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Unitary Method


So far we're feeling pretty confident about finding a certain percentage of an amount, but is there a way to find the total amount again after being given a percentage of it?

For instance, let's say $40%$40% of a dog's weight is $10$10kg, how heavy is the dog?

We can use what is called the unitary method here, which means finding out what one unit of something is first.

In our case we're going to find what one percent is first. Let's see how this works in the following diagram:

Can you see how there are three stages in this method? 

1. Start with the amount that we know and its related percentage

2. Convert both numbers (percentage and amount) to what $1%$1% would be

3. Multiply by $x$x to get $x$x%, which is however much you want to find


Adding On

Sometimes questions will involve starting amounts over $100%$100%.

For example, say we knew that a bank account was worth $\$770$$770 after $10%$10% interest was paid.

To find the original $100%$100% we would first need to figure out what the starting percentage is.

If $\$770$$770 is the amount after interest then it equals $100%+10%=110%$100%+10%=110%.

Then to find the total amount we would follow Step $2$2 above and divide everything by $110$110 to get $1%$1%, and then finally multiply by $100$100to get the whole amount.

So the original amount in the bank account would be

$\frac{770}{110}\times100$770110×100 $=$= $7\times100$7×100
  $=$= $\$700$$700

Wow, so the unitary method also works for amounts more than $100%$100%!

Worked Examples


Find the number if $\frac{1}{3}$13 of the number is $10$10.


$40%$40% of a quantity is $940$940.

  1. What is $1%$1% of the quantity?

  2. Hence find the total quantity.


$870%$870% of a number is $696$696. What is the number? Write your answer in simplest form.


A stockbroker bought $316$316 shares at $\$23.25$$23.25 per share and later sold the shares for a total of $\$15281$$15281. Calculate the percentage profit to two decimal places.

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