Practice

Question 1

1. Take one unit to be one share of the 2 : 1 at first, so Xin Yi is 2 and Daniel is 1.
2. Write what each of them ends with, still in those units.Xin Yi = 2 units, Daniel = 1 unit + 50
3. Xin Yi's amount never changes, so that bar is the same in both pictures.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.1 unit = 50
6. One unit is that shared out.50 ÷ 1 = 50
7. Xin Yi started with 2 units.2 × 50 = 100

Xin Yi had 100 entry tickets at first.

The pattern. Xin Yi's amount never changed, so that bar is the same in both pictures: 2 units and 1 unit + 50.

Question 2

1. Take one unit to be one share of the 5 : 2 at first, so Mei is 5 and Farid is 2.
2. Write what each of them ends with, still in those units.Mei = 5 units, Farid = 2 units + 150
3. Mei's amount never changes, so that bar is the same in both pictures.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = 150
6. One unit is that shared out.150 ÷ 3 = 50
7. Mei started with 5 units.5 × 50 = 250

Mei had 250 sweets at first.

The pattern. Mei's amount never changed, so that bar is the same in both pictures: 5 units and 2 units + 150.

Question 3

1. Take one unit to be one share of the 4 : 1 at first, so Priya is 4 and Suri is 1.
2. Write what each of them ends with, still in those units.Priya = 4 units − 90, Suri = 1 unit + 90
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = 180
6. One unit is that shared out.180 ÷ 3 = 60
7. Priya started with 4 units.4 × 60 = 240

Priya had 240 stickers at first.

The pattern. The two ending amounts are the same, written two ways: 4 units − 90 and 1 unit + 90.

Question 4

1. Take one unit to be one share of the 3 : 1 at first, so Nabila is 3 and Timothy is 1.
2. Write what each of them ends with, still in those units.Nabila = 3 units − $120, Timothy = 1 unit − $30
3. The new ratio is 2 : 1, so Nabila's ending amount is 2 parts and one part is Timothy's ending amount.
4. That gives Nabila's ending amount a second name, written in the same units.2 × (1 unit − $30) = 2 units − $60
5. Set the two names side by side and take the units they share off both.1 unit = $60
6. One unit is that shared out.$60 ÷ 1 = $60
7. Nabila started with 3 units.3 × 60 = $180

Nabila had $180 at first.

The pattern. Nabila's amount in the end has two names: 3 units − $120 and 2 units − $60.

Question 5

1. Take one unit to be one share of the 5 : 3 at first, so Nabila is 5 and Timothy is 3.
2. Write what each of them ends with, still in those units.Nabila = 5 units − $24, Timothy = 3 units + $24
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = $48
6. One unit is that shared out.$48 ÷ 2 = $24
7. Nabila started with 5 units.5 × 24 = $120

Nabila had $120 at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − $24 and 3 units + $24.

Question 6

1. Take one unit to be one share of the 5 : 2 at first, so Michelle is 5 and Zulfadli is 2.
2. Write what each of them ends with, still in those units.Michelle = 5 units, Zulfadli = 2 units − 30
3. Michelle's amount never changes, so that bar is the same in both pictures.
4. The new ratio is 5 : 1, so Michelle's ending amount is 5 parts and one part is Zulfadli's ending amount.
5. That gives Michelle's ending amount a second name, written in the same units.5 × (2 units − 30) = 10 units − 150
6. Set the two names side by side and take the units they share off both.5 units = 150
7. One unit is that shared out.150 ÷ 5 = 30
8. Michelle started with 5 units.5 × 30 = 150

Michelle had 150 postcards at first.

The pattern. Michelle's amount never changed, so that bar is the same in both pictures: 5 units and 10 units − 150.

Question 7

1. Take one unit to be one share of the 2 : 1 at first, so Michelle is 2 and Zulfadli is 1.
2. Write what each of them ends with, still in those units.Michelle = 2 units + 80, Zulfadli = 1 unit − 10
3. The new ratio is 4 : 1, so Michelle's ending amount is 4 parts and one part is Zulfadli's ending amount.
4. That gives Michelle's ending amount a second name, written in the same units.4 × (1 unit − 10) = 4 units − 40
5. Set the two names side by side and take the units they share off both.2 units = 120
6. One unit is that shared out.120 ÷ 2 = 60
7. Michelle started with 2 units.2 × 60 = 120

Michelle had 120 postcards at first.

