This is the marking copy and it carries every answer. Her sheet is the other file — same numbering, same margin codes, so you can mark straight down the page.
The table shows the boys and girls in four Primary Four classes, with some cells blank: 4A has 22 boys and 18 girls; 4B has 25 girls and 41 students in total; 4C has 13 boys; 4D has 20 girls. There are as many students in Primary 4A as in Primary 4C. How many girls are there in Primary 4C?
Class | Boys | Girls | Total ------+------+-------+------ 4A | 22 | 18 | 4B | | 25 | 41 4C | 13 | | 4D | | 20 |
answer 27
method Fill in 4A first: 22 + 18 = 40 students. We are told 4C has as many students as 4A, so 4C also has 40. Of those, 13 are boys, so the girls number 40 − 13 = 27.
watch Comparing 4A's BOYS with 4C's boys. The sentence says as many STUDENTS, which means the totals match, not the boys.
Using the same table, the four classes have a total of 160 students. How many students are from the class that has the least number of students?
4A = 22 + 18 = 40 4B = 41 (given) 4C = 40 (same as 4A) 4D = ? all four classes together = 160
answer 39
method Three of the classes are now known: 4A = 40, 4B = 41, 4C = 40, which is 121 students. So 4D = 160 − 121 = 39. Comparing 40, 41, 40 and 39, the smallest is 4D with 39 students.
watch Stopping once 4D is found without checking it really is the smallest, or picking 4C's 13 boys as "the least". The question asks about whole classes.
The mass of a container filled with 2 identical ping pong balls was 80 g. The mass of the same container filled with 3 identical marbles was 220 g. The mass of each marble was 3 times the mass of each ping pong ball. What was the mass of 1 ping pong ball?
answer 20
method Swap the marbles for ping pong balls: each marble weighs the same as 3 ping pong balls, so 3 marbles weigh the same as 9 ping pong balls. Now both weighings hold the same container: container + 2 balls = 80 g, and container + 9 balls = 220 g. Subtracting removes the container: 7 balls = 220 − 80 = 140 g, so one ball is 140 ÷ 7 = 20 g.
watch Dividing 80 by 2 and forgetting the container has a mass of its own. Subtracting the two weighings is what makes the container disappear.
Star Bookshop is having a promotion: notebooks are $5 each, and for every 2 notebooks bought the third notebook is free (buy 3 for the price of 2). Karen bought 25 notebooks. What is the least amount of money she has spent in total?
answer 85
method Group the notebooks in threes, because every group of 3 costs only 2 × $5 = $10. In 25 notebooks there are 25 ÷ 3 = 8 groups of three (24 notebooks) with 1 left over. The 8 groups cost 8 × $10 = $80, and the last notebook has no partners so it costs the full $5. Total = $80 + $5 = $85.
watch Paying full price for all 25 ($125), or counting the free ones as groups of 2. The deal gives 3 notebooks for the price of 2, so it is groups of THREE that matter.
Mr Tan has less than 40 sweets for a group of pupils. If he gives his pupils 4 sweets each, he will be short of 1 sweet. If he gives each pupil 5 sweets each, he will be short of 11 sweets. How many pupils does he have in the group?
answer 10
method Compare the two plans. Going from 4 sweets each to 5 sweets each means one extra sweet per pupil, and the shortfall grows from 1 to 11 — that is 10 more sweets needed. So there are 10 pupils. (Check: 10 pupils × 4 = 40, short of 1, so he has 39, which is indeed less than 40.)
watch Trying to find the number of sweets first. The difference between the two plans gives the number of PUPILS straight away, because each pupil accounts for exactly one extra sweet.
In a General Knowledge quiz, 5 marks were awarded for a correct answer but 2 marks were deducted for an incorrect answer. Mei Ling answered all 50 questions and was awarded a total of 166 marks. How many questions did she answer correctly?
answer 38
method Suppose every answer were correct: 50 × 5 = 250 marks. She actually scored 166, which is 84 marks less. Each wrong answer costs 5 marks not gained plus 2 marks deducted — 7 marks in all. So the number wrong is 84 ÷ 7 = 12, and the number correct is 50 − 12 = 38.
watch Counting a wrong answer as costing only 2 marks. It costs 7: the 5 she did not win AND the 2 taken away.
Ben was standing at Point X. He turned 225° anti-clockwise and then made another 1/4-turn in the clockwise direction. He ended up facing the Bus Bay. Which location was Ben facing at first?
