Fractions, decimals and percentages: interpret fractions and percentages as operators

The idea

A fraction or percentage acts as an operator when applied to a quantity to produce a new quantity. A fraction such as 3/4 means divide the amount into 4 equal parts and take 3, equivalent to multiplying by 3/4. A percentage such as 35% means 35 parts per 100, operating on an amount by multiplying by 0.35. For example, 35% of £240 is 0.35 × £240 = £84, and 3/4 of 240 g is 240 × 3/4 = 180 g. The operator can be reversed: if a value has been increased by 20%, the original is recovered by dividing by 1.2, because 1.2 is the multiplier for a 20% increase. Recognising the multiplier form makes percentage change, reverse percentage and fraction-of-amount questions a single consistent method.

What earns the marks

• Interprets a fraction as an operator by multiplying the quantity by the fraction, for example 3/4 of 240 = 240 × 3/4 = 180.
• Interprets a percentage as an operator by converting it to a decimal or fraction multiplier, for example 35% of £240 = 0.35 × £240 = £84.
• Uses a single multiplier for increase or decrease, for example multiplying by 1.2 for a 20% increase and by 0.8 for a 20% decrease.
• Reverses an operator to find an original amount, for example dividing a value after a 20% increase by 1.2.
• Selects the correct multiplier when the percentage is greater than 100 or when successive changes are applied.

Common mistakes and how to put them right

• Adding or subtracting the percentage of the original instead of multiplying by a single multiplier.

  Put it right: Use 1 + r for an increase and 1 − r for a decrease, where r is the decimal rate.

• Dividing by the percentage instead of multiplying by its decimal equivalent.

  Put it right: Convert the percentage to a decimal first, so 35% becomes 0.35.

• Reversing a percentage change by subtracting the same percentage from the new value.

  Put it right: Divide by the multiplier, because the percentage was taken from the original amount.

• Treating a fraction operator as division only, for example reading 3/4 of 240 as 240 ÷ 3/4.

  Put it right: Multiply by the fraction, 240 × 3/4.

Exam tips

• Convert every percentage to a decimal multiplier before calculating, so increase, decrease and reverse problems use one consistent method.
• For a reverse percentage, write the multiplier equation first, for example original × 1.2 = new value, then rearrange.
• Check the size of the answer against the original: a percentage increase must give a larger value and a decrease a smaller value.