Problems Where a Ratio Changes

Solving problems that give a ratio before and a ratio after: find what does not change, make it the same number of units in both ratios, then find 1 unit.

Before you start

In a bar model, a ratio of $3 : 5$ is a bar of 3 units over a bar of 5 units, and every unit is the same size. If the 3 units stand for 45 stickers, then 1 unit $= 45 \div 3 = 15$ stickers and the 5 units $= 5 \times 15 = 75$ stickers. In this lesson a "unit" is always one equal part of a bar.

Multiplying every term of a ratio by the same number gives an equivalent ratio: $2 : 3 = 8 : 12$ (every term $\times\ 4$ ). The bars are cut into more units, each one smaller. The quantities themselves do not change.

Two ratios that share a quantity are made to agree with the lowest common multiple (LCM). If $a : b = 2 : 3$ and $b : c = 4 : 5$ , then $b$ is 3 units in one ratio and 4 units in the other. The LCM of 3 and 4 is 12, so write $a : b = 8 : 12$ and $b : c = 12 : 15$ . Now $b$ is 12 units in both.

What Stays the Same?

In these problems you are given a ratio before something happens and another ratio after it. The two ratios cannot be used together as they stand, because 1 unit of the first ratio and 1 unit of the second are usually different sizes. The way in is to look for something that is the same before and after.

Read what happened, and ask what it leaves alone. There are three kinds of problem.

What happened	What stays the same	Where to find it in each ratio
Something is added to one quantity only, or taken from one quantity only	The other quantity	The term of the quantity that was left alone
Some is passed from one quantity to the other	The total	The sum of the terms
The same amount is added to both quantities, or taken from both (as when two people grow older by the same number of years)	The difference	The larger term minus the smaller term

Whatever stays the same is one amount, before and after. Once it is written as the same number of units in both ratios, 1 unit is worth the same in both. Only then can the units of the two ratios be compared, added and subtracted.

Rule: make what does not change the same number of units

Decide what does not change: one quantity, the total or the difference.
Write down its number of units in the "before" ratio and in the "after" ratio.
If the two numbers are different, find their LCM. Write each ratio as an equivalent ratio in which the thing that does not change is the LCM. Multiply every term of a ratio by the same number.
Every unit is now the same size. Match the amount you are told to its number of units, and find 1 unit.
Answer the question. Then check: the "before" amounts are in the "before" ratio, the "after" amounts are in the "after" ratio, and the change from one to the other is what the question says.

Do not say what 1 unit is worth until step 3 is done.

One Quantity Does Not Change

Something is added to one quantity, or taken from it, and the other quantity is left alone. The quantity left alone is the same amount before and after, so make its two terms the same with the LCM, just as when two ratios are combined. The change in the other quantity is then a whole number of units.

Example 1: More is added to one quantity

Siti and Mei Ling collect stickers. The ratio of the number of Siti's stickers to the number of Mei Ling's stickers was $2 : 3$ . Siti then bought 21 more stickers, and the ratio became $5 : 4$ . How many stickers did each girl have at first?

Solution

What does not change? Only Siti bought stickers. Mei Ling has the same number of stickers before and after.

Draw the two ratios as they are given. Mei Ling's bar is the same length both times, because her number of stickers did not change. Before, her bar is cut into 3 units. After, the same bar is cut into 4 units, so these units are smaller. The units of the two ratios are different sizes, and nothing can be worked out yet.

Make Mei Ling's terms the same. They are 3 and 4, and the LCM of 3 and 4 is 12. Multiply every term of $2 : 3$ by 4, and every term of $5 : 4$ by 3.

	Ratio given	Multiply every term by	Siti	Mei Ling
Before	$2 : 3$	4	8	12
After	$5 : 4$	3	15	12

Mei Ling is now 12 units before and 12 units after, so every unit is the same size. Draw the bars again with these units.

Siti went from 8 units to 15 units, so $15 - 8 = 7$ units were added. These 7 units are the 21 stickers she bought.

7 units $= 21$ stickers, so 1 unit $= 21 \div 7 = 3$ stickers.

At first Siti had 8 units $= 8 \times 3 = 24$ stickers and Mei Ling had 12 units $= 12 \times 3 = 36$ stickers.

