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### What is a ratio?

A ratio is used when we are comparing two different amounts. For example, if we were to mix a drink that was one part squash and five parts water, this could be shown to be in the ratio of 1:5.

This has a clear link to fractions as we can put 1:5 as  which then tells us the amount of the first part compared to the second. So in this case we have an amount of squash which is  of the amount of water. So by putting one value in a ratio over another we can find out how much more, as a fraction, we have of one part than another.

Ratios can be changed in similar ways to fractions: they can be simplified by finding a common factor. Therefore, the ratio 5:6 is exactly the same as 25:30 in the same way as  is the same as . These are what we call equivalent ratios.

#### Example

Simplify the following ratios:

a) 72:90

b) 42:70

a) Both 72 and 90 have a factor of 9, therefore when dividing both sides by 9 we get 8:10. This can then be simplified again since 8 and 10 can be divided by 2, so the final answer is 4:5.

b) 42 and 70 have a common factor of 7, therefore when dividing both sides by this number we get 6:10, which again can be further simplified to 3:5.

The rule we used when simplifying fractions was to do the same to both the top and bottom. The rule for ratios is to do the same calculation for both sides of the ratio, or for every side, as we will now see how ratios can have more than two parts.

### Ratios with more than two parts

Suppose that we were making a cake mix that had more than two ingredients; clearly a ratio that had only two parts would not be enough to work with as we must consider everything that is put into the mixture.

To do this we must make use of ratios that have more than two parts to them. For example, if we were to make a cake which was 2 parts butter to 3 parts flour and 1 part sugar, we could write the ratio as 2:3:1 in the form butter:flour:sugar.

This then has the same rules which apply to ratios with only two parts when we are trying to simplify – we must do the same operation to all parts so that the ratio is equal to the one we started with.

#### Example

Simplify the following ratios:

a) 6:12:54

b) 25:30:15:10

a) To solve this we must find a common factor of 6, 12, and 54. With a little bit of working we can find this to be 6, and so each part of the ratio can be divided by 6. This gives the result of 1:2:9

b) Again, we must find a common factor but this time of 25, 30, 15 and 10. This common factor is 5, and so dividing all four parts by 5 gives us a simplified ratio of 5:6:3:2

### Splitting an amount using ratios

Ratios can be used in a variety of different situations to split different parts of a whole into various sections. When we are asked to split a certain amount using a ratio, we must do the following steps:

1) Add the parts of the ratio together to get a total

2) Divide the amount that we need to split by this total

3) Multiply this amount by each individual ratio part to get the amount that is needed for each

#### Example

Share £100 in the ratio of 3:7

1) 3+7=10

2) £100 divided by 10, so each individual part is worth £10

3) 3:7 multiplied by £10 gives us the ratio £30:£70

We can then check this answer by adding the two amounts to make sure that they make the original amount. So £30 + £70 = £100, therefore we know that the original amount has been divided into the ratio 3:7.

#### Example

Split 1200 in the ratio of 13:11:6

1) 13+11+6=30

2) 1200 divided by 30 is equal to 40, so each value of 1 is equal to 40 of the original amount

3) Multiply 13, 11 and 6 by 40 to get the ratio 520:440:240

Again, we can check this answer by adding together 520, 440 and 240 to get 1200, which is the same as the original value which we were given in the question.

### Increase and decrease

If we wish to increase or decrease an amount by a ratio, then we treat the amount as either the smaller or larger number in the ratio. If we are looking to increase the amount then we act as if the value is the smaller of the two, so that when we find the other, the answer is larger. If we want to decrease, we act like the amount is the larger of the two so we end up with a smaller answer.

Basically, what we must do is to convert the ratio into a fraction and then multiply this by the value we wish to change. This is best seen with an example:

#### Example

Increase £90 in the ratio 3:5.

3:5 can be shown as the fraction  or , since we are looking to increase the amount of £90 we must use the larger of these two fractions to multiply. So our answer is £150.

#### Example

Decrease £200 in the ratio 5:8.

5:8 can be the fraction  or  but since we are looking to decrease, we will take the smaller of the two so we must calculate  £125.

### Use of ratios in everyday life

Ratios are used in all sorts of situations and you need to become used to being able to change a worded problem into one of the form we have already seen. Ratios are used in lots of things from cooking instructions to maps and a lot of the time a question will not appear to be a ratio problem when not in a mathematical form. Here are a few examples which are problems that can be solved using the methods we have seen in this lesson:

#### Example

It takes 6 men 15 days to build a wall. How long would it take for just 2 men to build the same wall?

Here we can see that the ratio for men is 6:2 and since it takes the 6 men 15 days, we must increase this value to find the answer for 2 men (since it will clearly take longer with less people to help). So, as said earlier, we work out 45 days.

An architect draws a plan for a building of height 24m and draws this on his plans as 75cm. What scale is he using in his plan?

Here we need to change the ratio of 75cm:24m into one which uses a 1 instead of 75cm. Therefore, we can use the fact that , this tells us that we can find an equivalent ratio that is 75cm:24m with both sides multiplied by . Giving us an answer of 1:32 being the scale in use.

5 plates cost £7.65. How much is it to buy 8 of the same plates?

This is a problem where we must increase the value \$7.65 by the ratio of 5:8. Using the rules outlined earlier this becomes .

Hopefully, you will now be able to find the key parts of a question that tells you what to look for and how to convert the problem into one which we know how to solve.

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