R9 - Percentages and percentage change

What one hundred per cent means

A club's membership grows from $40$ to $80$. The new membership is $200\%$ of the old membership. Has it increased by $200\%$? Keep the two claims separate: by the end you will be able to explain exactly what happened.

Imagine one full bottle holds $500\,\mathrm{mL}$. Half a bottle is $50\%$ of that capacity. One and a half bottles hold $150\%$. Nothing has gone wrong with the percentage: you have more than one whole bottle's worth.

The same reference amount must stay fixed while you interpret the percentage. A bottle cannot be more than completely full, but a jug can hold more than $100\%$ of that bottle's capacity.

[DIAGRAM: asset_name: Percentages on a bottle-capacity scale; asset_slug: aqa8300_r04_percentage_scale; caption: With 500 millilitres as the whole, 750 millilitres is 150 per cent; description: Three horizontal bars share a scale of 0 to 750 millilitres. The first blue bar reaches 250 millilitres and is labelled 50 per cent and one half. The second blue bar reaches 500 millilitres and is labelled 100 per cent and one whole. The third bar has a blue section to 500 millilitres and a sage extension to 750 millilitres, labelled 150 per cent and one and a half. A dashed vertical line at 500 millilitres marks the fixed reference amount.]

Multiplying by $1$ keeps an amount unchanged. A multiplier above $1$ makes a positive amount larger; a multiplier between $0$ and $1$ makes it smaller.

Zero per cent gives none of a positive reference amount. One hundred per cent gives exactly that amount. More than one hundred per cent means more than the amount, rather than an impossible percentage.

The next check separates a percentage of an amount from a number simply added to it.

Expressing and comparing amounts as percentages

To express one amount as a percentage of another, divide by the reference amount first.

Suppose $18$ of $24$ seedlings survive. The fraction surviving is $\frac{18}{24} = \frac{3}{4}$, so $75$ out of every $100$ would survive at that same rate. The percentage is $75\%$. Both amounts must use the same units: convert metres to centimetres, for example, before dividing them.

Percentages make comparisons fair when the groups have different sizes. A larger number of successful seedlings need not mean a larger success rate.

The next example keeps the two surviving counts but changes what they are compared with. The reference is now the other grower's surviving group, so the two percentages are different.

The phrase after “of” decides the denominator. Write it down before reaching for a calculator.

Use the same decision with two lengths, expressing the first length as a percentage of the second.

Increasing and decreasing an amount

An increase adds to the original whole; a decrease removes part of it.

Start with $\pounds 80$. An increase of $15\%$ adds $\pounds 12$, making $\pounds 92$. You now have $115\%$ of the original amount, or $1.15$ times it. A decrease of $15\%$ leaves $85\%$, or $0.85$ times it.