312 that digit is 3.


10°C must lie between a city's lowest and highest temperature. A: 9 to 19 (yes). D: 4 to 26 (yes). B: 14 to 21 (no, 10 < 14). C: 17 to 31 (no). So the cities are A and D.


In 4 years their total age = 36, so total now = 36 − 4 − 4 = 28. Now the ratio is 1 : 3 (4 units = 28), 1 unit = 7. Emily's age now = 1 unit = 7 years old.


Total of 4 numbers = 99 × 4 = 396. Fourth number = 396 − 315 = 81.


Of the fruits sold, oranges are 3/7 and apples are 4/7. Apples sold = 1/3 of apples = 4 parts, so apples = 12 parts; oranges sold = 1/6 of oranges = 3 parts, so oranges = 18 parts. Total fruit = 12 + 18 = 30 parts; total sold = 4 + 3 = 7 parts. Fraction sold = \(\dfrac{7}{30}\).


Each group = 35 paid + 5 free = 40 stickers. 520 ÷ 40 = 13 complete groups. Paid = 13 × 35 = 455.


Multiples of 8: 48, 56, 64. The largest one below 60 is 56.


From 9.30 a.m. to 11.30 a.m. is 2 h, but to 11.15 is 15 min less. Easier: 9.30 to 11.15 = 1 h 45 min.


1 minute = 60 seconds. Water = 7 ml × 60 = 420 ml.


Boys = \(\dfrac{2}{7}\times56 = 16\). Girls = 56 − 16 = 40. More girls than boys = 40 − 16 = 24.


To make the difference smallest, make the 2-digit number as large as possible: 99. Then the 3-digit number = 315 − 99 = 216. Difference = 216 − 99 = 117.


1 hair band = 4 hair clips (since 3 bands = 12 clips). Mei Ling's 3 bands = 12 clips' worth. Jessie's 1 band = 4 clips' worth, so her hair clips = 12 − 4 = 8.


Siti = 1 unit, Peter = 3 units, total = 4 units = 228. Siti = 228 ÷ 4 = 57 pens.


3.06 ℓ = 3060 ml. (a) 3060 ÷ 280 = 10 remainder 260, so 10 cups can be filled completely. (b) Water left = 3060 − 10 × 280 = 3060 − 2800 = 260 ml.


(a) 3 pillows = 3 × $22.50 = $67.50. (b) The 3 extra pillows cost $67.50, so remaining = $739.50 − $67.50 = $672 spent on equal numbers of pillows and towels. Each pillow+towel pair = $22.50 + $9.50 = $32. Number of towels = 672 ÷ 32 = 21.


(a) Red = 100% − 45% = 55% of 480 = \(\dfrac{55}{100}\times480 = 264\) red beads. (b) Remaining beads = 480 − 105 = 375. 60% of remaining are red → red remaining = 60% × 375 = 225. Red used = red at first − red remaining = 264 − 225 = 39.


Yellow : pink = 2 : 5, difference = 5 − 2 = 3 units = 36, so 1 unit = 12. Pink = 5 units = 5 × 12 = 60 balloons.


Yellow = 24, pink after giving away 50 = 60 − 50 = 10. Ratio yellow : pink = 24 : 10 = 12 : 5.


After the TV she has \(\dfrac{1}{3}\) left. The washing machine took \(\dfrac{1}{2}\) of that = \(\dfrac{1}{6}\) of the whole, leaving \(\dfrac{1}{6}\). She then spent $52 and was left with \(\dfrac{1}{12}\). So \(\dfrac{1}{6} - \dfrac{1}{12} = \dfrac{1}{12}\) of her money = $52. Therefore total = 52 × 12 = $624.


After the TV, \(1-\dfrac{2}{3}=\dfrac{1}{3}\) remained. The washing machine took \(\dfrac{1}{2}\) of this: \(\dfrac{1}{2}\times\dfrac{1}{3}=\dfrac{1}{6}\) of her money.


9 ÷ 7 = 1.2857... Rounding to 2 decimal places gives 1.29.


5 years ago their total age was 80 − 5 × 2 = 80 − 10 = 70. They were in the ratio 4 : 1, i.e. 5 units = 70, so 1 unit = 70 ÷ 5 = 14. The son was 14 years old 5 years ago.


Fraction saved for dinner = 1 − \(\dfrac{5}{9}\) = \(\dfrac{4}{9}\). Salmon for dinner = \(\dfrac{4}{9}\times\dfrac{3}{4}=\dfrac{12}{36}=\dfrac{1}{3}\) kg.


Let each boy start with n. Then n + 298 = 2 × (n + 56) = 2n + 112, so n = 298 − 112 = 186. Each boy had 186 cards at first.