Check. Before: $24 : 36 = 2 : 3$ (divide both terms by 12). After: Siti has $24 + 21 = 45$ stickers, and $45 : 36 = 5 : 4$ (divide both terms by 9). Both ratios are right.

At first Siti had 24 stickers and Mei Ling had 36 stickers.

The commonest mistake: mixing "before" units with "after" units

In Example 1 it is tempting to say: "Siti went from 2 units to 5 units, so 3 units are the 21 stickers, and 1 unit is 7 stickers." This is wrong, because the 2 and the 5 count units of different sizes.

Look at Mei Ling's 36 stickers. In $2 : 3$ they are 3 units, so 1 unit is $36 \div 3 = 12$ stickers. In $5 : 4$ the same 36 stickers are 4 units, so 1 unit is $36 \div 4 = 9$ stickers. Taking 2 units of one size from 5 units of another size is like comparing 2 ten-dollar notes with 5 two-dollar notes and saying that the second pile has 3 more: it has 3 more notes, but less money. The two numbers count things of different sizes.

The check shows that the wrong working fails. It gives Siti $2 \times 7 = 14$ stickers and Mei Ling $3 \times 7 = 21$ stickers at first. After buying 21 more, Siti would have $14 + 21 = 35$ stickers, and $35 : 21 = 5 : 3$ , not $5 : 4$ .

Compare, add or subtract numbers of units only after the two ratios have been made to agree. Until then, do not say what 1 unit is worth.

Example 2: Some is taken from one quantity

At Mr Tan's fruit stall, the ratio of the number of oranges to the number of apples was $4 : 5$ . Mr Tan sold 20 apples and no oranges, and the ratio became $6 : 5$ . How many oranges and apples were left at the stall altogether?

Solution

What does not change? Only apples were sold. The number of oranges is the same before and after.

The apples' term is 5 in both of the ratios given, and yet 20 apples were sold: 5 units before, 5 smaller units after. Decide what does not change from the story, never from the numbers in the ratios.

Make the oranges' terms the same. They are 4 and 6, and the LCM of 4 and 6 is 12. Multiply every term of $4 : 5$ by 3, and every term of $6 : 5$ by 2.

	Ratio given	Multiply every term by	Oranges	Apples
Before	$4 : 5$	3	12	15
After	$6 : 5$	2	12	10

The oranges are now 12 units before and 12 units after, so every unit is the same size.

The apples went from 15 units to 10 units, so $15 - 10 = 5$ units were sold. These 5 units are the 20 apples.

5 units $= 20$ apples, so 1 unit $= 20 \div 5 = 4$ fruits.

Left at the stall: $12 + 10 = 22$ units $= 22 \times 4 = 88$ fruits.

Check. Before: $12 \times 4 = 48$ oranges and $15 \times 4 = 60$ apples, and $48 : 60 = 4 : 5$ (divide both terms by 12). After: $60 - 20 = 40$ apples, and $48 : 40 = 6 : 5$ (divide both terms by 8). Altogether $48 + 40 = 88$ .

There were 88 oranges and apples left at the stall altogether.

The Total Does Not Change

When some is passed from one quantity to the other, both quantities change: one loses exactly what the other gains. Nothing comes in from outside and nothing goes out, so the total is the same before and after. The total is the sum of the terms of a ratio: in $3 : 1$ it is $3 + 1 = 4$ units. Make the two totals the same with the LCM.

Example 3: One gives some to the other

Ali and Ravi collect game cards. The ratio of the number of Ali's cards to the number of Ravi's cards was $3 : 1$ . Ali gave 18 of his cards to Ravi, and the ratio became $3 : 2$ . How many cards did each boy have at first?

Solution

What does not change? Ali has 18 cards fewer and Ravi has 18 cards more, so both numbers changed. The total number of cards is the same before and after. (Ali's term is 3 in both of the ratios given, but he has fewer cards after: his 3 units after are smaller than his 3 units before.)

Make the totals the same. Before, the total is $3 + 1 = 4$ units. After, it is $3 + 2 = 5$ units. The LCM of 4 and 5 is 20. Multiply every term of $3 : 1$ by 5, and every term of $3 : 2$ by 4.

	Ratio given	Multiply every term by	Ali	Ravi	Total
Before	$3 : 1$	5	15	5	20
After	$3 : 2$	4	12	8	20

The total is now 20 units before and 20 units after, so every unit is the same size.

