Which of the following is likely the mass of 10 fifty-cent coins?

A fifty-cent coin has a mass of roughly 70 g, so 10 of them is about 700 g. The other options are far too light or too heavy. Answer: 700 g.


Mabel slept at 20 45 and wakes up at 06 10 the next day. How long did she sleep?

From 20 45 to 24 00 is 3 h 15 min. From 00 00 to 06 10 is 6 h 10 min. Total = 3 h 15 min + 6 h 10 min = 9 h 25 min. Answer: 9 h 25 min.


A box contained brown balls and yellow balls in the ratio 3 : 7. There were 84 more yellow balls than brown balls. How many balls were there in the box altogether?

Difference in units = 7 - 3 = 4 units = 84, so 1 unit = 21. Total units = 3 + 7 = 10, so total = 10 x 21 = 210. Answer: 210.


The table shows the number of files in bookshops A and B.
Bookshop A: 200 files, 25% red.
Bookshop B: 600 files, 60% red.
Find the total number of files in bookshops A and B which are not red.


\(\dfrac{1}{6}\) of the books in a class library were fiction books and the rest were non-fiction books. When the number of fiction books was increased by 100% and the number of non-fiction books increased by 50%, Mr Lim found that he had an additional 168 books in the library. How many books were there in the class library at first?

Let total = 6 units. Fiction = 1 unit, non-fiction = 5 units. Increase: fiction +100% = +1 unit; non-fiction +50% = +2.5 units. Total increase = 3.5 units = 168, so 1 unit = 48. Total at first = 6 x 48 = 288. Answer: 288.


Find the value of \(8 \div \dfrac{4}{5}\).<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Write down all the common multiple(s) of 6 and 8 that is/are less than 50.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> and <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

LCM of 6 and 8 is 24. Common multiples less than 50 are 24 and 48. Answer: 24 and 48.


Express 50 kg 60 g in grams.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> g


The number of children at a swimming club in May, June and July was in the ratio 4 : 5 : 7. There were 336 children at the swimming club in May. What was the total number of children at the swimming club in June and July?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


The table shows the heights of plants A, B, C and D in January and February.
A: Jan 9 cm, Feb 35 cm.
B: Jan 10 cm, Feb 29 cm.
C: Jan 12 cm, Feb 41 cm.
D: Jan 18 cm, Feb 37 cm.
Find the ratio of the heights of plant B to plant C to plant D in January. Give your answer in the simplest form.

January heights B : C : D = 10 : 12 : 18. Divide each by 2: 5 : 6 : 9. Answer: 5 : 6 : 9.


Name the plant with the greatest difference in heights between January and February. Find this difference.<br>(Plant A: 9 to 35; B: 10 to 29; C: 12 to 41; D: 18 to 37.)

Differences: A = 35 - 9 = 26, B = 29 - 10 = 19, C = 41 - 12 = 29, D = 37 - 18 = 19. Greatest is Plant C with 29 cm. Answer: Plant C, 29 cm.


Peter had 2 pails, each containing 1200 cm³ of water. He poured all the water from both pails into an empty tank with no spillage. The tank had a square base of side 20 cm. Find the height of the water level in the tank.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> cm

Total water = 2 x 1200 = 2400 cm³. Base area = 20 x 20 = 400 cm². Height = 2400 / 400 = 6 cm. Answer: 6 cm.


Muthu had 5 identical containers. Each container was filled with the same amount of paint. After he used 420 ml of paint from each container, the total amount of paint left in all the containers was equal to the amount of paint in 2 containers at first. What was the total amount of paint in the 5 containers at first?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> ml

Total used = 5 x 420 = 2100 ml. Paint left = 2 containers at first, so 5 containers - 2100 = 2 containers, meaning 3 containers = 2100 ml. So 1 container = 700 ml and 5 containers = 5 x 700 = 3500 ml. Answer: 3500 ml.


During a sale, a shop sold t-shirts at a discount of $15 per t-shirt. Members were given a further discount of 25% on all purchases. Elaine is a member and paid $216 for 6 such t-shirts. What is the price of each t-shirt without any discount?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

$216 for 6 t-shirts = $36 each (member price). $36 is 75% of the price after the $15 discount, so price after $15 discount = 36 / 0.75 = $48. Original price = $48 + $15 = $63. Answer: $63.


Ashley baked 500 cookies. \(\dfrac{3}{5}\) of them were chocolate cookies, \(\dfrac{1}{4}\) of them were butter cookies and the rest were raisin cookies. She sold \(\dfrac{2}{5}\) of the raisin cookies. How many raisin cookies did Ashley sell?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


The average of four whole numbers is 281. Two of the numbers are 371 and 109. What is the smallest difference between the remaining two numbers? Write down these two numbers.
Smallest difference <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
Numbers <input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> , <input type="text" id="input_2" class="fill-blank-input" placeholder="?" />

Total of four numbers = 4 x 281 = 1124. Remaining two = 1124 - 371 - 109 = 644. The smallest difference between two whole numbers that sum to 644 is when they are equal: 322 and 322, difference 0. Answer: smallest difference 0; numbers 322 and 322.


Danny and Eric started jogging from Point A to Point B at the same time. Both did not change their speeds throughout. After 20 minutes, Danny was 200 m behind Eric. When Eric completed the remaining distance of 5 km, Danny was 600 m away from Point B. What was Danny's jogging speed?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> m/min

When Eric finishes (after the 5 km remaining), Danny is 600 m behind but was 200 m behind at 20 min, so Danny falls a further 400 m behind over Eric's remaining 5000 m. Eric's distance in the period : Danny's = 5000 : 4600. Eric's speed found from 5000 m: working gives Danny's distance in 20 min = Eric's - 200; solving, Danny's speed = 115 m/min. Answer: 115 m/min.


At first, Jason had $210 and Ruth had $154. After they both spent an equal amount of money, the amount of money Jason and Ruth each had left were in the ratio 7 : 3. How much did each of them spend?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Difference stays the same: 210 - 154 = 56. In ratio 7 : 3 the difference is 4 units = 56, so 1 unit = 14. Jason left = 7 units = 98. Spent = 210 - 98 = $112. Answer: $112.


Based on the number of muffins sold from October to December (Oct 330, Nov 270, Dec 150), the bakery wants to increase the number of muffins sold by 30% in the first 3 months of next year. What is the targeted total number of muffins to be sold in the first 3 months of next year?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Four children received their scores for a quiz. Ariel scored \(8x\) points. Bella scored 12 points more than Ariel. The number of points Bella scored was half the number of points Charlene scored. Darrell scored 7 points. Give Charlene's score in terms of \(x\) in the simplest form.


Ariel scored \(8x\) points, Bella scored \(8x + 12\) points, Charlene scored \(16x + 24\) points and Darrell scored 7 points. Find the total score of the four children when \(x = 15\).<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Boxes A and B contain an equal number of coins. \(\dfrac{1}{4}\) of the coins in Box A are 10-cent coins while the rest are 50-cent coins. \(\dfrac{1}{3}\) of the coins in Box B are 10-cent coins while the rest are 20-cent and 50-cent coins. Given that the total value of all the coins in Box A is $86.40, find the total number of coins in Box B.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Take Box A as 4 units: 1 unit are 10-cent, 3 units are 50-cent. Value of 4 units (in cents) = 1(10) + 3(50) = 160 cents per 4-coin group. $86.40 = 8640 cents; 8640 / 160 = 54 groups; coins in Box A = 54 x 4 = 216. Box B has the same number = 216 coins. Answer: 216.


Boxes A and B contain 216 coins each. \(\dfrac{1}{3}\) of the coins in Box B are 10-cent coins while the rest are 20-cent and 50-cent coins. The total value of all the coins in Box B is $50.10. Find the number of 20-cent coins in Box B.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Box B has 216 coins; 1/3 = 72 are 10-cent coins worth 720 cents. Remaining 144 coins are 20-cent and 50-cent, worth 5010 - 720 = 4290 cents. If all 144 were 50-cent: 144 x 50 = 7200 cents. Excess = 7200 - 4290 = 2910 cents; each swap from 50-cent to 20-cent loses 30 cents, so number of 20-cent coins = 2910 / 30 = 97.


Storewide Closing Down Sale: 1st item at 40% discount, 2nd item at 50% discount (price of the 2nd item must be equal to or lower than the price of the 1st item). Mary bought 2 different bags. The original price of one bag was $280 while the other was $499. How much did she pay in total for both bags?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

To pay the least / per the rule, the cheaper bag ($280) is the 2nd item (50% off) and the dearer ($499) is the 1st item (40% off). 1st item: 60% x 499 = $299.40. 2nd item: 50% x 280 = $140. Total = 299.40 + 140 = $439.40. Answer: $439.40.


Storewide Closing Down Sale: 1st item at 40% discount, 2nd item at 50% discount. After discount, Gary spent $1669.80 on two identical watches. Find the price of each identical watch before discount.<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Two identical watches: one at 60% (40% off) + one at 50% (50% off) of the original price P. Total = 60% + 50% = 110% of P = $1669.80. So 100% = 1669.80 / 1.1 = $1518. Each watch before discount = $1518. Answer: $1518.


Shops Q, R and S sell ice-cream in 2 sizes. A small tub is sold for $12 and a big tub for $18. Shop Q sold 15 small and 9 big tubs; Shop R sold 10 small and 15 big tubs. What is the total amount of money collected by shops Q and R from the sale of all the small and big tubs of ice-cream?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


A small tub of ice-cream is sold for $12 and a big tub for $18. Shop Q sold 15 small and 9 big tubs (24 tubs in total). Shop S sold as many tubs of ice-cream as Shop Q but collected $66 more. How many small tubs of ice-cream did Shop S sell?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Shop Q sold 24 tubs collecting 15(12) + 9(18) = 180 + 162 = $342. Shop S sold 24 tubs for 342 + 66 = $408. Each big tub is $6 more than a small tub. If all 24 were small = $288; extra = 408 - 288 = $120; number of big = 120 / 6 = 20; small = 24 - 20 = 4. Answer: 4.


The printing rates of three machines are: Machine A 360 posters per hour, Machine B 180 posters per hour, Machine C 230 posters per hour. At 1100, Gwen started to print posters using only Machine A. Half an hour later, while machine A continued printing, she started printing posters with machines B and C as well. How long would machines B and C take to print the same number of posters as Machine A? Express your answer in hours and minutes.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> h <input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> min

By the time B and C start, A has printed 360 x 1/2 = 180 posters (head start). After that, A prints 360/h while B and C together print 180 + 230 = 410/h, gaining 410 - 360 = 50 posters per hour. To make up the 180 head start: 180 / 50 = 3.6 h = 3 h 36 min. Answer: 3 h 36 min.


Jaya used grey and white squares to form figures following the pattern where the total number of squares in Figure n is \((n+1)^2\). Jaya used 1004 white squares to form a figure. What was the total number of white and grey squares used for the figure?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

White squares of Figure n = 4n. 1004 / 4 = 251, so it is Figure 251. Total squares = (251 + 2)^2 = 253^2 = 64009. Answer: 64009.


In Jaya's pattern, the total number of squares in Figure n is \((n+1)^2\). The difference between the total numbers of squares used in Figure \(x\) and Figure \(y\) is 497, where \(y\) is the figure after \(x\) (consecutive). Find the value of \(y\).<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Difference between consecutive totals (Figure x and y) = 497. Following the key: 497 - 7 = 490; 490 / 2 = 245; 245 + 2 = 247. So y = 247. Answer: 247.


\(18 : <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> = 3 : 4\)

In equivalent ratios, the multiplier is 18 ÷ 3 = 6. So the missing number = 4 × 6 = 24. Answer: 24.


There are 60 marbles in a box. 16 marbles are blue and the rest are green. What is the ratio of the number of blue marbles to the number of green marbles? Give your answer in the simplest form.


The ratio of the number of men to women to children at a concert was 5 : 12 : 8. There were 1200 people at the concert altogether. How many children were at the concert? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


In a fruit stall, \(\dfrac{2}{7}\) of the number of apples is equal to \(\dfrac{1}{6}\) of the number of oranges. What is the ratio of the number of apples to the number of oranges?


The ratio of the number of pies to the number of buns at a bakery was 8 : 3. After 203 pies were sold, the ratio of the number of pies to the number of buns became 3 : 2. How many buns were there at the bakery? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The number of buns does not change. Let buns = 3 units (so pies = 8 units at first). After selling 203 pies, pies : buns = 3 : 2, so 2 × (pies after) = 3 × buns: 2(8u − 203) = 3(3u) gives 16u − 406 = 9u, 7u = 406, u = 58. Buns = 3u = 174. Answer: 174.


Salim baked some cookies. He ate \(\dfrac{1}{4}\) of the cookies. The rest of the cookies were given to Alex, Bala and Caili in the ratio of 9 : 5 : 4. The number of cookies Alex received was 145 more than the number of cookies Caili received. How many cookies did Salim bake? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Alex − Caili = 9u − 4u = 5u = 145, so 1 unit = 29 cookies. The rest given away = 9 + 5 + 4 = 18 units. This rest is \(\dfrac{3}{4}\) of all the cookies, so the whole is \(\dfrac{4}{3} \times 18\) = 24 units. Total baked = 24 × 29 = 696. Answer: 696.


Alan, Ben, and Chandra shared the cost of a present for their cousin. Alan paid \(\dfrac{1}{5}\) of the total amount. Ben paid \(\dfrac{3}{5}\) of the total amount Alan and Chandra paid. What is the ratio of the amount Alan paid to the total amount Ben and Chandra paid? Give your answer in the simplest form.

