# Use Multiplicative Relationships

In this worksheet, students will express multiplicative relationships in terms of ratios and fractions, cancelling them into their simplest by finding the HCF.

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

There are a number of occasions when we may see a relationship expressed as a multiple.

For example, we could say, "this bottle holds three times as much as that one."

This is an example of a multiplicative relationship.

One of the essential skills you need to master to be successful with questions involving multiplicative relationships, is to know how to write them as either a ratio or a fraction.

Let's look at this in action now.

e.g. One jug of water is three times larger than another. Express this relationship as a ratio.

We know that all ratios are written as the form a:b, where a and b are whole numbers.

In this case, we can write this relationship as 1:3 or 3:1.

This question doesn't specify which one is bigger out of the jugs, so it is acceptable to use either order in our ratio here.

e.g. I have two pieces of wood. Piece A is 60 cm and piece B is 80 cm long.

a) Write the lengths of A:B as a ratio in its simplest form.

We start by writing each number simply as a ratio, in the order expressed in the question:

60:80

We can then cancel this ratio down into its simplest form by finding the Highest Common Factor (HCF) of both numbers.

Here the HCF is 20:

60 ÷ 20 = 3

80 ÷ 20 = 4

So the simplest possible form of this ratio is 3:4

b) Find the length of B as a fraction of A.

In this question, we are asked to find B as a fraction of A, so B must go on the top of the fraction and A on the bottom, like this:

 80 60

This fraction can then be cancelled down if we divide by a HCF of 20:

 80 ÷ 20 = 4 60 ÷ 20 3

In this activity, we will express multiplicative relationships in terms of ratios and fractions, cancelling them into their simplest by finding the HCF or using them to find missing variables.

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

Bag A contains 15 marbles and Bag B contains 10 marbles.

Express this relationship as a ratio in its simplest form.

Bag A contains 15 marbles and Bag B contains 10 marbles.

Express this relationship as a ratio A:B in its simplest form.

Match each multiplicative relationship on the left with its equivalent ratio on the right.

## Column B

3 times the size
2:1
4 times the size
1:3
Half the size
1:4
One tenth the size
10:1

It takes Lucy 20 minutes to walk to the shop.

If it takes James 15 minutes, what fraction of Lucy's time does James take?

Take care to only write one number in each of the spaces, as the fraction bar has already been provided for you.

## Column B

3 times the size
2:1
4 times the size
1:3
Half the size
1:4
One tenth the size
10:1

Two cans of fizzy drink contain 250 ml and 400 ml respectively.

What fraction of the smaller can does the larger can contain?

Take care to only write one number in each of the spaces, as the fraction bar has already been provided for you.

## Column B

3 times the size
2:1
4 times the size
1:3
Half the size
1:4
One tenth the size
10:1

Match each scenario on the left with its equivalent fraction in its simplest form on the right.

## Column B

6 minutes as a fraction of 10 minutes
1/2
6 minutes as a fraction of 20 minutes
3/5
100 ml as a fraction of 200 ml
3/10
2 hours as a fraction of 2 day
1/12

The length of two pieces of wood can be expressed in the ratio 3:5.

If the smaller piece of wood is 60 cm long, how long is the larger piece?

## Column B

6 minutes as a fraction of 10 minutes
1/2
6 minutes as a fraction of 20 minutes
3/5
100 ml as a fraction of 200 ml
3/10
2 hours as a fraction of 2 day
1/12

Jamal takes 4/5 of the time as his friend Richard to finish a race.

If Richard takes 1 hour to finish, how many minutes does Jamal take?

## Column B

6 minutes as a fraction of 10 minutes
1/2
6 minutes as a fraction of 20 minutes
3/5
100 ml as a fraction of 200 ml
3/10
2 hours as a fraction of 2 day
1/12

The contents of three milk cartons can be expressed using the ratio 2:3:5.

If the medium-sized carton contains 450 ml, how much do the smaller and larger ones contain?

Take care to only write numbers in each of the spaces, as the unit of measurement (ml) has already been provided for you.

 Contains (in ml): Smaller Larger
• Question 1

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

EDDIE SAYS
There's only two possible ways in which we can express a multiplicative relationship. They have to be either a fraction or a ratio. Review the Introduction now, if you need to refresh on how to express a multiplicative relationship in terms of a fraction or ratio, before you tackle the rest of this activity.
• Question 2

Bag A contains 15 marbles and Bag B contains 10 marbles.

Express this relationship as a ratio in its simplest form.

EDDIE SAYS
We know that all ratios are written as the form a:b, where a and b are whole numbers. In this case, we can write this relationship as 15:10 or 10:15. As this question doesn't specify which order to put the numbers in, it is acceptable to use either order in our ratio here. We can simplify this ratio by finding the HCF of both numbers. The HCF of 15 and 10 is 5: 15 ÷ 5 = 3 10 ÷ 5 = 2 So our simplest ratio is either 3:2 or 2:3. Which did you choose?
• Question 3

Bag A contains 15 marbles and Bag B contains 10 marbles.

