# Reduce a Ratio to its Simplest Form

In this worksheet, students will practise converting values into the same units and finding the HCF to simplify ratios with three or more elements into their simplest possible form.

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

Using ratio is a practice in maths which compares two or more quantities which have something in common.

For example, if I had 3 red balls,  9 blue balls and 10 green balls, the ratio of red to blue to green balls would be 3:9:10.

One thing that is absolutely vital when working with ratios is to remember that the order in which numbers are placed is very important.

Cancelling a Ratio

When we are trying to cancel a ratio into its simplest form, the same rules apply as if we are trying to cancel a fraction.

We need to find the Highest Common Factor (HCF) of all the numbers present and then divide every number by this.

Remember that the HCF simply means the largest number that is a factor of all of the numbers present.

Let's look at how to simplify a ratio now in some examples.

e.g. Cancel 9:3:6 to its simplest form.

Firstly, we need to ask ourselves: "What is the highest number which 9, 3 and 6 can all be divided by?"

The HCF of 9,3 and 6 is 3.

Now we need to divide every section of the ratio by this number:

9 ÷ 3 = 3

3 ÷ 3 = 1

6 ÷ 3 = 2

So 9:3:6 can be simplified to 3:1:2.

e.g. Simplify 25:85:15.

The HCF of 25, 85 and 15 is 5.

Now we need to divide every section of the ratio by this number:

25 ÷ 5 = 5

85 ÷ 5 = 17

15 ÷ 5 = 3

So 25:85:15 can be simplified to 5:17:3.

e.g. Write £2 to 30 p to 45 p as a ratio in its simplest form.

The key thing to notice here is that the units aren't the same so we need to convert these to the same format before we start.

There is 100p in £1, so £2 = 200 p.

Remember, as well, that the order of the ratio must be the same as in the question:

£2: 30 p:45 p

Now we have created a ratio with each number in the same format, we can remove the units and start to cancel it down:

200:30:45

The HCF of 200, 30 and 45 is 5.

Now we need to divide every section of the ratio by this number:

200 ÷ 5 = 40

30 ÷ 5 = 6

45 ÷ 5 = 9

So 200:30:45 can be simplified to 40:6:9.

In this activity, we will apply the process shown above of converting values into the same units and finding the HCF to simplify ratios with three or more elements into their simplest possible form.

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

When we are writing a ratio, the order in which we place the numbers is important.

Is the statement above true or false

True

False

Simplify 10:8:2 as much as possible.

True

False

Reduce 30:20:10 to its simplest form.

True

False

We are trying to reduce the ratio 28:35:42 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

15:20:5
1:3:4
7:21:28
4:10:1
48:120:12
3:4:1
9:10:11
9:10:11

Compare 200 m to 1 km to 750 m in a ratio in its simplest possible form.

## Column B

15:20:5
1:3:4
7:21:28
4:10:1
48:120:12
3:4:1
9:10:11
9:10:11

Write 42 minutes to 70 minutes to 2 hours as a ratio in its simplest possible form.

## Column B

15:20:5
1:3:4
7:21:28
4:10:1
48:120:12
3:4:1
9:10:11
9:10:11

Which of the ratios below are simplified versions of the ratio 21:63:42?

1:3:2

7:21:14

3:9:6

Simplify 180:360:720:900 as far as possible.

1:3:2

7:21:14

3:9:6

• Question 1

Type three words in the spaces 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 are writing a ratio, the order in which we place the numbers is important.

Is the statement above true or false

True
EDDIE SAYS
This statement is definitely true. If we write 1:2 and 2:1 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 1:2 or 2:1 to work out how many sweets you get in relation to pieces of fruit? If you like sweets, it would probably be 2:1, as this way round would mean you get more sweets and less fruit, whilst 1:2 would mean the reverse!
• Question 3

Simplify 10:8:2 as much as possible.

EDDIE SAYS
We need to find the HCF here which we can divide the numbers 10, 8 and 2 by. All of these numbers appear in the 2 times table, so this is our HCF in this case. If we divide each by 2, we reach: 10 ÷ 2 = 5 8 ÷ 2 = 4 2 ÷ 2 = 1 So 10:8:2 can be simplified to 5:4:1. We know this cannot be simplified further as it has a 1 in it, which is the smallest unit we can use without resorting to a decimal number.
• Question 4

Reduce 30:20:10 to its simplest form.

EDDIE SAYS
We need to find the HCF here which we can divide the numbers 30, 20 and 10 by. All 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: 30 ÷ 10 = 3 20 ÷ 10 = 2 10 ÷ 10 = 1 So 30:20:10 can be simplified to 3:2:1. Again, this ratio has a 1 in it, so it definitely cannot be simplified any further.
• Question 5

We are trying to reduce the ratio 28:35:42 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 three numbers in the ratio with no remainders or decimals. Lots of these options are factors of one or two of the numbers in the ratio (e.g. 35 with 5, 42 with 6, 28 and 35 with 7), but only two are factors of all three. The possible factors are 1 and 7, but 7 is the HCF. 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

15:20:5
3:4:1
7:21:28
1:3:4
48:120:12
4:10:1
9:10:11
9:10:11
EDDIE SAYS
Remember we are always looking for the HCF to simplify our ratios. The HCF of 15, 20 and 5 is 5, so this simplifies to 3:4:1. The HCF of 7, 21 and 28 is 7, so this simplifies to 1:3:4. The HCF of 48, 120 and 12 is 12, so this simplifies to 4:10:1. 9:10:11 cannot be simplified, as the three numbers do not have any common factors at all.
• Question 7

Compare 200 m to 1 km to 750 m 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 km = 1000 m Now, that we have all three values expressed in metres, we can remove the units: 200:1000:750 Next, we need to find the HCF which we can divide the numbers 200, 1000 and 750 by. All of these numbers appear in the 50 times table, so this is our HCF in this case. If we divide each by 50, we reach: 200 ÷ 50 = 4 1000 ÷ 50 = 20 750 ÷ 50 = 15 So our simplest ratio in this case is 4:20:15.
• Question 8

Write 42 minutes to 70 minutes to 2 hours 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 2 h = 120 m Now, that we have all three values expressed in minutes, we can remove the units: 42:70:120 Next, we need to find the HCF which we can divide the numbers 42, 70 and 120 by. All of these numbers appear in the 2 times table, so this is our HCF in this case. If we divide each by 2, we reach: 42 ÷ 2 = 21 70 ÷ 2 = 35 120 ÷ 2 = 60 So our simplest ratio in this case is 21:35:60.
• Question 9

Which of the ratios below are simplified versions of the ratio 21:63:42?

1:3:2
7:21:14
3:9:6
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 all three elements. e.g. For 1:3:2, we need to consider what 21:63:42 could have been divided by to reach this. The relationship between 21 and 1 would be '÷ 21'. Let's check that works in the other numbers: 63 ÷ 21 = 3 42 ÷ 21 = 2 So the ratio 1:3:2 is an equivalent ratio to 21:63:42. Did you spot that they were all equivalent ratios?
• Question 10

Simplify 180:360:720:900 as far as possible.

EDDIE SAYS
Does it matter that there are 4 numbers here? Not at all, it's just one more to divide by the HCF! The HCF of 180, 360, 720 and 900 is 180. Then we need to divide each number by this to find our simplest equivalent ratio of 1:2:4:5. Great job completing this activity! Now you can convert values into the same units and find the HCF to simplify ratios with three or more elements into their simplest possible form. If you are feeling confident, why not try the challenging Level 3 activity on this theme?
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