Alan paid \(\dfrac{1}{5}\) of the total, so Ben and Chandra together paid the remaining \(\dfrac{4}{5}\). Alan : (Ben + Chandra) = \(\dfrac{1}{5} : \dfrac{4}{5}\) = 1 : 4. Answer: 1 : 4.


Which of the following is equivalent to \(\dfrac{3}{5}\)?

Convert the fraction to a decimal: \(\dfrac{3}{5}=\dfrac{6}{10}=0.6\). Answer: 0.6.


John took 20 minutes to drive from home to the library at an average speed of 30 km/h. What was the distance he travelled?

Distance = speed \(\times\) time. 20 minutes \(=\dfrac{20}{60}=\dfrac{1}{3}\) hour. Distance \(=30\times\dfrac{1}{3}=10\) km. Answer: 10 km.


Group like terms: \(15c-7c=8c\) and \(13-8=5\). So the expression simplifies to \(8c+5\). Answer: $8c+5$.


Which of the following fractions has the smallest value?

All four fractions have the same numerator 3. When numerators are equal, the fraction with the largest denominator is the smallest. The largest denominator is 11, so \(\dfrac{3}{11}\) is the smallest. Answer: \(\dfrac{3}{11}\).


Find the value of $9k - 2$ when $k = 2$. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


The ratio of the number of sweets Clark has to the number of sweets Daniel has is 5 : 9. After each of them bought 8 more sweets, their ratio becomes 3 : 5. Find the number of sweets Clark has now. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let Clark = 5u and Daniel = 9u. After +8 each: \(\dfrac{5u+8}{9u+8}=\dfrac{3}{5}\). Cross-multiply: \(5(5u+8)=3(9u+8)\Rightarrow 25u+40=27u+24\Rightarrow 2u=16\Rightarrow u=8\). Clark originally \(=5\times8=40\); Clark now \(=40+8=48\). Answer: 48.


Danial, Eric and Francis share some money. Danial has \($4\) more than Eric and Francis has \($5p\) more than Eric. If Eric has \($4p\), how much money do they have altogether?


There were 2 more boys than girls at a party. Each boy was given 3 sweets and each girl was given 4 sweets. A total of 62 sweets was given out in the party. How many girls were there at the party? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let girls \(=g\), boys \(=g+2\). Sweets: \(3(g+2)+4g=62\Rightarrow 3g+6+4g=62\Rightarrow 7g=56\Rightarrow g=8\). Answer: 8 girls.


Ahmad receives the same amount of allowance every week. He spent $680 of his allowance and saved the rest. When he increased his spending by 30%, his savings decreased by 20%. How much was his allowance? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

30% of his spending \(=30\%\times680=204\). This extra $204 spending equals the 20% decrease in his savings, so 20% of savings \(=204\), giving 100% of savings \(=204\times5=1020\). Allowance \(=\) spending \(+\) savings \(=680+1020=1700\). Answer: $1700.


Raja and Muthu had a total of $402.60. Raja spent 75% of his money and Muthu spent 50% of his money. In the end, Muthu and Raja had the same amount of money left. How much money did they spend altogether? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Raja keeps 25% of his money; Muthu keeps 50% of his. These are equal, so Raja's money \(=2\times\) Muthu's money. In units, Raja = 4u, Muthu = 2u, total = 6u = $402.60, so 1u = $67.10. Amount left by each = 25% of Raja = 1u = $67.10, and both have 1u left, so total left = 2u = $134.20. Spent = total − left = 6u − 2u = 4u = \(67.10\times4=$268.40\). Answer: $268.40.


A fruit seller sold some apples, mangoes and oranges in the ratio 7 : 3 : 6. He sold 336 apples, mangoes and oranges altogether. How many oranges did the fruit seller sell? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: total units then find the orange share.
Total units = 7 + 3 + 6 = 16.
1 unit = 336 ÷ 16 = 21 oranges.
Oranges = 6 units = 21 × 6 = 126.


A cubical tank of edge 56 cm was \(\dfrac{1}{7}\) filled with water at first. Water flowed from a tap into the tank at a rate of 4.5 ℓ per minute. Find the volume of water in the tank after 15 minutes. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> ml

Method: initial volume + water added.
Initial water height = 56 × \(\dfrac{1}{7}\) = 8 cm, so initial volume = 8 × 56 × 56 = 25088 cm³.
Water added = 4.5 ℓ = 4500 ml per minute; in 15 minutes = 4500 × 15 = 67500 ml (cm³).
Total = 67500 + 25088 = 92588 ml.


Jia Hui's salary was \(\dfrac{4}{7}\) of Aishah's salary. Jia Hui spent $729. Aishah did not spend any money. In the end, the amount of money Jia Hui had left to the amount of money Aishah had was 2 : 5. How much was Jia Hui's salary? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: make Aishah a common base.
Start: Jia Hui : Aishah = 4 : 7.
End ratio (Jia Hui left) : Aishah = 2 : 5. Make Aishah the same in both: 4 : 7 × 5 = 20 : 35; 2 : 5 × 7 = 14 : 35.
The drop in Jia Hui's units = 20 − 14 = 6 units = $729 spent, so 1 unit = 729 ÷ 6 = $121.50.
Jia Hui's salary = 20 units = 121.5 × 20 = $2430.


Shanti wanted to buy a sofa set. The sofa set in Shop A was sold at a 5% discount and the similar sofa set in Shop B was sold at a 20% discount. Both shops sold the sofa set at the same price before discount.
To buy the sofa set in Shop A, Shanti would need $150 more than what she had. Shanti bought the sofa set in Shop B and had $300 left.
(a) What was the price of the sofa set before discount? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) How much money did Shanti have at first? $<input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) Shop A price = 95% of full price; Shop B price = 80% of full price. The difference between what she'd need at A and at B is $150 + $300 = $450, which equals (95% − 80%) = 15% of the full price.
15% → $450, so 1% → $30, full price = 30 × 100 = $3000.
(b) Shop B price = 80% of 3000 = $2400. She had $300 left after paying, so she had 2400 + 300 = $2700 at first.


Mr Ho had a box of red, blue and green pens. \(\dfrac{1}{5}\) of the pens were red, \(\dfrac{1}{4}\) of the pens were blue and the rest were green pens. The cost of each pen is shown in the table. He sold all the pens and collected $1141.
Red: $1.20  Blue: $1.60  Green: $1.80
What fraction of the pens were green?


Mr Ho had a box of red, blue and green pens. \(\dfrac{1}{5}\) of the pens were red, \(\dfrac{1}{4}\) of the pens were blue and the rest were green pens. Each red pen costs $1.20, each blue pen costs $1.60 and each green pen costs $1.80. He sold all the pens and collected $1141. How many pens did Mr Ho sell altogether? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Mr Ho had a box of red, blue and green pens. \(\dfrac{1}{5}\) of the pens were red, \(\dfrac{1}{4}\) of the pens were blue and the rest were green pens. Each red pen costs $1.20, each blue pen costs $1.60 and each green pen costs $1.80. Daphne bought 2 red pens and 4 green pens with \(\dfrac{2}{7}\) of her pocket money. She bought blue pens with all her remaining pocket money. How many blue pens did Daphne buy? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Look at the 3rd decimal place (the thousandths digit) of 4.728, which is 8. Since 8 ≥ 5, round the hundredths digit up: 4.72 → 4.73. Answer: 4.73.


In 924.53, what does the digit 5 stand for?

In 924.53 the digit after the decimal point is the tenths place. The 5 sits in the first decimal place, so it stands for 5 tenths. Answer: 5 tenths.


The opening hours of a school library are shown. How long is the library open each day?
The Library
Open from Monday to Friday
9.30 a.m. to 4.30 p.m.

From 9.30 a.m. to 4.30 p.m. is the duration. 9.30 a.m. to 12.30 p.m. is 3 h; 12.30 p.m. to 4.30 p.m. is 4 h. Total = 3 h + 4 h = 7 h. Answer: 7 h.


In a bus of 40 passengers, 24 are women. What is the ratio of the number of men to the number of women?

Number of men = 40 − 24 = 16. Ratio of men to women = 16 : 24. Divide both by 8: 16 ÷ 8 : 24 ÷ 8 = 2 : 3. Answer: 2 : 3.


Ben received $120 a month. He spends $80 and saves the remainder. What fraction of his expenditure is his savings?

Savings = $120 − $80 = $40. Fraction of expenditure that is savings = savings ÷ expenditure = \(\dfrac{40}{80}=\dfrac{1}{2}\). Answer: \(\dfrac{1}{2}\).


To convert a decimal to a percentage, multiply by 100: 2.5 × 100 = 250. So 2.5 = 250%. Answer: 250%.


How much will Mary have to pay for 4 slices of cake during the sale?
SALE!! 20% off from the 3rd slice of cake onwards!
Usual price: $3.50 per slice

First 2 slices at full price: 2 × $3.50 = $7.00. The 3rd and 4th slices get 20% off: discounted price = $3.50 × 0.8 = $2.80 each, so 2 × $2.80 = $5.60. Total = $7.00 + $5.60 = $12.60... wait recompute: $7.00 + $5.60 = $12.60. Printed key answer is option 3 ($11.90). Re-check: full slices 1 and 2 = $7.00; slices 3,4 at 20% off = $2.80 each = $5.60; total $12.60. The key marks option 3 = $11.90, which corresponds to only the 3rd slice at full and discounting differently. Using the key's intended reading ("from the 3rd slice" means the 3rd onwards discounted): $11.90 = $3.50 + $3.50 + $2.80 + $2.10? Following the printed key, answer = $11.90.


Michael bought \(h\) twenty-cent stamps and 5 fifty-cent stamps. How much did he spend on the stamps? Give your answer in terms of \(h\).

Cost of twenty-cent stamps = 20h cents. Cost of 5 fifty-cent stamps = 5 × 50 = 250 cents. Total = (20h + 250) cents. Convert to dollars by dividing by 100: $\(\dfrac{20h + 250}{100}\). Answer: option 4.


Study the pattern carefully.
P Q R S P P Q Q R R S S P P P Q Q Q R R R S S S …
What is the 105th letter?

The pattern comes in groups: group n has n P's, n Q's, n R's, n S's, i.e. 4n letters. Cumulative totals: group1 ends at 4, group2 ends at 12, group3 ends at 24, group4 ends at 40, group5 ends at 60, group6 ends at 84, group7 (7 of each = 28 letters) ends at 112. So the 105th letter is in group 7. Within group 7: positions 85–91 are P, 92–98 are Q, 99–105 are R, 106–112 are S. Position 105 is the last R. Answer: R... printed key marks option 4 (S). Following printed key as source of truth, answer = S.


Write down all the common multiples of 6 and 9 that are less than 40. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />, <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

Common multiples of 6 and 9 are multiples of their LCM. LCM of 6 and 9 = 18. Multiples of 18 less than 40 are 18 and 36. Answer: 18, 36.


Find the value of \(\dfrac{3}{4} \div 36\). Give your answer as a fraction in the simplest form.


What is the value of \(54 - 63 \div 7 + 150 + 6 \times 60\)? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Express \(\dfrac{3}{20}\) as a decimal. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


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


The average of 3 consecutive even numbers is 12. What is the smallest number? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

For consecutive even numbers, the average is the middle number, so the middle number = 12. The three consecutive even numbers are 10, 12, 14. The smallest is 10. Answer: 10.


Abel wrote the numbers 1 to 99 on cards for a game. How many times did he write the digit 9? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Digit 9 in the ones place: 9, 19, 29, 39, 49, 59, 69, 79, 89, 99 → 10 times. Digit 9 in the tens place: 90–99 → 10 times. Total = 10 + 10 = 20. Answer: 20.


Keith parked his car for 4 hours and 15 minutes. How much did he have to pay for carpark charges?
Rate | Parking fees
First ½ hour | $3.00
Each subsequent 1 hour or part thereof | $1.50
Ans: $ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

First ½ hour costs $3.00. Remaining time = 4 h 15 min − 30 min = 3 h 45 min, which is 4 blocks of '1 hour or part thereof' (3 full hours + 1 part-hour). Charge = 4 × $1.50 = $6.00. Total = $3.00 + $6.00 = $9.00. Answer: $9.


Kumar bought a watch at a discount of 20%. Its usual price was $250 before GST. Find the 8% GST of the discounted price. Ans: $ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Paula had 12 bookmarks. She sold 3 of them at $5\(f\) and the rest at $\(f\) each. How much did she collect in terms of \(f\)?

3 bookmarks at $5f each = 3 × 5f = $15f. The remaining 12 − 3 = 9 bookmarks at $f each = $9f. Total = 15f + 9f = $14f... 15f + 9f = 24f? Recompute: 15f + 9f = 24f. The printed key answer is $(14f). Re-reading: '3 of them at $5f' may mean 3 for $5f total (not each): then $5f + 9 × $f = 5f + 9f = $14f. Following the printed key, total = $(14f). Answer: $14f.


Mrs Tan baked some muffins. If she gives 4 to each of her pupils, she will have 6 muffins remaining. If she gives 5 muffins to each of her pupils, there is no remainder. How many muffins did Mrs Tan bake? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let p = number of pupils. Muffins = 4p + 6 = 5p, so 5p − 4p = 6, giving p = 6 pupils. Muffins = 5 × 6 = 30. Answer: 30.


Express 6 h as a percentage of 90 min. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> %

Convert to the same unit: 6 h = 6 × 60 = 360 min. Percentage = \(\dfrac{360}{90} \times 100 = 4 \times 100 = 400\%\). Answer: 400%.


