5.03 Find part or whole measures with ratios

Consider a bag containing 56 red and blue marbles where the ratio of red marbles to blue marbles is 3:5.

How many red marbles are there? How many blue marbles are there?

To find the number of red and blue marbles we would normally use equivalent ratios by increasing or decreasing the ratio by some multiple. However, in this case, we only know the total number of marbles so this method won't work. So what do we do?

To solve our problem we can make the total number of marbles a component in our ratio. To do this we need to know how many ratio parts correspond to the total number of marbles. Since there are 3 parts red marbles and 5 parts blue marbles, the total number of marbles is simply 3+5=8 parts.

Taking this information, our ratio of red to blue to the total number of marbles is written 3:5:8.

Now we can use equivalent ratios to solve our problem.

Since we get 56 from 8 by increasing by a multiple of 7, we can find the number of red and blue marbles by increasing their ratio parts by the same multiple.\begin{array}{c} &\text{Red} &\text{to} &\text{Blue} &\text{to} &\text{Total} \\ &3 &: &5 &: &8 \\ \\ &\times 7 & &\times 7 & &\times 7 \\ \\ &21 &: &35 &: &56 \end{array}

Using the equivalent ratio we find that there are 21 red marbles and 35 blue marbles.

We used the ratio 3:5:8 to represent the ratio of red to blue to the total number of marbles. This is an example of a part to whole ratio. A part to whole ratio is a ratio that shows the ratio of one component compared to the whole and is useful when considering what fraction of the total represents that component.

For example, we can write the part to whole ratio of blue marbles to the total as 5:8 which is equivalent to saying that \dfrac{5}{8} of the marbles are blue. This can then be used to find the number of blue marbles from the total with the calculation:

\text{Number of blue marbles}=56 \times \dfrac{5}{8}=35

This is equivalent to finding the number of marbles in one part, that is we divide the total, 56, by 8, and then multiplying by 5 to find the number of marbles in 5 parts. This is known as the unitary method.

\text{Number of blue marbles}=\dfrac{56}{8} \times 5=35

Notice that the only difference is which number we divide by 8.

We can use the ratio 3:8 to perform a similar calculation to find the number of red marbles.

A part to whole ratio is a ratio between the quantity of a component (or components) and the total quantity. We can find the quantity of that component by multiplying the total quantity by the fraction that is equivalent to the ratio.

Can we use these types of ratios to find the total quantities when we only know the quantity of the component?

Yes, we can do this by multiplying the quantity of the component by the reciprocal of the fraction equivalent to the ratio.

For example, if we know that there are 21 red marbles and the ratio of red to the total number of marbles is 3:8 we can find the total number of marbles with the calculation:\text{Total number of marbles}=21 \times \dfrac{8}{3}=56 which returns the expected answer.

We can also solve this using the unitary method.

If 21 red marbles is 3 parts, we can divide 21 by 3 to find the size of one part, and then multiply this amount by 8 to find the total number of marbles:\text{Total number of marbles}=\dfrac{21}{3} \times 8=56

Examples