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.
Siti is 17 years old and Julia is 5 years old. How old will Julia be when she is 3/5 as old as Siti?
answer 18
method The gap between their ages never changes: 17 − 5 = 12 years, now and forever. At the moment we want, Julia is 3 units and Siti is 5 units, so the gap is 5 − 3 = 2 units. So 2 units = 12 years, 1 unit = 6 years, and Julia (3 units) is 3 × 6 = 18.
watch Taking 3/5 of 17 straight away and getting 10.2. Both of them get older, so the fraction applies to their ages LATER, not to Siti's age now — the fixed gap is what pins the moment down.
The number of girls is 3/7 the number of boys at a National Kids’ Run competition. There are 960 more boys than girls. How many children took part in the competition?
answer 2400
method Draw boys as 7 units and girls as 3 units. The difference is 7 − 3 = 4 units = 960, so 1 unit = 960 ÷ 4 = 240. Altogether there are 7 + 3 = 10 units, so 10 × 240 = 2400 children.
watch Answering 1680 (the boys) or 720 (the girls). The question asks for everyone, which is all 10 units.
Jeff is 12 years older than Leon now. 10 years ago, the total age of Jeff and Leon was 78 years. How old is Leon now?
answer 43
method Ten years ago both were 10 years younger, so their ages today total 78 + 10 + 10 = 98. Jeff is 12 more than Leon, so take the 12 off and split what is left equally: (98 − 12) ÷ 2 = 86 ÷ 2 = 43. Leon is 43 now.
watch Adding only 10 to the 78 instead of 10 for EACH of them, or answering 55 (Jeff's age).
Melody baked some muffins. 2/7 of the muffins were blueberry muffins and the rest were chocolate muffins. There were 72 more chocolate muffins than blueberry muffins. How many muffins did she bake altogether?
answer 168
method Blueberry is 2 units out of 7, so chocolate is the other 5 units. The difference is 5 − 2 = 3 units = 72 muffins, so 1 unit = 24. Altogether there are 7 units: 7 × 24 = 168 muffins.
watch Treating 72 as one of the fractions rather than as the DIFFERENCE between them, or answering 120 (the chocolate ones).
Tania had 156 stickers and Amelia had 114 stickers. After each of them gave away an equal number of stickers, Tania had 4 times as many stickers as Amelia. How many stickers did each of them give away?
answer 100
method They give away the same number, so the gap between them never changes: 156 − 114 = 42. At the end Tania is 4 units and Amelia is 1 unit, so the gap is 3 units = 42, giving 1 unit = 14. So Amelia ended with 14 stickers, and she gave away 114 − 14 = 100.
watch Answering 14 (what Amelia has LEFT) instead of what she gave away. The unchanging gap is the key — it is what makes the 42 usable.
A bookshop had a total of 160 erasers and pencils. After 28 erasers and 24 pencils were sold, the number of erasers became thrice the number of pencils. How many pencils did the bookshop have at first?
answer 51
method After the sale, 160 − 28 − 24 = 108 items were left. At that point erasers were 3 units and pencils 1 unit, so 4 units = 108 and 1 unit = 27 — that is the pencils left. Before the sale the shop had 27 + 24 = 51 pencils.
watch Answering 27, the pencils remaining. The 24 sold pencils have to be added back to reach "at first".
John spilled some paint on his report card, hiding his Mathematics and Science scores. English was 86, Chinese was 80, and the total of all four subjects was 342. He scored 12 fewer marks in Science than in Mathematics. How many marks did he score for Mathematics?
Subject | Score -------------+------- English | 86 Chinese | 80 Mathematics | ##### Science | ##### Total | 342
answer 94
method Maths and Science together = 342 − 86 − 80 = 176. Science is 12 fewer than Maths, so take that 12 off and split the rest equally: (176 − 12) ÷ 2 = 164 ÷ 2 = 82 — that is Science. Maths = 82 + 12 = 94.
watch Splitting the 176 evenly into 88 and 88 and forgetting the 12, or adding the 12 to the wrong subject. Science is the SMALLER one.
Kate and Sam had the same number of books at first. Kate gave away 2/3 of her books while Sam gave away 1/6 of his books. The number of books that Kate gave away was 63 more than Sam. Find the number of books that each of them had at first.
answer 126
method They started with the same number, so both fractions are of the same amount. Kate gave 2/3 and Sam gave 1/6. In sixths that is 4/6 and 1/6, so Kate gave 3/6 — that is half — more than Sam. If half the books is 63, the whole is 63 × 2 = 126 books each.
watch Subtracting the fractions as 2/3 − 1/6 = 1/3 by taking the bottoms away too. Rewrite both in sixths before subtracting.