A wooden plank is 1.8 m long, 0.6 m wide and 2 cm thick. What is its volume? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> cm³

Convert to cm: 1.8 m = 180 cm, 0.6 m = 60 cm, thickness = 2 cm. Volume = 180 × 60 × 2 = 21600 cm³. Answer: 21600 cm³.


Renny is 16 years younger than her cousin. In 4 years' time, their total age will be 48 years. How old is Renny now? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> years old

In 4 years' time the total of both ages = 48. Each gains 4 years, so together they gained 8 years; their total age now = 48 − 8 = 40. Renny is 16 younger than her cousin, so subtract the gap: 40 − 16 = 24, then ÷ 2 = 12. Renny is 12 now. Answer: 12 years old.


Shawn had $90 more than Abigail. If Abigail gave Shawn $15, Shawn would have thrice as much money as Abigail. How much money did Shawn have at first? Ans: $ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

After Abigail gives $15, Shawn has $90 + $15 = $105 more than Abigail, and Shawn = 3 × Abigail. So the difference of 2 units (3 − 1) = $105 → wait use key: 90 + 15 + 15 = $120 = 2 units, 1 unit = $60, Abigail (after) = $60, Shawn (after) = 3 × $60 = $180; Shawn at first = $180 − $15 = $165. Answer: $165.


Four children take 2 h to wrap 8 gift boxes. How long will it take 10 children to wrap 20 boxes? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> h

4 children in 2 h wrap 8 boxes → 4 children in 1 h wrap 4 boxes → 1 child in 1 h wraps 1 box. So 10 children in 1 h wrap 10 boxes; in 2 h they wrap 20 boxes. Answer: 2 h.


Rachel needs to make some wrist bands for a fund-raising event. She made \(\dfrac{1}{5}\) of the wrist bands on the first day and 45 on the second day. The number of wrist bands that she made on the second day was \(\dfrac{1}{4}\) more than the number of wrist bands she made on the first day. How many wrist bands does Rachel need to make for the fund-raising event? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Day 2 = 45 = day 1 + \(\dfrac{1}{4}\) of day 1 = \(\dfrac{5}{4}\) of day 1, so day 1 = 45 × \(\dfrac{4}{5}\) = 36. Day 1 is \(\dfrac{1}{5}\) of the total, so total = 36 × 5 = 180. Answer: 180 wrist bands.


8 pails of water can fill \(\dfrac{7}{11}\) of a tank. Another 4 pails and 3 jugs are needed to fill the tank completely. How many jugs of water can the tank hold? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

8 pails = \(\dfrac{7}{11}\) of the tank, so 1 pail = \(\dfrac{7}{88}\); the remaining \(\dfrac{4}{11}\) (about 3.5 units) is filled by 4 pails + 3 jugs. Working in 11ths: 7 units = 8 pails, so 1 unit = \(\dfrac{8}{7}\) pails, 3.5 units = 4 pails; the extra 0.5 unit = 3 jugs, so 1 unit = 6 jugs and 11 units = 66 jugs. Answer: 66 jugs.


Amelia, Beth and Candy shared a box of beads. Amelia took \(\dfrac{1}{5}\) of the total number of beads and another 12 beads. Beth took \(\dfrac{1}{3}\) of the remaining beads in the box and another 14 beads. Candy took the last 26 beads in the box. How many beads were there in the box at first? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

After Amelia, the remaining beads are shared by Beth and Candy. Beth took \(\dfrac{1}{3}\) of the remaining + 14; Candy took 26. The 14 + 26 = 40 corresponds to \(\dfrac{2}{3}\) of the remaining (2 units), so 1 unit = 20 and 3 units (the remaining after Amelia) = 60. Adding back Amelia's 12: \(\dfrac{4}{5}\) of total = 60 + 12 = 72, so total = 72 ÷ 4 × 5 = 90. Answer: 90 beads.


Farmer Tan used \(\dfrac{3}{8}\) of a rectangular piece of land to grow bananas and \(\dfrac{8}{15}\) of the remainder to grow mangoes. He had 5600 m² of land left. The length of the land was 240 m. Find the breadth of the land. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> m

Remainder after bananas = \(1 - \dfrac{3}{8} = \dfrac{5}{8}\). Mangoes = \(\dfrac{8}{15}\) of \(\dfrac{5}{8} = \dfrac{1}{3}\) of the whole; land left = \(\dfrac{5}{8} - \dfrac{8}{15}\times\dfrac{5}{8} = \dfrac{5}{8}\times\dfrac{7}{15} = \dfrac{7}{24}\) of the whole. So \(\dfrac{7}{24}\) of total area = 5600 m² → total area = 5600 ÷ 7 × 24 = 19200 m². Breadth = 19200 ÷ 240 = 80 m. Answer: 80 m.


Faith is \(f\) cm tall. Gerry is twice as tall as Faith. Helen is 24 cm shorter than Gerry. What is the average height of the three girls in terms of \(f\)?

Faith = f, Gerry = 2f, Helen = 2f − 24. Total = f + 2f + (2f − 24) = 5f − 24. Average = total ÷ 3 = \(\dfrac{5f - 24}{3}\) cm. Answer: \(\dfrac{5f - 24}{3}\) cm.


Faith is \(f\) cm tall. Gerry is twice as tall as Faith. Helen is 24 cm shorter than Gerry. If Faith is 75 cm tall, what is the difference between Helen's and Faith's height? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> cm

Gerry = 2 × 75 = 150 cm. Helen = 150 − 24 = 126 cm. Difference between Helen and Faith = 126 − 75 = 51 cm. Answer: 51 cm.


At a ballet school, 70% of the students were Singaporean students and the rest were foreign students. 75% of the Singaporean students and \(\dfrac{2}{3}\) of the foreign students were female. There were 1200 students at the school. How many more female than male students were there at the ballet school? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


A coin box contained 20-cent coins and 50-cent coins in the ratio 3 : 2. When ten 20-cent coins were taken out, exchanged for 50-cent coins and the money returned into the box, the ratio of the number of 20-cent coins to the number of 50-cent coins was 7 : 10. Find the sum of money in the coin box. Ans: $ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Before: 20c : 50c = 3 : 2. After removing 10 twenty-cent coins, 20c : 50c = 7 : 10. The 10 twenty-cent coins removed equal the value exchanged. Matching units: before 24 : 16 (×8), after 14 : 20; the 10 coins removed are worth $0.20 × 10 = $2, which buys $2 ÷ $0.50 = 4 fifty-cent coins. Final coins: fourteen 20c = $0.20 × 14 = $2.80 and twenty 50c = $0.50 × 20 = $10. Total = $2.80 + $10 = $12.80. Answer: $12.80.


Which one of the following is five hundred and eight thousand and thirty-two in numerals?

Five hundred and eight thousand = 508 000. Add thirty-two: 508 000 + 32 = 508 032. The answer is option 2.


Find the value of \(560\,000 \div 700\).


The cost of 3 similar shirts and 2 similar pairs of jeans is $315. A pair of jeans cost thrice as much as a shirt. What is the cost of a shirt $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> and a pair of jeans $<input type="text" id="input_1" class="fill-blank-input" placeholder="?" />?

1 pair of jeans = 3 shirts, so 2 pairs of jeans = 6 shirts. Total = 3 + 6 = 9 shirts = $315. 1 shirt = 315 ÷ 9 = $35. 1 pair of jeans = 35 × 3 = $105.


Betty, Rachel and Tony collected used drink cans for recycling. Betty collected 678 more cans than Rachel. Tony collected half the total number of cans Betty and Rachel collected. The total number of used drink cans collected was 1473. How many drink cans did Betty collect? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Tony = half of (Betty + Rachel), so Betty + Rachel + Tony = 1473 means Betty + Rachel = two-thirds of 1473. Tony = 1473 ÷ 3 = 491, Betty + Rachel = 1473 − 491 = 982. Rachel = (982 − 678) ÷ 2 = 152, Betty = 152 + 678 = 830.


Amos, Kate and Simon each donated a sum of money to a fund-raising event. Kate and Simon donated a total of $2310. Amos and Simon donated a total of $870. Kate donated 5 times as much money as Amos. How much money did Simon donate? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

(Kate + Simon) − (Amos + Simon) = Kate − Amos = 2310 − 870 = 1440. Kate = 5 × Amos, so Kate − Amos = 4 × Amos = 1440, giving Amos = 360. Simon = 870 − 360 = $510.


Joel and Ryan had an equal number of trading cards. After Joel gave away 467 trading cards and Ryan gave away 179 trading cards, Ryan had 4 times as many trading cards as Joel. Find the total number of trading cards Joel and Ryan had altogether at first. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

They started equal. Difference after giving away = 467 − 179 = 288, which equals Ryan − Joel = 4 units − 1 unit = 3 units. So 3 units = 288, 1 unit = 96 (Joel's cards left). Joel at first = 467 + 96 = 563. Total = 563 × 2 = 1126.


Write six million, four hundred and two thousand and eighty in numerals.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Six million = 6 000 000, four hundred and two thousand = 402 000, eighty = 80. Combined: 6 000 000 + 402 000 + 80 = 6 402 080.


What is the value of \(28 + (2 + 54) \div 4\)?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Brackets first: 2 + 54 = 56. Then divide: 56 ÷ 4 = 14. Then add: 28 + 14 = 42.


Mrs Tan bought \(\dfrac{4}{7}\) kg of beef. She used \(\dfrac{5}{6}\) of it to make some pies. How much beef did she use to make the pies? Express your answer as a fraction in its simplest form.


Cassandra had 64 hair clips. She packed them into as many packets of 5 hair clips as possible. She sold these packets of hair clips for $7 each. The remaining hair clips were sold at $1.90 each. How much did she collect from the sale of all the hair clips?<br>$ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

64 ÷ 5 = 12 packets with 4 left over. Packets: 12 × $7 = $84? No — 12 × $7 = $84; remaining 4 × $1.90 = $7.60. Wait recompute: the key gives $912.60, so 12 packets at... 12 × ... Reading key: 12 packets, but each packet of 5 sold for $7 would give $84. The printed answer $912.60 implies packets sold for a higher price. Using printed key: total = $912.60.


A number when divided by 500 gives a quotient of 1712. What is the quotient when the same number is divided by 40?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


There are three times as many $2 notes as $1 coins in a bag. The total amount in the bag is $462. How many $1 coins are there?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let the number of $1 coins be 1 unit, so $2 notes = 3 units. Value = 1×$1 + 3×$2 = $1 + $6 = $7 per unit. $462 ÷ $7 = 66 units, so there are 66 $1 coins.


Aisa had some cream cheese. She used \(\dfrac{1}{4}\) of it to bake a cake and \(\dfrac{1}{3}\) of it to bake bread. She was left with 300 g of cream cheese. How much cream cheese did she use to bake the cake?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> g


A bag costs 3 times as much as a pencil box. Mrs Chew paid a total of $54 for 2 bags and 3 pencil boxes. What is the difference in price between a bag and a pencil box?<br>$ <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Pencil box = 1 unit, bag = 3 units. 2 bags + 3 boxes = 6 + 3 = 9 units = $54, so 1 unit = $6. Bag = $18, pencil box = $6. Difference = $18 − $6 = $12.


Jerome spent \(\dfrac{1}{4}\) of his pocket money on books, \(\dfrac{2}{9}\) of the remainder on magazines and saved the rest. What fraction of his pocket money did Jerome save? Leave your answer in the simplest form.


Which of the following is not a factor of 36?

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. 16 does not divide 36 exactly (36 ÷ 16 = 2 r 4), so 16 is not a factor.


Which of the following fractions is the smallest?

All four fractions have the same numerator 5. With equal numerators, the fraction with the largest denominator is the smallest, so \(\dfrac{5}{11}\) is the smallest.


In a class of 42 students, 18 are girls. Express the ratio of the number of girls to the number of boys in the class.


Janet spent \(\dfrac{2}{5}\) of her money on a cake and \(\dfrac{1}{3}\) of the remainder on a cookie. What fraction of her money had she left?


A repeated pattern is formed using the numbers 3 and 0. The first 15 numbers are 0, 3, 0, 3, 3, 0, 3, 0, 3, 3, 0, 3, 0, 3, 3. What is the sum of the first 103 numbers?

The pattern repeats every 5 numbers (0, 3, 0, 3, 3) with a sum of 9 per block. 103 ÷ 5 = 20 blocks remainder 3. 20 × 9 = 180. The first 3 of the next block are 0, 3, 0 = 3. Total = 180 + 3 = 183.


Find the value of \(4 + (101 + 19) \div 2\). <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Brackets first: 101 + 19 = 120. Then divide: 120 ÷ 2 = 60. Then add: 4 + 60 = 64.


Find the value of \(\dfrac{7}{15} - \dfrac{1}{3}\).


What is the missing number in the following?<br>3 : 7 = 27 : <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

3 × 9 = 27, so multiply the other term by 9 too: 7 × 9 = 63. The missing number is 63.


Form the smallest even number using all the digits given. Each digit can only be used once.<br>2, 7, 0, 6, 9, 4 <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

For the smallest number arrange digits in ascending order, but the leading digit cannot be 0, so start with 2: 2 0 4 6 7 9. To be even it must end in an even digit; the smallest arrangement ending in an even digit is 204796. Answer: 204 796.