Library N
| ^
General Office | Bus Bay |
\ | /
Computer Lab ------- X ------- Field
/ | \
Dental Clinic | Basketball Court
|
Hall(1) Hall (2) Field (3) Library (4) Computer Lab
answer (1) — Hall
method Work out the NET turn first: 225° anti-clockwise, then 90° clockwise, leaves 225 − 90 = 135° anti-clockwise. He finished facing the Bus Bay (north-east), so to find where he began, undo it — turn 135° CLOCKWISE from north-east. North-east → (90°) → south-east → (45° more) → south. South is the Hall.
watch Applying the turns forwards from the Bus Bay instead of backwards. The Bus Bay is where he ENDED, so the working has to run in reverse.
The table shows the favourite colours of 108 pupils in Primary Four. Some numbers have been blotted out by ink stains. There were twice as many pupils who chose green than yellow. There is a total of 38 pupils who like 2 of the colours. Which of the following options shows the two colours?
Colour | Number of Pupils -------+----------------- Green | ### stain ### Yellow | ### stain ### Blue | 22 Red | 38 108 pupils altogether · green = 2 × yellow
(1) Blue and Green (2) Yellow and Blue (3) Green and Red (4) Red and Yellow
answer (2) — Yellow and Blue
method Green and yellow together = 108 − 22 − 38 = 48. Green is twice yellow, so that is 3 equal shares: yellow = 48 ÷ 3 = 16 and green = 32. Now test the pairs: blue + green = 54, yellow + blue = 16 + 22 = 38 ✓, green + red = 70, red + yellow = 54.
watch Splitting the 48 evenly into 24 and 24. "Twice as many" makes THREE shares, not two.
A stall had a promotion: buy 1 muffin for $3 each, or buy 4 muffins at $11. Mandy wanted to buy 50 muffins. What was the least amount of money Mandy had to pay for all the 50 muffins?
Buy 1 muffin for $3 each Buy 4 muffins at $11
answer 138
method The 4-for-$11 deal works out at $2.75 a muffin, cheaper than $3, so take as many fours as possible: 50 ÷ 4 = 12 sets of four (48 muffins) with 2 left over. That is 12 × $11 = $132, plus 2 singles at $3 = $6. Total = $138. (Buying 13 sets would give 52 muffins for $143 — more muffins AND more money, so it is not cheaper.)
watch Buying 13 sets of four to avoid paying full price for the last two. Always price the leftover BOTH ways — here the singles win.
David paid $7374 for 3 laptops and 2 headphones. Kumar paid $5002 more than David for 5 laptops and 4 headphones. What was the cost of 1 headphone?
answer 129
method Kumar paid 7374 + 5002 = $12 376 for 5 laptops and 4 headphones. Now double David's order so the headphones match: 6 laptops and 4 headphones cost 2 × $7374 = $14 748. Comparing that with Kumar's 5 laptops and 4 headphones, the headphones cancel and the difference is exactly 1 laptop: 14 748 − 12 376 = $2372. So 3 laptops cost 3 × 2372 = $7116, leaving 7374 − 7116 = $258 for 2 headphones, and one headphone is $129.
watch Subtracting David's order from Kumar's directly (2 laptops + 2 headphones = $5002) and then guessing. Doubling one order first is what makes a pair of items cancel exactly.
Mr Lee had an apple, a papaya and a honeydew. He weighed them in pairs: apple + papaya = 730 g, apple + honeydew = 1960 g, honeydew + papaya = 2470 g. What was the mass of the honeydew, in grams?
Setup 1 apple + papaya = 730 g Setup 2 apple + honeydew = 1960 g Setup 3 honeydew + papaya = 2470 g
answer 1850
method Add all three weighings: 730 + 1960 + 2470 = 5160 g. Every fruit has been weighed exactly twice, so the three fruits together weigh 5160 ÷ 2 = 2580 g. The honeydew is the one missing from Setup 1, so honeydew = 2580 − 730 = 1850 g.
watch Subtracting two of the weighings and hoping. Adding all three and halving is what turns three pair-weights into the single total, and the rest is one subtraction.
At a bakery, muffins are sold at the prices shown. Nick wants to order 57 muffins for a birthday party. What is the least amount of money he will need to pay for the muffins?