Ali went from 15 units to 12 units, so he gave away $15 - 12 = 3$ units. Ravi went from 5 units to 8 units, which is the same 3 units. These 3 units are the 18 cards.

3 units $= 18$ cards, so 1 unit $= 18 \div 3 = 6$ cards.

At first Ali had 15 units $= 15 \times 6 = 90$ cards and Ravi had 5 units $= 5 \times 6 = 30$ cards.

Check. Before: $90 : 30 = 3 : 1$ . After: Ali has $90 - 18 = 72$ cards and Ravi has $30 + 18 = 48$ cards, and $72 : 48 = 3 : 2$ (divide both terms by 24). The total is 120 cards both times.

At first Ali had 90 cards and Ravi had 30 cards.

Example 4: The amount passed is not known

A bookcase has two shelves. The ratio of the number of books on the top shelf to the number of books on the bottom shelf was $1 : 2$ . Mrs Goh moved some books from the bottom shelf to the top shelf, and the ratio became $5 : 3$ . There are now 45 books on the top shelf. How many books did Mrs Goh move?

Solution

What does not change? Books were moved from one shelf to the other, so the total number of books on the two shelves is the same before and after.

Make the totals the same. Before, the total is $1 + 2 = 3$ units. After, it is $5 + 3 = 8$ units. The LCM of 3 and 8 is 24. Multiply every term of $1 : 2$ by 8, and every term of $5 : 3$ by 3.

	Ratio given	Multiply every term by	Top shelf	Bottom shelf	Total
Before	$1 : 2$	8	8	16	24
After	$5 : 3$	3	15	9	24

The total is now 24 units before and 24 units after, so every unit is the same size.

This time the amount you are told is not the change. It is the 45 books that are now on the top shelf, and they are 15 units.

15 units $= 45$ books, so 1 unit $= 45 \div 15 = 3$ books.

The top shelf went from 8 units to 15 units, so $15 - 8 = 7$ units were moved: $7 \times 3 = 21$ books.

Check. Before: $8 \times 3 = 24$ books on the top shelf and $16 \times 3 = 48$ on the bottom shelf, and $24 : 48 = 1 : 2$ . After: $24 + 21 = 45$ books on the top shelf and $48 - 21 = 27$ on the bottom shelf, and $45 : 27 = 5 : 3$ (divide both terms by 9).

Mrs Goh moved 21 books.

The Difference Does Not Change

When the same amount is added to both quantities, or taken from both, the gap between them stays as it was. If Hui Min has $10 more than Ali and each of them is given $5, Hui Min still has $10 more than Ali. So the difference is the same before and after, although both quantities change and their total changes too. Ages are the best-known case: two people grow older by the same number of years, so the difference between their ages never changes. The difference is the larger term of a ratio minus the smaller term: in $2 : 7$ it is $7 - 2 = 5$ units. Make the two differences the same with the LCM.

Example 5: Ages

The ratio of Mei Ling's age to her father's age is $2 : 7$ now. In 3 years' time the ratio will be $1 : 3$ . How old is each of them now?

Solution

What does not change? In 3 years' time each of them will be 3 years older, so both ages change and so does their total. The difference between their ages does not change: Mei Ling's father is always the same number of years older than she is.

Make the differences the same. Now, the difference is $7 - 2 = 5$ units. In 3 years' time it will be $3 - 1 = 2$ units. The LCM of 5 and 2 is 10. Multiply every term of $2 : 7$ by 2, and every term of $1 : 3$ by 5.

	Ratio given	Multiply every term by	Mei Ling	Father	Difference
Now	$2 : 7$	2	4	14	10
In 3 years' time	$1 : 3$	5	5	15	10

The difference is now 10 units both times, so every unit is the same size.

Mei Ling goes from 4 units to 5 units, and her father from 14 units to 15 units. Each has 1 unit more, and this 1 unit is the 3 years.

1 unit $= 3$ years.

Now Mei Ling is 4 units $= 4 \times 3 = 12$ years old and her father is 14 units $= 14 \times 3 = 42$ years old.

Check. Now: $12 : 42 = 2 : 7$ (divide both terms by 6). In 3 years' time they will be 15 and 45, and $15 : 45 = 1 : 3$ (divide both terms by 15). The difference is 30 years both times.