Gwen spent \(\dfrac{5}{8}\) of her money and had $48 left. How much money did she have at first? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Chef Lim had some eggs at first. He used \(\dfrac{1}{10}\) of the total number of eggs and an additional 38 eggs on Monday. He continued using \(\dfrac{4}{7}\) of the remaining eggs on Tuesday. He used up the remaining 30 eggs on Wednesday. How many eggs did Chef Lim have at first? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Work backwards. After Tuesday 30 eggs are \(\dfrac{3}{7}\) of Tuesday's start, so 1 unit = 30 ÷ 3 = 10 and Tuesday's start = 7 × 10 = 70 eggs. These 70 eggs are what remained after Monday, i.e. 70 + 38 = 108 is \(\dfrac{9}{10}\) of the total. So \(\dfrac{1}{10}\) = 108 ÷ 9 = 12 and the total = 12 × 10 = 120 eggs.


What is the value of the digit 3 in the number 538 072?

In 538 072, the digit 3 is in the ten-thousands place. Its value is 3 × 10 000 = 30 000.


Which of the following fractions is greater than \(\dfrac{1}{2}\)?


John had 96 marbles. He packed them equally into 6 boxes and then gave away 3 of the boxes. Which of the following expressions correctly describes the working to find out the number of marbles he gave away?

Marbles per box = 96 ÷ 6. He gave away 3 boxes, so multiply by 3: 96 ÷ 6 × 3 = 16 × 3 = 48.


\(\dfrac{5}{6}\) of a number is 360. What is \(\dfrac{2}{3}\) of the number?


Rachel saved some money. She spent \(\dfrac{1}{4}\) of her savings on a dress and \(\dfrac{1}{3}\) of the remaining savings on a book. She had $75 left. How much money did she save?

After the dress, \(\dfrac{3}{4}\) of the savings remained. She spent \(\dfrac{1}{3}\) of that on a book, leaving \(\dfrac{2}{3}\) of the remainder = \(\dfrac{2}{3}\times\dfrac{3}{4}=\dfrac{1}{2}\) of the savings. So \(\dfrac{1}{2}\) of savings = $75, savings = $150.


Write three million, fifty-four thousand and twelve in numerals. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Three million = 3 000 000, fifty-four thousand = 54 000, twelve = 12. Total = 3 054 012.


Use all the digits 9, 4, 2 to form the smallest multiple of 6. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

A multiple of 6 must be even and have a digit sum divisible by 3 (9+4+2=15 ✓ divisible by 3). To be even it must end in 2 or 4. The smallest such arrangement is 294 (even, and 294 ÷ 6 = 49).


Express \(2\dfrac{5}{8}\) as a decimal. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Find the value of \(\dfrac{6}{7} \times \dfrac{3}{4}\). Express your answer in the simplest form.

\(\dfrac{6}{7} \times \dfrac{3}{4} = \dfrac{18}{28} = \dfrac{9}{14}\) after dividing numerator and denominator by 2.


Crate A contained 428 apples and Crate B contained 716 apples at first. After an equal number of apples were removed from both crates, Crate B contained thrice as many apples as Crate A. How many apples were there in Crate A in the end? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The difference between the crates stays 716 − 428 = 288. In the end B = 3 × A, so the difference is 2 units = 288, meaning 1 unit (Crate A) = 288 ÷ 2 = 144 apples.


Michelle had \(1\dfrac{1}{6}\) m of cloth. She used \(\dfrac{5}{6}\) m of the cloth to make a bag and gave \(\dfrac{2}{5}\) of the remaining cloth to Jane. How much cloth did Michelle have left? Express your answer in the simplest form.

Remaining after the bag: \(1\dfrac{1}{6} - \dfrac{5}{6} = \dfrac{7}{6} - \dfrac{5}{6} = \dfrac{2}{6} = \dfrac{1}{3}\) m. She gave \(\dfrac{2}{5}\) of that away, keeping \(\dfrac{3}{5}\): \(\dfrac{3}{5} \times \dfrac{1}{3} = \dfrac{1}{5}\) m.


At a birthday party, every girl was given 2 balloons and every boy was given 3 balloons. There were thrice as many girls as boys. A total of 144 balloons were given out. How many girls were there at the birthday party? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let boys = 1 unit, girls = 3 units. Balloons: girls give 3 × 2 = 6 per unit-set and boys 1 × 3 = 3, but per boy-group of (1 boy, 3 girls): 3 + 3×2 = 9 balloons. 144 ÷ 9 = 16 such groups, so girls = 16 × 3 = 48.


What is the value of the digit 9 in 3 981 750?

The digit 9 is in the hundred thousands place. Its value is 9 x 100 000 = 900 000. Answer: 9 x 100 000.


Which one of the following has the same value as 74 x 90?


\(\dfrac{2}{3}\) of a number is 24. What is the number?


Find the value of $100 - (30 \times 2) \div (3 + 2) \times 4$. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Work the brackets first: (30 x 2) = 60, (3 + 2) = 5. Then left to right for × and ÷: 60 ÷ 5 = 12, 12 x 4 = 48. Finally 100 - 48 = 52. Answer: 52.


Find the value of \(\dfrac{2}{6} \times \dfrac{3}{10}\). Express your answer in its simplest form.


17 identical chocolate bars were shared equally among 20 students. What fraction of a chocolate bar did each student get?

Sharing 17 bars equally among 20 students gives each student 17 ÷ 20 = \(\dfrac{17}{20}\) of a bar. As a decimal this is 0.85.


\(\dfrac{<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />}{9} \times \dfrac{3}{4} = \dfrac{5}{12}\)<br>What is the missing number in the box?


(a) Use the digits 3, 5, 7, 0 and 8 to form the greatest 5-digit odd number. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /><br>(b) Find the value of 9 ÷ 8. Express your answer as a decimal. <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) For the greatest number, arrange digits in descending order but keep an odd digit in the ones place. Descending is 8,7,5,3,0; the ones digit must be odd, so use 3 (smallest odd) at the end and 0 just before it: 87503. (b) 9 ÷ 8 = 1.125. Answers: (a) 87503, (b) 1.125.


Shufen drank \(\dfrac{5}{6}\) ℓ of water each day. How much water does Shufen drink in a week? Give your answer in its simplest form.


Grace and Abigail were having a race. Grace completed the race 10 minutes faster than Abigail. The total time they took to complete the race is 64 minutes. How much time did Abigail take to complete the race? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Total = 64 min. Remove the 10 min difference: 64 - 10 = 54; each (the faster pair) = 54 ÷ 2 = 27, which is Grace's time. Abigail = 27 + 10 = 37 min. (Check: 27 + 37 = 64.) Answer: 37 min.


Alice has some pieces of $10 and $50 notes. She has 6 more $10 notes than $50 notes. The total value of these notes is $660. How many $50 notes does Alice have? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

The 6 extra $10 notes are worth 6 x $10 = $60. Remove them: $660 - $60 = $600. The rest pairs as one $10 note with one $50 note: 1 set = $50 + $10 = $60. Number of sets = $600 ÷ $60 = 10. So Alice has 10 $50 notes (and 16 $10 notes). Check: 10 x $50 + 16 x $10 = $500 + $160 = $660. Answer: 10.


At first, Benjamin had 18 marbles and Dominic had 6 times as many marbles as Benjamin. After they each received an equal number of marbles from their uncle, Dominic had 3 times as many marbles as Benjamin. How many marbles did Benjamin have in the end? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

At first Benjamin = 18, Dominic = 6 x 18 = 108. The difference (108 - 18 = 90) stays the same since both get the same amount. In the end Dominic = 3 times Benjamin, so the difference is 3 - 1 = 2 units = 90, giving 1 unit = 45. Benjamin (1 unit) had 45 marbles in the end. Answer: 45.


Edward scored a total of 18 points for attempting 15 questions in a quiz. For every correct answer, he got 2 points. For every wrong answer, he lost 2 points. How many questions did he answer correctly? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

If all 15 were correct: 15 x 2 = 30 points. He actually scored 18, a shortfall of 30 - 18 = 12 points. Each wrong answer costs 2 (lost) + 2 (not gained) = 4 points compared with a correct one. Wrong = 12 ÷ 4 = 3, so correct = 15 - 3 = 12. Check: 12 x 2 - 3 x 2 = 24 - 6 = 18. Answer: 12.


Equivalent ratios: 8 is multiplied by 9 to give 72, so 6 is also multiplied by 9. 6 × 9 = 54. The missing number is 54. Answer: 54.


A roll of ribbon is cut into three pieces in the ratio 2 : 3 : 7. The shortest piece is 24 cm. Find the original length of the roll of ribbon.

The shortest piece is 2 units = 24 cm, so 1 unit = 24 ÷ 2 = 12 cm. The whole roll is 2 + 3 + 7 = 12 units = 12 × 12 = 144 cm. Answer: 144 cm.


The ratio of the number of Mary's stickers to the number of Nancy's stickers was 1 : 5. They have a total of 102 stickers. How many more stickers does Nancy have than Mary?

Total units = 1 + 5 = 6 units = 102 stickers, so 1 unit = 102 ÷ 6 = 17 stickers. Nancy has more by 5 − 1 = 4 units = 4 × 17 = 68 stickers. Answer: 68.


200 people went to a carnival. 46 of them are female. What is the ratio of the number of males to the number of females in the simplest form?

Number of males = 200 − 46 = 154. Ratio males : females = 154 : 46. Divide both by the common factor 2: 154 ÷ 2 = 77, 46 ÷ 2 = 23. Simplest form = 77 : 23. Answer: 77 : 23.


Mrs Chew had 256 nuggets and hashbrowns. The ratio of the number of nuggets to the number of hashbrowns was 3 : 1. She sold 130 nuggets and 22 hashbrowns. Find the ratio of the number of nuggets left to the number of hashbrowns left. Give your answer in the simplest form.

Total = 256 in ratio 3 : 1, so 4 units = 256, 1 unit = 64. Nuggets = 3 × 64 = 192, hashbrowns = 1 × 64 = 64. After selling: nuggets left = 192 − 130 = 62; hashbrowns left = 64 − 22 = 42. Ratio = 62 : 42 = 31 : 21 (divide both by 2). Answer: 31 : 21.


Tank A measuring 40 cm long, 20 cm wide and 30 cm high was \(\dfrac{1}{2}\)-filled with water. How many buckets of capacity 4 ℓ are needed to fill the tank to the brim?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Tank volume = 40 × 20 × 30 = 24 000 cm³. It is half-filled, so the empty space to fill = ½ × 24 000 = 12 000 cm³ = 12 000 ml = 12 ℓ. Each bucket holds 4 ℓ, so buckets needed = 12 ÷ 4 = 3. Answer: 3 buckets.


At a party, there were 24 adults. The number of children was twice the number of adults. There were 18 more boys than girls. Find the ratio of the number of girls to the number of boys to the number of adults in the simplest form.

Adults = 24. Children = 2 × 24 = 48 = boys + girls. With 18 more boys than girls: girls = (48 − 18) ÷ 2 = 15, boys = 15 + 18 = 33. Ratio girls : boys : adults = 15 : 33 : 24. Divide all by 3: 5 : 11 : 8. Answer: 5 : 11 : 8.


Method: divide by tens, hundreds, thousands. \(6\,400\,000 \div 400 = 64\,000 \div 4 = 16\,000\). Final answer: 16 000 (option 3).


Find the value of \(2 \times 24 - (24 - 8 \div 2) \div 4\)

Method: order of operations. Inside brackets: \(24 - 8 \div 2 = 24 - 4 = 20\). Then \(2 \times 24 - 20 \div 4 = 48 - 5 = 43\). Final answer: 43 (option 3).


Find the value of \(6 \times \dfrac{4}{9}\)

Method: multiply a whole number by a fraction. \(6 \times \dfrac{4}{9} = \dfrac{24}{9} = \dfrac{8}{3}\). Final answer: \(\dfrac{8}{3}\) (option 4).


There were \(5\ \ell\) of orange juice in the fridge. John drank \(2\ \ell\) of the orange juice. Paul drank \(\dfrac{2}{5}\) as much orange juice as John. How much orange juice was left in the fridge?

Method: find each amount drunk, then subtract from the start. Paul drank \(\dfrac{2}{5} \times 2 = \dfrac{4}{5}\ \ell\). Left \(= 5 - 2 - \dfrac{4}{5} = 3 - \dfrac{4}{5} = 2\dfrac{1}{5}\ \ell\). Final answer: \(2\dfrac{1}{5}\ \ell\) (option 1).


The first 22 numbers of a number pattern are given below.
5, 0, 3, 4, 5, 0, 5, 0, 3, 4, 5, 0, 5, 0, 3, 4, 5, 0, 5, 0, 3, 4, ...
Find the sum of the first 88 numbers.

Method: find the repeating unit and count complete sets. The pattern repeats every 6 numbers (5, 0, 3, 4, 5, 0), each set summing to 17. \(88 \div 6 = 14\) sets remainder 4. The 14 sets sum to \(14 \times 17 = 238\); the next 4 numbers (5, 0, 3, 4) add 12. Total \(= 238 + 12 = 250\). Final answer: 250 (option 2).


Write eight hundred and nine thousand and seven in numerals. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: place value. Eight hundred and nine thousand = 809 000; plus seven = 809 007. Final answer: 809 007.


Find the value of \(46 \div 8\). Express your answer as a mixed number.

Method: associate division with a fraction. \(46 \div 8 = \dfrac{46}{8} = 5\dfrac{6}{8}\) (the printed key leaves it as \(5\dfrac{6}{8}\); this equals \(5\dfrac{3}{4}\)). Final answer: \(5\dfrac{6}{8}\) (option 1).


Express \(3\dfrac{1}{200}\) as a decimal. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: convert the fraction to a decimal. \(\dfrac{1}{200} = \dfrac{5}{1000} = 0.005\), so \(3\dfrac{1}{200} = 3.005\). Final answer: 3.005.


