Percentage Change

\( \%\,\text{change}=\frac{\text{new}-\text{old}}{\text{old}}\times100\% \)
Percentages GCSE
Question 4 of 20
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Weight decreases from 80 to 72. % change?

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
(new-old)/old*100.

Explanation

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Statement

The percentage change formula measures how much a value increases or decreases compared to its original value. It is defined as:

\[ \% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\% \]

If the result is positive, it represents a percentage increase. If negative, it represents a percentage decrease.

Why it’s true

  • The numerator \((\text{new} - \text{old})\) gives the absolute change.
  • Dividing by the old value compares the change relative to the original amount.
  • Multiplying by 100 converts the ratio into a percentage form.

Recipe (how to use it)

  1. Identify the original (old) and new values.
  2. Calculate the difference: new − old.
  3. Divide this difference by the old value.
  4. Multiply the result by 100 to express as a percentage.

Spotting it

Percentage change is commonly used in finance, shopping discounts, population growth, exam scores, and everyday comparisons of “before and after.”

Common pairings

  • Percentage increase and decrease problems.
  • Compound growth and depreciation.
  • Profit, loss, and discounts.

Mini examples

  1. Given: Price rises from £40 to £50. Find: percentage change. Answer: \((50-40)/40 \times 100 = 25\%\).
  2. Given: Population decreases from 200 to 160. Find: percentage change. Answer: \((160-200)/200 \times 100 = -20\%\), i.e. a 20% decrease.

Pitfalls

  • Dividing by the new value instead of the old one.
  • Forgetting to multiply by 100.
  • Not recognising that negative results mean decreases.

Exam strategy

  • Clearly identify which value is “old” and which is “new.”
  • Check the sign — positive for increase, negative for decrease.
  • Simplify fractions before multiplying by 100 when possible.

Summary

The percentage change formula compares the difference between new and old values relative to the old value, then converts this into a percentage. It is vital in handling real-world percentage problems, from shopping discounts to population growth and finance.

Worked examples

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  1. Price rises from £40 to £50. Find percentage change.
    1. \( Difference=50-40=10 \)
    2. \( Divide by old=10/40=0.25 \)
    3. \( 0.25*100=25% \)
    Answer: 25%
  2. Population decreases from 200 to 160. Find percentage change.
    1. \( Difference=160-200=-40 \)
    2. \( Divide by old=-40/200=-0.2 \)
    3. \( -0.2*100=-20% \)
    Answer: -20%
  3. A book price drops from £25 to £20. Find percentage change.
    1. \( 20-25=-5 \)
    2. \( -5/25=-0.2 \)
    3. \( -0.2*100=-20% \)
    Answer: -20%
  4. Exam score rises from 60 to 75. Find percentage change.
    1. \( 75-60=15 \)
    2. \( 15/60=0.25 \)
    3. \( 0.25*100=25% \)
    Answer: 25%
  5. Weight drops from 80kg to 72kg. Find percentage change.
    1. \( 72-80=-8 \)
    2. \( -8/80=-0.1 \)
    3. \( -0.1*100=-10% \)
    Answer: -10%
  6. Price rises from £120 to £138. Find percentage change.
    1. \( 138-120=18 \)
    2. \( 18/120=0.15 \)
    3. \( 0.15*100=15% \)
    Answer: 15%
  7. Car value decreases from £8000 to £6800. Find percentage change.
    1. \( 6800-8000=-1200 \)
    2. \( -1200/8000=-0.15 \)
    3. \( -0.15*100=-15% \)
    Answer: -15%
  8. Investment grows from £500 to £600. Find percentage change.
    1. \( 600-500=100 \)
    2. \( 100/500=0.2 \)
    3. \( 0.2*100=20% \)
    Answer: 20%
  9. Attendance falls from 350 to 280. Find percentage change.
    1. \( 280-350=-70 \)
    2. \( -70/350=-0.2 \)
    3. \( -0.2*100=-20% \)
    Answer: -20%
  10. Wages increase from £15/hour to £18/hour. Find percentage change.
    1. \( 18-15=3 \)
    2. \( 3/15=0.2 \)
    3. \( 0.2*100=20% \)
    Answer: 20%