Mei Ling is 12 years old and her father is 42 years old.

Example 6: The same amount is taken from both

Hui Min and Siti had savings in the ratio $7 : 4$ . Each girl then spent $20 on a present for her mother, and the ratio of the money they had left was $3 : 1$ . How much money did each girl have left?

Solution

What does not change? Each girl spent the same amount, $20. Both amounts went down, and their total went down by $40. The difference between the two amounts did not change: Hui Min has as much more than Siti as she had before.

Make the differences the same. Before, the difference is $7 - 4 = 3$ units. After, it is $3 - 1 = 2$ units. The LCM of 3 and 2 is 6. Multiply every term of $7 : 4$ by 2, and every term of $3 : 1$ by 3.

	Ratio given	Multiply every term by	Hui Min	Siti	Difference
Before	$7 : 4$	2	14	8	6
After	$3 : 1$	3	9	3	6

The difference is now 6 units before and 6 units after, so every unit is the same size.

Hui Min went from 14 units to 9 units, and Siti from 8 units to 3 units. Each has 5 units fewer, and these 5 units are the $20 she spent.

5 units = $20, so 1 unit = $4 (because $20 \div 5 = 4$ ).

Hui Min had 9 units left, and $9 \times 4 = 36$ , so she had $36. Siti had 3 units left, and $3 \times 4 = 12$ , so she had $12.

Check. Before: Hui Min had $36 + $20 = $56 and Siti had $12 + $20 = $32, and $56 : 32 = 7 : 4$ (divide both terms by 8). After: $36 : 12 = 3 : 1$ . The difference is $24 both times.

Hui Min had $36 left and Siti had $12 left.

Common Mistakes

Mixing units of different sizes. From $2 : 3$ to $5 : 4$ is not "3 units more" for the first quantity. Subtract numbers of units only after the two ratios have been made to agree.
Judging by the numbers in the ratios. In Example 2 the apples' term is 5 in $4 : 5$ and 5 in $6 : 5$ , yet 20 apples were sold. What stays the same is decided by what happened in the story.
Keeping the wrong thing the same. When one gives some to the other, neither quantity is unchanged: the total is. When both gain or lose the same amount, the total changes: the difference does not.
Multiplying only one term. To make the 3 in $2 : 3$ into 12, multiply both terms by 4 to get $8 : 12$ . Writing $2 : 12$ changes the ratio.
Answering for the wrong time. Read whether the question asks for the amount "at first" or the amount "left" or "now", and use the units of that row.
Leaving out the check. Work out the amounts before and after, and see that each pair is in its ratio. A wrong choice of what stays the same nearly always shows up here.

Practice

In every question, decide first what does not change: one quantity, the total or the difference. Work each question on paper before you open its solution.

Question 1

For each change, say what stays the same: one of the quantities, the total or the difference.
(a) Ali gives 5 of his marbles to Ravi.
(b) A mother and her son are both 4 years older.
(c) Water is added to a drink made of syrup and water.
(d) Two sisters each spend $10 of their savings.
(e) Six students move from Class 1A to Class 1B.
(f) Hui Min buys more stickers and Siti buys none.

ShowHide solution

(a) Marbles pass from one boy to the other, so the total number of marbles does not change.

(b) Both ages go up by the same number of years, so the difference between their ages does not change.

(d) The same amount is taken from both, so the difference between their savings does not change.

(e) Students pass from one class to the other, so the total number of students in the two classes does not change.

(f) Only Hui Min's number changes, so Siti's number of stickers does not change.

(a) The total
(b) The difference
(c) One quantity: the syrup
(d) The difference
(e) The total
(f) One quantity: Siti's stickers

Question 2

Write each pair of ratios as equivalent ratios that agree.
(a) Before $3 : 4$ , after $5 : 6$ . The first quantity does not change.
(b) Before $2 : 3$ , after $1 : 1$ . The total does not change.
(c) Before $5 : 2$ , after $3 : 1$ . The difference does not change.

ShowHide solution

(a) The first terms are 3 and 5, and the LCM of 3 and 5 is 15. $3 : 4 = 15 : 20$ (every term $\times\ 5$ ) and $5 : 6 = 15 : 18$ (every term $\times\ 3$ ).

