GCSE Maths Practice: fractions

Understanding Division of Fractions

Dividing fractions often looks more complicated than it really is. The simple rule is: keep, change, flip. This means you keep the first fraction, change the division sign to multiplication, and flip (find the reciprocal of) the second fraction. Once you’ve done that, the question becomes a straightforward multiplication of fractions, which you can simplify easily.

Why “Keep, Change, Flip” Works

Dividing by a number means finding how many times that number fits into another. For example, dividing by \(\frac{1}{2}\) asks, “How many halves are in the quantity?” Since there are two halves in one whole, dividing by \(\frac{1}{2}\) is equivalent to multiplying by 2. The reciprocal rule generalises this concept for all fractions, ensuring consistent and accurate results.

Step-by-Step Method

1. Keep the first fraction the same.
2. Change the division sign to multiplication.
3. Flip the second fraction (swap numerator and denominator).
4. Multiply the numerators together and the denominators together.
5. Simplify the resulting fraction to its lowest terms.

Using these steps: \(\frac{3}{4}\div\frac{1}{2}=\frac{3}{4}\times\frac{2}{1}=\frac{6}{4}=\frac{3}{2}=1\tfrac{1}{2}\).

Worked Examples

• Example 1: \(\frac{2}{3}\div\frac{4}{5}\). Flip the second → \(\frac{5}{4}\). Then \(\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}=\frac{5}{6}\).
• Example 2: \(\frac{7}{8}\div\frac{1}{2}\). Flip the second → \(\frac{2}{1}\). Then \(\frac{7}{8}\times\frac{2}{1}=\frac{14}{8}=\frac{7}{4}=1\tfrac{3}{4}\).
• Example 3: \(\frac{5}{6}\div\frac{5}{12}\). Flip the second → \(\frac{12}{5}\). Multiply → \(\frac{5}{6}\times\frac{12}{5}=\frac{60}{30}=2\).
• Example 4 (whole number): \(2\div\frac{1}{3}=2\times3=6\).

Common Mistakes to Avoid

• Forgetting to flip the second fraction: This is the most common mistake. You must use the reciprocal.
• Dividing straight across: Division of fractions is not done by dividing top and bottom separately.
• Not simplifying the final answer: Always reduce to simplest form or write as a mixed number when appropriate.
• Flipping the wrong fraction: Only the second fraction (the divisor) is inverted.

Real-Life Applications

Dividing fractions is used in cooking, measurements, and scaling problems. For example, if a recipe calls for three-quarters of a litre of milk and you only want half the recipe, you calculate \(\frac{3}{4}\div2=\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\) litres. In physics and engineering, dividing fractions helps determine proportions and rates, such as velocity per unit time or chemical concentrations.

FAQs

Q1: What is a reciprocal?
A1: A reciprocal is what you get when you flip a fraction — for example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

Q2: Can you divide by a whole number?

A2: Yes. Write the whole number as a fraction over 1 and apply the same rule. For instance, \(\frac{3}{4}\div2=\frac{3}{4}\times\frac{1}{2}=\frac{3}{8}\).

Q3: What happens if you divide by 1?

A3: Dividing by 1 doesn’t change the value: \(\frac{3}{4}\div1=\frac{3}{4}\).

Study Tip

Remember the phrase “Keep, Change, Flip.” Write it at the top of your exam page if you often forget the rule. Practise dividing a mix of proper, improper, and whole-number fractions until it feels automatic. Strong fluency with this skill supports algebraic manipulation and ratio work throughout GCSE Maths.

By mastering fraction division, you’ll build confidence for more complex problems involving algebraic fractions, proportions, and real-world applications.