Multiple Ratios

How do I combine two ratios to make a 3-part ratio?

Sometimes you may be given two separate ratios, that link together in some way, so that you can form a 3-part ratio
Suppose that on a farm with 85 animals

The ratio of cows to sheep is 2:3
The ratio of sheep to pigs is 6:7
We want to find the number of each animal on the farm

We can’t just share 85 in the ratio 2:3 or 6:7, because these ratios don’t account for all the animals on the farm on their own

We need to find a combined, 3-part ratio that shows the relative portions of all the animals together

Notice that sheep appear in both ratios, so we can use sheep as the link between the two

C:S = 2:3 and S:P = 6:7
We can multiply both sides of the C:S ratio by 2, so that both ratios are comparing relative to 6 sheep
C:S = 4:6 and S:P = 6:7
These can now be joined together
C:S:P = 4:6:7

We can now use this to share the 85 animals in the ratio 4:6:7

There are 17 parts in total (4 + 6 + 7 = 17)
Each part is worth 5 animals (85 ÷ 17 = 5)
There are 20 cows (4 × 5), 30 sheep (6 × 5), and 35 pigs (7 × 5)

How do I answer a 'changed ratio' question?

A question might start by saying something like 'a bag of 28 sweets contains only lemon and blackcurrant sweets, and the ratio of lemon to blackcurrant sweets is 4:3'
- At this point we can figure out how many of each type of sweet there are
  - 4+3=7, so the ratio divides the sweets into 7 parts
  - 28÷7=4, so each part contains 4 sweets
  - Therefore there are 4×4=16 lemon sweets, and 3×4=12 blackcurrant sweets
  - So far, so 'I'm smashing ratios'!
But a harder question might then go on to say: 'Natasha adds some blackcurrant sweets to the bag, so that the ratio of lemon to blackcurrant sweets is now 2:5. Determine how many blackcurrant sweets Natasha added to the bag.'
- You can still use all your usual ratio 'tricks' here, the only difference is that more steps will be involved
- For this sort of changed ratio question first make sure you know how many of each thing there was to start with
  - The question may give you this info
  - If it doesn't, you will need to work it out from the info given (like we just did above)
- Then bring in the 'tricks'
  - We know there were 16 lemon sweets in the bag, and that now those only represent 2 parts in the ratio
  - So now one part is equal to 16÷2=8 sweets
  - And blackcurrant sweets are now 5 parts in the ratio
  - So there are now 5×8=40 blackcurrant sweets in the bag
  - And originally there were 12 blackcurrant sweets in the bag
  - Therefore Natasha added 40-12=28 blackcurrant sweets to the bag
So really a changed ratio question is just a series of simple ratio questions joined together
- Your job in answering is to break the question down into those simple bits
If you are not given any amounts and only the ratios then identify which of the actual amounts have remained the same and make this part of the ratio the same for both ratios
- e.g. The ratio of circles to squares is 3 : 4
- 3 circles are removed and the ratio now becomes 3 : 5
- The number of squares has remained the same so change the "4" and "5" into a common multiple
- You may need to try a few different multiples until you get the numbers in the other part of the ratio to work with the scenario
  - 3 : 4 = 15 : 20 and 3 : 5 = 12 : 20
  - So there were 15 circles and 20 squares originally

Examiner Tip

Adding labels to your ratios will help make your working clearer and help you remember which number represents which quantity e.g. $\begin{matrix} A & : & B \\ 3 & : & 4 \end{matrix}$

Worked example

In Jamie’s sock drawer there are only black socks, striped socks, and white socks. The ratio of black socks to striped socks is 5:2, and the ratio of striped socks to white socks 6:7.

Given that there are 18 striped socks in the drawer, determine the total number of socks that are in the drawer.

Because the 'striped socks' number is different in the two ratios, we can't use those ratios yet to talk about all the socks in the drawer.
However, 'striped socks' appears in both ratios, so we can use that to link the two ratios together into one overall ratio.
If we 'scale up' the black : striped ratio by multiplying it by 3, then the 'striped socks' number will be the same in both ratios.

$\begin{array}{rcl} black : striped & = & 5 : 2 \\ = & 5 \times 3 : 2 \times 3 \\ = & 15 : 6 \end{array}$

Now that the 'striped socks' number is the same in both ratios, we can join them together into a 3-part ratio.

$black : striped : white = 15 : 6 : 7$

Now we can say that 'striped socks' represent 6 parts of the overall ratio.
And there are 18 striped socks overall.
So divide 18 by 6 to find the size of 1 part.

$18 \div 6 = 3 socks$

In the overall ratio, black socks are 15 parts and white socks are 7 parts.
Multiply those by 3 to find how many of each type of sock there are.

$3 \times 15 = 45 black socks 3 \times 7 = 21 white socks$

Finally add all those numbers together to find the total number of socks.

$45 + 18 + 21 = 84$

84 socks in total

Worked example

At Antonio's llamas and quokkas sanctuary there are originally 12 llamas, and the ratio of llamas to quokkas is 3:7.

After some new quokkas arrive at the sanctuary, the ratio of llamas to quokkas changes to 4:13.

Given that the number of llamas has not changed, determine the number of quokkas that arrived at the sanctuary.

First figure out the number of quokkas that were there originally.
12 llamas represents 3 parts in the original ratio.
So divide 12 by 3 to find the size of one part.

$12 \div 3 = 4 animals$

Quokkas represent 7 parts in the original ratio.
So multiply 4 by 7 to find the original number of quokkas.

$7 \times 4 = 28 quokkas (start)$

After the new quokkas arrive, there are still 12 llamas.
But now those 12 llamas represent 4 parts in the changed ratio.
So divide 12 by 4 to find the size of one part.

$12 \div 4 = 3 animals$

And now quokkas represent 13 parts in the changed ratio.
So multiply 3 by 13 to find the number of quokkas after the new arrivals.

$13 \times 3 = 39 quokkas (after arrivals)$

And finally subtract 28 from 39 to find out how many quokkas arrived.

$39 - 28 = 11$

11 quokkas arrived at the sanctuary

Multiple Ratios (AQA GCSE Maths)

Revision Note

Multiple Ratios

How do I combine two ratios to make a 3-part ratio?

How do I answer a 'changed ratio' question?

Examiner Tip

Worked example

Worked example

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Author: Roger