(b) The totals are $2 + 3 = 5$ units and $1 + 1 = 2$ units, and the LCM of 5 and 2 is 10. $2 : 3 = 4 : 6$ (every term $\times\ 2$ ) and $1 : 1 = 5 : 5$ (every term $\times\ 5$ ). Both totals are 10 units.

(c) The differences are $5 - 2 = 3$ units and $3 - 1 = 2$ units, and the LCM of 3 and 2 is 6. $5 : 2 = 10 : 4$ (every term $\times\ 2$ ) and $3 : 1 = 9 : 3$ (every term $\times\ 3$ ). Both differences are 6 units.

(a) $15 : 20$ and $15 : 18$
(b) $4 : 6$ and $5 : 5$
(c) $10 : 4$ and $9 : 3$

Question 3

The ratio of the number of red beads to the number of blue beads in a box was $3 : 7$ . Hui Min put 12 more red beads into the box, and the ratio became $5 : 7$ . How many red beads are in the box now?

ShowHide solution

Only red beads were added, so the number of blue beads does not change.

The blue beads are 7 units in both ratios already, so the two ratios agree as they stand and nothing needs to be multiplied.

The red beads went from 3 units to 5 units, so $5 - 3 = 2$ units were added. 2 units $= 12$ beads, so 1 unit $= 12 \div 2 = 6$ beads.

Red beads now: 5 units $= 5 \times 6 = 30$ beads.

Check. Before: $3 \times 6 = 18$ red and $7 \times 6 = 42$ blue, and $18 : 42 = 3 : 7$ . After: $18 + 12 = 30$ red, and $30 : 42 = 5 : 7$ .

There are 30 red beads in the box now.

Question 4

The ratio of the volume of water in Tank A to the volume of water in Tank B was $5 : 1$ . After 20 litres of water was pumped from Tank A into Tank B, the ratio became $5 : 4$ . How much water was in each tank at first?

ShowHide solution

Water passed from one tank to the other, so the total volume of water does not change.

Before, the total is $5 + 1 = 6$ units. After, it is $5 + 4 = 9$ units. The LCM of 6 and 9 is 18.

	Ratio given	Multiply every term by	Tank A	Tank B	Total
Before	$5 : 1$	3	15	3	18
After	$5 : 4$	2	10	8	18

Tank A went from 15 units to 10 units, so $15 - 10 = 5$ units were pumped out. 5 units $= 20$ litres, so 1 unit $= 20 \div 5 = 4$ litres.

At first: Tank A had 15 units $= 15 \times 4 = 60$ litres and Tank B had 3 units $= 3 \times 4 = 12$ litres.

Check. Before: $60 : 12 = 5 : 1$ . After: $60 - 20 = 40$ litres and $12 + 20 = 32$ litres, and $40 : 32 = 5 : 4$ . The total is 72 litres both times.

At first Tank A had 60 litres of water and Tank B had 12 litres.

Question 5

The ratio of Mrs Goh's age to her daughter's age is $3 : 1$ now. Five years ago it was $4 : 1$ . How old is each of them now?

ShowHide solution

Five years ago each of them was 5 years younger, so the difference between their ages does not change.

Now, the difference is $3 - 1 = 2$ units. Five years ago it was $4 - 1 = 3$ units. The LCM of 2 and 3 is 6.

	Ratio given	Multiply every term by	Mrs Goh	Daughter	Difference
Five years ago	$4 : 1$	2	8	2	6
Now	$3 : 1$	3	9	3	6

Mrs Goh went from 8 units to 9 units, and her daughter from 2 units to 3 units. Each has 1 unit more, and this 1 unit is the 5 years. So 1 unit $= 5$ years.

Now: Mrs Goh is 9 units $= 9 \times 5 = 45$ years old and her daughter is 3 units $= 3 \times 5 = 15$ years old.

Check. Now: $45 : 15 = 3 : 1$ . Five years ago they were 40 and 10, and $40 : 10 = 4 : 1$ . The difference is 30 years both times.

Mrs Goh is 45 years old and her daughter is 15 years old.

Question 6

The ratio of Ravi's savings to Ali's savings was $5 : 4$ . After Ravi spent $28 on a pair of football boots, the ratio became $2 : 3$ . How much money did Ravi have left?

ShowHide solution

Only Ravi spent money, so Ali's savings do not change.

