1MA1 · N2, N8, N10–N12NumberFoundation

Fractions

A fraction shows a part of a whole. Once you can find equivalent fractions, the four operations all become straightforward — the golden rule for adding and subtracting is a common denominator.

55 min Video by Maths Genie Pearson Edexcel GCSE Maths (1MA1)

The big picture

A fraction is simultaneously a number, a division and an operator. That is why fractions appear throughout ratio, probability, algebra and percentage problems. Fluency comes from understanding which representation makes the next step easiest—not from memorising disconnected tricks.

Adding and Subtracting FractionsWatch the walkthrough, then read the notes below.

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What you'll learn

Simplify fractions and find equivalent fractions
Add and subtract fractions
Multiply and divide fractions
Work with mixed numbers

Equivalent fractions

Multiplying or dividing top and bottom by the same number gives an equal fraction: $\frac{1}{2} = \frac{2}{4} = \frac{5}{10}$ . Dividing to the smallest whole numbers is called simplifying.

Why this works

Equivalent fractions name the same point on a number line. Multiplying numerator and denominator by the same non-zero value changes the name, not the value.

Adding and subtracting

You can only add or subtract fractions when the bottoms (denominators) are the same. Rewrite each fraction with a common denominator, then add or subtract the tops.

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

Common denominator, then add the numerators.

1Common denominator is 12.

\frac{1}{3} = \frac{4}{12}

\frac{1}{4} = \frac{3}{12}

\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

Answer

\frac{7}{12}

Why this works

Addition needs equal-sized parts, so denominators must match. Multiplication does not: it scales one fraction by another.

Tip — For multiplying, just multiply tops and bottoms. To divide, “keep, flip, multiply” (multiply by the reciprocal).

Fractions as operators in exam problems

“Find $rac{3}{5}$ of 40” means multiply: $rac{3}{5} imes40$ . Divide by the denominator to find one fifth, then multiply by the numerator.

When a fraction of an amount is known, reverse the operation to find the whole.

rac{3}{8}

corresponds to 18, so

rac{1}{8}=18div3=6

2The whole is

rac{8}{8}=8 imes6

Answer

48

students

Why this works

Finding one fractional part first is a unitary method. It remains reliable when the numbers become awkward or the context is unfamiliar.

Think like an examiner

Give exact fractional answers unless the question requests a decimal or a particular degree of accuracy.
Simplify the final fraction fully. A correct unsimplified value can lose the final accuracy mark when simplest form is required.
For mixed numbers, convert to improper fractions before multiplying or dividing, then convert back only if the context requires it.

Remember these

\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}

Multiply tops and bottoms.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Divide = multiply by the reciprocal.

Watch out for these

Adding fractions by adding tops and bottoms:

\frac{1}{3} + \frac{1}{4} = \frac{2}{7}

Use a common denominator first:

\frac{4}{12} + \frac{3}{12} = \frac{7}{12}

Flipping the wrong fraction when dividing.

Keep the first fraction, flip the second, then multiply.

Stretch yourself

A tank is $rac{2}{5}$ full. After 21 litres are added it is $rac{3}{4}$ full. Find the tank’s capacity.

Hint — The added 21 litres represents the difference between the two fractions.

Questions students ask

Why do I need a common denominator when adding but not multiplying?

Addition combines equal-sized parts, so the pieces must have the same size. Multiplication scales a quantity, so numerators and denominators can be multiplied directly.

Should I convert fractions to decimals?

Only when it makes the comparison or requested answer clearer. Fractions preserve exact values, while some decimal forms recur forever.

Key takeaways

Equivalent fractions: multiply/divide top and bottom by the same number.
Add/subtract only with a common denominator.
Multiply across; divide by flipping the second fraction.

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