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Ratios before and after a change

Key idea: Before comparing two ratios, find what stays the same and give it the same number of units in both ratios.

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  • MOE teaching syllabus · Primary 6 Mathematics (taught from 2026)
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Check your understanding 1

Look for the amount that does not change

Only Ravi's stamps change. Whose amount stays the same?

Ravi and Siti collect stamps. At first, Ravi : Siti = 3 : 5. Ravi then buys more stamps. Siti does not buy or give away any stamps.

Which amount is the same before and after?
Show answerHide answer

Only Ravi’s amount changes, so Siti’s number of stamps is the same before and after. Ravi’s purchase makes the total larger, and it changes the ratio.

Find the amount that stays the same

At first, Ravi : Siti = 3 : 5. Ravi buys 28 stamps. Now Ravi : Siti = 7 : 5. How many stamps did each of them have at first?

Siti has 5 units in both ratios and her stamps do not change. So one unit has the same size before and after.

Siti’s five units do not change

Each full strip is one equal-sized whole.

Each row uses the same seven-unit scale. Ravi has three hatched units at first. The four dotted units are the 28 stamps he buys. Siti has five hatched units before and after.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Ravi at firstmarked; marked; marked; empty; empty; empty; empty
Ravi nowmarked; marked; marked; added; added; added; added
Siti, before and aftermarked; marked; marked; marked; marked; empty; empty
Show workingHide answer
7-3 = 4 units; 4 units = 28 stamps; 1 unit = 28÷4 = 7 stamps.

Ravi had 3 × 7 = 21 stamps and Siti has 5 × 7 = 35 stamps. Check: Ravi now has 21 + 28 = 49 stamps, and 49:35 = 7:5.

Can you use the same unit count for the unchanged blue beads?

Match the units for the unchanged amount

At first, red beads : blue beads = 5 : 4. Mei adds 12 red beads. The blue beads do not change. Now red : blue = 3 : 2.

Red seems to fall from 5 units to 3 units, even though Mei added red beads. The units in the two ratios have different sizes. Rewrite the new ratio so that blue has 4 units again.

Which ratio is equal to 3 : 2 and gives blue 4 units?

Blue has four units in both ratios

Each full strip is one equal-sized whole.

Each row uses the same six-unit scale. Red has five hatched units at first and one dotted unit added. Blue has four hatched units before and after.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Red at firstmarked; marked; marked; marked; marked; empty
Red nowmarked; marked; marked; marked; marked; added
Blue, before and aftermarked; marked; marked; marked; empty; empty
Show workingHide answer
before 5:4, after 3:2 = 6:4; 6-5 = 1 unit = 12 beads.

Red had 5 × 12 = 60 beads and blue has 4 × 12 = 48 beads. Check: 60 + 12 = 72 and 72:48 = 3:2. Comparing 5 units with 3 units would compare units of different sizes.

Does moving money between two people change their total?

Move some from one group to the other

At first, Amir : Bala = 7 : 3 in money. Amir gives Bala $24. Then they have the same amount of money. Nothing is spent or earned, so the total stays the same: 10 units. “The same amount” is 1 : 1. With a total of 10 units, that is 5 : 5.

Ten units before and after

Each full strip is one equal-sized whole.

Each row uses the same seven-unit scale. Amir starts with seven hatched units and gives two crossed units to Bala. Bala starts with three hatched units and receives two dotted units.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Amir at firstmarked; marked; marked; marked; marked; marked; marked
Amir after givingmarked; marked; marked; marked; marked; removed; removed
Bala at firstmarked; marked; marked; empty; empty; empty; empty
Bala after receivingmarked; marked; marked; added; added; empty; empty
Show workingHide answer
1:1 = 5:5 (10 units, like 7:3); 7-5 = 2 units = $24; 1 unit = $12.

Amir had 7 ×12=$84and Bala had3\times$12=$36. Check: after the gift, Amir has$84-$24=$60and Bala has$36+$24=$60$.

Check your understanding 2

Make the totals match

At first, the marbles in Jar A : Jar B = 5 : 3. Lina moves 45 marbles from Jar A to Jar B. Now Jar A : Jar B = 1 : 3. No marbles are added or taken away.