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.
Mrs Sim wanted to buy 20 glass bowls but she was short of $8. She then bought 19 glass bowls and had $6 left. Each glass bowl cost the same. How much money did Mrs Sim have at first?
answer 272
method Compare the two plans. Buying one bowl fewer took her from being $8 short to having $6 spare — a swing of 8 + 6 = $14. That swing is the price of exactly one bowl, so a bowl costs $14. She bought 19 of them and had $6 left, so she started with 19 × $14 + $6 = $266 + $6 = $272.
watch Subtracting the 8 and the 6 instead of adding them. Being short and having spare are on opposite sides of zero, so the gap between the two plans is their sum.
Colin, Dylan and Eden shared some sweets. Colin took 1/6 of the sweets. Dylan took the remaining sweets and shared it equally with Eden. Eden ate 12 of his sweets and has 18 sweets left. How many sweets were there altogether?
answer 72
method Work backwards for Eden: he has 18 left after eating 12, so he had 12 + 18 = 30. Colin took 1/6, leaving 5/6, and Dylan and Eden split that equally — so Eden got half of 5/6, which is 5/12 of the whole. If 5/12 is 30 sweets, then 1/12 is 6, and the whole is 12 × 6 = 72.
watch Taking Eden's share as half of everything (1/2) instead of half of what was LEFT (5/12).
Yiping had 100 fewer stickers than Joanne. Joanne had 4 times as many stickers as Wendy. Wendy had 40 stickers. How many stickers did they have altogether?
answer 260
method Start from the one you know. Wendy = 40. Joanne = 4 × 40 = 160. Yiping = 160 − 100 = 60. Altogether = 40 + 160 + 60 = 260.
watch Working from Yiping first, or adding the 100 to Joanne instead of taking it off. "100 fewer than Joanne" means Yiping is the smaller one.
John bought 2 crates of oranges. Crate A contained 200 more oranges than Crate B. After he transferred 50 oranges from Crate B to Crate A, there were twice as many oranges in Crate A than in Crate B. How many oranges were there in Crate A at first?
answer 550
method After the move, Crate B has 50 fewer and Crate A has 50 more, so the gap between them grows from 200 to 200 + 50 + 50 = 300. At that point A is twice B, so the gap is exactly one B: B after = 300. So B at first = 300 + 50 = 350, and A at first = 350 + 200 = 550.
watch Thinking the gap stays at 200. Moving oranges from one crate to the other changes BOTH, so the gap shifts by twice the number moved.
Uncle Lee had some pears in his shop. He sold 36 of them in the morning and threw away 5 bad ones. He sold 10 pears in the afternoon and packed half of the remaining pears into 10 bags. Each bag contained 7 pears. How many pears did he have at first?
answer 191
method Work backwards. The 10 bags hold 10 × 7 = 70 pears, and that was HALF of what remained, so 140 pears remained after the afternoon sale. Before that he sold 10, so there were 150. Before the morning he had also lost 36 sold and 5 thrown away: 150 + 10 is already counted, so add back 36 + 5 = 41 to the 150, giving 191.
watch Taking the 70 bagged pears as everything that was left. It is half of it, so the remainder is 140, not 70.
Mother bought 5 more cups than plates. Each cup cost $8 and each plate cost $3. She paid $183 altogether. How many cups did she buy?
answer 18
method Set the 5 extra cups aside first: they cost 5 × $8 = $40. That leaves $183 − $40 = $143 for equal numbers of cups and plates. One cup and one plate together cost $8 + $3 = $11, so there are $143 ÷ $11 = 13 pairs. So there are 13 plates and 13 + 5 = 18 cups.
watch Answering 13 — that is the number of PLATES (and of the matched cups). The five extra cups still have to be added on.