Ali's terms are 4 and 3, and the LCM of 4 and 3 is 12.

	Ratio given	Multiply every term by	Ravi	Ali
Before	$5 : 4$	3	15	12
After	$2 : 3$	4	8	12

Ravi went from 15 units to 8 units, so he spent $15 - 8 = 7$ units. 7 units = $28, so 1 unit = $4 (because $28 \div 7 = 4$ ).

Ravi had 8 units left, and $8 \times 4 = 32$ , so he had $32.

Check. Before: Ravi had $32 + $28 = $60, and Ali had 12 units, which is $48 (because $12 \times 4 = 48$ ). $60 : 48 = 5 : 4$ . After: $32 : 48 = 2 : 3$ .

Ravi had $32 left.

Question 7

Box A and Box B hold exercise books. The ratio of the number of books in Box A to the number of books in Box B was $2 : 3$ . Mr Tan moved 18 books from Box B to Box A, and the ratio became $4 : 3$ . How many books were in Box A at first?

ShowHide solution

Books passed from one box to the other, so the total number of books does not change.

Before, the total is $2 + 3 = 5$ units. After, it is $4 + 3 = 7$ units. The LCM of 5 and 7 is 35.

	Ratio given	Multiply every term by	Box A	Box B	Total
Before	$2 : 3$	7	14	21	35
After	$4 : 3$	5	20	15	35

Box A went from 14 units to 20 units, so $20 - 14 = 6$ units were moved into it. 6 units $= 18$ books, so 1 unit $= 18 \div 6 = 3$ books.

At first Box A had 14 units $= 14 \times 3 = 42$ books.

Check. Before: Box B had $21 \times 3 = 63$ books, and $42 : 63 = 2 : 3$ . After: $42 + 18 = 60$ books and $63 - 18 = 45$ books, and $60 : 45 = 4 : 3$ . The total is 105 books both times.

There were 42 books in Box A at first.

Question 8

Mei Ling and her brother Wei Jie had savings in the ratio $4 : 1$ . At Chinese New Year their grandmother gave each of them $35, and the ratio became $5 : 3$ . How much money did Mei Ling have at first?

ShowHide solution

Each of them was given the same amount, so the difference between their savings does not change.

Before, the difference is $4 - 1 = 3$ units. After, it is $5 - 3 = 2$ units. The LCM of 3 and 2 is 6.

	Ratio given	Multiply every term by	Mei Ling	Wei Jie	Difference
Before	$4 : 1$	2	8	2	6
After	$5 : 3$	3	15	9	6

Mei Ling went from 8 units to 15 units, and Wei Jie from 2 units to 9 units. Each has 7 units more, and these 7 units are the $35. 7 units = $35, so 1 unit = $5 (because $35 \div 7 = 5$ ).

Mei Ling had 8 units at first, and $8 \times 5 = 40$ , so she had $40.

Check. Before: Wei Jie had 2 units, which is $10 (because $2 \times 5 = 10$ ), and $40 : 10 = 4 : 1$ . After: $40 + $35 = $75 and $10 + $35 = $45, and $75 : 45 = 5 : 3$ . The difference is $30 both times.

Mei Ling had $40 at first.

Question 9

Mrs Goh mixed orange syrup and water in the ratio $3 : 8$ to make a drink. The drink was too sweet, so she added 250 ml of water, and the ratio of syrup to water became $2 : 7$ . How much drink is there now?

ShowHide solution

Only water was added, so the volume of syrup does not change.

The syrup's terms are 3 and 2, and the LCM of 3 and 2 is 6.

	Ratio given	Multiply every term by	Syrup	Water
Before	$3 : 8$	2	6	16
After	$2 : 7$	3	6	21

The water went from 16 units to 21 units, so $21 - 16 = 5$ units were added. 5 units $= 250$ ml, so 1 unit $= 250 \div 5 = 50$ ml.

The drink is now $6 + 21 = 27$ units $= 27 \times 50 = 1350$ ml.

Check. Before: $6 \times 50 = 300$ ml of syrup and $16 \times 50 = 800$ ml of water, and $300 : 800 = 3 : 8$ . After: $800 + 250 = 1050$ ml of water, and $300 : 1050 = 2 : 7$ (divide both terms by 150). Altogether $300 + 1050 = 1350$ ml.

