Percentage Change

Percentage Change

You may be given the original amount and the value after a percentage increase or decrease, and will need to find what the percentage change was.

\text{\textcolor{blue}{Percentage Change}} = \dfrac{\text{\textcolor{Red}{Change}}}{\text{\textcolor{green}{Original}}} \times \textcolor{black}{100}

If the answer is positive then the change was an increase. If the answer is negative then it was a decrease.

Example: Calculate the percentage change of the value of a boat when it goes down from \textcolor{blue}{£8,000} to \textcolor{blue}{£6,000}.

Firstly, calculate the difference, which is:

\textcolor{blue}{£6,000} - \textcolor{blue}{£8,000} = \textcolor{Red}{-£2,000}

Then, use the formula above:

\text{\textcolor{blue}{Percentage Change}} = \dfrac{\textcolor{Red}{-2000}}{\textcolor{green}{8000}} \times \textcolor{black}{100} = \textcolor{red}{-25\%}

So, the value of the boat has decreased by \textcolor{blue}{25\%}.

Percentage Increase

For a percentage increase, the decimal that you multiply the amount by will be greater than \bf{1}.

Example: Sue owns a vintage car. Last year it was worth \textcolor{blue}{£6,500}.

This year, the value of the car has increased by \textcolor{green}{5\%}.

What will be the value of the car this year?

Step 1: Add the percentage increase to \textcolor{blue}{100\%} and convert to a decimal to find the multiplier:

5\% + 100\% = 105\% = 105 \div 100 = \textcolor{green}{1.05}

Step 2: Multiply the original amount by the multiplier to find the new increased value:

\textcolor{blue}{£6,500} \times \textcolor{green}{1.05} = \textcolor{blue}{£6,825}

So,

the value of the car this year is \textcolor{blue}{£6,825}

Note: If you find it easier, you can find 5\% of £6,500:

\dfrac{5}{100} \times £6,500 = 5 \div 100 \times £6,500 = £325

and then add this on:

£6,500 + £325 = \textcolor{black}{£6,825}

Percentage Decrease

For a percentage decrease, the decimal that you multiply the amount by will be less than \bf{1}.

Example: Last year, a company made a profit of \textcolor{blue}{£30,000}. This year, the profits are down by \textcolor{red}{10\%}.

How much profit does the company make this year?

Step 1: Subtract the percentage decrease from \textcolor{blue}{100\%} and convert to a decimal to find the multiplier:

100\% - 10\% = 90\% = \textcolor{red}{0.90}

Step 2: Multiply the original amount by the multiplier to find the new decreased value:

\textcolor{blue}{£30,000} \times \textcolor{red}{0.90} = \textcolor{blue}{£27,000}

So,

this year, the company makes a profit of \textcolor{blue}{£27,000}.

Note: Like for percentage increase, you may find it easier to find the percentage of the number and then add this on.

Reverse Percentages

Sometimes we are given the result of a percentage change and have to work backwards to find the original value.

Example: Sheila buys a jacket in a sale. It is reduced by \textcolor{red}{30\%} down to a price of \textcolor{blue}{£56}. Work out the original price of the jacket.

Step 1: Calculate the cost of the jacket as a percentage of its original value. We know it has \textcolor{red}{30\%} off so:

100\% - \textcolor{red}{30\%} = \textcolor{blue}{70\%}

Step 2: Divide the cost by 70\% to find 1\% of the original value:

\begin{aligned} (\div \, 70) \, \, \, \, \, 70\% &= \textcolor{blue}{£56}  \,\,\,\,\, (\div \, 70) \\ 1\% &= \textcolor{blue}{£0.80} \end{aligned}

Step 3: Multiply by 100 to get 100\% and the original value:

\begin{aligned} (\times \, 100) \, \, \, \, \, 1\% &= £0.80  \,\,\,\,\, (\times \, 100) \\ 100\% &= \textcolor{blue}{£80} \end{aligned}