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3.3 I) Combing Ratios
3.3 I) Combing Ratios
Often in the exam you will be given two different ratios and will be asked a question that will need you to combine both of the ratios. It may even be the case that you are given the second ratio in the form of a fraction or a percentage rather than a ratio (we will be given a fraction in the second example).
Example 1
There are 135 sweets and they are split between 3 different individuals; Gavin, Ben & Holly. The ratio of sweets between Gavin and Ben is 3 : 4. The ratio of sweets between Ben and Holly is 2 : 1. How many sweets do each of the individuals get?
There are 135 sweets and they are split between 3 different individuals; Gavin, Ben & Holly. The ratio of sweets between Gavin and Ben is 3 : 4. The ratio of sweets between Ben and Holly is 2 : 1. How many sweets do each of the individuals get?
The question gives us two different ratios. We are told the ratio between Gavin and Ben, and the ratio between Ben and Holly. The best way to answer this question is to combine the two ratios together so that we have one ratio. When we have one ratio, we are then able to answer the question as a fairly straightforward ratio question.
Let’s use the initial ratio between Gavin and Ben to start off with; the ratio is.
Let’s use the initial ratio between Gavin and Ben to start off with; the ratio is.
The second ratio that we are given is between Ben and Holly and it is 2 : 1. This means that for every 2 sweets that Ben gets, Holly will get 1 sweet. Another way of viewing this is that Ben gets double as many sweets as Holly. In the ratio above, Ben is represented as 4 parts. We know from the second ratio that if ben gets 4 parts, Holly will have 2 parts because Holly will have half as many parts. Therefore, the ratio between Gavin, Ben & Holly become.
We are now able to solve this question like a normal ratio question. The first step is to add all of the parts together. When we do this, we see that the ratio contains 9 parts (3 + 4 + 2). The next step is to see how much 1 part of the ratio represents. There are 135 sweets to be shared out and there are 9 parts to the ratio. Therefore, we divide 135 by 9, which tells us that each part represents 15 sweets. The final step is to multiply the number of parts that each individual has by the number of sweets that each part represents (15):
We should always check that these 3 amounts of sweets add up to the number of sweets that we started with. We started with 135 sweets and 45 + 60 + 30 does add up to 135. Therefore, all the sweets have been shared out.
The final answer to this question is that Gavin gets 45 sweets, Ben gets 60 sweets and Holly gets 30 sweets.
- Gavin – 3 x 15 = 45 sweets
- Ben – 4 x 15 = 60 sweets
- Holly – 2 x 15 = 30 sweets
We should always check that these 3 amounts of sweets add up to the number of sweets that we started with. We started with 135 sweets and 45 + 60 + 30 does add up to 135. Therefore, all the sweets have been shared out.
The final answer to this question is that Gavin gets 45 sweets, Ben gets 60 sweets and Holly gets 30 sweets.
Example 2
This is going to be another sweet question, but we are not going to be given any ratios in this question. Instead we will need to make the ratios ourselves.
96 sweets are shared between 3 individuals; Sarah, Anya and Emily. Sarah receives 4 times as many sweets as Anya. Anya receives 3 times as many sweets as Emily. How many sweets does each individual get?
Let’s start by representing the information that we are given in the question as ratios. We are told that Sarah receives 4 times as many sweets as Anya; for every 4 sweets that Sarah gets, Anya gets 1. Therefore, we can give the ratio of sweets between Sarah and Anya as:
This is going to be another sweet question, but we are not going to be given any ratios in this question. Instead we will need to make the ratios ourselves.
96 sweets are shared between 3 individuals; Sarah, Anya and Emily. Sarah receives 4 times as many sweets as Anya. Anya receives 3 times as many sweets as Emily. How many sweets does each individual get?
Let’s start by representing the information that we are given in the question as ratios. We are told that Sarah receives 4 times as many sweets as Anya; for every 4 sweets that Sarah gets, Anya gets 1. Therefore, we can give the ratio of sweets between Sarah and Anya as:
The next step is to bring Emily into the ratio. We are told that Anya receives 3 times as many sweets as Emily; for every 3 sweets that Anya receives, Emily receives 1 sweet. Another way of viewing this statement is that Emily receives 3 times less sweets than Anya. In the above ratio, Anya receives 1 sweet. We know that Emily will receive 3 times less sweets that Anya, which means that if Anya receives 1 sweet, Emily will receive 1/3 of a sweet. Our ratio between Sarah, Anya and Emily is:
We would usually give ratios as whole numbers. The only component of the ratio that is not a whole number is the final term, 1/3. We are able to make it into a whole number by multiplying by the denominator of the fraction, which is 3. Therefore, we multiply the whole of the ratio by 3. The ratio becomes:
We now know the ratio and the number of sweets that are being shared between the individuals. There are 96 sweets being shared and there are 16 parts to the ratio (12 + 3 + 1). To find the value of 1 part of the ratio, we divide the number of sweets by the number of parts in the ratio (96 ÷ 16). This tells us that 1 part in the ratio is equal to 6. The next step is to multiply the number of parts that individual has in the ratio by the number of sweets per part.
Let’s just check that these sweets add up to the total number of sweets that we started with. 72 + 18 + 3 is 96 and this is the number of sweets that we started with. Also, we were told in the question that Sarah receives 4 times as many sweets as Anya. Anya receives 18 sweets, so Sarah should receive 72 sweets (4 x 18), which is what we found. In addition, we were told that Anya received 3 times as many sweets as Emily. Emily received 6 sweets and Anya received 18 sweets (which is 3 times more than Emily).
Therefore, the answer to this question is that Sarah receives 72 sweets, Anya receives 18 sweets and Emily receives 6 sweets.
Note: you would have been able to work out the answer to the question without modifying the ratio so that all of the components were whole numbers. However, the maths is slightly harder, so I would always recommend getting your ratio with no decimals or fractions.
- Sarah – 12 x 6 = 72 sweets
- Anya – 3 x 6 = 18 sweets
- Emily – 1 x 6 = 6 sweets
Let’s just check that these sweets add up to the total number of sweets that we started with. 72 + 18 + 3 is 96 and this is the number of sweets that we started with. Also, we were told in the question that Sarah receives 4 times as many sweets as Anya. Anya receives 18 sweets, so Sarah should receive 72 sweets (4 x 18), which is what we found. In addition, we were told that Anya received 3 times as many sweets as Emily. Emily received 6 sweets and Anya received 18 sweets (which is 3 times more than Emily).
Therefore, the answer to this question is that Sarah receives 72 sweets, Anya receives 18 sweets and Emily receives 6 sweets.
Note: you would have been able to work out the answer to the question without modifying the ratio so that all of the components were whole numbers. However, the maths is slightly harder, so I would always recommend getting your ratio with no decimals or fractions.