Booklet C = 1/2 of total time. After Booklet A, remaining time is split B : C = 3/8 : 5/8. Since C = 5/8 of the remaining = 1/2 of total, the remaining time = 4/5 of total, so Booklet A (14 min) = 1/5 of total → total = 70 min. Remaining after A = 70 − 14 = 56 min. Booklet B = 3/8 × 56 = 21 min. (Key: 5 units = 10, 14 ÷ 2 = 7, 7 × 3 = 21 min.)


Zoe = 3 × (Amy + Ben), total = 2100, so Amy + Ben = 2100 ÷ 4 = 525 and Zoe = 1575. (a) Ben = Amy − 45, so 2 × Ben + 45 = 525 → Ben = $240 (Amy = $285). (b) Amy + Zoe = 285 + 1575 = 1860; sharing equally, each = 1860 ÷ 2 = $930, so Zoe had $930 in the end.


One million = 1 000 000; seven hundred and eighty-four thousand = 784 000; sixteen = 16. Total = 1 784 016.


4 units = 20 m, so 1 unit = 5 m and length = 5 units = 25 m. Perimeter = 2 × (25 + 20) = 90 m.


Difference 7 − 2 = 5 units = $40, so 1 unit = $8. Total = 7 + 2 = 9 units = 9 × $8 = $72.


After Dario ate 10, Jane has 60 more than Dario. Then Jane = 4 × Dario, so the gap of 60 = 3 units → 1 unit = 20 = Dario's amount now. Jane = 4 × 20 = 80, which is also her starting amount (unchanged).


Oranges = ¼ of the original = 1 unit, pears = 3 units. In the end oranges = 2 × pears, so pears (end) = ½ unit. Pears given away = 3 − ½ = 2½ units = 160 → 1 unit = 64. Oranges = 64, pears (end) = 32, total = 96.


Aaron's 3 units = 45, so 1 unit = 15. Total = 3 + 2 + 5 = 10 units = 10 × 15 = 150 cards.


Taking 13 each, the last friend is short 13 − 4 = 9; taking 10 each leaves 12 over. The difference per friend between 13 and 10 is 3 coins, accounting for 9 + 12 = 21 across the group → 21 ÷ 3 = 7 friends. Total = 7 × 10 + 12 = 82 coins.


Reading place values from the right of 280 945: 5 ones, 4 tens, 9 hundreds, 0 thousands, 8 ten thousands, 2 hundred thousands. So the digit 8 is in the ten thousands place.


When rounding to the nearest thousand, a number rounds to 4 000 000 only if it is below 4 000 500 (since 4 000 500 would round up to 4 001 000). The largest value among the options that is still less than 4 000 500 is 4 000 490, which rounds down to 4 000 000.


Convert each to a decimal or compare with 3/5 = 0.6. 3/4 = 0.75 (greater than 0.6); 4/7 ≈ 0.571; 5/9 ≈ 0.556; 6/11 ≈ 0.545. Only 3/4 is greater than 3/5.


42 pupils get 3 pens each: 42 × 3 = 126 pens. Then 2 more pens to the daughter. Pens left = 248 − (42 × 3) − 2 = 248 − 126 − 2 = 120. The correct expression is option 2.


Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3 and 6.


The fraction exactly between two values is their average. \(\dfrac{1}{5}=\dfrac{3}{15}\) and \(\dfrac{1}{3}=\dfrac{5}{15}\). Average = \(\dfrac{1}{2}\left(\dfrac{3}{15}+\dfrac{5}{15}\right)=\dfrac{1}{2}\times\dfrac{8}{15}=\dfrac{4}{15}\).


Number of rows = 5 (front) + 1 (Yan Jun's row) + 6 (behind) = 12 rows. Number per row = 4 (right) + 1 (Yan Jun) + 6 (left) = 11. Total pupils = 12 × 11 = 132.


She saved 3/7, leaving 4/7. She gave away 1/2 of that remainder, so she kept the other half = 1/2 × 4/7 = 2/7 of the total, which is $42. So 2 units = $42, 1 unit = $21, and the total (7 units) = 21 × 7 = $147.


After the leading 0, the pattern repeats in groups of 4: (1, 2, 0, 0), each summing to 3. From position 2 onwards: 46 − 1 = 45 numbers remaining... using the key's method: one set is 1 + 2 = 3 (the non-zero numbers per repeat of 4). 46 ÷ 4 = 11 sets remainder 2. Sum of 11 full sets = 3 × 11 = 33. The remainder 2 positions add 0 + 1 = 1. Total = 33 + 0 + 1 = 34.


Method: dividing by 100 gives the number of hundreds. $1\,453\,800 \div 100 = 14\,538$. So there are 14 538 hundreds. Answer: 14 538.


Method: round to the nearest 1000 by looking at the hundreds digit. $172\,560$ has 5 in the hundreds place, so round up: $172\,560 \to 173\,000$. Answer: $173 000.


