 # Apply Ratios Involving Changes

In this worksheet, students solve ratio problems involving changing amount by using algebra. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Ratio, Proportion and Rates of Change

Curriculum subtopic:   Ratio, Proportion and Rates of Change, Calculations with Ratio

Difficulty level:   ### QUESTION 1 of 10

A lot of the time, when you are dealing with ratio at the higher levels of the syllabus, you have to be able to deal with amount within a ratio changing (which will change the ratio).

Example :

James and John had marbles in the ratio 7:3

John gives James 3 marbles

They now have marbles in the ratio 5:2

How many marbles did john have originally?

We know that the marbles were in the ratio 7:3 but we don't know how many marbles there wer in total so we write the ratio as

7x : 3x

John gives James 3 marbles so the ratio is now

7x + 3 : 3x - 3

We also know that the marbles are now in the ratio 5:2 so we can say that...

7x + 3 : 3x - 3 = 5 : 2

If two ratios are equal, we can say that the first part of one divided by the first part of the other will be equal to the second part of one divide by the second part of the other.

This gives the equation...

 7x + 3 5
=
 3x - 3 2

We can multiply through to get...

2(7x + 3) = 5 (3x - 3)

14x + 6 = 15x - 15

x = 21

We know that John has 7x marbles originally, so he must have started with 147 (7 x 21) marbles,

A ratio of 1:1 means that we have...

When we are dealing with changing amount in a ratio question, we often have to...

Red and blue balls are in a bag in the ratio of 1:4.

If 5 red balls are added, the ratio changes to 1:3.

How many blue balls are in the bag?

Derek and Doris's ages are in the ratio 3:4

In 7 years, their ages will be in the ratio 4:5

How old are they now?

 Age now Derek Doris

Rick and Rob own marbles in the ratio 5:6. Rick wins 2 more.

The ratio of marbles is now 7:8

How many marbles do they have now?

 Age now Derek Doris

Bag A and Bag B contain counters in the ratio 7:3

3 counters are moved from Bag A to Bag B.

The ratio is now 5:3

How many balls are now in bag A?

Box A and box B have pens in the ratio 7:8.

I move 8 pens from box A to box B, the ratio is now 4: 5

How many pens are now in box A?

I only own red and white socks. In my sock drawer, I have red socks to white socks in the ratio 9:5

If I remove 6 pairs of red socks, this ratio changes to 3:2.

How many pairs of socks do I own?

2 mobile phones have prices in the ratio of 1:b

If I increase the price of both phones by 20, the ratio becomes 5:2

If I decrease the prices by £5, the ratio becomes 5:1

What is the ratio of a:b in its simplest form.

a : b

25 : 6

6 : 25

31 : 25

I start with two bags that contain the same amount of marbles.

It I add 2 marbles to bag A and 4 to bag B, the ratio will be the same as if I added 6 to bag A and 9 to bag B.

How many marbles were in either bag at the start.

• Question 1

A ratio of 1:1 means that we have...

EDDIE SAYS
Where the numbers in a ratio are the same (1:1, 2:2, 3:3 etc) this means that the same amount of each are present.
• Question 2

When we are dealing with changing amount in a ratio question, we often have to...

EDDIE SAYS
Whenever you see something that you don't know that is combined with another topic, your first thought should be. 'Can I create an equation to solve this?'
• Question 3

Red and blue balls are in a bag in the ratio of 1:4.

If 5 red balls are added, the ratio changes to 1:3.

How many blue balls are in the bag?

60
EDDIE SAYS

The original ratio is 1:4

This can be written as 1x:4x

Adding 5 red balls changes this to 1x + 5: 4x

The new ratio is 1:3

This gives 1x + 5: 4x = 1:3

 x + 3 1
=
 4x 3

3x + 15 = 4x

15 = x

Blue balls are represented by 4x

• Question 4

Derek and Doris's ages are in the ratio 3:4

In 7 years, their ages will be in the ratio 4:5

How old are they now?

 Age now Derek Doris
EDDIE SAYS

The original ratio is 3:4

This can be written as 3x:4x

In 7 years this will be 3x + 7 : 4x + 7

The new ratio is 4 : 5

This gives 3x + 7 : 4x + 7 = 4:5

 3x + 7 4
=
 4x + 7 5

15x + 35 = 16x + 28

7 = x

• Question 5

Rick and Rob own marbles in the ratio 5:6. Rick wins 2 more.

The ratio of marbles is now 7:8

How many marbles do they have now?

 Age now Derek Doris
EDDIE SAYS

The original ratio is 5 : 6

This can be written as 5x : 6x

In 7 years this will be 5x + 2 : 6x

The new ratio is 7 : 8

This gives 5x + 2 : 6x = 7:8

 5x + 2 7
=
 6x 8

40x + 16 = 42x

2x = 16

x = 8

• Question 6

Bag A and Bag B contain counters in the ratio 7:3

3 counters are moved from Bag A to Bag B.

The ratio is now 5:3

How many balls are now in bag A?

25
EDDIE SAYS
Creating a ratio with the changes gives,,, This gives 7n - 3: 3n + 3 = 5:3 This can be used to create the equation 21n - 9 = 15n + 15 6n = 24 n = 4
• Question 7

Box A and box B have pens in the ratio 7:8.

I move 8 pens from box A to box B, the ratio is now 4: 5

How many pens are now in box A?

160
EDDIE SAYS
Creating a ratio with the changes gives,,, This gives 7x - 8: 8x + 8 = 4 : 5 This can be used to create the equation 35x - 40 = 32x + 32 3x = 72 x = 24
• Question 8

I only own red and white socks. In my sock drawer, I have red socks to white socks in the ratio 9:5

If I remove 6 pairs of red socks, this ratio changes to 3:2.

How many pairs of socks do I own?

56
EDDIE SAYS
Creating a ratio with the changes gives,,, This gives 9x - 6:5x = 3:2 This can be used to create the equation 18x - 12 = 15x 3x = 12 x = 4
• Question 9

2 mobile phones have prices in the ratio of 1:b

If I increase the price of both phones by 20, the ratio becomes 5:2

If I decrease the prices by £5, the ratio becomes 5:1

What is the ratio of a:b in its simplest form.

25 : 6
EDDIE SAYS
This question should be screaming one thing at you. There are two things we need to find out. This can only be done by using simultaneous equations If we take the adding 20 situation and apply what we have done before, we get the equation 2a - 5b = 80 If we take the subtracting 5 situation and apply it, we get the equation a - 5b = -20 We can then solve these simultaneously to get a = 100 and b = 24 All we have to do is make these into a ratio and cancel
• Question 10

I start with two bags that contain the same amount of marbles.

It I add 2 marbles to bag A and 4 to bag B, the ratio will be the same as if I added 6 to bag A and 9 to bag B.

How many marbles were in either bag at the start.

6
EDDIE SAYS
The hardest part of this is setting up the equation We start with the same amount in each so the ratio is 1:1 which we can write as n : n It I add 2 marbles to bag A and 4 to bag B n+2 : n + 4 If I add 6 to bag A and 9 to bag B. n+6 : n + 9 We know these ratios will be the same so n+2 : n + 4 = n+6 : n + 9 Create an equation as before and solve for n.
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