Percentages

Lesson

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

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%!

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

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

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

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.