# Reduce a Ratio to its Simplest Form

In this worksheet, student will cancel ratios down into their simplest possible forms using the method of converting amounts into the same units then finding the Highest Common Factor.

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

Ratios are used in maths to compare two (or more) quantities which have something in common.

For example, if I had 3 red balls and 9 blue balls in a bag, the ratio of red to blue balls would be 3:9.

One thing that is absolutely vital when working with ratios is to remember that the order in which the numbers are placed is very important, as it determines what is being compared to what.

Cancelling a Ratio

When we are trying to cancel a ratio, the rules are the same as if we were trying to cancel a fraction.

Just like with fractions, we want to ensure our ratios are expressed in their simplest possible forms, so that they can be applied as easily as possible in simple calculations.

To simplify a ratio, we need to find the Highest Common Factor (HCF) and then divide all elements by this number to find an equivalent.

Let's look at this process in action.

e.g. Cancel the ratio 9:3 into its simplest form.

We need to ask ourselves: "What is the Highest Common Factor between 9 and 3?"

Remember, the Highest Common Factor (HCF) of two (or more) numbers is simply the highest number which all values can be divided by, which does not leave any remainders or decimals.

The HCF of 9 and 3 is 3, as 3 × 3 = 9 and 3 × 1 = 3.

So we need to divide both sides by 3:

9:3 ÷ 3  -->  3:1

e.g. Simplify the ratio 25:85 as far as possible.

The HCF of 25 and 85 is 5, so we need to divide both sides by 5:

25:85 ÷ 5  -->  5:17

e.g. Write a ratio which compares £2 to 30 p in its simplest possible form.

The key thing to notice here is that the units which our amounts are expressed in <b>are not</b> the same, so we have to convert these into the same units before we start.

There is 100 p in £1, so £2 = 200 p

Remember that the order for our ratio must be the same as in the question:

200 p:30 p

As both our units are now the same, we can remove them:

200:30

Now we can cancel this ratio down by finding the HCF.

The HCF of 200 and 30 is 10, so we need divide both sides by 10:

200:30 ÷ 10  -->  20:3

In this activity, we will cancel ratios down into their simplest possible forms using the method shown above of converting amounts into the same units then finding the Highest Common Factor.

Type three words into the space to complete the sentence below.

When we write a ratio, the order in which the numbers are placed is important.

Is the statement above true or false

True

False

Simplify 12:8 as far as possible.

True

False

Reduce 25:15 to its simplest form.

True

False

We are trying to reduce the ratio 48:120 into its simplest form.

For each of the numbers below, state if it is a common factor, the HCF or not a factor at all

Match each uncancelled ratio on the left with its simplest form on the right.

## Column B

75:25
3:1
7:21
1:3
48:120
9:10
9:10
2:5

Compare £2.50 to 20 p in a ratio in its simplest possible form.

## Column B

75:25
3:1
7:21
1:3
48:120
9:10
9:10
2:5

Write 4 hours to 24 minutes as a ratio in its simplest possible form.

## Column B

75:25
3:1
7:21
1:3
48:120
9:10
9:10
2:5

Which of the ratios below are simplified versions of the ratio 28:40?

14:20

7:10

7:20

Simplify 28:35:42 as far as possible.

14:20

7:10

7:20

• Question 1

Type three words into the space to complete the sentence below.

EDDIE SAYS
Did you recall this key fact from the Introduction? Just the same as when we are working with fractions, we need to find the Highest Common Factor (HCF) of all parts of a ratio in order to simplify it. Review the Introduction now if you need to before you move on to give the rest of this activity a go.
• Question 2

When we write a ratio, the order in which the numbers are placed is important.

Is the statement above true or false

True
EDDIE SAYS
This statement is definitely true. If we write 1:3 and 3:1 these are not the same. Let's apply this ratio to a real life scenario, to help illustrate this point. If we apply these two ratios to sweets and fruit, would you rather use the ratio 1:3 or 3:1 to work out how many sweets you get in relation to pieces of fruit? If you like sweets, it would probably be 3:1, as this way round would mean you get more sweets and less fruit, whilst 1:3 would mean the reverse!
• Question 3

Simplify 12:8 as far as possible.

