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[P6] Constant Part, Total, Difference

Example 1 — Constant Total Concept Nancy had $222 more than Shawn. After Nancy gave $30 to Shawn, Nancy had 4 times as much money as Shawn. How much money did Nancy have at first?

Simplify the problem: Nancy gave Shawn money. [Internal transfer]

What has changed: Nancy’s money decreased. Shawn’s money increased.

What remained unchanged: The total amount of money both of them had altogether.

How to start?

4 times

Draw a model.

Remember Shawn received $30, and Nancy had 4 times the amount he had.

And don’t forget the $30 Nancy gave away.

After

Nancy $30 $30 $30 $30 $30

Shawn $30

Before

Nancy $222

Shawn

3 units = $222 – 5 × $30 = $222 – $150 = $72 1 unit = $72 ÷ 3 = $24 Amount of money Nancy had at first = $24 + $222 = $246 (Ans)

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▲ Same But Different

Nancy had $222 less than Shawn. After Nancy gave $30 to Shawn, Shawn had 4 times as much money as Nancy. How much money did Nancy have at first?

Simplify the problem: Nancy gave Shawn money [Internal transfer]

What has changed: Nancy’s money decreased. Shawn’s money increased.

What remained unchanged: The total amount of money both of them had altogether.

After

Nancy

Shawn

Before

Nancy $30

Shawn $30

3 units = $30 + $222 + $30 = $282 1 units = $282 ÷ 3 = $94 Amount of money Nancy had at first = $94 + $30 = $124 (Ans)

$222

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Example 2 — Constant Part Concept Miko had some stamps. The ratio of Singapore stamps to Malaysia stamps was 4 : 7. After Miko bought more Singapore stamps, the ratio became 2 : 1. Miko had 420 Singapore stamps in the end. How many Singapore stamps did he buy? Unitary method 2 units = 420 1 unit = 420 ÷ 2 = 210 Number of Malaysia stamps = 210 7 parts = 210 1 part = 210 ÷ 7 = 30 4 parts = 4 × 30 = 120 Number of Singapore stamps at first = 120 Number of Singapore stamps he bought = 420 – 120 = 300 (Ans)

After Singapore : Malaysia

2 : 1

Before Singapore : Malaysia

4 : 7

Common mistake: Confusing parts and units. Before

Singapore part

Malaysia

After

Singapore

Malaysia unit

420

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Ratio method (Recommended)

Simplify the problem: Before S : M = 4 : 7 After S : M = 2 : 1

What has changed: The number of Singapore stamps increased.

What remained unchanged: The number of Malaysia stamps.

How to start?

Malaysia stamps remained unchanged.

Use equivalent ratios.

Make the units of Malaysia stamps the same.

Singapore : Malaysia

Before 4 : 7

After 2 × 7 : 1 × 7

14 : 7

14 units = 420 1 unit = 420 ÷ 14 = 30 10 units = 10 × 30 = 300 Number of Singapore stamps he bought = 300 (Ans)

Difference = 14 – 4 = 10 units

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▲ Same But Different

Miko had some stamps. The ratio of Singapore stamps to Malaysia stamps was 4 : 7. After Miko gave away some Singapore stamps, the ratio became 1 : 2. Miko had 420 Singapore stamps in the end. How many Singapore stamps did he give away? Ratio method

Singapore : Malaysia

Before 4 × 2 : 7 × 2

8 : 14

After 1 × 7 : 2 × 7

7 : 14

7 units = 420 1 unit = 420 ÷ 7 = 60 Number of Singapore stamps he gave away = 60 (Ans)

Difference = 8 – 7 = 1 unit

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Example 3 — Constant Difference Concept Understanding the concept

Number of carrots = 5 Number of potatoes = 3

Difference = 2

Add the same number of carrots and potatoes. Add 3 carrots. Add 3 potatoes.

Number of carrots = 8 Number of potatoes = 6

Difference = 2

When the same value is added or taken away, the difference remains unchanged. Activity

Year 2020

My age: 12 years old Jason: 18 years old

Difference = 6 years

Year 2016

My age: 8 years old Jason: 14 years old

Difference = 6 years

Year 2025

My age: 17 years old Jason: 23 years old

Difference = 6 years

When years passed, everyone adds the same number of years to their age. Therefore, the difference in age between two people remains unchanged.