What stays the same when marbles move from Jar A to Jar B?
Show answerHide answer

5:3 has 8 units and 1:3 has 4 units. The total does not change, so write 1:3 = 2:6. Jar A changes from 5 units to 2 units: 3 units are 45 marbles, so 1 unit is 15 marbles. Jar A had 5 × 15 = 75 marbles and the total is 8 × 15 = 120 marbles. Check: Jar A now has 75-45 = 30 and Jar B has 45 + 45 = 90; 30:90 = 1:3.

Add the same number to both boxes. What happens to their difference?

Add the same amount to both groups

At first, the pens in Box A : Box B = 3 : 1. Then 15 pens are put into each box. Now Box A : Box B = 2 : 1.

Both boxes gain the same number of pens, so the difference between them stays the same. In 3 : 1 the difference is 2 units. In 2 : 1 it is only 1 unit, so those units are larger.

Which ratio is equal to 2 : 1 and has a difference of 2 units?

Each box gains one unit

Each full strip is one equal-sized whole.

Each row uses the same four-unit scale. Box A has three hatched units and Box B has one. Each box gains one dotted unit of 15 pens, so the difference stays at two units.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Box A at firstmarked; marked; marked; empty
Box A nowmarked; marked; marked; added
Box B at firstmarked; empty; empty; empty
Box B nowmarked; added; empty; empty
Show workingHide answer
2:1 = 4:2 (difference 2 units, like 3:1); 4-3 = 1 unit = 15 pens.

Box A had 3 × 15 = 45 pens and Box B had 15 pens. Check: now Box A has 60 pens and Box B has 30 pens; 60:30 = 2:1. Adding the same number to both amounts kept their difference of 30 pens. It did not keep the ratio 3 : 1.

Which amount stays the same as both people get older?

Choose a method that fits

The ratio of Aunt Mei’s age to her niece’s age is 4 : 1. In 12 years, the ratio will be 2 : 1. How old is each of them now?

Both of them grow 12 years older, so the difference in their ages never changes. Choose a way to work it out. Each method gives the same answer.

Use units

Quick check

Match the difference in units

The age difference stays three units

Each full strip is one equal-sized whole.

Each row uses the same six-unit scale. Aunt Mei has four hatched units now and her niece has one. Each gains two dotted units in 12 years, so the difference stays three units.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Aunt nowmarked; marked; marked; marked; empty; empty
Aunt in 12 yearsmarked; marked; marked; marked; added; added
Niece nowmarked; empty; empty; empty; empty; empty
Niece in 12 yearsmarked; added; added; empty; empty; empty

Now the difference is 4-1 = 3 units. In 2:1 it is 1 unit, so write 2:1 = 6:3, also a difference of 3 units. The niece goes from 1 unit to 3 units: 2 units = 12 years, so 1 unit = 6 years.

Use a table

Quick check

List ages that keep 4 : 1

List ages that are in the ratio 4 : 1 now. Add 12 years to both and test whether the aunt is then twice as old.

Niece nowAunt nowNiece in 12 yearsAunt in 12 yearsTwice the niece’s age then
312152430
416162832
520173234
624183636

Only the last row makes the aunt’s age in 12 years equal to twice the niece’s age then. The gap between the last two columns falls by 2 each row. Listing in order helps you find the answer.

Use a letter

Quick check

Let a letter stand for one age

Let the niece be n years old now. Aunt Mei is 4n years old, so the aunt is 4n-n = 3n years older. This difference never changes.

In 12 years the niece is n + 12 years old. The aunt will be twice as old, so she will be older by exactly the niece’s age then:

3n = n + 12; 2n = 12; n = 6.

Which items stay in the box without changing?

Check your understanding 3

Decide what stays the same, then solve

A box holds pens and pencils in the ratio 3 : 5. Mr Tan removes 12 pencils. The pens stay in the box. Now pens : pencils = 2 : 3.

Which amount is the same before and after?
Show answerHide answer

Pens do not change. Give them 6 units in both ratios: 3:5 = 6:10 and 2:3 = 6:9. Pencils fall from 10 units to 9 units, so 1 unit is 12. There are 6 × 12 = 72 pens and there were 10 × 12 = 120 pencils. Check: 120-12 = 108 pencils remain, and 72:108 = 2:3.

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