There is 1350 ml of drink now.

Question 10

Siti is 12 years old and her mother is 40 years old. In how many years' time will the ratio of Siti's age to her mother's age be $1 : 3$ ?

ShowHide solution

Both grow older by the same number of years, so the difference between their ages does not change. It is always $40 - 12 = 28$ years.

In the ratio $1 : 3$ the difference is $3 - 1 = 2$ units. So 2 units $= 28$ years, and 1 unit $= 28 \div 2 = 14$ years.

Siti will then be 1 unit $= 14$ years old. She is 12 now, so this is in $14 - 12 = 2$ years' time.

Check. In 2 years' time Siti will be 14 and her mother will be 42, and $14 : 42 = 1 : 3$ .

The ratio will be $1 : 3$ in 2 years' time.

Question 11

Ali and Hui Min had some marbles in the ratio $7 : 3$ . Ali gave some of his marbles to Hui Min, and the ratio became $8 : 7$ . Ali now has 6 more marbles than Hui Min. How many marbles did Ali give to Hui Min?

ShowHide solution

Marbles passed from one to the other, so the total number of marbles does not change.

Before, the total is $7 + 3 = 10$ units. After, it is $8 + 7 = 15$ units. The LCM of 10 and 15 is 30.

	Ratio given	Multiply every term by	Ali	Hui Min	Total
Before	$7 : 3$	3	21	9	30
After	$8 : 7$	2	16	14	30

The amount you are told is how many more Ali has now: $16 - 14 = 2$ units. 2 units $= 6$ marbles, so 1 unit $= 6 \div 2 = 3$ marbles.

Ali went from 21 units to 16 units, so he gave away $21 - 16 = 5$ units $= 5 \times 3 = 15$ marbles.

Check. Before: $21 \times 3 = 63$ and $9 \times 3 = 27$ marbles, and $63 : 27 = 7 : 3$ . After: $63 - 15 = 48$ and $27 + 15 = 42$ marbles, and $48 : 42 = 8 : 7$ . Ali has $48 - 42 = 6$ more.

Ali gave 15 marbles to Hui Min.

Question 12

A tray at a hawker stall held chicken curry puffs and sardine curry puffs in the ratio $4 : 1$ . The stallholder took 12 chicken curry puffs off the tray and put 12 sardine curry puffs in their place. The ratio became $1 : 1$ . How many curry puffs were on the tray?

ShowHide solution

Twelve puffs came off the tray and twelve went on, so the total number of puffs does not change. It is as if 12 chicken puffs had been turned into sardine puffs.

Before, the total is $4 + 1 = 5$ units. After, it is $1 + 1 = 2$ units. The LCM of 5 and 2 is 10.

	Ratio given	Multiply every term by	Chicken	Sardine	Total
Before	$4 : 1$	2	8	2	10
After	$1 : 1$	5	5	5	10

The chicken puffs went from 8 units to 5 units, so $8 - 5 = 3$ units were taken off. 3 units $= 12$ puffs, so 1 unit $= 12 \div 3 = 4$ puffs.

On the tray: 10 units $= 10 \times 4 = 40$ puffs.

Check. Before: $8 \times 4 = 32$ chicken and $2 \times 4 = 8$ sardine, and $32 : 8 = 4 : 1$ . After: $32 - 12 = 20$ chicken and $8 + 12 = 20$ sardine, and $20 : 20 = 1 : 1$ .

There were 40 curry puffs on the tray.

Key Points

A unit of the "before" ratio and a unit of the "after" ratio are usually different sizes. Do not compare them until the two ratios agree.
First find what does not change, from what happened in the story.
Something is added to or taken from one quantity only: the other quantity does not change. Make its two terms the same.
Some is passed from one quantity to the other: the total does not change. Make the two sums of the terms the same.
The same amount is added to or taken from both, as with ages: the difference does not change. Make the two differences of the terms the same.
Use the LCM, and multiply every term of a ratio by the same number. Then match the amount you are told to its number of units and find 1 unit.
Check the amounts against both ratios.

When both quantities change, and neither the total nor the difference stays the same (as when one gains 5 and the other gains 3), there is nothing unchanged to make the same number of units, so the method of this lesson cannot be used. Such problems are usually solved with an equation, which is taught in the chapter on linear equations.