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Repeated Percentage Change


Finding a percentage of an amount is a useful skill when shopping as well as when calculating any changes in productivity, profit or investments. 

Amounts can change more than once. If you are going sale shopping you can often see that a product has had its price reduced and then reduced again. A sale that offers $20%$20% off and then $50%$50% off sounds great, right? Is it any different to offering $50%$50% then $20%$20% off?

Repeated percentage change can seem a bit misleading. If I wanted to buy a TV for $\$1000$$1000 that had been reduced by $20%$20% and then $50%$50% it would be easy to think you save $20%$20% of $1000$1000 ($\$200$$200) and then $50%$50% of $1000$1000 ($\$500$$500), which would be a saving of $\$700$$700. But this isn't actually correct! 

The reduction happens successively. So at first you save the $\$200$$200. So the TV was then priced at $\$800$$800. You can then save $50%$50%  of $\$800$$800 - which is $\$400$$400. So in total this is a saving of $\$600$$600.  

Of course amounts can increase or decrease.


Reduction/decreased/contracted means you need to subtract

Decreasing by $\left(x\right)%$(x)% means calculate $\left(100-x\right)%$(100x)%

Increase/grew/recovered means you need to add

Increasing by $\left(x\right)%$(x)% means calculate $\left(100+x\right)%$(100+x)%

Worked Examples

question 1

The price of a phone was increased by $40%$40% and then again by $40%$40%.

  1. What was the overall percentage increase?

  2. The overall percentage increase when the price is increased by $40%$40% twice is $2\times40%$2×40%. True or false?






A country's GDP (Gross Domestic Product) contracted by $4%$4% one year due to a drought, but then grew by $6%$6% the next year when the weather returned to normal.

  1. Express the country's GDP during the drought as a percentage of the previous year's GDP.

  2. Express the country's GDP in the year after the drought as a percentage of the drought affected GDP.

  3. Calculate the new GDP as a percentage of the original.

  4. Hence, evaluate the percentage change over the two years.

  5. Is this change an overall increase or decrease?





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