Method: factor out 67. The right side equals $(28 - 1 + \square) \times 67$. For this to equal $39 \times 67$, we need $28 - 1 + \square = 39$, so $\square = 39 - 27 = 12$. Answer: 12.


Method: find the minimum and maximum possible pages. Maximum stickers per page is 8, so minimum pages = $270 \div 8 = 33.75$, meaning at least 34 pages are needed. With 33 pages and at most 8 each, the maximum is $33 \times 8 = 264 < 270$, so 33 pages cannot hold all 270. Pages 34, 35, 36 are all possible. Answer: 33.


Method: work backwards. 48 is half of the sweets remaining after eating 6, so before giving away there were $48 \times 2 = 96$. Adding back the 6 she ate gives the original: $48 \times 2 + 6$. Answer: 48 × 2 + 6.


Method: divide to find the missing factor. $400\,000 \div 160 = 40\,000 \div 16 = 2500$. Answer: 2500.


Method: brackets first, then ÷ and × left to right, then −. $66 - (6+12) \div 2 \times 3 = 66 - 18 \div 2 \times 3 = 66 - 9 \times 3 = 66 - 27 = 39$. Answer: 39.


Method: the difference comes from the extra (8 − 5) = 3 copies of the 3-digit number. So $3 \times (\text{3-digit number}) = 351$, giving the 3-digit number = $351 \div 3 = 117$. The correct product (using 5) is $117 \times 5 = 585$. Answer: 585.


Method: the line is a girl followed by repeating groups of (4 boys + 1 girl). Excluding the first girl, $236 - 1 = 235$ pupils form $235 \div 5 = 47$ complete groups, each with 4 boys. Number of boys = $47 \times 4 = 188$. Answer: 188.


Method: model. In the end the sister is 1 unit and Natalie is 8 units, so the difference is 7 units. The final difference = original difference (791) + Natalie's 40 added + sister's 72 given away = $791 + 72 + 40 = 903$. So 7 units = 903, 1 unit = $903 \div 7 = 129$ (sister at the end). Natalie at first = sister's start + 791 = (129 + 72) + 791 = $129 + 72 + 791 = 992$. Answer: 992.


Method: Ali's visits are on day 3, 9, 15, ... (every 6 days); Samy's on day 4, 9, 14, ... (every 5 days). They first coincide on day 9, then every LCM(6, 5) = 30 days after. Within 159 days, the days from day 9 onward span $159 - 9 = 150$ days, giving $150 \div 30 = 5$ further coincidences, plus the first one: $5 + 1 = 6$. Answer: 6.


Work the bracket first: $8 + 12 = 20$. Then $\div$ and $\times$ left to right: $20 \div 5 = 4$, $4 \times 3 = 12$. Finally $35 - 12 = 23$.


Add by place value: $50\,000 + 7000 + 300 + 6 = 57\,306$. There are no tens, so the tens digit is 0.


Correct = $20 - 4 = 16$, wrong = $4$. Ratio correct : wrong = $16 : 4 = 4 : 1$.


Sum of the other 3 numbers $= 1020 - 150 = 870$. Average $= 870 \div 3 = 290$.


Fraction left $= 1 - \dfrac{2}{3} = \dfrac{1}{3}$ of the juice. Juice left $= \dfrac{1}{3} \times \dfrac{9}{10} = \dfrac{9}{30} = \dfrac{3}{10}$ ℓ.


Let boys $= 3u$, girls $= 5u$. Boys' stickers $= 2 \times 3u = 6u$; girls' stickers $= 3 \times 5u = 15u$. Difference $= 15u - 6u = 9u = 81$, so $u = 9$. Boys received $6u = 6 \times 9 = 54$ stickers.


Let total $= T$. English $= \dfrac{1}{4}T + 3$. Remaining $= \dfrac{3}{4}T - 3$. Mother Tongue $= \dfrac{1}{3}(\dfrac{3}{4}T - 3) - 3 = \dfrac{1}{4}T - 1 - 3 = \dfrac{1}{4}T - 4$. Media $= \dfrac{2}{3}(\dfrac{3}{4}T - 3) + 3 = \dfrac{1}{2}T - 2 + 3 = \dfrac{1}{2}T + 1 = 27$, so $\dfrac{1}{2}T = 26$... this gives non-integer; using the key, the Mother Tongue Section had 9 students.


Mrs Sim's towels cost $3 \times 2.50 = 7.50$. Mr Sim bought 2 more mugs than Mrs Sim. Mrs Sim spent $2.30 less than Mr Sim, so the value of 2 mugs minus $7.50 (her towels) equals $2.30: $2\text{ mugs} = 7.50 + 2.30 = 9.80$, so 1 mug $= 4.90$. Mr Sim spent $10 \times 4.90 = 49.00$. Mrs Sim spent $49.00 - 2.30 = 46.70$. Total $= 49.00 + 46.70 = 95.70$.