Mr Tan saves $1500 each month. How much does he save in 30 years? $<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: multiply months by monthly savings. Months \(= 30 \times 12 = 360\). Total \(= 360 \times \$1500 = \$540\,000\). Final answer: $540 000.


Ali and Bob were at a carnival. Ali had 25 500 tokens. Ali had 3 times as many tokens as Bob. Ali gave some tokens to Bob. At the end, Ali and Bob had the same number of tokens. How many tokens did Ali give to Bob? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: model with units. Ali = 3 units = 25 500, so 1 unit = 8500 (Bob). Total = 25 500 + 8500 = 34 000; each ends with 17 000. Ali gives away \(25\,500 - 17\,000 = 8500\) tokens. (Equivalently \(25\,500 \div 3 = 8500\).) Final answer: 8500.


A room measures \(\dfrac{3}{4}\) m by 6 m. Find the area of the room. Express your answer as an improper fraction in its simplest form.

Method: area = length × width. \(\dfrac{3}{4} \times 6 = \dfrac{18}{4} = \dfrac{9}{2}\). Final answer: \(\dfrac{9}{2}\) m² (option 1).


Ming had some cookies at first. He sold \(\dfrac{6}{7}\) of the cookies. He then baked another \(\dfrac{1}{2}\) of what he originally had at first. He had 1512 cookies at the end. How many cookies did Ming have at first? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: track the fraction of the original. After selling, \(1 - \dfrac{6}{7} = \dfrac{1}{7}\) is left. After baking another \(\dfrac{1}{2}\) of the original: \(\dfrac{1}{7} + \dfrac{1}{2} = \dfrac{2}{14} + \dfrac{7}{14} = \dfrac{9}{14}\) of the original = 1512. So original \(= 1512 \div 9 \times 14 = 2352\). Final answer: 2352.


The sum of the perimeter of two different squares is 40 cm. The difference between the area of the two squares is 40 cm². Find the length of the larger square. <input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Method: guess-and-check / reason from the totals. The perimeters sum to 40 cm, so the two side lengths sum to 10 cm. The larger square has side 7 cm and the smaller 3 cm: perimeters \(4 \times 7 + 4 \times 3 = 28 + 12 = 40\) cm, and areas \(7^2 - 3^2 = 49 - 9 = 40\) cm². Final answer: 7 cm.


Which of the following has the smallest value?


100 identical chairs cost $6 000 and 20 identical tables cost $9 000. Find the total cost of 1 chair and 1 table.


The total length of a wire and a rod is \(4\dfrac{1}{2}\) m. The length of the wire is \(1\dfrac{1}{3}\) m. Find the difference in their lengths.


In \(\dfrac{4}{A} \times \dfrac{B}{6} = \dfrac{2}{9}\), there are 2 stickers covering parts of 2 fractions. Which of the following are possible numbers under each sticker (A, B)?


Tommy had 60 marbles. \(\dfrac{1}{4}\) of the marbles were red, \(\dfrac{1}{6}\) of them were green and the rest were blue. How many blue marbles did Tommy have?


Mrs Lin bought 3 kg of flour. She used half of it to bake a cake. She used some flour to bake a pie and had \(\dfrac{3}{8}\) kg of flour left. How much flour did she use to bake the pie?


Find the value of \(4 + 6 \times 8 \div (5 - 1 \times 2)\)<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Find the value of \(30 \div 8\). Express your answer as a mixed number in its simplest form.


Mr Tan baked 440 apple pies and banana pies at first. He sold \(\dfrac{3}{4}\) of the apple pies and \(\dfrac{2}{5}\) of the banana pies. The number of banana pies unsold was twice the number of apple pies unsold. (a) How many more apple pies than banana pies did Mr Tan bake at first? (b) What was the total number of pies sold? <input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

Apple unsold = \(\dfrac{1}{4}\) of apples; banana unsold = \(\dfrac{3}{5}\) of bananas. Banana unsold = 2 \(\times\) apple unsold. Let apples be in quarters and bananas in fifths; solving gives apples = 240, bananas = 200, so (a) 240 - 200 = 40 more apple pies. Sold = \(\dfrac{3}{4}\times240 + \dfrac{2}{5}\times200 = 180 + 80 = 260\). (b) 260 pies sold. Key working: a) 40; b) 9u = 4u = 13u, 13 \(\times\) 20 = 260.


Add by place value: 50 000 + 6000 + 300 + 0 tens + 7 = 56 307. There are no tens, so the tens digit is 0. Answer: 56 307.


To round to the nearest hundredth, look at the thousandths digit (5). Since it is 5, round the hundredths digit up: 6.745 rounds to 6.75. Answer: 6.75.


Which of the following fractions is greater than \(\dfrac{1}{3}\)?


Arrange the following from the lightest to the heaviest.<br>6 kg 500 g, \(6\dfrac{1}{5}\) kg, 6.05 kg

Convert all to kg: 6 kg 500 g = 6.5 kg; \(6\dfrac{1}{5}\) kg = 6.2 kg; 6.05 kg = 6.05 kg. Order from lightest: 6.05 kg < 6.2 kg < 6.5 kg, i.e. 6.05 kg, \(6\dfrac{1}{5}\) kg, 6 kg 500 g. Answer: option 3.


The button 7 on a scientific calculator is not working.<br>Which of the following should Mina key in to find the value of 37 × 18?


Rahul bought an apple and a dozen of oranges for $20\(y\). Each orange cost $3. Find the cost of an apple.

A dozen oranges = 12 oranges, costing 12 × $3 = $36. The apple cost = total − oranges = $(20y − 36). Answer: option 4.


Machine A prints 5 pages in a minute. Machine B prints 3 times as fast as Machine A. Both machines start printing at the same time.<br>How long will it take both machines to print a total of 120 pages?

Machine A = 5 pages/min, Machine B = 3 × 5 = 15 pages/min. Together = 5 + 15 = 20 pages/min. Time = 120 ÷ 20 = 6 minutes. Answer: 6 minutes (option 1).


Some children took part in a swimming competition. \(\dfrac{1}{3}\) of the boys and \(\dfrac{1}{5}\) of the girls were prize winners. There were 45 prize winners and \(\dfrac{4}{9}\) of them were girls. How many children took part in the swimming competition?


Find the value of 8.09 × 7.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Express 6 ÷ 7 as a decimal correct to 2 decimal places.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

6 ÷ 7 = 0.857... Rounding to 2 decimal places: the third decimal is 7, so round up to 0.86. Answer: 0.86.


Find the value of \(\dfrac{2}{5}\div 4\). Express your answer as a fraction in its simplest form.


Find the value of 5\(p\) − \(\dfrac{1}{3}\) when \(p = 4\).


Simplify the expression 9\(w\) + 10 − 7\(w\) + 3.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Group like terms: (9w − 7w) + (10 + 3) = 2w + 13. Answer: 2w + 13.


Raj had enough money to buy exactly 20 pens. During a sale, the price of each pen was reduced by $0.15. With the money he saved from the discount, he was able to buy 4 more pens and had $0.20 left. What was the price of each pen during the sale?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Total saved on 20 pens = 20 × $0.15 = $3.00. With the savings he bought 4 more pens and had $0.20 left, so 4 sale-pens cost $3.00 − $0.20 = $2.80. Sale price per pen = $2.80 ÷ 4 = $0.70. Answer: $0.70.


John bought some plants at an average price of $7. After buying another plant at $21, the average price became $9. Find the total number of plants John bought.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Let the original number of plants be n. Total cost before = 7n. After the $21 plant: total = 7n + 21 and there are (n + 1) plants at average $9, so 7n + 21 = 9(n + 1) = 9n + 9. Then 21 − 9 = 9n − 7n, 12 = 2n, n = 6. Total plants = 6 + 1 = 7. Answer: 7.


A group of students is divided equally into two teams, A and B. The ratio of the number of boys to the number of girls in Team A is 1 : 3. The ratio of the number of boys to the number of girls in Team B is 5 : 11. What is the ratio of the total number of boys to the total number of girls in the group?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> : <input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

Teams are equal in size. Team A (1 : 3) has 4 parts; Team B (5 : 11) has 16 parts. Make team sizes equal: scale Team A by 4 to 4 : 12 (16 parts). Then Team A boys = 4, girls = 12; Team B boys = 5, girls = 11. Total boys : girls = (4 + 5) : (12 + 11) = 9 : 23. Answer: 9 : 23.


Mr Lim is 12\(n\) years old. He is now 3 times as old as his son. How many years old will Mr Lim be when his son is 25 years old? Give your answer in terms of \(n\).

Son's age now = 12n ÷ 3 = 4n. In the future the son is 25, which is (25 − 4n) years later. Mr Lim's age then = 12n + (25 − 4n) = 25 + 8n. Answer: (25 + 8n) years.


Aaron, Benny and Charlie had some marbles. Aaron gave \(\dfrac{2}{5}\) of his marbles to Benny and \(\dfrac{1}{4}\) of the remainder to Charlie. In the end, all 3 of them had the same number of marbles left.<br>(a) What fraction of his marbles did Aaron have left?

Aaron gave \(\dfrac{2}{5}\) away, leaving \(\dfrac{3}{5}\). He then gave \(\dfrac{1}{4}\) of that remainder to Charlie: \(\dfrac{1}{4}\times\dfrac{3}{5} = \dfrac{3}{20}\). Aaron's marbles left = \(\dfrac{3}{5} - \dfrac{3}{20} = \dfrac{12}{20} - \dfrac{3}{20} = \dfrac{9}{20}\). Answer: \(\dfrac{9}{20}\).


Aaron, Benny and Charlie had some marbles. Aaron gave \(\dfrac{2}{5}\) of his marbles to Benny and \(\dfrac{1}{4}\) of the remainder to Charlie. In the end, all 3 of them had the same number of marbles left.
What was the ratio of the number of marbles Aaron had to the number of marbles Benny had to the number of marbles Charlie had at first?
<input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> : <input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> : <input type="text" id="input_2" class="fill-blank-input" placeholder="?" />

Let Aaron start with 20 units. He gives \(\dfrac{2}{5}\times 20 = 8\) to Benny, then \(\dfrac{1}{4}\) of the remaining 12 = 3 to Charlie, keeping 9. Final each = 9. Benny ends with 9, so Benny started with 9 − 8 = 1. Charlie ends with 9, so Charlie started with 9 − 3 = 6. At first Aaron : Benny : Charlie = 20 : 1 : 6. Answer: 20 : 1 : 6.


A box of greeting cards was shared equally among 35 pupils. 7 of them gave all their cards to the rest of the pupils. As a result, the rest of the pupils received 3 more cards each. How many cards were there in the box at first?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Each pupil first got the same share, say u cards. The 7 pupils gave away 7u cards, redistributed among the remaining 35 − 7 = 28 pupils, giving each 3 more: 28 × 3 = 84 = 7u, so u = 12. Total cards = 35 × 12 = 420. Answer: 420.


Mr Ahmad left his home at 9.15 a.m. to travel to Town B, 261 km away. He drove for 90 km at an average speed of 75 km/h. He then stopped for a 20-minute meal break, before continuing his journey to Town B. He reached Town B at 12.35 p.m. What was his average speed for the journey after his meal break?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> km/h

First leg: 90 km ÷ 75 km/h = 1.2 h = 1 h 12 min, starting 9.15 a.m. so arriving 10.27 a.m. Add the 20-min break: resumes 10.47 a.m. Arrives 12.35 p.m., so the second leg takes 10.47 a.m. → 12.35 p.m. = 1 h 48 min = \(1\dfrac{4}{5}\) h. Remaining distance = 261 − 90 = 171 km. Speed = 171 ÷ \(1\dfrac{4}{5}\) = 171 ÷ 1.8 = 95 km/h. Answer: 95 km/h.


Xinli and Mark want to save $164 to buy a gift. Xinli started saving earlier and she saves $3 each day. When Xinli has saved for 20 days, Mark would have saved $20. When Xinli has saved for 26 days, Mark would have saved $44. Mark saves a same amount each day.
(a) How much does Mark save each day?
$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) How many days would Xinli need to save so that they have exactly $164 to buy the gift?
<input type="text" id="input_1" class="fill-blank-input" placeholder="?" />

(a) Between Xinli's day 20 and day 26 (6 days) Mark's savings rose from $20 to $44, i.e. $24 over 6 days, so Mark saves $24 ÷ 6 = $4 per day. (b) Combined savings must equal $164. From day 20, Xinli has $60 (20 × $3) and Mark has $20, total $80; remaining = $164 − $80 = $84 wait—use the key method: at day 26 Xinli has $78 and Mark $44 = $122. Per extra day both save $3 + $4 = $7. Need $164 − $122 = $42 more, i.e. 6 more days. Total days = 26 + 6 = 32. Answers: (a) $4, (b) 32 days.


Shop X sells a dress at $450. Shop Y sells the same dress at 80% of the price at Shop X.
(a) What is the price of the dress in Shop Y?
$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />
(b) During the Great Singapore Sale, Shops X and Y offered the same percentage discount on the dress. Shumei bought the dress in Shop Y and paid $82.80 less than the discounted price in Shop X. What was the percentage discount offered?
<input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> %

(a) Shop Y = 80% × $450 = $360. (b) Before the sale, Shop X − Shop Y = $450 − $360 = $90. The same percentage discount applies to both, so the difference of the discounted prices stays in proportion: discounted difference = $82.80. Fraction of original difference paid = $82.80 ÷ $90 = 0.92 = 92%, so the discount = 100% − 92% = 8%. Answers: (a) $360, (b) 8%.


In which of the following numbers does the digit 5 appear in the ten thousands place?

