Percentage Problems

Percentage Problems

Percentages express a quantity as parts per hundred. They appear in three core problem types at GCSE: calculating a percentage of a quantity, finding a percentage change, and working backwards from a changed value to find the original (reverse percentage). A unified approach using multipliers handles all three efficiently.

The Multiplier Method

Any percentage operation corresponds to multiplying by a decimal:

  • Find 35% of a value: multiply by 0.35.
  • Increase by 20%: multiply by \( 1 + 0.20 = 1.20 \).
  • Decrease by 15%: multiply by \( 1 - 0.15 = 0.85 \).
  • Reverse a 20% increase (find the original): divide by 1.20.
  • Reverse a 15% decrease: divide by 0.85.

Percentage Change

\[ \text{Percentage change} = \frac{\text{change}}{\text{original}} \times 100 \]

The denominator is always the original value, never the new value.

Simple Interest

\[ \text{Interest} = \frac{P \times R \times T}{100} \]

where \( P \) = principal, \( R \) = rate (%), \( T \) = time (years). Simple interest adds the same fixed amount each year — it does not compound.

Worked Examples

Find 37.5% of £640. Find the value after a 12% increase on £850.
\[ 0.375 \times 640 = £240 \] \[ 1.12 \times 850 = £952 \]
Calculate the simple interest on £2,000 invested at 4.5% per annum for 3 years.
\[ \text{Interest} = \frac{2000 \times 4.5 \times 3}{100} = \frac{27{,}000}{100} = £270 \]
A jacket costs £80, reduced to £56. Find the percentage decrease. A salary rises from £24,000 to £26,400. Find the percentage increase.
\[ \text{Decrease} = \frac{80-56}{80} \times 100 = \frac{24}{80} \times 100 = 30\% \] \[ \text{Increase} = \frac{2400}{24000} \times 100 = 10\% \]
After a 30% increase, a price is £91. Find the original price. After a 25% reduction, a price is £135. Find the original.
\[ \text{Original} = 91 \div 1.30 = £70 \] \[ \text{Original} = 135 \div 0.75 = £180 \]
Common error: do NOT subtract 30% from £91 to find the original. The 30% was applied to the original, not to £91. Always divide by the multiplier.

 Key Takeaways

  • Multiplier for an increase of \( r\% \): \( 1 + r/100 \). Decrease: \( 1 - r/100 \).
  • Percentage change: \( \frac{\text{change}}{\text{original}} \times 100 \). The original is always the denominator.
  • Reverse percentage: divide the final value by the multiplier — do not subtract the percentage from the final value.
  • Simple interest: the same amount added each year — no compounding. Compound interest (R12) multiplies by the multiplier repeatedly.
  • Percentage profit/loss is always calculated as a percentage of the cost price (original), not the selling price.