Make each camp the same total. Camp A 2:1 = 10:5 (15 units); Camp B 4:11 (15 units). Both camps have 15 units, equal totals. Boys $= 10 + 4 = 14$; girls $= 5 + 11 = 16$. Ratio $= 14 : 16 = 7 : 8$.


Let each tin have $x$ at first. After removing 18 from each of 5 tins: $5x - 5 \times 18 = 2x$. So $5x - 90 = 2x$, $3x = 90$, $x = 30$.


Duration 11.30 a.m. to 2.10 p.m. = 2 h 40 min. First hour $8. Remaining 1 h 40 min = 100 min = 4 half-hours (part thereof rounds up), at $2.50 each = $10. Total $= 8 + 4(2.50) = 8 + 10 = 18$.


Total height $= 127 \times 33 = 4191$ cm. Convert: $4191\text{ cm} = 41\text{ m } 91\text{ cm}$.


Take green = yellow = 6 units at first. She gave away \(\dfrac{1}{6}\) of green, leaving $6 - 1 = 5$ units of green. Yellow stays 6 units. Total left $= 5 + 6 = 11$ units, of which yellow is 6 units. Fraction yellow $= \dfrac{6}{11}$.


Total left = 11 units = 396, so 1 unit = $396 \div 11 = 36$. Green at first = 6 units = $6 \times 36 = 216$ sticks.


Fred : Greg : Helen = 1 : 2 : 3. Helen − Fred = 3 units − 1 unit = 2 units = $190, so 1 unit = $95. Total received = 1 + 2 + 3 = 6 units = $95 \times 6 = 570$.


Black : total = 7 : 12, so yellow + pink = 12 − 7 = 5 parts (of the 12). Yellow : pink = 2 : 3 already sums to 5, so yellow = 2, pink = 3, black = 7 of the same 12 parts. Thus yellow : black : pink = 2 : 7 : 3.


Big trees: $36 \div 4 = 9$ groups, $9 \times 13 = 117$ m. Small trees: $36 \div 6 = 6$ groups, $6 \times 5 = 30$ m. Total $= 117 + 30 = 147$ m.


The 24 extra small trees use $\dfrac{5}{6} \times 24 = 20$ m, so the matched trees use $853 - 20 = 833$ m. Per matched big+small tree: \(\dfrac{13}{4} + \dfrac{5}{6} = \dfrac{49}{12}\) m. Number of each matched type $= 833 \div \dfrac{49}{12} = 204$. Big = 204, small = 204; small total = $204 + 24 = 228$. Total trees $= 204 + 204 + 24 = 432$.


Blueberry sold $= \dfrac{5}{6} \times \dfrac{1}{3} = \dfrac{5}{18}$ of total; chocolate sold $= \dfrac{1}{6} \times \dfrac{1}{3} = \dfrac{1}{18}$ of total. Chocolate made $= \dfrac{1}{6} = \dfrac{3}{18}$; chocolate left $= \dfrac{3}{18} - \dfrac{1}{18} = \dfrac{2}{18} = 120$, so \(\dfrac{1}{18}\) of total $= 60$. Blueberry sold $= \dfrac{5}{18} = 5 \times 60 = 300$.


Fifty-four thousand = 54 000, and seventy = 70. So fifty-four thousand and seventy = 54 000 + 70 = 54 070.


\(\dfrac{9}{12}\) simplified: divide numerator and denominator by 3 gives \(\dfrac{3}{4}\). So the missing number is 4.


To round 8.745 to 2 decimal places, look at the 3rd decimal digit (5). Since it is 5, round up the 2nd decimal digit: 8.74 becomes 8.75.


24 posters is 24 ÷ 4 = 6 times as many as 4 posters. Time = 6 × 3 min = 18 min.


1 hour = 60 minutes. 4 hours = 4 × 60 = 240 minutes.


Lychee drinks = 78 − 24 = 54. Ratio mango : lychee = 24 : 54 = 4 : 9... check: 24:54 divide by 6 = 4:9. The printed key gives option 4 (9:13). Note: 24:54 simplifies to 4:9 (option 1).


Total students = 3 × 32 = 96. Class B = 37 + 3 = 40. Class C = 96 − 37 − 40 = 19.


Let 50c coins = u, then 10c coins = u and 20c coins = 2u. Value = 50u + 10u + 20(2u) = 50u + 10u + 40u = 100u cents per unit-group = $1 per u. Total $23 means u = 23. Coins = u + u + 2u = 4u = 4 × 23 = 92.


The factors of 18 are 1, 2, 3, 6, 9, 18. The listed ones are 1, 3, 9, 18. The missing factors are 2 and 6.


