3.3 H) Fractions from Ratios
Whenever we are given a ratio, we are able to work out what fraction of the total amount each component of the ratio gets. We find the fraction by dividing the number of parts that our component has in the ratio by the number of parts in the ratio.
After we obtain our fraction, we then need to make sure that we have given the fraction in it’s simplest form.
Working out fractions of ratios will make a lot more sense after we have looked through a few examples.
Example 1
70 sweets are shared between Ryan and Sam in the ratio 2 : 5.
a) What fraction of the 70 sweets does Ryan get?
B) What fraction of the 70 sweets does Sam get?
Part a of this question asks us to work out the fraction of sweets that Ryan gets. Therefore, the numerator of our fraction will be the number of parts that Ryan has in the ratio. In our ratio, Ryan has 2 parts, and this means that the numerator will be 2.
The denominator of the fraction is the total number of parts in the whole ratio. We find the total number of parts in the whole ratio by adding all of the different components of the ratio together; we therefore add 2 and 5, which gives us 7.
We now need to check whether our fraction is able to be simplified. We check whether a fraction can be simplified by looking for common factors in the numerator and the denominator. We then divide both the numerator and the denominator by any common factors that we find. There are no common factors between 2 and 7, which means that our fraction is already in its simplest form. Therefore, Ryan gets 2/7 of the sweets.
We are now onto part b of the question, which is to find the fraction of the sweets that Sam gets. The numerator of this fraction will be the number of parts that Sam has in the ratio, which is 5. The denominator of the fraction will be the total number of parts in the ratio, which will be the same as before (it will be 7).
Like before, we need to check whether the fraction can be simplified, which it can’t because there are no common factors between 5 (the numerator) and 7 (the denominator). Therefore, Sam gets 5/7 of the sweets.
Example 2
I have a bracelet that has 3 different coloured stones. The colours are green, orange and blue and they are in the ratio 5 : 3 : 2
a) What fraction of the stones in the bracelet are blue?
b) What fraction of the stones in the bracelet are orange?
c) What fraction of the stones in the bracelet are green?
Let’s start by working out part a, which is the fraction of the stones that are blue. The numerator for the fraction is going to be the number of parts in the ratio that represent blue stones, which is 2.
The denominator is going to be the total number of parts in the ratio. We work this out by adding all of the different components of the ratio. The total number of parts in the ratio is 10 (5 + 3 + 2 = 10).
We now need to check whether our fraction can be simplified. Both the numerator and the denominator are even. This means that they both have a common factor of 2. Therefore, we can simplify the fraction by dividing both the numerator and the denominator by 2.
We now check again for common factors between the new numerator and denominator. This time there are no common factors, which means that we have our fraction in its simplest form.
Therefore, 1/5 of the stones in the bracelet are blue.
We now move on to part b, which is to work out the fraction of the stones that are orange. This time the numerator of the fraction is going to be the number of parts in the ratio that represent orange, which is 3. The denominator will be the total number of parts in the ratio, which will still be 10.
There are no common factors in the numerator and the denominator, which means that our fraction is in it’s simplest form. Therefore, 3/10 of the stones will be orange.
Part c asks us to find the fraction of the stones that are green. This time the numerator is going to be the number of parts of the ratio that are green, which is 5. The denominator of the fraction will still be the total number of parts in the ratio, which is 10.
The numerator and the denominator of the fraction are both multiples of 5. Therefore, the fraction can be simplified by dividing both the numerator and the denominator by 5.
There are no more common factors between the numerator and the denominator, which means that the fraction is in it’s simplest form. Therefore 1/2 of the stones in the bracelet are green.