Percentages

Video coming soon3 worked examples

Theory

The Golden Rule: understanding the interrelationships between fractions, decimals, and percentages lets you choose the most efficient route to a solution, saving you vital time in the non-calculator paper.

1. Equivalences (Fractions, Decimals, and Percentages)

You must be able to convert equivalences between fractions, decimals, and percentages seamlessly. For example, you are expected to know that calculating 75% of an amount is exactly the same as finding $\frac{3}{4}$ of it. You must commit the following standard equivalences to memory:

$50\% = \frac{1}{2}$
$25\% = \frac{1}{4}$
$75\% = \frac{3}{4}$
$20\% = \frac{1}{5}$
$10\% = \frac{1}{10}$
$33 \frac{1}{3}\% = \frac{1}{3}$
$66 \frac{2}{3}\% = \frac{2}{3}$

2. Calculating Percentages of a Quantity (Non-Calculator)

More complex percentages can be worked out without a calculator by finding 10% or 1% first as "building blocks".

To find 10%, divide the amount by 10.
To find 1%, divide the amount by 100.
To find 5%, simply halve your 10% value.

By combining these blocks, you can find any amount. For example, to find 7.5%, you can calculate 5% (half of 10%) and 2.5% (half of 5%), and then add them together.

3. Expressing a Quantity as a Percentage

You will frequently need to express one quantity as a percentage of another. To find the percentage, there are three steps: write the quantities as a fraction, change it to a decimal by dividing, and then change it to a percentage by multiplying by 100.

Formula: $\frac{\text{Amount}}{\text{Total}} \times 100$

Without a calculator: Try to simplify the fraction or use equivalent fractions to multiply/divide the top and bottom until you obtain the number 100 on the bottom. The number left on the top is your percentage.

⚠️ Common Examiner Traps

The "Original Value" Trap: In questions asking for percentage increase, decrease, profit, or loss, you always have to work out the percentage based on the original amount. Candidates frequently drop marks by successfully finding the difference in price, but then mistakenly dividing it by the new/selling price.
The "Inefficient Route" Trap: In Paper 1, if you are asked to calculate $33 \frac{1}{3}\%$ or $66 \frac{2}{3}\%$ of an amount, do NOT attempt to find 1% and multiply. You are expected to spot the equivalence and simply divide the amount by 3 (and multiply by 2 for the latter).
The "Subtract from 100" Trap: A question may state how many items passed a quality check, but then ask you to calculate the percentage that were rejected. Always read the bold words carefully to ensure you are calculating the percentage of the correct group.

Worked examples

Example 1

A local theatre has a maximum capacity of 840 seats. For a Wednesday matinee performance, the theatre was only 17.5% full. Calculate exactly how many seats were occupied. (3 marks)

Step 1: Break 17.5% down into easily calculable blocks (10% + 5% + 2.5%).

Step 2: Calculate each block of 840.

$10\% = 840 \div 10 = 84$
$5\% = \text{Half of } 10\% = 42$
$2.5\% = \text{Half of } 5\% = 21$

Step 3: Add the building blocks together.

84 + 42 + 21 = 147\text{ seats occupied}

Example 2

A vet clinic treated 320 animals in one week. 144 of these animals were dogs. Calculate the percentage of the treated animals that were dogs. (2 marks)

Step 1: Write the quantities as a fraction.

\frac{144}{320}

Step 2: Convert to a decimal by dividing, and multiply by 100 to find the percentage.

(144 \div 320) \times 100 = 45\%

Example 3

A gamer buys a vintage games console online for £450. One year later, he decides to sell it to a second-hand shop for £387. Calculate his loss as a percentage of the original price. (3 marks)

Step 1: Calculate the actual financial loss.

\pounds450 - \pounds387 = \pounds63

Step 2: Express the loss as a fraction of the original purchase price, NOT the selling price.

\frac{63}{450}

Step 3: Convert to a percentage.

(63 \div 450) \times 100 = 14\%\text{ loss}

Ready to practise?

10 Percentages questions with answers and video solutions