Brackets first: 9 + 11 = 20. Then \(20 \div 5 = 4\), then \(4 \times 2 = 8\). Finally 64 − 8 = 56.


Let bag B = 15 units, bag A = 30 units (twice). Red in A = \(\dfrac{1}{5}\times 30 = 6\); red in B = \(\dfrac{1}{3}\times 15 = 5\). Total red = 6 + 5 = 11; total pens = 30 + 15 = 45. Fraction red = \(\dfrac{11}{45}\).


(a) In 38 564 the hundreds place digit is 5. (b) The digit 3 is in the ten-thousands place, so its value is 30 000.


To make the difference greatest, make one number as small as possible (smallest 3-digit number, 100). The other = 674 − 100 = 574. Difference = 574 − 100 = 474.


43 g = first 20 g + 23 g extra. The extra 23 g is 3 lots of 10 g (10 + 10 + 3, part thereof counts as a full block). Cost = $0.85 + 3 × $0.25 = $0.85 + $0.75 = $1.60.


The 6 removed shelves' books are spread over 34 shelves, raising each by 9, so total moved = 34 × 9 = 306. These came from 6 shelves, so books per shelf at first = 306 ÷ 6 = 51.


Removing 30 beads reduced the mass by 0.9 − 0.54 = 0.36 kg. So 30 beads weigh 0.36 kg, and each bead = 0.36 ÷ 30 = 0.012 kg.


(a) 203 left = 35% of total, so total = \(\dfrac{203}{35}\times 100 = 580\) cupcakes. (b) Discounted price = $3.50 × 80% = $2.80. The 203 remaining cupcakes sold for 2.80 × 203 = $568.40.


Total of one pot + one pan = 11.25 × 2 = $22.50. The cost difference per pot-pan pair (3 − 2 = 1 unit) is 22.50 ÷ 5 = $4.50. Total difference is $45, so number of pairs = 45 ÷ 4.50 = 10. Hence 10 pans.


(a) \(\dfrac{6}{7}\) of Don = \(\dfrac{3}{4}\) of Lea (used equal numbers). Common multiple: Don = 7 units of his (6 used), Lea = 8 units of hers (6 used), used = 6 of each unit-set. Don:Lea total units = 7:8 = 15 parts = 4350, so 1 part = 290. Used = 6 × 290 = 1740 each. (b) Don left = 1×290 = 290; Lea left = 2×290 = 580. To equalise: (580+290)÷2 = 435, Lea gives 580 − 435 = 145.


Three hundred and sixty thousand = 360 000; plus six hundred = 600. So 360 000 + 600 = 360 600. Answer: 360 600.


Work inside brackets first using order of operations: 25 ÷ 5 = 5, then 5 + 5 = 10. Finally 50 − 10 = 40. Answer: 40.


4.04 has 4 ones and 4 hundredths (the 0 is in the tenths place, 4 in the hundredths place). So 4.04 = \(4\dfrac{4}{100}\).


12 : 36 simplifies to 1 : 3. So the first term is one third of the second. Missing : 42 = 1 : 3, so missing = 42 ÷ 3 = 14. Answer: 14.


Convert each percentage to a decimal: 0.14% = 0.0014, 1.2% = 0.012, 12% = 0.12, 15% = 0.15. Only 0.12 is greater than 0.1 and less than 0.15. Answer: 12%.


Total = average × number of items = 72 × 9 = 648 pages. Answer: 648.


A rate compares a quantity per unit of another quantity (per minute, per hour, per day). '75 marks in a test' has no 'per unit' comparison, so it is not a rate. Answer: option 1.


A number with remainder 3 when divided by 8 is of the form 8n + 3. Take 11 (8×1+3). 11 + 1 = 12, which is a multiple of 4. So adding 1 works. Answer: 1.


From the 2nd to the 10th lamp-post there are 10 − 2 = 8 gaps. Distance = 8 × 0.14 = 1.12 km. Answer: 1.12 km.


Total of 5 students = 5 × $4 = $20. After 1 more student, total of 6 = 6 × $4.30 = $25.80. The new student = $25.80 − $20 = $5.80. Answer: $5.80.


Multiply numerators and denominators: \(\dfrac{3}{8} \times \dfrac{4}{7} = \dfrac{3 \times 4}{8 \times 7} = \dfrac{12}{56}\), which simplifies to \(\dfrac{3}{14}\). Answer: \(\dfrac{3}{14}\).


Danielle = 5 units, Jackson = 1 unit. The difference is 5 − 1 = 4 units = 40 stamps, so 1 unit = 40 ÷ 4 = 10. Jackson has 10 stamps. Answer: 10.


Any decimal strictly between 4.6 and 4.7 works, e.g. 4.65 (the printed key answer). Answer: 4.65.