Express this relationship as a ratio A:B in its simplest form.

EDDIE SAYS
This is the same question as previously, with one crucial difference. This time, we are asked to express the ratio in the form Bag A:Bag B, which means that the number relating to Bag A must appear first in our ratio. So, on this occasion, the only acceptable ratio is 3:2.
• Question 4

Match each multiplicative relationship on the left with its equivalent ratio on the right.

## Column B

3 times the size
1:3
4 times the size
1:4
Half the size
2:1
One tenth the size
10:1
EDDIE SAYS
This is all about key words. 'Half' means the second amount is '× 0.5 / ÷ 2' the first. Whilst 'one tenth' means that the second amount is '× 0.1 / ÷ 10' the first. Let's look at 'Half the size' as an example. If we write this as a ratio using the info above, it should be: 1 : 0.5 However, we are not allowed to use decimals in ratios of this type, so we need to convert the second number into a whole number. We can achieve this using × 2, which we also then need to apply to the first number also: 1 : 0.5 × 2 = 2:1 Can you use this example and your own knowledge of ratio to match the other pairs successfully?
• Question 5

It takes Lucy 20 minutes to walk to the shop.

If it takes James 15 minutes, what fraction of Lucy's time does James take?

Take care to only write one number in each of the spaces, as the fraction bar has already been provided for you.

EDDIE SAYS
The trick here is to take note of the words 'fraction of Lucy's time'. This means that Lucy's time has to go on the bottom of our fraction (as the denominator), whilst James' time needs to go on the top (as the numerator). We can then cancel this fraction by finding the HCF, which in this case is 5:
 15 ÷ 5 = 3 20 ÷ 5 4
• Question 6

Two cans of fizzy drink contain 250 ml and 400 ml respectively.

What fraction of the smaller can does the larger can contain?

Take care to only write one number in each of the spaces, as the fraction bar has already been provided for you.

EDDIE SAYS
Once again, we need to look at the phrase 'fraction of the smaller can'. This means that 250 goes on the bottom of our fraction, whilst 400 goes on the top. Then we can simplify by finding the HCF, which in this case is 50:
• Question 7

Match each scenario on the left with its equivalent fraction in its simplest form on the right.

 400 ÷ 50 = 8 250 ÷ 50 5

## Column B

6 minutes as a fraction of 10 min...
3/5
6 minutes as a fraction of 20 min...
3/10
100 ml as a fraction of 200 ml
1/2
2 hours as a fraction of 2 day
1/12
EDDIE SAYS
Let's look at '6 minutes as a fraction of 10 minutes' as an example. If we write this as a fraction using the info above, we know that 6 needs to be on top of 10, like this:
 6 10
We can simplify this fraction by finding the HCF of both 6 and 10, which is 2:
 6 ÷ 2 = 3 10 ÷ 2 5
Can you use this example and your own knowledge of fractions to match the other pairs successfully?
• Question 8

The length of two pieces of wood can be expressed in the ratio 3:5.

If the smaller piece of wood is 60 cm long, how long is the larger piece?

EDDIE SAYS
This is actually a hidden ratio question. We need to apply what we know to find a missing length, rather than calculate a ratio or fraction. If the smaller piece of wood is 60 cm, this represents 3 parts of our overall total length. We can use this fact to find that one part is worth 20 cm. The longer piece of wood represents 5 parts of our total length, and each part is worth 20 cm: 5 × 20 = 100 cm
• Question 9

Jamal takes 4/5 of the time as his friend Richard to finish a race.

If Richard takes 1 hour to finish, how many minutes does Jamal take?

EDDIE SAYS
This question may sound complex, but all it is asking us to do is to find 4/5 of 1 hour. As 1 hour is 60 minutes, we need to find 4/5 of 60 minutes. Remember that to find a fraction of an amount, we should divide by the bottom, then multiply by the top: 60 ÷ 5 × 4 = 48 minutes
• Question 10

The contents of three milk cartons can be expressed using the ratio 2:3:5.

If the medium-sized carton contains 450 ml, how much do the smaller and larger ones contain?

Take care to only write numbers in each of the spaces, as the unit of measurement (ml) has already been provided for you.

 Contains (in ml): Smaller Larger
EDDIE SAYS
Another hidden ratio question to end this activity. If the medium-sized carton contains 450 ml, this represents 3 parts of our overall total capacity. We can use this fact to find that one part is worth 150 ml. The largest carton represents 5 parts of our total capacity, and each part is worth 150 ml: 5 × 150 = 750 ml The smallest carton represents 2 parts of our total capacity, and each part is worth 150 ml: 2 × 150 = 300 ml Great job completing this activity! You can now express multiplicative relationships in terms of ratios and fractions, cancelling them into their simplest by finding the HCF or using them to find missing variables.
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