Apply Ratios Involving Changes

In this worksheet, students will solve ratio problems involving changing scenarios 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

When we are working with ratio problems, it is common (particularly in more challenging questions) to need to use changing ratios.

A ratio will change when the scenario on which it has been built changes, e.g. items being removed from a bag, food being eaten, etc.

Let's look at a changing ratio in action now in an example.

e.g. James and John have a group of 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 originally the marbles are distributed in the ratio 7:3, but we don't know how many marbles there are so we write this ratio as:

7x : 3x

We know that John gives James 3 marbles, so that our ratio now becomes:

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 these 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, divided by the second part of the other:

 7x + 3 5
=
 3x - 3 2

Using the regular laws of algebra, we can multiply both sides as opposites to reach:

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

14x + 6 = 15x - 15

x = 21

Returning to our initial statements, we know that John had 7x marbles originally.

If we substitute '21' for x in this equation, we reach:

7 × 21 = 147 marbles

In this activity, we will apply the algebraic method shown above to changing situations to find starting values, unknown variables or the changing ratio.

Type a word into the space to complete the sentence below.

Type two words into the spaces to complete the sentence below.

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 can be related in the ratio 3:4.

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

How old are Derek and Doris now?

 Current Age: Derek Doris

Rick and Rob own a number of marbles in the ratio 5:6.

After a game of marbles, Rick wins 2 more.

The ratio of marbles which Rick and Rob now own is 7:8.

 Total number of marbles: Rick Rob

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

3 counters are moved from Bag A and put into Bag B.

The ratio of counters in the bags is now 5:3.

How many counters are now in bag A?

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

I move 8 pens from Box A into Box B, and the ratio is now 4:5.

How many pens are now in Box A?

A girl only owns red and white socks.

In her sock drawer, she has pairs of red socks to pairs of white socks in the ratio 9:5.

She throws away 6 pairs of red socks as they are worn out, and this ratio changes to 3:2 as a result.

How many pairs of socks of each colour socks does she now own?

 Total of pairs: Red White

2 mobile phones have prices in the ratio of a:b.

If the price of both phones increases by £20, the ratio becomes 5:2.

If the prices of both phones decreases by £5, the ratio becomes 5:1.

Which ratio expresses the relationship between the two phones in its simplest possible form?

a : b

25 : 6

6 : 25

31 : 25

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

If 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

Type a word into the space to complete the sentence below.

EDDIE SAYS
Where the numbers in a ratio are the same (e.g. 1:1, 2:2, 3:3, etc.) this means that exactly the same amount of each variable are present. Remember this key fact to support you in the rest of this activity.
• Question 2

Type two words into the spaces to complete the sentence below.

EDDIE SAYS
Whenever we see a challenging maths question which is tricky to express in words, our first thought should be: "Can I create an equation to solve this?" If the answer is yes, then you are on to a winner! Remember than an equation must have an equals sign (=), which is the key feature which allows it to be solved.
• 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
Let's follow our process from the Introduction. The original ratio is 1:4, which can be written as: 1x : 4x Adding 5 red balls changes this ratio to: 1x + 5 : 4x The new ratio, after the balls have been added, is 1:3, which gives us: 1x + 5 : 4x = 1 : 3 As these two ratios are equal, we can say that:
 1x + 5 1
=
 4x 3
We can now multiply both sides as opposites to reach: 3(1x + 5) = 1(4x) Let's solve this equation to find x: 3x + 15 = 4x 15 = 4x - 3x 15 = x So if x = 15 and we know that the original total of blue balls is represented by 4x: 4 × 15 = 60 So there were 60 blue balls originally.
• Question 4

Derek and Doris's ages can be related in the ratio 3:4.

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

How old are Derek and Doris now?

 Current Age: Derek Doris
EDDIE SAYS
Let's follow our process again. The original ratio is 3:4, which can be written as: 3x : 4x Adding 7 years changes this ratio to: 3x + 7 : 4x + 7 The new ratio, in 7 years time, is 4:5, which gives us: 3x + 7 : 4x + 7 = 4 : 5 As these two ratios are equal, we can say that:
 3x + 7 4
=
 4x + 7 5
We can now multiply both sides as opposites to reach: 5(3x + 7) = 4(4x + 7) Let's solve this equation to find x: 15x + 35 = 16x + 28 35 - 28 = 16x - 15x 7 = x So if x = 7 and we know that our original ratio was 3x : 4x then: Derek = 3x = 3 × 7 = 21 years Doris = 4x = 4 × 7 = 28 years
• Question 5

Rick and Rob own a number of marbles in the ratio 5:6.

After a game of marbles, Rick wins 2 more.

The ratio of marbles which Rick and Rob now own is 7:8.

