Most tested M3.8

Calculating with Percentages

This topic covers percentages as a way to represent parts of a whole, and their application in calculating changes, comparisons, and interest. Mastering non-calculator percentage arithmetic is crucial for solving problems involving finance, statistics, and changes in physical quantities.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • A percentage is fundamentally a fraction out of 100. To perform calculations, always convert percentages to fractions or decimals.
    e.g., 40% = 40/100 = 2/5
    e.g., 40% = 0.4
  • Percentage changes are multiplicative. A 15% increase corresponds to multiplying by 1.15, while a 15% decrease means multiplying by 0.85.
  • To find the original value after a percentage change (reverse percentages), you must divide by the multiplier. For example, if a value was increased by 20% to give 120, the original is 120 / 1.20 = 100.
  • Percentages can be greater than 100%. For instance, if a company's profit is 250% of its initial investment, it means the profit is 2.5 times the investment.
  • Successive percentage changes do not simply add or subtract. A 10% increase followed by a 10% decrease results in a net decrease from the original value, as the second change is calculated on a larger base.

Formulae

Percentage Change = (New Value - Original Value) / Original Value × 100

To find the percentage increase or decrease between two values. A positive result is an increase, a negative result is a decrease.

New Value = Original Value × (1 + (Percentage Increase / 100))

To find the final amount after a percentage increase.

Original Value = New Value / (1 + (Percentage Change / 100))

To find the original amount before a percentage change occurred (reverse percentage problems).

Interest = Principal × Rate × Time

For simple interest calculations, where 'Rate' is the annual interest rate as a decimal (e.g., 5% = 0.05) and 'Time' is in years.

Definitions

Percentage
A representation of a number as a fraction of 100. The symbol '%' means 'per hundred'.
Percentage Change
The difference between a new and an old value, expressed as a percentage of the old value.
Simple Interest
Interest calculated only on the principal (original) amount of a loan or deposit, not on the interest accrued in previous periods.

Worked example

A research lab's budget was increased by 25%. After one year, due to cuts, the new budget was decreased by 30%. What was the overall percentage change in the budget from its original value?

  1. 1

    Let the original budget be B.

  2. 2

    The first change is a 25% increase.

    The multiplier is 1 + (25/100) = 1.25.

    The new budget is B × 1.25.

  3. 3

    The second change is a 30% decrease on this new amount.

    The multiplier is 1 - (30/100) = 0.70.

  4. 4

    The final budget is (B × 1.25) × 0.70.

  5. 5

    Calculate the combined multiplier:

    1.25 × 0.70.

    It's easier to use fractions:

    (5/4) × (7/10) = 35/40 = 7/8.

  6. 6

    Convert the final multiplier back to a decimal:

    7/8 = 0.875
  7. 7

    This multiplier of 0.875 means the final budget is 87.5% of the original.

    The change is 100% - 87.5% = 12.5%.

  8. 8

    Since the final value is smaller, this is a 12.5% decrease.

Answer: A 12.5% decrease

Common mistakes

  • ×Applying a percentage change to the wrong base value. In multi-step problems, remember each new percentage change applies to the most recent value, not the original one.
  • ×Incorrectly reversing a percentage change. To reverse a 20% increase, you must divide by 1.20, not multiply by 0.80. These are not inverse operations.
  • ×Making arithmetic errors with fraction-to-decimal conversions. For example, mixing up 1/6 (approx 16.7%) and 1/8 (12.5%) under time pressure.
  • ×Confusing percentage change with absolute change. A question might ask for the percentage profit, but a student might calculate only the monetary profit.

No-calculator tips

  • ✓Break down complex percentages into simpler 'building blocks'. To find 35% of a number, find 10%, multiply it by 3, then find 5% (half of 10%) and add them together.
  • ✓Convert percentages to their simplest fraction form before multiplying. Calculating (3/8) × 120 is much easier than 0.375 × 120.
  • ✓When comparing quantities, you don't always need the exact percentage. Simplifying the fractions (e.g., 27/200 vs 42/150) might be enough to see which is larger.
  • ✓For reverse percentages, use proportions. If a price after a 20% increase is £96, then 120% corresponds to £96. Find 10% by dividing by 12 (£8), then find the original 100% by multiplying by 10 (£80).

Read this topic in the official UAT-UK ESAT guide →

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