FR10, HR10 - Percentage change and original values

FR10, HR10 - Percentage change and original values

Use a multiplier to increase or decrease an amount, measure a change against its original value, and work backwards from a changed amount. FR10 and HR10 connect these three uses of percentages so that you can choose the correct reference amount.

Increase and decrease

A £100100 price is reduced by 20%20\%, then increased by 20%20\%. Does it return to £100100? Each percentage applies to the amount at that stage, so we need to keep track of the changing whole.

An increase of 12%12\% retains the original 100%100\% and adds 12%12\%: the new amount is 112%=1.12112\%=1.12 of the original. A decrease of 12%12\% retains 88%=0.8888\%=0.88.

Percentage change multipliers

For original amount AA and percentage rr:

after an increase=A(1+r100)after a decrease=A(1−r100)\begin{aligned} \text{after an increase}&=A\left(1+\frac r{100}\right)\\ \text{after a decrease}&=A\left(1-\frac r{100}\right) \end{aligned}

For ordinary non-negative quantities, a decrease has 0≤r≤1000\le r\le100.

Not in the formula booklet: learn it.

The multiplier for an increase is above 11; for a decrease below 100%100\%, it is between 00 and 11. A 0%0\% change uses 11; a 100%100\% decrease uses 00. A 200%200\% increase means 300%300\% of the original, so its multiplier is 33, not 22.

Worked example 1: an increase

Question. Increase £250250 by 12%12\%.

Plan. Keep 112%112\% of the starting amount by multiplying by 1.121.12.

250×1.12=280250\times1.12=280

Answer. £280280.

Check. 12%12\% of £250250 is £3030; adding it gives £250+30=250+30= £280280.

Keep the starting amount and rate, but change the direction.

Worked example 2: a decrease

Question. Decrease £250250 by 12%12\%.

Plan. Keep 88%88\% by multiplying by 0.880.88.

250×0.88=220250\times0.88=220

Answer. £220220.

Check. Subtracting the same £3030 percentage amount gives £250−30=250-30= £220220. The result is below the original.

If a calculator is used, show the mathematical multiplier and calculation. Round a final money answer to the required penny accuracy, keeping full values through any intermediate steps.

Watch out

Multiplying by 0.120.12 finds the amount of a 12%12\% change. It does not find the new total. Use 1.121.12 for an increase and 0.880.88 for a decrease.

Find a new amount after a fresh reduction.

Percentage change

To find the percentage change, compare the amount gained or lost with the original amount. The original is the value before this change, even when a previous change has already taken place.

Percentage change

For a positive original amount:

percentage change=new amount−original amountoriginal amount×100%\text{percentage change} =\frac{\text{new amount}-\text{original amount}}{\text{original amount}}\times100\%

A positive result is an increase; a negative result is a decrease. You can instead calculate the positive size of a decrease and name its direction.

The reference amount matters: the same cash difference can be a different percentage when the direction is reversed.

Worked example 3: finding a percentage increase

Question. A price rises from £8080 to £100100. Work out the percentage increase.

Plan. Divide the increase by the original £8080.

increase=100−80=20percentage increase=2080×100%=25%\begin{aligned} \text{increase}&=100-80=20\\ \text{percentage increase}&=\frac{20}{80}\times100\%=25\% \end{aligned}

Answer. A 25%25\% increase.

Check. 80×1.25=10080\times1.25=100.

Reverse the same two prices. The size of the cash difference stays £2020, but the denominator changes.

Worked example 4: finding a percentage decrease

Question. A price falls from £100100 to £8080. Work out the percentage decrease.

Plan. Divide the reduction by the original £100100.

reduction=100−80=20percentage decrease=20100×100%=20%\begin{aligned} \text{reduction}&=100-80=20\\ \text{percentage decrease}&=\frac{20}{100}\times100\%=20\% \end{aligned}

Answer. A 20%20\% decrease.

Check. 100×0.8=80100\times0.8=80. Using the signed formula gives −20%-20\%, the same decrease.

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