The pattern. Michelle's amount in the end has two names: 2 units + 80 and 4 units − 40.

Question 8

1. Take one unit to be one share of the 5 : 3 at first, so Ali is 5 and Billy is 3.
2. Write what each of them ends with, still in those units.Ali = 5 units − $12, Billy = 3 units + $12
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = $24
6. One unit is that shared out.$24 ÷ 2 = $12
7. Ali started with 5 units.5 × 12 = $60

Ali had $60 at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − $12 and 3 units + $12.

Question 9

1. Take one unit to be one share of the 4 : 1 at first, so Ali is 4 and Billy is 1.
2. Write what each of them ends with, still in those units.Ali = 4 units − $75, Billy = 1 unit + $75
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = $150
6. One unit is that shared out.$150 ÷ 3 = $50
7. Ali started with 4 units.4 × 50 = $200

Ali had $200 at first.

The pattern. The two ending amounts are the same, written two ways: 4 units − $75 and 1 unit + $75.

Question 10

1. Take one unit to be one share of the 3 : 2 at first, so Ali is 3 and Billy is 2.
2. Write what each of them ends with, still in those units.Ali = 3 units − $84, Billy = 2 units − $60
3. The new ratio is 2 : 1, so Ali's ending amount is 2 parts and one part is Billy's ending amount.
4. That gives Ali's ending amount a second name, written in the same units.2 × (2 units − $60) = 4 units − $120
5. Set the two names side by side and take the units they share off both.1 unit = $36
6. One unit is that shared out.$36 ÷ 1 = $36
7. Ali started with 3 units.3 × 36 = $108

Ali had $108 at first.

The pattern. Ali's amount in the end has two names: 3 units − $84 and 4 units − $120.

Question 11

1. Take one unit to be one share of the 3 : 2 at first, so Wan Ting is 3 and Iskandar is 2.
2. Write what each of them ends with, still in those units.Wan Ting = 3 units, Iskandar = 2 units − 35
3. Wan Ting's amount never changes, so that bar is the same in both pictures.
4. The new ratio is 5 : 1, so Wan Ting's ending amount is 5 parts and one part is Iskandar's ending amount.
5. That gives Wan Ting's ending amount a second name, written in the same units.5 × (2 units − 35) = 10 units − 175
6. Set the two names side by side and take the units they share off both.7 units = 175
7. One unit is that shared out.175 ÷ 7 = 25
8. Wan Ting started with 3 units.3 × 25 = 75

Wan Ting had 75 sheets of paper at first.

The pattern. Wan Ting's amount never changed, so that bar is the same in both pictures: 3 units and 10 units − 175.

Question 12

1. Take one unit to be one share of the 5 : 3 at first, so Wan Ting is 5 and Iskandar is 3.
2. Write what each of them ends with, still in those units.Wan Ting = 5 units − 40, Iskandar = 3 units + 40
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 80
6. One unit is that shared out.80 ÷ 2 = 40
7. Wan Ting started with 5 units.5 × 40 = 200

Wan Ting had 200 sheets of paper at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − 40 and 3 units + 40.

Question 13

1. Take one unit to be one share of the 5 : 3 at first, so Wan Ting is 5 and Iskandar is 3.
2. Write what each of them ends with, still in those units.Wan Ting = 5 units − 20, Iskandar = 3 units + 20
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 40
6. One unit is that shared out.40 ÷ 2 = 20
7. Wan Ting started with 5 units.5 × 20 = 100

Wan Ting had 100 sheets of paper at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − 20 and 3 units + 20.

Question 14

1. Take one unit to be one share of the 2 : 1 at first, so Kai Xuan is 2 and Aaron is 1.
2. Write what each of them ends with, still in those units.Kai Xuan = 2 units, Aaron = 1 unit − $12
3. Kai Xuan's amount never changes, so that bar is the same in both pictures.
4. The new ratio is 3 : 1, so Kai Xuan's ending amount is 3 parts and one part is Aaron's ending amount.
5. That gives Kai Xuan's ending amount a second name, written in the same units.3 × (1 unit − $12) = 3 units − $36
6. Set the two names side by side and take the units they share off both.1 unit = $36
7. One unit is that shared out.$36 ÷ 1 = $36
8. Kai Xuan started with 2 units.2 × 36 = $72

Kai Xuan had $72 at first.

The pattern. Kai Xuan's amount never changed, so that bar is the same in both pictures: 2 units and 3 units − $36.