 Total number of marbles: Rick Rob
EDDIE SAYS
The original ratio here is 5:6, which can be written as: 5x : 6x Rick winning 2 marbles changes this ratio to: 5x + 2 : 6x - 2 The new ratio, after the marbles have been won, is 7:8, which gives us: 5x + 2 : 6x - 2 = 7 : 8 As these two ratios are equal, we can say that:
 5x + 2 7
=
 6x - 2 8
We can now multiply both sides as opposites to reach: 8(5x + 2) = 7(6x - 2) Let's solve this equation to find x: 40x + 16 = 42x - 14 16 + 14 = 42x - 40x 30 = 2x --> x = 15 So if x = 15 and we know that the original ratio was 5x : 6x then: Rick = 5x = 5 × 15 = 75 Rob = 6x = 6 × 15 = 90
• Question 6

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

3 counters are moved from Bag A and put into Bag B.

The ratio of counters in the bags is now 5:3.

How many counters are now in bag A?

25
EDDIE SAYS
The original ratio here is 7:3, which can be written as: 7x : 3x 3 counters being removed from Bag A and added to Bag B changes this ratio to: 7x - 3 : 3x + 3 The new ratio, after this change, is 5:3, which gives us: 7x - 3 : 3x + 3 = 5 : 3 As these two ratios are equal, we can say that:
 7x - 3 5
=
 3x + 3 3
We can now multiply both sides as opposites to reach: 3(7x - 3) = 5(3x + 3) Let's solve this equation to find x: 21x - 9 = 15x + 15 21x - 15x = 15 + 9 6x = 24 --> x = 4 So if x = 4 and we know that the new total in Bag A is represented by 5x then: Bag A = 5 × 4 = 20 counters Did you spot that we were asked for the new total of counters in Bag A, not the starting total?
• Question 7

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

I move 8 pens from Box A into Box B, and the ratio is now 4:5.

How many pens are now in Box A?

160
EDDIE SAYS
Original ratio: 7x : 8x This changes to: 7x - 8 : 8x + 8 New ratio is 4:5, which gives us: 7x - 8 : 8x + 8 = 4 : 5
 7x - 8 4
=
 8x + 8 5
5(7x - 8) = 4(8x + 8) 35x - 40 = 32x + 32 35x - 32x = 32 + 40 3x = 72 --> x = 24 If x = 24 and the new total in Box A is represented by 7x - 8 then: Box A = (7 × 24) - 8 = 160 pens
• Question 8

A girl only owns red and white socks.

In her sock drawer, she has pairs of red socks to pairs of white socks in the ratio 9:5.

She throws away 6 pairs of red socks as they are worn out, and this ratio changes to 3:2 as a result.

How many pairs of socks of each colour socks does she now own?

 Total of pairs: Red White
EDDIE SAYS
Original ratio: 9x : 5x This changes to: 9x - 6 : 5x New ratio is 3:2, which gives us: 9x - 6 : 5x = 3 : 2
 9x - 6 3
=
 5x 2
2(9x - 6) = 3(5x) 18x - 12 = 15x 18x - 15x = 0 + 12 3x = 12 --> x = 4 If x = 4 and the new ratio is 9x - 6 : 5x then: Red = (9 × 4) - 6 = 30 pairs of socks Blue = 5 × 4 = 20 pairs of socks
• Question 9

2 mobile phones have prices in the ratio of a:b.

If the price of both phones increases by £20, the ratio becomes 5:2.

If the prices of both phones decreases by £5, the ratio becomes 5:1.

Which ratio expresses the relationship between the two phones in its simplest possible form?

25 : 6
EDDIE SAYS
There are two things we need to find out here - the values of a and b. These values can only be found using simultaneous equations. If we take the first scenario (increasing by £20), then we can create the equation: 2a - 5b = 80. If we take the second scenario (decreasing by £5), then we can create the equation: a - 5b = -20. We can then solve these simultaneously: 2a - 5b = 80 a - 5b = -20 a = 100 2a - 5b = 80 (2 × 100) - 5b = 80 200 = 80 + 5b 120 = 5b --> b = 24 Finally, we just need to express these values as a ratio and simplify them to their lowest form: 100:24 100:24 ÷ 4 25:6
• Question 10

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

If 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 question is setting up the equation. We start with the same amount in each bag, so the starting ratio is 1:1 which we can write as n:n as this is currently an unknown value. 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:
 n + 2 n + 6
=
 n + 4 n + 9
(n + 9)(n + 2 ) = (n + 6)(n + 4) n2 + 11n + 18 = n2 + 10n + 24 n = 6 Well done for reaching the end of this activity - there were some tricky questions here! You can now use algebra to define changing situations to find starting values, unknown variables or the changing ratio.
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