Percentage change
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Percentage change is one of the most common concepts tested in Quantitative Reasoning. You will likely have to calculate a percentage change several times on Test Day.
Percentage change is the rate of change in a value over time compared to the original value.
This means you must find the difference in values and compare this part to the whole, which is the original value.
You will use a three-part formula to solve for percentage change:
The difference is the actual increase or decrease, which equals the final value minus the original value.
Remember to divide the difference by the original value. If you mistakenly divide by the final value – which is an easy mistake when working quickly in the timed exam – you are likely to fall for a wrong answer trap.
Note that the ‘× 100’ is not a variable – it is a fixed step to convert the fraction into a percentage. You may prefer to move the decimal point two places to the right in your head, but take care to remember this step to avoid wrong answer traps.
The original value is normally the earlier value chronologically.
The final value is usually the later value chronologically.
When in doubt, ask yourself: Which value is the question starting from?
When you solve, you must start from the original value in order to relate it to the final value and to calculate the percentage change.
You can solve for several unknown values by using the percentage change formula:
- Percentage change = the formula’s result
- Difference = Final value – Original value
- Original value = the earlier number that you start from
- Final value = the later number that you end on
Notice that you need two of these numbers in order to solve using the formula to find any of the others by rearranging the formula.
This makes the percentage change formula extremely powerful for Quantitative Reasoning. You can use it widely and sometimes subtly on Test Day.
A key challenge is training yourself to recognise when to use the percentage change formula.
Most commonly, you will see an explicit reference to percentage change in the question stem. This could specify percentage increase or percentage decrease.
The original and final values may be positive or negative.
If the final value is more than the original value, it will be a percentage increase.
If the final value is less than the negative value, it will be a percentage decrease.
More subtly, you will see percentages as answer choices with question wording that implies a percentage change. Such wording could take many forms.
Remember to check the answer choice format as you could have the same wording with actual numbers – not percentages – as the answers. In such cases, you would simply add or subtract the actual numbers to find the absolute change.
By contrast, percentage change is a relative change. Be alert to this distinction to avoid maths errors on Test Day.
Use your notebook to keep track of your work as you can only save one number in the calculator memory.
If a percentage change question requires multiple steps, you may find it helps to write down intermediate figures with a brief notation in case you need to use them again later in the question.
Walkthrough question
Percentage change
Let's try a QR question using the percentage change formula.
Use the percentage change formula to solve for:
- The percentage change using the difference and the original.
- The original value using the percentage change and the difference.
- The difference using the percentage change and the original.
However, there is an extra step when you have the final value.
- You can subtract the original value from the final value to find the difference.
- You can subtract the difference from the final value to find the original value.
Be ready to rearrange the percentage change formula as needed, and watch out for any extra steps to ensure you do the right maths with the correct values
Sometimes, you must use the final value to solve for:
- The original value using the percentage change.
- The difference using the percentage change.
Use the two alternative formulae which convert the percentage change formula to make it faster to start with the final value.
Follow these essential guidelines for using these alternative formulae correctly:
- Convert the percentage change to decimal form by moving the decimal point two places to the left.
- Use the percentage increase formula when the final value is higher than the original value or when the question specifies a percentage increase.
- Use the percentage decrease formula when the final value is lower than the original value or when the question indicates a percentage decrease.
These alternative formulae internalise the algebra that you would otherwise need to use to solve using the standard percentage change formula.
You can always use algebra in Quantitative Reasoning, but you will improve pacing if you can find shortcuts as you practise.
The alternative formulae allow you to skip the algebra as it is already built into the formulae. Try to use these formulae in your practice so they become quick and familiar.
Walkthrough question
Percentage change with alternative formula
Now attempt this QR question with a large percentage change.
The percentage change formulae are powerful because you can use two of four possible values to solve for either of the other two:
- The original value
- The final value
- The difference between the original and final values
- The percentage change
Remember to use the percentage change in decimal form whenever you substitute into a formula.
Take care to move the decimal place appropriately.
Sometimes, QR will challenge you with a complex question involving percentage change.
You may have to solve for more than one percentage change, or find a percentage change as an intermediate step in solving towards a different goal.
As always, keep an eye on the goal to avoid falling for a wrong answer trap that omits the final step.
Walkthrough question
Complex calculation with percentage change
Let's work through a complex question using the percentage change formula.
Pace yourself as you practise percentage change questions.
Try to improve your speed and accuracy at using the percentage change formulae to set up before you calculate.
Aim to answer every percentage change correctly in 30 seconds or less. At first, this may be very challenging, but it can pay off with more marks in less time on Test Day.
Timed practice
Percentage change
Try to answer this percentage change question within 30 seconds.