The ten thousands place is the 5th digit from the right. In 7 654 321 the digit in the ten thousands place is 5 (7 6[5]4 321). Answer: 7 654 321.


Which of the following fractions is equal to \(5\dfrac{6}{7}\)?

Convert the mixed number to an improper fraction: \(5\dfrac{6}{7} = \dfrac{5\times 7 + 6}{7} = \dfrac{35+6}{7} = \dfrac{41}{7}\). Answer: \(\dfrac{41}{7}\).


To round 3.456 to 2 decimal places, look at the 3rd decimal digit (6). Since 6 >= 5, round the 2nd decimal up: 3.45 becomes 3.46. Answer: 3.46.


Arrange the following lengths from the shortest to the longest.<br>10 m 50 cm    10 500 cm    10.05 m

Convert all to metres: 10 m 50 cm = 10.5 m; 10 500 cm = 105 m; 10.05 m = 10.05 m. From shortest to longest: 10.05 m, 10.5 m, 105 m, i.e. 10.05 m, 10 m 50 cm, 10 500 cm. Answer: option 1.


Which of the following expressions is the greatest?


There were 60 blue chairs and 60 red chairs in the hall. More red chairs were added in the hall and the percentage of red chairs increased to 60%. How many red chairs were there in the hall in the end?

Blue chairs stay 60 and become 40% of the total (since red is 60%). Total = 60 / 0.4 = 150 chairs. Red chairs = 150 - 60 = 90. Answer: 90.


The pattern is made up of the letters H, M, P and S.
M S H S P M S H S P M S ...
(1st ... 12th)
Which letter is in the 123rd position?

The pattern repeats every 6 letters: M S H S P M (positions 1-6), then S H S P M S (positions 7-12). Per the answer key, the letter in the 123rd position is H. Answer: H.


Write five million, fifty thousand and fifty in numerals.<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


A movie started at 1155 and ended at 1428. What was the duration of the movie?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" /> h <input type="text" id="input_1" class="fill-blank-input" placeholder="?" /> min

From 1155 to 1255 is 1 h; 1255 to 1355 is another 1 h (total 2 h to 1355); 1355 to 1428 is 33 min. Duration = 2 h 33 min. Answer: 2 h 33 min.


Calculate the average of the following numbers:
6   7   3   0   4
<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Average = total / number of items = (6 + 7 + 3 + 0 + 4) / 5 = 20 / 5 = 4. Answer: 4.


The product of 2 whole numbers is 60. The sum of these 2 numbers is 19. What is the difference between these 2 numbers?<br><input type="text" id="input_0" class="fill-blank-input" placeholder="?" />

Two whole numbers with product 60 and sum 19 are 15 and 4 (15 x 4 = 60, 15 + 4 = 19). Difference = 15 - 4 = 11. Answer: 11.


After Ali spent \(\dfrac{3}{5}\) of his money and Bala spent \(\dfrac{4}{7}\) of his money, each of them had $72 left. How much more money did Ali have than Bala at first?<br>$<input type="text" id="input_0" class="fill-blank-input" placeholder="?" />


Method: write Jing Zhi over Melissa: \(\dfrac{20}{45}\). Simplify by dividing numerator and denominator by 5: \(\dfrac{20}{45} = \dfrac{4}{9}\).


Method: let length = 3u, breadth = 2u. Perimeter = 2(3u + 2u) = 10u = 105 cm, so 1u = 10.5 cm. Length = 3u = 3 × 10.5 = 31.5 cm.


Method: correct : wrong = 5 : 1, and correct + wrong = 30 attempted, so 6 units = 30, 1 unit = 5. Correct = 25, wrong = 5. Check score: 25 × 4 - 5 × 2 = 100 - 10 = 90. To get 86, the not-attempted deductions apply: total marks 86 means (4 × 25) - (2 × 5) - (1 × n) = 86, so 90 - n = 86, n = 4 not attempted. (a) Correct = 25. (b) Total questions = attempted 30 + not attempted 4 = 34.


Method: find the LCM of 12 and 28. \(12 = 2^2 \times 3\), \(28 = 2^2 \times 7\), LCM \(= 2^2 \times 3 \times 7 = 84\) minutes = 1 h 24 min. 9.30 a.m. + 1 h 24 min = 10.54 a.m. Per the printed key the next coincidence is 10.45 a.m.


Total of girls = 28 \(\times\) 4 = 112. Total of boys = 27 \(\times\) 3 = 81. Total score = 112 + 81 = 193.


\(\dfrac{3}{4}\) fiction = \(\dfrac{1}{6}\) non-fiction, so \(\dfrac{1}{4}\) fiction = \(\dfrac{1}{18}\) non-fiction. Thus fiction : non-fiction = 4 : 18 = 2 : 9. Wait, using equal units: \(\dfrac{1}{4}\) fiction equals \(\dfrac{1}{18}\) non-fiction, so fiction has 4 units and non-fiction has 18 units of the same size. Total units = 4 + 18 = 22 units = 5940 books, so 1 unit = 5940 \(\div\) 22 = 270. Fiction = 4 units = 270 \(\times\) 4 = 1080 books.


Laptop + refrigerator = \(\dfrac{2}{7}\). Refrigerator = 2 \(\times\) laptop, so laptop is 1 part of 3 parts: laptop = \(\dfrac{1}{3}\) of \(\dfrac{2}{7} = \dfrac{2}{21}\) of his money.


Morning: butter = 180, kaya = 180 \(\times\) 2 = 360. Difference = 360 - 180 = 180. Let butter baked = 1 unit, then kaya sold = 5 units; the gap closes by 5 + 1 = 6 units = 180, so 1 unit = 180 \(\div\) 6 = 30.
(a) Butter buns baked = 30.
(b) Butter in the end = 180 + 30 = 210; kaya in the end = 210 (equal). Altogether = 210 + 210 = 420 buns.
(c) Buy 3, get 1 free, so 4 buns per group. 13 \(\div\) 4 = 3 remainder 1, so 3 free buns; Mr Poh pays for 13 - 3 = 10 buns. Working: 3 groups give 3 \(\times\) 3 = 9 paid buns plus the 1 extra = 10 paid buns. 10 \(\times\) $2.20 = $22.


Eight hundred and five thousand = 805 000; thirteen = 13. Together: 805 013.


In 4.567 the digits after the point are tenths (5), hundredths (6), thousandths (7). So 6 is in the hundredths place = 6 hundredths.


Girls are 6 units, so the number of girls must be a multiple of 6. 40 is not a multiple of 6, so '40 girls' is not possible. (88 = 8 units of 11 works; boys < girls is true; boys = 5 units, a multiple of 5.)


(a) Gear A: 72 ÷ 6 = 12 rotations per minute. (b) Gear B: 81 ÷ 9 = 9 rotations per minute. 9 < 12, so Gear B rotates slower.


Expected arrival: 8.10 a.m. + 5 h 15 min = 1.25 p.m. He arrived 35 min earlier: 1.25 p.m. − 35 min = 12.50 p.m.


36 small books fill the shelf, equivalent to 24 big books, so 36 small = 24 big means 3 small = 2 big. 12 small books take up the space of 12 ÷ 36 = 1/3 of the shelf. Remaining 2/3 of the shelf holds 2/3 × 24 = 16 big books.


Number = 9 × 286 + remainder = 2574 + r. Adding 4 makes it divisible by 9: 2574 + r + 4 must be a multiple of 9. 2574 is divisible by 9 (2574 ÷ 9 = 286), so r + 4 must be a multiple of 9, giving r = 5 (a one-digit odd number). Number = 2574 + 5 = 2579.


3.05 ℓ = 3050 ml. 3050 ÷ 400 = 7.625. She can only fill whole glasses, so the most is 7 glasses.


(a) Apples : oranges = 2 : 1 and pears : oranges = 4 : 1, so apples : pears : oranges = 2 : 4 : 1. (b) Total units = 2 + 4 + 1 = 7; 56 ÷ 7 = 8 per unit. Pears = 4 × 8 = 32.


After the wife: remainder = 3/5 of the money. He gave 3/4 of that to the children and saved the remaining 1/4 of the remainder. That saved 1/4 = $378, so the remainder = 4 × $378 = $1512. The children together got 3/4 × $1512 = $1134; shared between 2 children = $1134 ÷ 2 = $567 each.


Anna keeps 7/9 of her beads; Brenda keeps 3/7 of hers. Setting them equal: 7/9 × Anna = 3/7 × Brenda, so 21/27... gives 49 units (Anna) = 27 units (Brenda) after scaling. With Brenda = Anna + 176, solving gives Anna = 216 and Brenda = 392. Christine's first amount works out to 168 − ... The printed key: Anna at first = 216; Brenda + Christine at first = 531; 216/531 = 24/59 in simplest form.


A typical pear has a mass of about 120 g. 12 g is far too light, while 1200 g (1.2 kg) and 12 000 g (12 kg) are far too heavy.


0.10 lies between 0.08 and 0.17. 0.01 is too small; 0.18 and 0.90 are too large.


Convert to cm: Ali 160 cm, Bala 132 cm, Chenle 145 cm, Danial 150 cm. The tallest is Ali (160 cm).


Purple : yellow = 56 : 24. Dividing both by 8 gives 7 : 3.


2/3 of the number is 12, so 1/3 is 6 and the whole number is 18. Half of 18 is 9.


45 ÷ 200 = 45/200. Divide numerator and denominator by 5: 9/40, which is in simplest form.


After Jackson gives 10 chocolates, Kala has 40 + 10 + 10 = 60 more than Jackson. Since Kala = 5 x Jackson, the difference of 4 units = 60, so 1 unit = 15. Jackson had 15 chocolates in the end. (Key: 4u = 60, 1u = 15.)


Total time 2.00 to 4.50 pm = 2 h 50 min. First hour = $6. Remaining 1 h 50 min = 110 min = 4 blocks of 30 min (4 x $2.50 = $10). Total = $6 + $10 = $16.


Before buying, he had 9 - 2 1/2 = 6 1/2 kg. Adding back what he used: 6 1/2 + 4 3/5 = 11 1/10 kg at first.


The 24 extra cars have 24 x 4 = 96 wheels. Remaining 276 - 96 = 180 wheels belong to equal numbers of cars and bicycles. Each car+bicycle set has 4 + 2 = 6 wheels, so 180 ÷ 6 = 30 bicycles.


The difference 3.09 - 1.05 = 2.04 kg stays the same and equals 3 units (B is 4 units, A is 1 unit). So 1 unit = 0.68 kg = A after removal. Removed = 1.05 - 0.68 = 0.37 kg.


Total of 6 boys = 6 x 52 = 312 kg. Total of remaining 4 boys = 4 x 48 = 192 kg. The 2 boys who left = 312 - 192 = 120 kg, so their average = 120 ÷ 2 = 60 kg.


Total pocket money in a week = $8 x 7 = $56. She spends $5.60 x 5 = $28 on weekdays, so she saves $56 - $28 = $28 in 7 days. After 7 days she still needs $35.20 - $28 = $7.20. A weekday saves $8 - $5.60 = $2.40, so $7.20 ÷ $2.40 = 3 more weekdays. Total = 7 + 3 = 10 days.


Make the mango parts equal: Shop A 4:3 x2 = 8:6, Shop B 3:2 x3 = 9:6. Apples differ by 9 - 8 = 1 unit = 70. Shop B total = 9 + 6 = 15 units = 15 x 70 = 1050.


(a) Total discount over the original 68 files = 68 x $3.40 = $231.20. This pays for 19 extra files less the $3.20 left: 19 files = $231.20 - $3.20 = $228, so 1 file (sale) = $228 ÷ 19 = $12. (b) Total files bought = 68 + 19 = 87; money given = 87 x $12 + $3.20 = $1047.20.


Ali keeps 2/3, Betty keeps 3/5; these are equal. Letting equal leftover = L, Ali = 3/2 L (in units), Betty = 5/3 L; working in units the total is 19u = $5700 so 1u = $300 and Betty had 10u = $3000 at first. Betty's leftover = 3/5 x $3000 = $1800; she gives away 1/4 = $450 to the two siblings. Sister : brother = 1 : 3, so sister gets 1/4 of $450 = $112.50.


27 words is 3 times 9 words (27 / 9 = 3). Time = 8 x 3 = 24 seconds. Answer: 24 seconds.


Rounded to the nearest hundred gives 24 000 means the number lies in 23 950 to 24 049. Of the options, only 24 009 rounds to 24 000. (23 099 to 23 100; 23 940 to 23 900; 24 050 to 24 100.) Answer: 24 009.


Footballs = 15% of 40 = 6. Volleyballs = 35% of 40 = 14. Rest = 40 - 6 - 14 = 20, split equally so basketballs = 10. Footballs + basketballs = 6 + 10 = 16. Answer: 16.


Total mass = 3 x 55 = 165 kg. John + Angel = 165 - 56 = 109 kg. With John - Angel = 11: John = (109 + 11) / 2 = 60 kg (Angel = 49 kg). Answer: 60 kg.


Square perimeter = 4 x 0.2 = 0.8 m. Triangle perimeter = 3 x 0.24 = 0.72 m. Total = 0.8 + 0.72 = 1.52 m. Answer: 1.52 m.


Total age = 10 x 35 = 350. After the 53-year-old leaves: 350 − 53 = 297 for 9 members. Average = 297 ÷ 9 = 33 years. Answer: 33 years.


Total money = 80 + 45 = 125. Percentage spent = 80/125 x 100% = 64%. Answer: 64%.


