Solution to TU1:
Raphael and Leonardo had the same number of stamps. After Raphael had given 80 stamps to Leonardo, Leonardo had twice as many stamps as Leonardo. How many stamps did Leonardo have in the end?
We can use models or units to solve this question. Let's try using the units method:
|
|
Raphael |
Leonardo |
Total |
|
|
|
|
|
|
|
|
|
–80 |
+80 |
|
|
|
After |
1u |
2u |
3u |
(´2) |
We know the 2 boys had the same number of stamps at first, so we need the total units to be divisible by 2. We do that by multiplying the total units by 2 and we get:
|
|
Raphael |
Leonardo |
Total |
|
|
|
|
|
|
|
|
|
–80 |
+80 |
|
|
|
After |
1u |
2u |
3u |
(´2) |
|
|
2m |
4m |
6m |
|
After that, we divide 6m by 2 to get 3m, and we can complete the table:
|
|
Raphael |
Leonardo |
Total |
|
|
|
3m |
3m |
6m |
|
|
|
–80 |
+80 |
|
|
|
After |
1u |
2u |
3u |
(´2) |
|
|
2m |
4m |
6m |
|
From the table, we can see that 3m – 2m = 1m
This means 1m à 80
and 4m à 80 ´ 4 = 320 Ans: 320
Solution to TU2:
Bala and Danny had money in the ratio 5 : 4. After Bala gave Danny $40, they had an equal amount of money. How much money did Danny have in the end?
In this case, we know the total is 9 units. We multiply all the units by 2 to get the total units to become 18 units:
|
|
Bala |
Danny |
Total |
|
|
5u |
4u |
9u |
(´2) |
|
|
10m |
8m |
18m |
|
|
|
–40 |
+40 |
|
|
|
After |
9m |
9m |
18m |
|
10m – 9m = 1m
1m à 40
9m à 40 ´ 9 = 360 Ans: $360
Solution to TU3:
Mark had 20% more stickers than Sam. After receiving some stickers from Sam, the ratio of Mark's stickers to Sam's stickers became 3:1. If Mark had 132 stickers in the end, find the number of stickers Mark received from Sam.
We know 20% is one fifth, so the ratio of Mark's stickers to Sam's is 6 : 5.
|
|
Mark |
Sam |
Total |
|
|
Before |
6a |
5a |
11a |
|
|
|
+ ? |
– ? |
|
|
|
After |
3b |
1b |
4b |
|
3b à 132
1b à 132 ÷ 3 = 44
4b à 44 ´ 4 = 176
Total remains the same so:
11a à 176
1a à 176 ÷ 11 = 16
6a à 16 ´ 6 = 96
132 – 96 = 36 Ans: 36
Of course, since 11a = 4b, we can merge all the units together and work out the answer that way. It's up to you which method you want to use.
Solution to TU4:
Lucy had 2 boxes of sweets. After transferring 5⁄9 of the sweets in Box B to Box A, the ratio of the number of sweets in Box A to the number of sweets in Box B became 3 : 2. Find the ratio of the number of sweets in Box A to the number of sweets in Box B at first.
This is a question where you have to put in the given information one piece at a time until the complete picture appears. We begin with the 5 units transferred from Box B to Box A:
|
|
Box A |
Box B |
Total |
|
|
Before |
|
9a |
|
|
|
|
+5a |
–5a |
|
|
|
After |
|
4a |
|
|
This means Box A should be 6a, so:
|
|
Box A |
Box B |
Total |
|
|
Before |
1a |
9a |
10a |
|
|
|
+5a |
–5a |
|
|
|
After |
6a |
4a |
10a |
|
From the table, we can see that the answer is 1:9
Solution to TU5:
Helen is reading a book. The ratio of the number of pages she has read to the number of pages she has not read is 2:5. If she reads another 50 pages, the number of pages she has read will become 75% of the number of pages she has not read. How many pages are there in the book?
It is easier if we change 75% to 3⁄4
|
|
Pages read |
Pages unread |
Total |
|
|
Before |
2u |
5u |
7u |
|
|
|
–50 |
+50 |
|
|
|
After |
3u |
4u |
7u |
|
3u – 2u = 1u
1u à 50
7b à 50 ´ 7 = 350 Ans: 350 pages
Solution to TU6:
The ratio of the number of stamps that Diana had to the number of stamps that Eileen had was 3:8. Eileen gave 60 stamps to Diana. After that, they had the same number of stamps. How many stamps had Diana at first?
|
|
Diana |
Eileen |
Total |
|
|
Before |
3m |
8m |
11m |
(´2) |
|
Before |
6u |
16u |
22u |
|
|
Transfers |
+60 |
–60 |
|
|
|
After |
11u |
11u |
22u |
|
11u – 6u = 5u
5u à 60
1u à 60 ÷ 5 = 12
6u à 12 ´ 6 = 72 Ans: 72 stamps
Note: We want 11m to be divisible by 2, so we multiply the whole row by 2.
Solution to TU7:
At first, Yeti had twice the number of laser guns that Zork had. Zork gave 4 guns to Yeti. Subsequently, Yeti had thrice the number of laser guns that Zork had. How many laser guns did Yeti have in the end?
|
|
Yeti |
Zork |
Total |
|
|
Before |
2a |
1a |
3a |
(´4) |
|
After |
3b |
1b |
4b |
(´3) |
so we multiply the Before row by 4 and the After row by 3.
After that, we get:
|
|
Yeti |
Zork |
Total |
|
|
Before |
8u |
4u |
12u |
|
|
Transfers |
+4 |
–4 |
|
|
|
After |
9u |
3u |
12u |
|
9u – 8u = 1u
1u à 4
9u à 4 ´ 9 = 36 Ans: 36 guns
Solution to TU8:
The ratio of the number of chickens to the number of
ducks on a farm was 8:5 at first. The farmer bought some more ducks and sold
some of the chickens. In the end, the total number of chickens and ducks on the
farm was unchanged. If the number of ducks increased by 20%, what percentage of
the chickens was sold?
|
|
Chickens |
Ducks |
Total |
|
|
Before |
8u |
5u |
13u |
|
|
|
–1u |
+1u |
|
|
|
After |
7u |
6u |
13u |
|
The total is unchanged, so that means the number of chickens decreased by 1 u.
1⁄8 = 1⁄8 ´ 100% = 12.5% Ans: 12.5%
Solution to TU9:
Peter had 2 aquariums. He transferred 9⁄17 of the fish in Aquarium A to Aquarium B. After that, the number of fish in Aquarium B was 1.5 times the number of fish in Aquarium A. What was the ratio of the number of fish in Aquarium A to the number of fish in Aquarium B at first?
|
|
Aquarium A |
Aquarium B |
Total |
|
|
Before |
17u |
3u |
20u |
|
|
|
–9u |
+9u |
|
|
|
After |
8u |
12u |
20u |
|
= 1 : 1.5 = 8 :12 Ans: 17:3
Solution to TU10:
Alison has thrice as many books as Lorna. How many books must Alison give to Lorna so that each of them would have 62 books?
|
|
Alison |
Lorna |
Total |
|
|
Before |
3u |
1u |
4u |
|
|
|
–1u |
+1u |
|
|
|
After |
2u |
2u |
4u |
|
2u à 62
1u à 62 ÷ 2 = 31 Ans: 31 books