Sandy and Tracy had $600 altogether. Sandy gave 1/5 of her money to Tracy and then Tracy gave 1/4 of the money she then had to Sandy. In the end, both of them had the same amount of money. How much money did Sandy have at first?
answer 250
method Work backwards from the end: they finished equal, so each had $300. The last move was Tracy giving away 1/4 of her money, keeping 3/4 — and 3/4 of Tracy's money was $300, so Tracy had $400 just before, and gave $100 to Sandy. So before that move Sandy had $300 − $100 = $200. That $200 was Sandy's money after giving away 1/5, so it is 4/5 of what she started with: 1/5 = $50, and Sandy began with 5 × $50 = $250.
watch Taking 1/4 of Tracy's ORIGINAL money instead of the amount she held after Sandy's gift. Each fraction refers to the money at that moment.
The total cost of 6 skirts and 3 blouses was $144. Three blouses cost as much as 2 skirts. Find the total cost of one skirt and one blouse.
answer 30
method Swap the blouses for skirts: 3 blouses cost the same as 2 skirts, so 6 skirts + 3 blouses costs the same as 6 skirts + 2 skirts = 8 skirts. So 8 skirts = $144, and one skirt = $18. Then 3 blouses = 2 skirts = $36, so one blouse = $12. One skirt and one blouse = $18 + $12 = $30.
watch Assuming a blouse costs the same as a skirt, or dividing $144 by 9 items. The swap is what makes everything one kind of thing.
Ashley is 10 years old and her mother is 46 years old. In how many years’ time will Ashley’s mother be 4 times as old as Ashley?
answer 2
method The gap between their ages never changes: 46 − 10 = 36 years, now and always. At the moment we want, Ashley is 1 unit and her mother is 4 units, so the gap is 4 − 1 = 3 units = 36, giving 1 unit = 12. So Ashley will be 12, which is 12 − 10 = 2 years from now.
watch Answering 12 — that is Ashley's AGE at that time, not how many years away it is. Read the last line again before writing.
There were 163 ribbons in Box A and 115 ribbons in Box B. Some ribbons were moved from Box A to Box B. In the end, there were 26 more ribbons in Box A than Box B. How many ribbons were there in Box A in the end?
answer 152
method Moving ribbons between the boxes does not change the TOTAL: 163 + 115 = 278 ribbons, before and after. At the end A is 26 more than B, so take that 26 off and split the rest equally: (278 − 26) ÷ 2 = 252 ÷ 2 = 126 — that is Box B. Box A = 126 + 26 = 152.
watch Trying to work out how many ribbons were moved first. You never need to know — the unchanged total plus the final difference is enough.
Julia had some beads in a jar. 1/4 of the beads were red and the rest were green. After Julia put another 318 red beads into the jar, the fraction of green beads in the jar became 3/7. What was the total number of beads in the jar at first?
answer 424
method The GREEN beads never change — only red ones were added. At first green was 3/4 of the jar; afterwards green is 3/7 of the bigger jar. Say the green beads number G. Then at first the jar held G ÷ 3 × 4 beads, and at the end it held G ÷ 3 × 7. The jar grew by exactly the 318 red beads added, so G ÷ 3 × 7 − G ÷ 3 × 4 = 318, that is G ÷ 3 × 3 = 318, so G = 318. The jar at first = 318 ÷ 3 × 4 = 424 beads.
watch Trying to track the red beads, which change. Spotting the quantity that STAYS THE SAME — the green ones — is what makes the two fractions comparable.
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.
The total mass of a box and a watermelon was 7 kg 34 g. When some strawberries were added into the box, the total mass became 8600 g. The watermelon was 3 times as heavy as all the strawberries added. Find the mass of the box, in grams.
answer 2336
method Work in grams: 7 kg 34 g = 7034 g. Adding the strawberries took the total from 7034 g to 8600 g, so the strawberries weigh 8600 − 7034 = 1566 g. The watermelon is 3 times that: 3 × 1566 = 4698 g. The box and watermelon together were 7034 g, so the box = 7034 − 4698 = 2336 g.
watch Reading 7 kg 34 g as 7340 g. It is 7000 + 34 = 7034 g — the 34 fills the ones and tens columns, not the hundreds.
A spider was climbing to the top of a garden wall, starting from the bottom. After climbing up 1/3 of the wall, it began to rain. During the rain the spider slipped down 30 cm and stayed there until the rain stopped. Then it climbed up the remaining 5/6 of the height of the wall to reach the top. What was the total height the spider had climbed before and after the rain, in cm?
answer 210
method Let the wall be 1 whole. The spider got to 1/3, slipped down 30 cm, then climbed 5/6 of the wall to finish at the top. So (1/3 of the wall) − 30 cm + (5/6 of the wall) = the whole wall. In sixths, 1/3 + 5/6 = 2/6 + 5/6 = 7/6, so 7/6 of the wall minus 30 cm equals 1 wall — meaning the extra 1/6 of the wall is exactly the 30 cm it slipped. So the wall is 6 × 30 = 180 cm. It climbed 1/3 of 180 = 60 cm before the rain and 5/6 of 180 = 150 cm after, a total of 60 + 150 = 210 cm.
watch Answering 180 — that is the height of the WALL. The question asks how far the spider CLIMBED, which is more, because it had to re-climb the 30 cm it slipped.