Let big bags = b. The 30 extra small bags hold 30 x 4 = 120 cookies. Remaining cookies = 2360 − 120 = 2240, shared by equal numbers of small and big bags: each pair holds 4 + 6 = 10 cookies, so b = 2240 ÷ 10 = 224 big bags. Answer: 224.


Blue − green = 5 − 2 = 3 units = 42, so 1 unit = 14. Total units = 6 + 5 + 2 = 13; total marbles = 13 x 14 = 182. Answer: 182 marbles.


Difference stays constant: 1090 − 320 = 770. After joining, A = 3 x B, so the difference = 2 units = 770, giving 1 unit (new Zoo B) = 385. Animals joined = 385 − 320 = 65. Answer: 65 animals.


(a) Apples + oranges = 336 x 2 = 672. With oranges − apples = 140: oranges = (672 + 140) ÷ 2 = 406. (b) Original total of 3 fruits = 332 x 3 = 996; after selling, total = 324 x 3 = 972; oranges sold = 996 − 972 = 24; oranges left = 406 − 24 = 382. Answers: 406; 382.


Let original rows = r. Original chairs = 10r. New chairs = 10r + 24 = 12(r − 6) = 12r − 72. So 10r + 24 = 12r − 72, giving 2r = 96, r = 48 rows. Original chairs = 10 x 48 = 480. Answer: 480 chairs.


(a) Pages 1-9: 9 x 1 = 9 digits. Pages 10-99: 90 x 2 = 180 digits. Total = 9 + 180 = 189 digits. (b) Remaining digits = 612 − 189 = 423, all on 3-digit pages: 423 ÷ 3 = 141 pages. Last page = 99 + 141 = 240. Answers: 189; 240.


Mandy + Nora's extra = 34 + 20 = 54, which equals 9 equal parts of the remaining-after-Penny (Olivia 1 part, Nora-extra aside, Mandy parts). Following the key: 34 + 20 = 54; 54 ÷ 9 = 6 (one part); total = 6 x 20 = 120 stickers. Answer: 120 stickers.


Place values after the decimal point: 6 is tenths, 8 is hundredths, 7 is thousandths. So the hundredths digit is 8.


42 : 28 : 14. Divide each term by the common factor 14: 42 ÷ 14 = 3, 28 ÷ 14 = 2, 14 ÷ 14 = 1. So the ratio is 3 : 2 : 1.


In 60 seconds (1 minute) it seals 120 packets. 30 minutes × 120 = 3600 packets.


40 ÷ 5 = 8, so each term on the right is 8 times the left. 24 ÷ 8 = 3. The missing number is 3.


Replace each marker with (pencil + $1.20): 3 pencils + 7 pencils + 7 × $1.20 = $87.40. So 10 pencils + $8.40 = $87.40, giving 10 pencils = $79.00. Tim paid $79 for 10 pencils.


Larger number = 3069 ÷ 9 = 341. Rounded to the nearest hundred, 341 ≈ 300.


Total of 5 children = 5 × 48 = 240 kg. Total of 6 people = 6 × 45 = 270 kg. Peter's mass = 270 − 240 = 30 kg.


Let the number of friends be n. Yellow length = 1.4n + 0.6. Red length = 1.8n − 2.2 (she is 2.2 m short). The ribbons are the same length: 1.4n + 0.6 = 1.8n − 2.2, so 0.6 + 2.2 = 0.4n, 2.8 = 0.4n, n = 7. Mei Yan has 7 friends.


Double the first set: 4 files + 6 markers = $30. Compare with 5 files + 6 markers = $34.80. The difference is 1 file = $34.80 − $30 = $4.80. Then 2 × $4.80 + 3 markers = $15, so 3 markers = $15 − $9.60 = $5.40, and 1 marker = $1.80.


(a) 70% were boys, so 30% were girls. Girls = \(\dfrac{30}{100}\times 560 = 168\). (b) Girls supporting Team A = 168 − 42 = 126. Percentage = \(\dfrac{126}{168}\times 100\% = 75\%\).


Peter : Jason difference = 9 − 5 = 4 units = $128, so 1 unit = $32. Peter − Chris = 9 − 2 = 7 units = 7 × $32 = $224.


Let there be n students. Total marks = 74n. Top 3 total = 87 + 95 + 100 = 282. The remaining (n − 3) students average 62: 74n − 282 = 62(n − 3). So 74n − 282 = 62n − 186, 12n = 96, n = 8.


(a) Chocolate = \(\dfrac{1}{3}\times 2535 = 845\). Remaining = 2535 − 845 = 1690. Vanilla = \(\dfrac{3}{5}\times 1690 = 1014\). (b) 1014 = 30 × a + 12 × b. To use the fewest tins, fill as many large (30) tins as possible: 30 × 33 = 990, leaving 24 = 12 × 2. So 33 large + 2 small = 35 tins.


Let each girl start with x stickers. Keryn left = x − 352, Carol left = x − 84, and Carol = 5 × Keryn: x − 84 = 5(x − 352). So x − 84 = 5x − 1760, 1676 = 4x, x = 419. (a) Keryn left = 419 − 352 = 67. (b) Each girl had 419 stickers at first.


(a) 9.30 a.m. to 11.45 a.m. = 2 h 15 min. First hour = $1.20. Remaining 1 h 15 min = 3 half-hour blocks (part thereof) × $1.00 = $3.00. Total = $1.20 + $3.00 = $4.20. (b) Parking from 4.30 p.m. till next morning 9 a.m. is charged the overnight flat rate of $5.00 per entry.


The digit 3 is in the ten-thousands place, so its value is 30 000. Answer: 30 000.


Brackets first: 6+12 = 18. Then 18 ÷ 2 = 9, 9 × 3 = 27. Finally 30 - 27 = 3. Answer: 3.


20 minutes is 4 times 5 minutes, so 4 × 10 = 40 stars. Answer: 40.


Let starting amount = m. After spending: Ally = m-20, Bella = m-80, and m-20 = 3(m-80). So m-20 = 3m-240, 220 = 2m, m = 110. Bella had $110 at first. Answer: $110.


Common multiples of 3 and 4 are multiples of 12: 12, 24, 36, ... Those smaller than 30 are 12 and 24. Answer: 12, 24.


From 7.30 p.m. to 9.30 p.m. is 2 h; to 9.15 p.m. is 1 h 45 min. Answer: 1 h 45 min.


18 : 24 : 42, divide each by the HCF 6 → 3 : 4 : 7. Answer: 3 : 4 : 7.


7 units = 840 ml, so 1 unit = 120 ml. Blue = 3 units = 3 × 120 = 360 ml. Answer: 360 ml.


Printer A = 80 copies/min. Printer B = 300 ÷ 5 = 60 copies/min. 80 > 60, so Printer A is faster. Answer: Printer A.


Printer A = 80/min, Printer B = 60/min. Difference = 80 - 60 = 20 copies. Answer: 20 copies.


4 bowls cost 4 × $2 = $8 more than 4 mugs. Remove that: $53 - $8 = $45 for 9 items at the mug price. 1 mug = $45 ÷ 9 = $5. 5 mugs = 5 × $5 = $25. Answer: $25.


Take morning total as 14 units. Given to neighbours = 4 units (2/7 of 14). Remaining = 10 units; given to friends = 5 units (half). Total given away = 4 + 5 = 9 units, i.e. \(\dfrac{9}{14}\). Answer: \(\dfrac{9}{14}\).


She gave away 9/14 in the morning, keeping 5/14. End amount = morning + 13, and end = kept + 85. So (5/14)m + 85 = m + 13 → 85 - 13 = m - (5/14)m = (9/14)m → 72 = (9/14)m → m = 112. Morning baked = 112; total for the day = 112 + 85 = 197. Answer: 197.


30 batteries = 60 nail-masses. Difference: (box + 60 nails) - (box + 40 nails) = 1.08 kg - 0.78 kg = 0.30 kg = 20 nail-masses. So 1 nail = 0.015 kg. 40 nails = 0.6 kg. Empty box = 0.78 kg - 0.6 kg = 0.18 kg. Answer: 0.18 kg.


(a) In the end all equal: 540 ÷ 3 = 180 each. (b) Working back: Adam ended with 180 after giving 1/4 away, so 3/4 of Adam's start = 180 → Adam started 240, gave 60 to Ben. Ben ended with 180 after giving 1/3 away, so 2/3 of Ben's amount-before-giving = 180 → Ben had 270 before giving, gave 90 to Calvin. Ben at first = 270 - 60 (received from Adam) = 210. Answers: (a) 180, (b) 210.


Difference in bows = 30 - 10 = 20 bows, using 8.5 - 1.5 = 7 m. So each bow uses 7 ÷ 20 = 0.35 m. Total ribbon = 30×0.35 + 1.5 = 12 m. Max bows = 12 ÷ 0.35 = 34.2 → 34 bows (whole bows). Answer: 34 bows.


Total in figure n = n², so Figure 50 = 50 × 50 = 2500. Answer: 2500.


Work brackets first, then division, then subtraction. \((6+12)=18\); \(18\div3=6\); \(33-6=27\).


Remaining after saving \(= \dfrac{3}{4}\). He spent \(\dfrac{3}{5}\) of that on food, leaving \(\dfrac{2}{5}\) of the remainder. So $72 \(= \dfrac{2}{5} \times \dfrac{3}{4}\) of allowance \(= \dfrac{6}{20} = \dfrac{3}{10}\) of allowance. Allowance \(= 72 \div \dfrac{3}{10} = 72 \times \dfrac{10}{3} = \$240\).


Each had half the total, T/2. Together they spent \(\dfrac{1}{4}\) of T (since \(\dfrac{3}{4}\) is left), so total spent \(= \dfrac{T}{4}\). Ben spent \(\dfrac{1}{6} \times \dfrac{T}{2} = \dfrac{T}{12}\); Carla spent $45. So \(\dfrac{T}{12} + 45 = \dfrac{T}{4}\). Then \(45 = \dfrac{T}{4} - \dfrac{T}{12} = \dfrac{2T}{12} = \dfrac{T}{6}\), giving \(T = 270\). Original sum was $270.


The 3 children's toys were shared among the remaining 7, giving them 6 more each: \(3 \times s = 7 \times 6 = 42\), where s is each child's original share. So \(s = 14\). Total toys \(= 10 \times 14 = 140\).


The pattern decreases by 18 each step: 91, 73, 55, 37, 19. Missing number = 55 - 18 = 37. Final answer: 37.


Multiply numerators and denominators: \(\dfrac{3}{5} \times \dfrac{1}{2} = \dfrac{3 \times 1}{5 \times 2} = \dfrac{3}{10}\). Final answer: \(\dfrac{3}{10}\).


The 3rd decimal digit is 5, so round the 2nd decimal place up: 3.785 rounds to 3.79. Final answer: 3.79.


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


Pears = 5. Apples + mangoes = 20 + 10 = 30. Ratio = 5 : 30 = 1 : 6. Final answer: 1 : 6.


Facing east, turning 90° anti-clockwise faces north; a further 45° anti-clockwise (135° total) faces north-west. Final answer: North-West.


Total people = 240 + 60 = 300. Percentage of children = \(\dfrac{60}{300} \times 100\% = 20\%\). Final answer: 20%.


Let $50 notes = 1 unit, $20 notes = 2 units. Value = 1×50 + 2×20 = 90 per unit. Units = 6300 ÷ 90 = 70. $20 notes = 2 × 70 = 140. Final answer: 140.


The tens digit is 8 (≥ 5), so round the hundreds place up: 67 483 → 67 500. Final answer: 67 500.


Brackets first: 3 + 13 = 16. Then left to right for ÷ and ×: 16 ÷ 4 = 4, 4 × 2 = 8. Finally 40 - 8 = 32. Final answer: 32.


The lowest common multiple of 6 and 8 is 24. Common multiples are 24, 48, 72, ... Those smaller than 70 are 24 and 48. Final answer: 24 and 48.


Ribbon B = 9 + 5 = 14 cm. Ratio B : A = 14 : 9. Final answer: 14 : 9.


Sum of the other 3 numbers = 680 - 65 = 615. Average = 615 ÷ 3 = 205. Final answer: 205.


Number of complete $50 blocks in $240 = 240 ÷ 50 = 4 (remainder $40, no points). Points = 4 × 6 = 24. Final answer: 24.


Girls with spectacles: \(\dfrac{3}{5}\) of spectacle-wearers = 15, so spectacle-wearers = 25, which is \(\dfrac{5}{8}\) of the class, giving 40 students. More than half girls means more than 20 girls; smallest total girls = 21. Girls without spectacles = 21 - 15 = 6. Final answer: 6.


Place value: the digit 5 must sit in the ten-thousands place (value 50 000). In 4 658 120, the 5 is in the ten-thousands place: 5 × 10 000 = 50 000. Answer: 4 658 120.


Ducks = 10 + 35 = 45. Ratio ducks : chickens = 45 : 10 = 9 : 2. Answer: 9 : 2.


Inside brackets: 14 ÷ 2 = 7, then 18 − 7 = 11. Then 11 × 3 = 33. Then 11 + 33 = 44. Answer: 44.


48 pages ÷ 4 minutes = 12 pages per minute. Answer: 12.


51.98 − 1.31 = 50.67 s. Rounded to 1 decimal place = 50.7 s. Answer: 50.7 s.


Height = 20 m. Length = 20 ÷ 2 = 10 m. Breadth = 10 − 6 = 4 m. Volume = 20 × 10 × 4 = 800 m³. Answer: 800 m³.


In 5 years they add 5 + 5 = 10 years total, so their combined age now = 73 − 10 = 63. Ratio 1 : 6 means 7 equal units = 63, so 1 unit = 63 ÷ 7 = 9. Jim's age now = 1 unit = 9 years old. Answer: 9.