Question 15

1. Take one unit to be one share of the 5 : 2 at first, so Kai Xuan is 5 and Aaron is 2.
2. Write what each of them ends with, still in those units.Kai Xuan = 5 units − $18, Aaron = 2 units + $18
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = $36
6. One unit is that shared out.$36 ÷ 3 = $12
7. Kai Xuan started with 5 units.5 × 12 = $60

Kai Xuan had $60 at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − $18 and 2 units + $18.

Question 16

1. Take one unit to be one share of the 5 : 2 at first, so Kai Xuan is 5 and Aaron is 2.
2. Write what each of them ends with, still in those units.Kai Xuan = 5 units − $30, Aaron = 2 units − $30
3. Both amounts move by $30, so the gap between them never changes.
4. The new ratio is 5 : 1, so Kai Xuan's ending amount is 5 parts and one part is Aaron's ending amount.
5. That gives Kai Xuan's ending amount a second name, written in the same units.5 × (2 units − $30) = 10 units − $150
6. Set the two names side by side and take the units they share off both.5 units = $120
7. One unit is that shared out.$120 ÷ 5 = $24
8. Kai Xuan started with 5 units.5 × 24 = $120

Kai Xuan had $120 at first.

The pattern. Both amounts moved by $30, so the gap between them never changed. Kai Xuan's amount in the end has two names: 5 units − $30 and 10 units − $150.

Question 17

1. Take one unit to be one share of the 2 : 1 at first, so Ravi is 2 and Ben is 1.
2. Write what each of them ends with, still in those units.Ravi = 2 units − 20, Ben = 1 unit + 20
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.1 unit = 40
6. One unit is that shared out.40 ÷ 1 = 40
7. Ravi started with 2 units.2 × 40 = 80

Ravi had 80 marbles at first.

The pattern. The two ending amounts are the same, written two ways: 2 units − 20 and 1 unit + 20.

Question 18

1. Take one unit to be one share of the 3 : 1 at first, so Ravi is 3 and Ben is 1.
2. Write what each of them ends with, still in those units.Ravi = 3 units − 12, Ben = 1 unit + 12
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 24
6. One unit is that shared out.24 ÷ 2 = 12
7. Ravi started with 3 units.3 × 12 = 36

Ravi had 36 marbles at first.

The pattern. The two ending amounts are the same, written two ways: 3 units − 12 and 1 unit + 12.

Question 19

1. Take one unit to be one share of the 5 : 2 at first, so Ravi is 5 and Ben is 2.
2. Write what each of them ends with, still in those units.Ravi = 5 units + 25, Ben = 2 units + 25
3. Both amounts move by 25, so the gap between them never changes.
4. The new ratio is 2 : 1, so Ravi's ending amount is 2 parts and one part is Ben's ending amount.
5. That gives Ravi's ending amount a second name, written in the same units.2 × (2 units + 25) = 4 units + 50
6. Set the two names side by side and take the units they share off both.1 unit = 25
7. One unit is that shared out.25 ÷ 1 = 25
8. Ravi started with 5 units.5 × 25 = 125

Ravi had 125 marbles at first.

The pattern. Both amounts moved by 25, so the gap between them never changed. Ravi's amount in the end has two names: 5 units + 25 and 4 units + 50.

Question 20

1. Take one unit to be one share of the 4 : 1 at first, so Ravi is 4 and Ben is 1.
2. Write what each of them ends with, still in those units.Ravi = 4 units − 45, Ben = 1 unit + 45
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = 90
6. One unit is that shared out.90 ÷ 3 = 30
7. Ravi started with 4 units.4 × 30 = 120

Ravi had 120 marbles at first.

The pattern. The two ending amounts are the same, written two ways: 4 units − 45 and 1 unit + 45.

Question 21

1. Take one unit to be one share of the 5 : 2 at first, so Nur Aisyah is 5 and Marcus is 2.
2. Write what each of them ends with, still in those units.Nur Aisyah = 5 units + 18, Marcus = 2 units + 18
3. Both amounts move by 18, so the gap between them never changes.
4. The new ratio is 2 : 1, so Nur Aisyah's ending amount is 2 parts and one part is Marcus's ending amount.
5. That gives Nur Aisyah's ending amount a second name, written in the same units.2 × (2 units + 18) = 4 units + 36
6. Set the two names side by side and take the units they share off both.1 unit = 18
7. One unit is that shared out.18 ÷ 1 = 18
8. Nur Aisyah started with 5 units.5 × 18 = 90

Nur Aisyah had 90 stamps at first.