EDDIE SAYS
We need to find the HCF here which we can divide the numbers 12 and 8 by. Both of these numbers appear in the 4 times table, so this is our HCF in this case. If we divide each by 4, we reach: 12 ÷ 4 = 3 8 ÷ 4 = 2 So 12:8 can be simplified to 3:2. We know this cannot be simplified further as there is a difference of only 1, which is the smallest difference we can see between ratios.
• Question 4

Reduce 25:15 to its simplest form.

EDDIE SAYS
We need to find the HCF here which we can divide the numbers 25 and 15 by. Both of these numbers appear in the 5 times table, so this is our HCF in this case. If we divide each by 5, we reach: 15 ÷ 5 = 3 25 ÷ 5 = 5 So 25:15 can be simplified to 5:3.
• Question 5

We are trying to reduce the ratio 48:120 into its simplest form.

For each of the numbers below, state if it is a common factor, the HCF or not a factor at all

EDDIE SAYS
Remember that a viable factor can be used to divide each of the two numbers in the ratio with no remainders or decimals. 5 and 48 are factors of one of the numbers present, but not both. The possible factors of both numbers are 2, 3, 4, 8, 12 and 24, but the HCF is 24. Remember, for any group of numbers in a ratio, there can only ever be one HCF.
• Question 6

Match each uncancelled ratio on the left with its simplest form on the right.

## Column B

75:25
3:1
7:21
1:3
48:120
2:5
9:10
9:10
EDDIE SAYS
Remember we are always looking for the HCF to simplify our ratios. The HCF of 75 and 25 is 25, so this simplifies to 3:1. The HCF of 7 and 21 is 7, so this simplifies to 1:3. The HCF of 48 and 120 is 24, so this simplifies to 2:5. 9:10 cannot be simplified, as these two numbers do not have any common factors at all.
• Question 7

Compare £2.50 to 20 p in a ratio in its simplest possible form.

EDDIE SAYS
The values here are not in the same format, so we need to address this first. £1 = 100 p so £2.50 = 250 p Now, that we have both values expressed in pence, we can remove the units: 250:20 Next, we need to find the HCF which we can divide the numbers 250 and 20 by. Both of these numbers appear in the 10 times table, so this is our HCF in this case. If we divide each by 10, we reach: 250 ÷ 10 = 25 20 ÷ 10 = 2 So our simplest ratio in this case is 25:2.
• Question 8

Write 4 hours to 24 minutes as a ratio in its simplest possible form.

EDDIE SAYS
Similar to the last question, the values here are not in the same format, so we need to address this first. 1 hour = 60 minutes so 4 h = 240 m Now, that we have both values expressed in minutes, we can remove the units: 240:24 Next, we need to find the HCF which we can divide the numbers 240 and 24 by. Both of these numbers appear in the 24 times table, so this is our HCF in this case. If we divide each by 24, we reach: 240 ÷ 24 = 10 24 ÷ 24 = 1 So our simplest ratio in this case is 10:1.
• Question 9

Which of the ratios below are simplified versions of the ratio 28:40?

14:20
7:10
EDDIE SAYS
It's important to notice here that the question doesn't ask for the simplest option, but a 'simplified version'. To test each option, we need to find the divisor and check it if works in both elements. e.g. For 14:20, we need to consider what 28:40 could have been divided by to reach this. The relationship between 28 and 14 would be '÷ 2'. Let's check that works in the other number: 40 ÷ 2 = 20 So the ratio 14:20 is an equivalent ratio to 28:40. Did you spot that the two equivalent ratios present within these options?
• Question 10

Simplify 28:35:42 as far as possible.

EDDIE SAYS
Does it matter that there are 3 numbers here? Not at all, it's just one more to divide by the HCF! The HCF of 28, 35 and 42 is 7. Then we need to divide each number by this to find our simplest equivalent ratio of 4:5:6. Great job completing this activity! Now you can convert values into the same units and find the HCF to reduce ratios into their simplest possible form. If you are feeling confident, why not try another activity to practise the concept of ratio more?
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