+-------------------------------------------------+ | Muffins on Sale! | | | | 1 for $1.40 6 for $5.70 10 for $8.90 | +-------------------------------------------------+
(1) $34.20 (2) $51.60 (3) $79.80 (4) $94.70
answer (2) — $51.60
method Work out the value of each deal first: 10 for $8.90 is 89c each, 6 for $5.70 is 95c each, and a single is $1.40. So take as many 10-packs as possible: 5 × 10 = 50 muffins for 5 × $8.90 = $44.50. The remaining 7 are cheapest as one 6-pack plus one single: $5.70 + $1.40 = $7.10. Total = $44.50 + $7.10 = $51.60.
watch Buying six 10-packs (60 muffins for $53.40) or paying for the last 7 as singles ($9.80). Both are more than $51.60 — always price the leftover two ways.
Among Jim and two other friends, John is the lightest and Jack is the heaviest. The table below shows the total weight of two boys weighing themselves each time. Find Jim's weight.
1st reading | 56 kg 2nd reading | 70 kg 3rd reading | 54 kg John < Jim < Jack
(1) 36 kg (2) 34 kg (3) 22 kg (4) 20 kg
answer (2) — 34 kg
method The three readings are the three possible pairs. Since John is lightest and Jack is heaviest, the LIGHTEST pair is John + Jim = 54 and the HEAVIEST pair is Jim + Jack = 70, leaving John + Jack = 56. Adding all three readings counts every boy exactly twice: 54 + 56 + 70 = 180, so the three boys together weigh 90 kg. Jim = 90 − (John + Jack) = 90 − 56 = 34 kg.
watch Assuming the readings are listed in a helpful order (that the 1st reading is John + Jim). Sort them smallest to largest first — it is the smallest and largest pairs that the clue pins down.
The table shows the number of customers at a restaurant over 4 days; part of it was covered by a stain. The number of customers on Day 3 is equal to the total number of customers on Day 1 and Day 2. The total number of customers on all 4 days is 240 when rounded to the nearest ten. What is the greatest possible number of customers on Day 4?
Day | Number of customers ----+-------------------- 1 | 30 2 | 45 3 | ##### stain ##### 4 | ##### stain #####
answer 94
method Day 3 = Day 1 + Day 2 = 30 + 45 = 75. So Days 1, 2 and 3 come to 30 + 45 + 75 = 150. The 4-day total rounds to 240 to the nearest ten, so the real total is anywhere from 235 to 244. The question wants the GREATEST Day 4, so take the greatest total: 244 − 150 = 94 customers.
watch Taking the total as exactly 240 and answering 90. "Rounds to 240" means a range (235 to 244), and the largest number in that range is what makes Day 4 largest.
At a bento shop in Tokyo, three friends ordered: Hana bought 4 bento sets and 2 bottles of tea for $35. Ken bought 2 bento sets, 3 mochi cakes and 1 bottle of tea for $35.50. Miki bought 2 bento sets, 1 mochi cake and 2 bottles of tea for $27.50. 2 mochi cakes cost the same as 3 bottles of tea. What is the cost of 1 bento set?
answer 6.75
method First swap every mochi for tea: 2 mochi = 3 tea, so 1 mochi = 1½ tea. Ken becomes 2 bento + (3 × 1½) tea + 1 tea = 2 bento + 5½ tea = $35.50. Miki becomes 2 bento + 1½ tea + 2 tea = 2 bento + 3½ tea = $27.50. Both have 2 bento sets, so subtracting cancels them: 2 tea = $35.50 − $27.50 = $8, so 1 tea = $4. Put that back into Miki: 2 bento + 3½ × $4 = $27.50 → 2 bento = $27.50 − $14 = $13.50 → 1 bento = $6.75. (Check with Hana: 4 × 6.75 + 2 × 4 = 27 + 8 = $35 ✓)
watch Starting from Hana's line. Ken's and Miki's both contain 2 bento sets, so subtracting THOSE two removes the bento and leaves only tea — choosing which two lines to subtract is the whole trick.
In a restaurant, a rectangular table can seat 10 people and a round table can seat 5 people. During lunch time, all the tables in the restaurant were occupied. The number of each type of table in the restaurant is more than 10 and less than 15. If there were 205 customers, how many rectangular and round tables altogether were there for the 205 customers?
answer 27
method "More than 10 and less than 15" means each type is 11, 12, 13 or 14. Seats: 10 × rectangular + 5 × round = 205, and dividing everything by 5 gives 2 × rectangular + round = 41. Now test each value: 11 rectangular → round = 41 − 22 = 19 ✗; 12 → 17 ✗; 13 → 15 ✗ (15 is not less than 15); 14 → 41 − 28 = 13 ✓. So 14 rectangular and 13 round, giving 14 + 13 = 27 tables.
watch Reading "more than 10 and less than 15" as including 10 and 15, or finding one pair that makes 205 without checking that BOTH counts sit inside the range.