In a library, there were some story books at first. Betty, the librarian, added another 17 story books and removed 36 of them. Then, Betty received 3 times as many new story books as what were left on the shelves. She then arranged all the story books equally between 2 sections. Each section had 458 story books in the end. How many story books were in the library at first?
answer 248
method Work backwards. Two sections of 458 make 2 × 458 = 916 books at the end. Just before that, the shelves held some number and Betty received 3 times as many again, so the 916 is 1 + 3 = 4 equal shares: one share = 916 ÷ 4 = 229 books on the shelves. That 229 came after adding 17 and removing 36, a net loss of 19. So at first there were 229 + 19 = 248 books.
watch Reading "3 times as many new books as what were left" as making the total 3 times bigger. It ADDS 3 shares to the 1 already there, giving 4 shares in all.
Susan and Tina had the same number of beads at first. After Susan threw away 18 beads and Tina bought another 72 beads, Tina had 4 times as many beads as Susan. How many beads did Susan have at first?
(1) 48 beads (2) 54 beads (3) 70 beads (4) 90 beads
answer (1) — 48 beads
method Draw Susan's beads at the end as 1 unit and Tina's as 4 units. They started equal, then Susan lost 18 and Tina gained 72, so the gap between them grew by 18 + 72 = 90 beads. That gap is 4 units − 1 unit = 3 units, so 1 unit = 90 ÷ 3 = 30 beads. Susan has 30 left, and she threw away 18, so she started with 30 + 18 = 48 beads.
watch Answering 30 — that is what Susan has LEFT at the end, not what she started with. Read the question again before writing the number down.
At a bakery, the price of a cupcake was $2 and the price of a tart was $5. Mrs Lim paid $38 to buy a total of 13 cupcakes and tarts. How many more cupcakes than tarts did Mrs Lim buy?
answer 5
method Suppose all 13 were cupcakes: that would cost 13 × $2 = $26. She actually paid $38, which is $12 more. Changing one cupcake into a tart adds $5 − $2 = $3 to the bill, so the number of tarts is 12 ÷ 3 = 4. Then cupcakes = 13 − 4 = 9. The question asks how many MORE: 9 − 4 = 5.
watch Answering 9 — that is the number of cupcakes, not the DIFFERENCE the question asked for. Circle the words "how many more" before you start.
At first, Mary had $166 and Larry had $304. Each of them bought 5 similar plates, and each plate was the same price. After buying the plates, Larry had 4 times as much money as Mary had left. What was the price of one plate?
answer 24
method They spend exactly the same amount, so the GAP between them never changes: 304 − 166 = $138, before and after. At the end Larry has 4 units and Mary has 1 unit, so the gap is 4 − 1 = 3 units. 3 units = $138, so 1 unit = $46 — that is Mary's money left. Mary spent 166 − 46 = $120 on 5 plates, so one plate costs 120 ÷ 5 = $24.
watch Not noticing that the difference stays the same when both people spend the same amount. That one idea turns a hard problem into two divisions.
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.
Mary had 48 stalks of flowers in a basket. 1/3 of them were roses and the rest were lilies and daisies. There were 4 more lilies than daisies. How many daisies should Mary buy such that the number of daisies would be 1/2 of the total number of flowers in the basket?
answer 20
method Roses = 1/3 of 48 = 16, so lilies and daisies together = 48 − 16 = 32. There are 4 more lilies than daisies, so take the 4 off and share the rest equally: (32 − 4) ÷ 2 = 14 daisies, and 14 + 4 = 18 lilies. Now she buys more daisies. Wanting daisies to be HALF the total is the same as wanting daisies to equal everything else put together — that is roses + lilies = 16 + 18 = 34. She already has 14, so she must buy 34 − 14 = 20 daisies. (Check: 34 daisies out of 48 + 20 = 68 flowers, and 34 is half of 68 ✓)
watch Taking "half the total" as half of 48 (= 24) and answering 10. The total GROWS with every daisy she buys, which is why it is easier to match the daisies against everything else.
