4.05 Finding the whole

When looking at percentages, the 'whole' is our reference number that our percentage amount refers to.

For example: 14 is equal to half of 28, so 14 is 50\% of 28.

But also: 14 is equal to double of 7, so 14 is 200\% of 7.

In both cases our amount is 14. The numbers that change are the percentage value and our reference number - the whole.

Examples

Let's consider the case where we don't know the whole, or reference number:

14 is 50\% of some number. What is this number?

To work out the whole, we can use the percentage value to figure out how much of the whole we

currently have. Since 50\% is equal to \dfrac{1}{2}\,when written as a fraction, we know that 14 is half of our

missing number. If we let x represent our missing number, we can express this information as the equation:14=x\times \frac{1}{2}

We can solve for x by multiplying both sides of the equation by 2, and find that x=28, so we know the 'whole' we are looking for is 28.

We can apply this same technique to any amount and percentage.

Examples

If 39 is 30\% of some number, what is this number?

An approach to solving this would be to find 1\% of the number first. From there we can find the whole, or indeed any other percentage amount.

We know that 30\% of the number is 39, so if we divide 39 by 30, we now know 1\% of the number. Multiplying this amount by 100 will then tell us 100\% or the whole of the amount.x=\frac{39}{30}\times 100=1.3\times 100 = 130

This is known as the unitary method.

We can find the whole by:

• Dividing the given amount by the percentage number. This finds 1\%of the whole.

• Multiplying the result by 100. This will give us 100\% of the whole which is the number we are looking for.

Since the order of operations allows us to perform either multiplication and division in any order, we can do whichever operation is easier first.

Examples