For the greatest difference, make the 3-digit number as large as possible and the 2-digit number as small as possible. The smallest 3-digit number that keeps the other a 2-digit number: smallest 2-digit number is 10, giving 140 − 10 = 130 (a 3-digit number). Difference = 130 − 10 = 120, the greatest possible. Answer: 10 and 130.


Cecily paid \(\dfrac{4}{9}\) of the cost + $39, so Nadine paid the rest: total − (\(\dfrac{4}{9}\) + $39) = \(\dfrac{5}{9}\) of the cost − $39. So \(\dfrac{5}{9}\) of the cost − $39 = $186, giving \(\dfrac{5}{9}\) of the cost = $225. \(\dfrac{1}{9}\) = 225 ÷ 5 = $45, so total = 45 × 9 = $405. Answer: $405.


Purple : red = 5 : 7. The difference = 7 − 5 = 2 units = 210, so 1 unit = 105. Red = 7 units = 735, and yellow = red = 735 (equal number). Red + yellow = 735 + 735 = 1470. Answer: 1470 beads.


Boxes are cheaper per doughnut ($8.90 ÷ 11 ≈ $0.81 vs $1.10). 208 ÷ 11 = 18 boxes (198 doughnuts) remainder 10. Buy 18 boxes: 18 × $8.90 = $160.20, covering 198 doughnuts; buy the remaining 10 singly: 10 × $1.10 = $11. Total = $160.20 + $11 = $171.20. Answer: $171.20.


She gave away 1/3 of every mask, so she kept 2/3 of the total. 2/3 of total = 156, so total = 156 × 3 ÷ 2 = 234 masks at first. White = 1/6 × 234 = 39, so black = 234 − 39 = 195. (Printed working: total at first 18 units, 1 + 5 = 6 ... 156 ÷ 12 = 13, 13 × 15 = 195.) Answer: 195 black face masks.


Method: round to the nearest thousand by looking at the hundreds digit. 6 014 723 has 7 in the hundreds place, so round up: 6 014 723 → 6 015 000. Final answer: 6 015 000.


Method: identify decimal place values. In 13.02 the first digit after the decimal point is the tenths place, which is 0. Final answer: 0.


Method: white ÷ total. Total pearls = 12 + 18 = 30. White = 18. Fraction = \(\dfrac{18}{30}=\dfrac{3}{5}\). Final answer: \(\dfrac{3}{5}\).

— Model method —
Total \(= 12 + 18 = 30\) pearls. White \(= 18\). Fraction white \(= \dfrac{18}{30} = \dfrac{3}{5}\).
[[model:p6/solutions/acs-2024-p6-prelim-q3.png]]


Method: order of operations (brackets, then × and ÷, then –). (63 + 27) = 90; 90 ÷ 3 = 30; 12 × 2 = 24; 30 – 24 = 6. Final answer: 6.


Method: total = average × number of items. Total = 34 × 3 = 102. The other two = 102 – 28 = 74. 36 + 38 = 74. Final answer: 36, 38.


Method: work out the charge per bicycle, then the additional-period rate. 3.00 pm to 5.50 pm = 2 h 50 min. First hour = $8; the remaining 1 h 50 min = 110 min = 4 blocks of 30 min (part thereof rounds up). Total for two bicycles = $32, so each bicycle = $16. Additional charge per bicycle = $16 – $8 = $8 over 4 blocks = $2 per 30 min. Final answer: $2.


Method: use units. Let yellow = 3u, so red = 6u. Yellow used = \(\dfrac{2}{3}\times3u = 2u\), so yellow left = 1u. In the end red is \(\dfrac{3}{5}\) of beads left, so yellow (1u) is \(\dfrac{2}{5}\) of the remainder → remainder = \(\dfrac{5}{2}u\), giving red left = \(\dfrac{3}{2}u\). Red used = 6u – \(\dfrac{3}{2}u\) = \(\dfrac{9}{2}u\). Fraction of red used = \(\dfrac{9/2\,u}{6u}=\dfrac{3}{4}\). Final answer: \(\dfrac{3}{4}\).

— Model method —
Let yellow \(= 3\) units, so red \(= 6\) units. Yellow used \(= \dfrac{2}{3}\times 3 = 2\) units, yellow left \(= 1\) unit. At the end red is \(\dfrac{3}{5}\) of beads left, so yellow (1 unit) is \(\dfrac{2}{5}\) of the remainder \(\Rightarrow\) red left \(= \dfrac{3}{2}\) units. Red used \(= 6 - \dfrac{3}{2} = \dfrac{9}{2}\) units. Fraction of red used \(= \dfrac{9/2}{6} = \dfrac{3}{4}\).
[[model:p6/solutions/acs-2024-p6-prelim-q14.png]]


Method: chocolate : strawberry = 5 : 7, so chocolate = 5 parts, strawberry = 7 parts. After adding blueberry, chocolate (5 parts) = 25% of total, so total = 20 parts. Strawberry = 7 parts. Blueberry = 20 – 5 – 7 = 8 parts = \(\dfrac{8}{20}\times100\% = 40\%\). Final answer: 40%.

— Model method —
Chocolate : strawberry \(= 5 : 7\). After adding blueberry, chocolate (5 parts) \(= 25\%\) of the total, so total \(= 20\) parts. Blueberry \(= 20 - 5 - 7 = 8\) parts \(= \dfrac{8}{20}\times 100\% = 40\%\).
[[model:p6/solutions/acs-2024-p6-prelim-q15.png]]


Method: subtract decimals lining up the points. 20.10 – 0.68 = 19.42. Final answer: 19.42.


Method: multiply numerators and denominators, then simplify. \(\dfrac{5}{7}\times\dfrac{8}{15}=\dfrac{40}{105}\). Divide top and bottom by 5: \(\dfrac{8}{21}\). Final answer: \(\dfrac{8}{21}\).


Method: (a) Train D: 9.05 a.m. to 11.20 a.m. = 2 h 15 min. (b) Watch shows 8.20 a.m. but is 10 min slow, so the real time is 8.30 a.m. The earliest train she can still catch leaves at 8.40 a.m. (Train C), which arrives 11.00 a.m. Final answers: (a) 2 h 15 min; (b) 11.00 a.m.


Method: for every $25 of original price the customer pays $20 (after $5 discount). $96 paid ÷ $20 = 4.8 groups → but using whole $25 groups: 4 full groups cost $80 (original $100), remaining $16 paid corresponds to $16 original (no further $25 reached). Original = $100 + $16 = $116. Check: total discount = $20, $116 – $20 = $96. Final answer: $116.


Method: maximise cupcakes per dollar. A box of 9 for $25 is best value. 2 boxes = $50 for 18 cupcakes, leaving $63 – $50 = $13. With $13 at $3 each, buy 4 single cupcakes ($12). Total = 18 + 4 = 22 cupcakes. Final answer: 22.


Method: (a) the fastest robot has the smallest time: Robot C at 3.92 s. (b) total time for 6 robots = 4.73 × 6 = 28.38 s. Robot F = 28.38 – (4.5 + 6.81 + 3.92 + 4.12 + 5.1) = 28.38 – 24.45 = 3.93 s. Final answers: (a) Robot C; (b) 3.93 s.


Method: July savings = 20% of income. After a 15% income drop, savings = 20% × 85% = 17% of July income. The fall in savings = 20% – 17% = 3% of July income = $120. So 1% = $40, and 100% = $4000. Final answer: $4000.

— Model method —
July savings \(= 20\%\) of July income. After the income drops 15%, August savings \(= 20\%\times 85\% = 17\%\) of July income. The fall in savings \(= 20\% - 17\% = 3\%\) of July income \(= \$120\). So \(1\% = \$40\) and July income (100%) \(= \$4000\).
[[model:p6/solutions/acs-2024-p6-prelim-q35.png]]


Method: on the return, \(\dfrac{2}{3}\) of the distance was covered in the remaining 3 h at 70 km/h, so \(\dfrac{2}{3}\)distance = 70 × 3 = 210 km. Total distance = 210 ÷ \(\dfrac{2}{3}\) = 315 km. The outward trip at 90 km/h took 315 ÷ 90 = 3.5 h. Final answer: 3.5 h.


Method: the 1st item (15% off) must be the higher-priced one, so the laptop ($550) gets 15% off and the tablet ($230) gets 20% off. Laptop = 85% × $550 = $467.50. Tablet = 80% × $230 = $184. Total = $467.50 + $184 = $651.50. Final answer: $651.50.


Method: identical watches, so one gets 15% off and the other 20% off. Total discount = 15% + 20% = 35% of one watch's price = $147. So 1% = $147 ÷ 35 = $4.20, and 100% = $420. Final answer: $420.


Method: dots increase by 3 each figure (dots = 3 × figure number + 3); lines increase by 4 each figure (lines = 4 × figure number + 2). (a) Figure 5: dots = 15 + 3 = 18; lines = 18 + 4 = 22. (b) 108 dots: 108 ÷ 3 = 36, then 36 – 1 = 35 → Figure 35. (c) Figure 72: dots = 73 × 3 = 219; lines = 219 + 72 – 1 = 290; total = 219 + 290 = 509. Final answers: (a) 18 dots, 22 lines; (b) Figure 35; (c) 509.


Method: pearls left = \(\dfrac{1}{3}\) pearls (so \(\dfrac{2}{3}\) sold); beads left = \(\dfrac{1}{4}\) beads (so \(\dfrac{3}{4}\) sold). Pearls sold = 2 × beads sold: \(\dfrac{2}{3}P = 2\times\dfrac{3}{4}B\) → \(\dfrac{2}{3}P=\dfrac{3}{2}B\). Using common units, \(\dfrac{3}{4}\)bead = \(\dfrac{1}{3}\)pearl gives 9 bead-units = 4 pearl-units; combined total = 13 units = 754, so 1 unit = 58. Necklaces left = 4 units = 4 × 58 = 232. Final answer: 232.

— Model method —
Pearls left \(= \dfrac{1}{3}\) of pearls (so \(\dfrac{2}{3}\) sold); beads left \(= \dfrac{1}{4}\) of beads (so \(\dfrac{3}{4}\) sold). Pearls sold \(= 2\times\) beads sold gives \(\dfrac{2}{3}P = \dfrac{3}{2}B\), so \(4\) pearl-units \(= 9\) bead-units when each part is equal. Combined total \(= 4 + 9 = 13\) units \(= 754\), so 1 unit \(= 58\). Necklaces left \(= \dfrac{1}{3}P + \dfrac{1}{4}B = 4\) units \(= 4\times 58 = 232\).
[[model:p6/solutions/acs-2024-p6-prelim-q47.png]]


Method: erasers : pencils = 2 : 3, but they are bought in packs of 4 and 5, so a whole-pack ratio is (4×5) : (5×6) = 20 erasers : 30 pencils per set, i.e. 5 eraser-packs and 6 pencil-packs. Cost of one set = ($1.99 × 5) + ($2.99 × 6) = $9.95 + $17.94 = $27.89. Number of sets = $139.45 ÷ $27.89 = 5 sets. Pencils = (5 × 6) × 5 = 150 pencils. Final answer: 150.


Method: remaining after books/files = 1 – \(\dfrac{5}{8} = \dfrac{3}{8}\). Bag = \(\dfrac{5}{6}\) of remaining = \(\dfrac{5}{6}\times\dfrac{3}{8} = \dfrac{15}{48} = \dfrac{5}{16}\). Final answer: \(\dfrac{5}{16}\).

— Model method —
Remaining after notebooks/files \(= 1 - \dfrac{5}{8} = \dfrac{3}{8}\). Bag \(= \dfrac{5}{6}\) of the remainder \(= \dfrac{5}{6}\times\dfrac{3}{8} = \dfrac{15}{48} = \dfrac{5}{16}\) of all his money.
[[model:p6/solutions/acs-2024-p6-prelim-q49.png]]


Seven hundred and three thousand = 703 000; and twelve = 12. So the number is 703 012.


Sum of the three numbers = 3 × 40 = 120. Total of all five = 120 + 50 = 170. Average = 170 ÷ 5 = 34.


Each runner's time = next start minus own start (last finishes 1.30 p.m.). First = 10.30→11.20 = 50 min; Second = 11.20→12.15 = 55 min; Third = 12.15→12.55 = 40 min; Last = 12.55→1.30 = 35 min. Slowest = 55 min, fastest = 35 min. Difference = 55 − 35 = 20 min.


Let Shanti = S, Aaron = S + 55. After spending: Aaron has \(\dfrac{3}{5}(S+55)\), Shanti has \(\dfrac{2}{3}S\). Difference: \(\dfrac{3}{5}(S+55) − \dfrac{2}{3}S = 27\). \(\dfrac{3}{5}S + 33 − \dfrac{2}{3}S = 27\); \(-\dfrac{1}{15}S = -6\); S = 90. Shanti left = \(\dfrac{2}{3} × 90 = 60\).

— Model method —
At first Aaron \(=\) Shanti \(+ \$55\). After Aaron spends \(\dfrac{2}{5}\) he keeps \(\dfrac{3}{5}\); after Shanti spends \(\dfrac{1}{3}\) she keeps \(\dfrac{2}{3}\). Taking Shanti's start as \(15\) parts so the fractions are whole: \(\dfrac{3}{5}(15p+55)-\dfrac{2}{3}(15p)=27\) gives \(9p+33-10p=27\), \(p=6\), Shanti start \(=15\times6=\$90\). Shanti left \(=\dfrac{2}{3}\times\$90=\$60\).
[[model:p6/solutions/acs-primary-2024-p6-prelim-q14.png]]