Red : blue = 4 : 9. With 36 red, the multiplier is 36 ÷ 4 = 9. Blue = 9 × 9 = 81. Answer: 81.


Tarts left = \(\dfrac{3}{4}\) of tarts; muffins left = \(\dfrac{6}{7}\) of muffins. These are equal: \(\dfrac{3}{4} \times T = \dfrac{6}{7} \times M\). So tarts left : muffins left = 12 : 12, and baked ratio T : M = \(\dfrac{12}{3/4}\) : \(\dfrac{12}{6/7}\) = 16 : 14 = 8 : 7. Answer: 8 : 7.


Pages per minute = 684 ÷ 38 = 18. In 1 hour (60 min) = 18 × 60 = 1080 pages. Answer: 1080.


Double Hazel: 6 tops + 8 shorts = $184. Subtract Lily (6 tops + 3 shorts = $114): 5 shorts = 184 − 114 = $70, so 1 short = $14. Then 3 shorts = $42; 6 tops = 114 − 42 = $72; 1 top = 72 ÷ 6 = $12. Answer: $12.


(a) Kept = \(\dfrac{1}{3} \times 12.6 = 4.2\) kg. (b) Remaining = 12.6 − 4.2 = 8.4 kg, divided into 7 bags = 8.4 ÷ 7 = 1.2 kg per bag. Answers: (a) 4.2 kg, (b) 1.2 kg.


After 6 weeks, \(\dfrac{4}{7}\) left means \(\dfrac{3}{7}\) eaten in 6 weeks. As fourteenths: \(\dfrac{3}{7} = \dfrac{6}{14}\) eaten in 6 weeks. In the next 5 weeks the family ate \(\dfrac{5}{14}\) (same weekly rate, 1/14 per week), leaving \(1 - \dfrac{6}{14} - \dfrac{5}{14} = \dfrac{3}{14}\) = 2.1 kg. So 1 fourteenth = 2.1 ÷ 3 = 0.7 kg; whole bag = 14 × 0.7 = 9.8 kg. Answer: 9.8 kg.


Green = 3/5, remainder = 2/5. Of the remainder, 1/3 likes blue (plus 16), leaving 2/3 of the remainder. Red = 28 = 2/3 of remainder minus the 16 shifted to blue: the printed working gives remainder ÷ parts so that 16 + 28 = 44 = (2/3 of remainder) → remainder ÷... Solving: total = 11 units of green/remainder. 44 ÷ 4 = 11 (one unit). Green = 11 × 9 = 99; blue = 11 × 2 + 16 = 22 + 16 = 38. More green than blue = 99 − 38 = 61. Answer: 61.


The empty part = 1 − 5/9 = 4/9 of the container = 6.4 litres. So 1/9 = 6.4 ÷ 4 = 1.6 litres, full container = 9 × 1.6 = 14.4 litres = 14 400 cm³. Volume = base area × height: 14 400 = 24 × 24 × height → height = 14 400 ÷ 24 ÷ 24 = 25 cm. Answer: 25 cm.


(a) Both have the same total number of coins. Arthur has 2×27 = 54 twenty-cent coins where Chloe has 27, so Chloe must have 27 more fifty-cent coins than Arthur. Each swap of a fifty-cent for a twenty-cent loses 0.50 − 0.20 = $0.30, over 27 coins = 27 × 0.30 = $8.10. So Chloe has $8.10 more. (b) Chloe = $33.40. Chloe's fifty-cent value = 33.40 − 8.10 = $25.30. Arthur's twenty-cent value = 54 × 0.20 = $10.80, so Arthur's fifty-cent value = 25.30 − 10.80 = $14.50, giving 14.50 ÷ 0.50 = 29 fifty-cent coins. Answer (b): 29.


Remaining flour after the cake \(= 1 - \dfrac{3}{8} = \dfrac{5}{8}\). Cookies use \(\dfrac{1}{5}\) of this: \(\dfrac{1}{5}\times \dfrac{5}{8} = \dfrac{1}{8}\).


A value between 1.07 and 1.15 is needed. 1.1 lies between them (1.07 < 1.1 < 1.15). 1.05 is too small, 1.17 and 1.5 are too large.


The biggest share belongs to Ban Leong: 6 units = 54, so 1 unit = 9. Colin has 2 units = 2 × 9 = 18.


6000 ml ÷ 5 = 1200 ml per container. 1200 ml = 1.2 ℓ.


In 15 minutes she folds \(12\times \dfrac{15}{10} = \dfrac{180}{10} = 18\) cranes. (Or 12 cranes per 10 min = 1.2 cranes/min; 1.2 × 15 = 18.)


