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 box below contains a mixture of circles and stars: 4 stars and 6 circles. ?/5 of the shapes in the box are stars. What is the missing number?
+--------------------------+ | * O O * O | | O O * O * | +--------------------------+ 4 stars, 6 circles, 10 shapes in all
answer 2
method Count first: 4 stars out of 10 shapes, so the fraction is 4/10. The answer must be written in fifths, so simplify by dividing top and bottom by 2: 4/10 = 2/5. The missing number is 2.
watch Answering 4 — that is the number of stars, not the numerator once the fraction is written in fifths.
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.
Mrs Thompson baked 175 tarts. She sold 3/5 of them. Each tart was sold for $4. How much money did she collect in total?
answer 420
method Find how many she sold: 1/5 of 175 is 35, so 3/5 is 3 × 35 = 105 tarts. Each sold for $4, so she collected 105 × $4 = $420.
watch Multiplying all 175 tarts by $4. She only sold three fifths of them — the unsold ones brought in nothing.
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).
There are 256 balls in a box. 1/4 of the balls are red and 5/8 of them are white. The rest are black. How many more white balls than black balls are there?
answer 128
method Red = 1/4 of 256 = 64. White = 5/8 of 256 = 5 × 32 = 160. Black is what is left: 256 − 64 − 160 = 32. So there are 160 − 32 = 128 more white balls than black.
watch Answering 160 (the white balls) instead of the difference, or adding 1/4 and 5/8 as 6/12 by adding tops and bottoms.
Charles bought 9 bags of sweets. Each bag had 48 sweets. He gave away 5/6 of all the sweets. How many sweets did he have left?
answer 72
method Find the total first: 9 × 48 = 432 sweets. He gave away 5/6, so he kept the other 1/6: 432 ÷ 6 = 72 sweets.
watch Working out 5/6 of 432 (= 360) and giving that as the answer. That is what he gave AWAY; the question asks what is left.
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.
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).
Ken had some money. He used 1/2 of it to buy a shirt and 1/5 of it to buy a pair of pants. He had $180 left. How much money did he have at first?
answer 600
method He spent 1/2 + 1/5 of his money. Using tenths: 5/10 + 2/10 = 7/10. So what is left is 10/10 − 7/10 = 3/10, and that is $180. One tenth is 180 ÷ 3 = $60, so the whole is 10 × $60 = $600.
watch Adding 1/2 and 1/5 as 2/7 by adding tops and bottoms. Fractions need a common denominator before they can be added.
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.
How many one-thirds are there in 4 wholes?
(A) 3/4 (B) 1 3/4 (C) 3 (D) 12
answer (D) — 12
method One whole holds 3 thirds, so 4 wholes hold 4 × 3 = 12 thirds.
watch Answering 3 — that is how many thirds are in ONE whole. Or dividing 3 by 4 instead of multiplying.
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.
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.
How many sixths are there in 2 wholes?
(1) 6 (2) 12 (3) 18 (4) 24
answer (2) — 12
method One whole holds 6 sixths, so 2 wholes hold 2 × 6 = 12 sixths.
watch Answering 6 — that is how many sixths are in ONE whole, not two.
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.
ELSA had 40 stickers. She gave 3/8 of them to her sister. How many stickers does she have left?
(A) 5 (B) 15 (C) 24 (D) 25 (E) 35
answer (D) — 25
method 40 ÷ 8 = 5, so 3/8 of 40 is 3 × 5 = 15 given away. The question asks what is LEFT: 40 − 15 = 25.