Percentage Multiplier

Statement

Percentage multipliers provide a quick way to increase or decrease a value by a given percentage in one calculation. Instead of first finding the percentage amount and then adding or subtracting it, we multiply directly by a multiplier:

• Increase by \(r\%\): Multiply by \(\left(1 + \frac{r}{100}\right)\).
• Decrease by \(r\%\): Multiply by \(\left(1 - \frac{r}{100}\right)\).

Why it’s true

• An increase of \(r\%\) means adding \(\frac{r}{100}\) of the original to itself. This is equivalent to multiplying by \(1 + \frac{r}{100}\).
• A decrease of \(r\%\) means subtracting \(\frac{r}{100}\) of the original. This is equivalent to multiplying by \(1 - \frac{r}{100}\).

Recipe (how to use it)

1. Identify the percentage change required (increase or decrease).
2. Convert it to a multiplier:
   • Increase: \(1 + \frac{r}{100}\).
   • Decrease: \(1 - \frac{r}{100}\).
3. Multiply the original value by this multiplier.

Spotting it

Whenever a question asks for a “new value after a percentage increase/decrease,” percentage multipliers are the quickest method. They are also essential for repeated changes such as compound interest and depreciation.

Common pairings

• Shopping discounts.
• Profit and tax calculations.
• Compound growth and decay.

Mini examples

1. Given: Increase £200 by 10%. Answer: \(200 \times 1.1 = 220\).
2. Given: Decrease £80 by 25%. Answer: \(80 \times 0.75 = 60\).

Pitfalls

• Adding the percentage amount instead of multiplying.
• Confusing increase with decrease (1+r/100 vs 1−r/100).
• Forgetting to express the percentage as a decimal fraction before using it.

Exam strategy

• Always convert percentage changes into multipliers first.
• For multiple successive changes, multiply all the multipliers together.
• Check your result: increases must be larger than the original, decreases smaller.

Summary

Percentage multipliers provide a one-step method to apply percentage increases and decreases. Use \(1+\frac{r}{100}\) for increases and \(1-\frac{r}{100}\) for decreases. This approach is fast, reliable, and essential for compound percentage problems.