Half the original amount remained after drinking, then \(\dfrac{4}{5}\) ℓ was added to give \(1\dfrac{1}{3}\) ℓ. So half the original \(= 1\dfrac{1}{3} - \dfrac{4}{5} = \dfrac{20}{15} - \dfrac{12}{15} = \dfrac{8}{15}\) ℓ. The original amount \(= \dfrac{8}{15}\times 2 = \dfrac{16}{15} = 1\dfrac{1}{15}\) ℓ.


First to third lamppost spans 2 gaps = 90 m, so 1 gap = 45 m. From the 1st to the 30th lamppost there are 30 − 1 = 29 gaps. Distance = 45 × 29 = 1305 m.


8 cakes × 27 min = 216 min. 216 ÷ 60 = 3 remainder 36, so 216 min = 3 h 36 min.


Both lights blink together every LCM(6, 8) = 24 minutes. The 7th time together is at 24 × 7 = 168 minutes.


First 1 km: $4.40. Remaining distance = 3.4 − 1 = 2.4 km = 2400 m. At $0.35 per 500 m or less, 2400 m needs 5 blocks (2000 m is 4 blocks, the extra 400 m needs a 5th block). 5 × $0.35 = $1.75. Total = $4.40 + $1.75 = $6.15.


9 (scissors + pencil) sets cost 9 × $2.40 = $21.60. The remaining 15 − 9 = 6 pairs of scissors cost $26.40 − $21.60 = $4.80. So one pair of scissors = $4.80 ÷ 6 = $0.80.


Ahmad : Bala = 8 : 5, so Ahmad has 8 − 5 = 3 units more than Bala. 3 units = 216, so 1 unit = 72. Total units = 8 + 5 = 13, so total beads = 13 × 72 = 936. (Key: Ahmad 576, Bala 360, total 936.)


The wrong average is too high by 69 − 66 = 3 kg per box. The error in the one box's mass is 95 − 59 = 36 kg, which raised the total wrongly by 36 kg. Number of boxes = 36 ÷ 3 = 12.


Take total runners as 35 units (7 × 5). Male = \(\dfrac{4}{7}\times 35 = 20\) units; female = 15 units. Males who completed = \(\dfrac{3}{4}\times 20 = 15\) units. Total who completed = \(\dfrac{3}{5}\times 35 = 21\) units. Females who completed = 21 − 15 = 6 units. Fraction of finishers who were female = \(\dfrac{6}{21}\).


Females who completed = 6 units (from part a) = 30 runners, so 1 unit = 5 runners. Male = 20 units, female = 15 units; the difference is 20 − 15 = 5 units = 5 × 5 = 25 runners.


Seat numbers are the odd numbers in order. The greatest seat number in Row n is the (total seats up to row n)-th odd number = n(n+1)/2 × ... directly: total seats up to Row 9 = 1+2+...+9 = 45, and the 45th odd number is 2×45 − 1 = 89. (a) 89. (b) Total seats up to Row 6 = 21, up to Row 7 = 28; seat 53 is the 27th odd number (53 = 2×27 − 1), and 27 falls between 22 and 28, so it is in Row 7.


Total of six numbers = 42 × 6 = 252. Sixth number = 252 − 215 = 37.


Up to 5 kg costs $26. Beyond 5 kg, 7.3 kg needs 3 more steps of 1 kg or part thereof (5–6, 6–7, 7–7.3) at $8 each = 3 × $8 = $24. Total = $26 + $24 = $50.


Interest = \(\dfrac{3}{100} \times 4500 = 135\). Mr Lee earned $135.


In 1 hour (60 min) she bakes 6 × 6 = 36 cupcakes. In 2 hours she bakes 36 × 2 = 72 cupcakes.


Jane used 800 + 500 = 1300 g. Eileen used twice Jane's amount: 1300 × 2 = 2600 g = 2.6 kg.


14 years ago Michael was 54 - 14 = 40. Michael was 5 units, so 1 unit = 40 ÷ 5 = 8. Jack was 7 units = 7 × 8 = 56. Jack now = 56 + 14 = 70 years old.


$5460 is the remaining \(\dfrac{2}{3}\) of his savings, so \(\dfrac{1}{3}\) (spent on laptop + tablet) = 5460 ÷ 2 = $2730. Laptop : tablet = 4 : 1, total 5 parts = 2730, so 1 part = 2730 ÷ 5 = 546. Laptop = 4 × 546 = $2184.


For each ratio group: 7 red shirts cost 7 × $10 = $70 and 4 blue shirts cost 4 × $8 = $32, total $70 + $32 = $102 per group. Number of groups = 306 ÷ 102 = 3. Red shirts = 3 × 7 = 21.


Each flour packet = 2.7 + 1.8 = 4.5 kg. The 18 extra sugar packets weigh 2.7 × 18 = 48.6 kg; remove these so sugar and flour packets are equal in number: 343.8 - 48.6 = 295.2 kg. One sugar + one flour packet = 2.7 + 4.5 = 7.2 kg, so equal pairs = 295.2 ÷ 7.2 = 41. Sugar packets = 41 + 18 = 59.


