These questions give you a value after a percentage rise or fall and ask for the value before the change. The essential choice is to work backwards from the final value without treating it as the percentage base.
Turn the change into an equation
First decide what fraction of the original remains in the final value. After an increase of p%, the final value is original × (1 + p/100). After a decrease of p%, it is original × (1 - p/100). Then rearrange the equation: original = final ÷ multiplier. For example, if a value is 154 after a 10% rise, the multiplier is 1.10 and the original is 154 ÷ 1.10 = 140. If a value is 126 after a 10% fall, the multiplier is 0.90 and the original is 126 ÷ 0.90 = 140.
| Signal | Form to use |
|---|---|
| Final value after a p% rise | Original = final ÷ (1 + p/100) |
| Final value after a p% fall, discount or deduction | Original = final ÷ (1 - p/100) |
Use a quick direction check before finishing. Reversing a rise must produce an original smaller than the final value. Reversing a fall must produce an original larger than the final value. Finally, apply the stated change to your result; it should reproduce the given final value exactly.
The traps
| Tempting route | How to reject it |
|---|---|
| Add the stated percentage of the final value after a fall | The fall was calculated from the larger original value, so a percentage of the smaller final value cannot replace what was removed. Divide by the remaining multiplier instead. |
| Multiply the final value by the forward change multiplier | That applies another change to the final value. Reversing multiplication requires division by the same multiplier. |
| Subtract a percentage of the final value to reverse a rise | The original percentage rise used the smaller original base. Subtracting a percentage based on the enlarged final value goes too far. |
| Divide by p/100 rather than by the remaining or enlarged multiplier | The percentage rate describes only the changed part. The final value represents 100 + p percent after a rise or 100 - p percent after a fall. |
| Calculate only p% of the final value | That produces a percentage amount, not the original whole. Build the final = original × multiplier equation before calculating. |
Try it
A cinema's Saturday attendance rose by 20% to 360 people. What was the attendance before the rise? Enter the number of people.
A water tank holds 560 litres after its capacity was reduced by 30%. What was the capacity before the reduction? Enter the number of litres.
A bakery now makes 150 rolls per batch after increasing its batch size by 25%. How many rolls were in the earlier batch? Enter the number of rolls.