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Mei had $14 and Charles had $2 at first. After receiving an equal amount of money from their father, the ratio of Mei’s money to Charles’ money became 3 : 1. How much money did each of them receive from their father? Model method

Mei

from father

Charles

2 units = $14 – $2 = $12 1 unit = $12 ÷ 2 = $6 Amount of money each of them received = $6 – $2 = $4 (Ans)

$2

$14

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Ratio method (Recommended)

Simplify the problem: Before M : C = 14 : 2 = 7 : 1 ‼️ [Difference = 6 units] After M : C = 3 : 1 ‼️ [Difference = 2 units]

What has changed: Both of them received the same amount of money.

What remained unchanged: The difference between the amount of money both of them had.

How to start?

‼️ Difference remained unchanged.

Use equivalent ratios.

Make the units of difference the same.

Mei : Charles Difference

Before 7 : 1 6

After 3 × 3 : 1 × 3 2 × 3

9 : 3 6

7 units = $14 1 unit = $14 ÷ 7 = $2 2 units = 2 × $2 = $4 Amount of money each of them received = $4 (Ans)

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[P6] Constant Part, Total, Difference

1. The ratio of Max’s age to Leon’s age is 2 : 5. In 4 years’ time, the ratio will become 1 : 2. How old is Max now?

Ans: _______________

2. Alice had as many apples as Stan. After Stan gave Alice of his apples, Alice had

160 more apples than Stan. How many apples did they have altogether? [Constant total]

Draw a before-after model to visualise. After that, try this question again using the ratio method.

Ans: _______________

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3. The number of boys at a party was of the number of girls. 12 girls joined the party

and the number of girls became twice the number of boys. How many children were at the party in the end?

Ans: _______________ 4. Joanne baked some buns. Of the buns she baked, 60% were raisin buns and the

rest were cranberry buns. After baking another 40 raisin buns and 40 cranberry buns, 45% of all the buns were cranberry buns in the end. How many buns did she bake at first?

Express the percentages into ratio in the simplest form.

Ans: _______________

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5. Timothy had 168 toy blocks and Shannon had 3588 toy blocks. After buying an equal number of toy blocks, Shannon had 7 times as many toy blocks as Timothy. How many toy blocks did Timothy buy?

Ans: _______________ 6a. Hanna had 5 times as many stickers as Ashley. After Hanna gave 84 of her stamps

to Ashley, they had an equal number of stamps. How many stamps did Hanna have at first?

Ans: _______________

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▲ Same But Different

6b. Hanna had 5 times as many stickers as Ashley. After Hanna gave 84 of her stamps to Ashley, the ratio of the number of stickers Hanna had to the number Ashley had became 1 : 3. How many stamps did Hanna have at first?

Ans: _______________

7. Mary’s current age is of Darwin’s age. In 10 years’ time, the ratio of Mary’s age to

Darwin’s age is 11 : 19. How old is Mary now?

Ans: _______________

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8. The number of muffins in Bakery A was of the number of muffins in Bakery B. 14

muffins were transferred from Bakery B to Bakery A. In the end, Bakery A had of

the number of muffins in Bakery B. How many muffins were there in Bakery A at first?

Ans: _______________

9. There is an equal number of lollipops in two boxes. The ratio of the number of strawberry lollipops to grape lollipops in the first box was 1 : 2 and 5 : 1 in the second box. What fraction of all the lollipops were grape lollipops?

Ans: _______________

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10a. The ratio of the number of $2 notes to $5 notes Willy had was 5 : 3. After receiving $90 worth of $5 notes, the ratio became 2 : 3. How much money did Willy have at first altogether?

Ans: _______________

▲ Same But Different

10b. The ratio of the value of $2 notes to $5 notes Willy had was 5 : 3. After receiving $90 worth of $5 notes, the ratio became 2 : 3. How many dollar notes did Willy have at first altogether?

Ans: _______________