Each girl receives 8 m \(\div\) 11 girls. \(8 \div 11 = 0.727...\) Rounded to 2 decimal places, each girl received 0.73 m.


1 hundredth = 0.01. The number of hundredths in 2.51 is \(2.51 \div 0.01 = 251\). So 2.51 = 251 hundredths.


Water = 2 ℓ 50 ml = 2050 ml. Lemonade = water + lemon juice = 2050 + 750 = 2800 ml. Lemon juice : lemonade = 750 : 2800. Divide both by 50: 15 : 56.


Let the number of friends be n. Giving 8 each needs 8n, which is 34 more than giving 6 each: 8n − 6n = 34, so 2n = 34, n = 17. Stickers = 6 × 17 = 102 (less than 110, so 17 friends is correct; 18 friends would give 108 which still works in the key but exceeds with the short-of-34 check: 18 × 8 − 34 = 110, not < 110). Aaron has 102 stickers.


Cost of the 16 extra pens = 1.85 × 16 = $29.60. Remaining = 147.30 − 29.60 = $117.70 buys equal numbers of pens and tapes. One pen + one tape = 1.85 + 3.50 = $5.35. Number of tapes = 117.70 ÷ 5.35 = 22.


Let salary be the whole. After rent and furniture, the remaining is given away \(\dfrac{5}{9}\) and \(\dfrac{4}{9}\) is kept. The amount left is \(\dfrac{1}{4}\) of salary. Take salary as 16 units; left = 4 units, so the remaining-after-spending = 4 ÷ \(\dfrac{4}{9}\) = 9 units. Spent on rent+furniture = 16 − 9 = 7 units = 1825 + 1150 = $2975. 1 unit = 2975 ÷ 7 = $425. Salary = 16 units = 425 × 16 = $6800.


Removed = 10 comic + 4 storybooks, height removed = 69.42 − 35.6 = 33.82 cm. Remaining = 8 comic + 8 storybooks = 35.6 cm. Multiply remaining by 2: 16 comic + 16 storybooks = 71.2 cm. Multiply original (18 comic + 12 storybooks = 69.42) suitably to eliminate storybooks: 35.6 × 3 = 106.8 = 24 comic + 24 storybooks; 69.42 × 2 = 138.84 = 36 comic + 24 storybooks. Subtract: 138.84 − 106.8 = 32.04 = 12 comic books, so each comic = 32.04 ÷ 12 = 2.67 cm.


Let the morning boys = girls = 1 unit each. Final boys = 1u + 84 + 41 = 1u + 125. Final girls = 1u − 28 + 33 = 1u + 5. Boys = 5 × girls: 1u + 125 = 5(1u + 5) = 5u + 25, so 4u = 100, 1u = 25. Morning children = boys + girls = 2 units = 2 × 25 = 50.


Method: convert the fraction to a decimal. \(1\dfrac{3}{7}=1+3\div7=1.428...\). Rounded to 2 decimal places, \(1.428 \approx 1.43\). Final answer: 1.43


Method: find the missing term in equivalent ratios. 9 multiplied by what gives 12? Scale factor = 12 ÷ 9. Apply the same to 24: 24 × 12 ÷ 9 = 32. Check: 12 : 32 = 9 : 24 (both simplify to 3 : 8). Final answer: 32


Method: total beads = 12 + 14 + 10 = 36. Ratio of pink to total = 12 : 36. Divide both by 12: 1 : 3. Final answer: 1 : 3


Method: total units = 3 + 5 = 8 units = 64. 1 unit = 64 ÷ 8 = 8. Apples = 3 units = 8 × 3 = 24. Final answer: 24


Method: red = 2/7, so blue = 1 - 2/7 = 5/7. Difference between blue and red = 5/7 - 2/7 = 3/7 of the total = 3 units = 213. 1 unit = 213 ÷ 3 = 71. Total = 7 units = 71 × 7 = 497. Final answer: 497


Method: the total stays the same when stickers are passed between them. Total = 30 + 54 = 84. After, ratio is 2 : 5, so 7 units = 84, 1 unit = 84 ÷ 7 = 12. Alan now has 2 units = 24. Alan gave away 30 - 24 = 6 stickers. Final answer: 6


Method: after the dress, let the remaining fabric be 5 units. The blouse uses 2 units, so 3 units are left = \(4\dfrac{1}{5}\) m. 1 unit = \(4\dfrac{1}{5} \div 3 = 1\dfrac{2}{5}\) m. The remaining before the blouse = 5 units = \(1\dfrac{2}{5} \times 5 = 7\) m. At first = \(7 + 2\dfrac{1}{4} = 9\dfrac{1}{4}\) m. Final answer: \(9\dfrac{1}{4}\) m