John divided the corridor of a school building into equal parts of length 4 m, and placed 2 potted plants in each part. For the same corridor, he divided it into equal parts of length 6 m and hung 5 lanterns in each part. If there were 24 more lanterns than potted plants, how long was the corridor?
Figure 1 |<-- 4 m -->| 2 potted plants in each part Figure 2 |<--- 6 m --->| 5 lanterns in each part
answer 72
method The two patterns use different part lengths, so compare them over a length both divide — 12 m. In 12 m there are 12 ÷ 4 = 3 parts of plants, giving 3 × 2 = 6 plants, and 12 ÷ 6 = 2 parts of lanterns, giving 2 × 5 = 10 lanterns. That is 10 − 6 = 4 more lanterns for every 12 m. We need 24 more, so the corridor is 24 ÷ 4 = 6 lots of 12 m = 72 m. (Check: 72 m gives 18 × 2 = 36 plants and 12 × 5 = 60 lanterns, and 60 − 36 = 24 ✓)
watch Comparing 2 plants against 5 lanterns directly and using a difference of 3. The parts are different LENGTHS, so the counts can only be compared over the same distance.
Ben has twice as many marbles as Cara. If Ben gives Cara 6 marbles, they will have the same number. How many marbles does Ben have at first?
(A) 12 (B) 18 (C) 24 (D) 30 (E) 36
answer (C) — 24
method Draw Cara as 1 unit and Ben as 2 units. Ben is 1 unit more than Cara, and moving 6 marbles across closes a gap of 12 — so 1 unit = 12. Ben has 2 units = 24. (Check: 24 and 12, then 18 and 18.)
Mai has 3 times as many stickers as Linh. Altogether they have 24 stickers. How many stickers does Mai have?
(A) 6 (B) 8 (C) 12 (D) 18 (E) 21
answer (D) — 18
method Draw Linh as 1 unit and Mai as 3 units. Altogether that is 4 units, so 4 units = 24 and 1 unit = 24 ÷ 4 = 6. Mai is 3 units = 3 × 6 = 18. (Check: 18 and 6 add up to 24, and 18 is 3 times 6.)
Duc has 5 more sweets than Linh. Duc then gives Linh 2 sweets. How many more sweets does Duc have than Linh now?
(A) 1 (B) 2 (C) 3 (D) 5 (E) 7
answer (A) — 1
method The gap starts at 5. Giving 2 away does two things at once: Duc drops by 2 AND Linh climbs by 2, so the gap closes by 2 + 2 = 4, not by 2. The gap is now 5 − 4 = 1. (Check with real numbers: 10 and 5 becomes 8 and 7.)
Nam has twice as many stamps as Hoa. If Nam gives Hoa 4 stamps, they will have the same number. How many stamps does Nam have at first?
(A) 8 (B) 12 (C) 16 (D) 20 (E) 24
answer (C) — 16
method Draw Hoa as 1 unit and Nam as 2 units, so the gap is 1 unit. Moving 4 across closes the gap by 4 + 4 = 8, and it has to close all the way — so 1 unit = 8. Nam is 2 units = 16. (Check: 16 and 8, then 12 and 12.)
An has 3 times as many marbles as Bao. If An gives Bao 10 marbles, they will have the same number. How many marbles does An have at first?
(A) 15 (B) 20 (C) 24 (D) 30 (E) 40
answer (D) — 30
method Draw Bao as 1 unit and An as 3 units, so the gap is 2 units. Moving 10 across closes the gap by 10 + 10 = 20, so 2 units = 20 and 1 unit = 10. An is 3 units = 30. (Check: 30 and 10, then 20 and 20.)
Linh has 4 times as many beads as Mai. If Linh gives Mai 9 beads, they will have the same number. How many beads do the two girls have altogether?
(A) 6 (B) 15 (C) 24 (D) 30 (E) 36
answer (D) — 30
method Draw Mai as 1 unit and Linh as 4 units, so the gap is 3 units. Moving 9 across closes the gap by 9 + 9 = 18, so 3 units = 18 and 1 unit = 6. Linh has 24 and Mai has 6. Now read the question again: it asks for the TOTAL, not Linh's share — 24 + 6 = 30. (Moving beads from one girl to the other never changes the total.)