The pattern. Both amounts moved by 18, so the gap between them never changed. Nur Aisyah's amount in the end has two names: 5 units + 18 and 4 units + 36.

Question 22

1. Take one unit to be one share of the 5 : 2 at first, so Nur Aisyah is 5 and Marcus is 2.
2. Write what each of them ends with, still in those units.Nur Aisyah = 5 units − 90, Marcus = 2 units + 90
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.3 units = 180
6. One unit is that shared out.180 ÷ 3 = 60
7. Nur Aisyah started with 5 units.5 × 60 = 300

Nur Aisyah had 300 stamps at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − 90 and 2 units + 90.

Question 23

1. Take one unit to be one share of the 3 : 1 at first, so Siti is 3 and Kenneth is 1.
2. Write what each of them ends with, still in those units.Siti = 3 units − 60, Kenneth = 1 unit + 60
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 120
6. One unit is that shared out.120 ÷ 2 = 60
7. Siti started with 3 units.3 × 60 = 180

Siti had 180 coupons at first.

The pattern. The two ending amounts are the same, written two ways: 3 units − 60 and 1 unit + 60.

Question 24

1. Take one unit to be one share of the 3 : 1 at first, so Siti is 3 and Kenneth is 1.
2. Write what each of them ends with, still in those units.Siti = 3 units − 45, Kenneth = 1 unit + 45
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 90
6. One unit is that shared out.90 ÷ 2 = 45
7. Siti started with 3 units.3 × 45 = 135

Siti had 135 coupons at first.

The pattern. The two ending amounts are the same, written two ways: 3 units − 45 and 1 unit + 45.

Question 25

1. Take one unit to be one share of the 5 : 3 at first, so Yi Ting is 5 and Haziq is 3.
2. Write what each of them ends with, still in those units.Yi Ting = 5 units − 60, Haziq = 3 units + 60
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 120
6. One unit is that shared out.120 ÷ 2 = 60
7. Yi Ting started with 5 units.5 × 60 = 300

Yi Ting had 300 erasers at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − 60 and 3 units + 60.

Question 26

1. Take one unit to be one share of the 3 : 2 at first, so Wei Jie is 3 and Farhan is 2.
2. Write what each of them ends with, still in those units.Wei Jie = 3 units − 10, Farhan = 2 units + 10
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.1 unit = 20
6. One unit is that shared out.20 ÷ 1 = 20
7. Wei Jie started with 3 units.3 × 20 = 60

Wei Jie had 60 canteen tokens at first.

The pattern. The two ending amounts are the same, written two ways: 3 units − 10 and 2 units + 10.

Question 27

1. Take one unit to be one share of the 5 : 3 at first, so Zi Xuan is 5 and Ryan is 3.
2. Write what each of them ends with, still in those units.Zi Xuan = 5 units − 50, Ryan = 3 units + 50
3. Nothing enters or leaves, so the two amounts still come to the same total.
4. They end up equal, so the two endings are the same amount written two ways.
5. Set the two names side by side and take the units they share off both.2 units = 100
6. One unit is that shared out.100 ÷ 2 = 50
7. Zi Xuan started with 5 units.5 × 50 = 250

Zi Xuan had 250 badges at first.

The pattern. The two ending amounts are the same, written two ways: 5 units − 50 and 3 units + 50.

Question 28

1. Take one unit to be one share of the 5 : 2 at first, so Zi Xuan is 5 and Ryan is 2.
2. Write what each of them ends with, still in those units.Zi Xuan = 5 units − 25, Ryan = 2 units − 25
3. Both amounts move by 25, so the gap between them never changes.
4. The new ratio is 4 : 1, so Zi Xuan's ending amount is 4 parts and one part is Ryan's ending amount.
5. That gives Zi Xuan's ending amount a second name, written in the same units.4 × (2 units − 25) = 8 units − 100
6. Set the two names side by side and take the units they share off both.3 units = 75
7. One unit is that shared out.75 ÷ 3 = 25
8. Zi Xuan started with 5 units.5 × 25 = 125

Zi Xuan had 125 badges at first.

The pattern. Both amounts moved by 25, so the gap between them never changed. Zi Xuan's amount in the end has two names: 5 units − 25 and 8 units − 100.