1MA1 · N2, N8, N12NumberFoundation & HigherGrades 2–6

Fractions

Fractions are exact in a way decimals often are not: $\frac{1}{3}$ is precisely one third, while $0.333$ is only close. That exactness is why the non-calculator paper leans on them heavily — adding mixed numbers, dividing fractions, and working backwards from a fraction of an unknown amount.

55 min Number Fundamentals

The big picture

Almost everything with fractions rests on one idea: multiplying or dividing the numerator and denominator by the same number gives an equivalent fraction. That is how you simplify, how you find common denominators, and how you compare.

Edexcel likes fractions in context — a tank that is two-fifths full, a share of a prize — and especially reverse problems, where you know what a fraction of something is worth and must find the whole.

What you'll learn

Simplify fractions and convert between mixed numbers and improper fractions
Add, subtract, multiply and divide fractions and mixed numbers
Find a fraction of an amount and express one quantity as a fraction of another
Solve reverse fraction problems

Equivalent fractions and mixed numbers

Divide the top and bottom by their HCF to simplify: $\frac{18}{24} = \frac{3}{4}$ .

A $mixed number$ like $2 \frac{3}{5}$ has a whole part and a fraction part. As an $improper fraction$ it is $\frac{2 \times 5 + 3}{5} = \frac{13}{5}$ .

To compare fractions, write them with a common denominator: $\frac{5}{8} = \frac{15}{24}$ and $\frac{2}{3} = \frac{16}{24}$ , so $\frac{2}{3}$ is larger.

Tip — Convert mixed numbers to improper fractions before multiplying or dividing. It removes a whole category of errors.

Adding and subtracting

Fractions can only be added or subtracted once they have the same denominator. Use the lowest common multiple of the denominators, convert, then combine the numerators.

With mixed numbers, either convert to improper fractions or deal with the whole parts and fraction parts separately — but take care when the fraction part would go negative.

1Improper fractions:

\frac{13}{4} - \frac{11}{6}

2LCM of 4 and 6 is 12:

\frac{39}{12} - \frac{22}{12} = \frac{17}{12}

3Convert back.

Answer

1 \frac{5}{12}

Your turnSame method, new numbers — try it before moving on.

Work out $2 \frac{1}{3} + 1 \frac{3}{4}$ . Give your answer as a mixed number.

Your answer

Type a mixed number like 3 5/12.

Multiplying and dividing

To multiply, multiply the numerators and multiply the denominators. Cancel common factors first to keep numbers small: $\frac{4}{9} \times \frac{15}{16} = \frac{1}{3} \times \frac{5}{4} = \frac{5}{12}$ .

To divide, keep the first fraction, change $\div$ to $\times$ , and flip the second fraction (use its reciprocal).

1Improper fractions:

\frac{8}{3} \div \frac{8}{7}

2Flip and multiply:

\frac{8}{3} \times \frac{7}{8}

3Cancel the 8s:

\frac{7}{3}

Answer

2 \frac{1}{3}

Your turnSame method, new numbers — try it before moving on.

Work out $1 \frac{3}{5} \times 2 \frac{1}{2}$ .

Your answer

Why this works

Dividing by $\frac{8}{7}$ asks "how many $\frac{8}{7}$ s fit into this?". Multiplying by $\frac{7}{8}$ gives the same answer because dividing by a number and multiplying by its reciprocal are the same operation.

Fractions of amounts and reverse problems

To find $\frac{3}{8}$ of $56$ : divide by the denominator to find one eighth ( $7$ ), then multiply by the numerator ( $21$ ).

To write one quantity as a fraction of another, put the part over the whole and simplify, using the same units: 45 minutes as a fraction of 2 hours is $\frac{45}{120} = \frac{3}{8}$ .

In a $reverse$ problem you know the value of a fraction and want the whole. Divide by the numerator to find one part, then multiply by the denominator.

1The 45 litres fill

\frac{4}{5} - \frac{2}{7} = \frac{28}{35} - \frac{10}{35} = \frac{18}{35}

of the tank.

2So

\frac{1}{35}

of the tank is

45 \div 18 = 2.5

litres.

3The whole tank is

35 \times 2.5

Answer

87.5

litres

Your turnSame method, new numbers — try it before moving on.

$\frac{3}{8}$ of a number is 45. What is the number?

Your answer

Tip — In reverse problems, write " $\frac{18}{35}$ of the tank $= 45$ litres" as a line of working. It shows the method clearly and prevents dividing by the wrong number.

Think like an examiner

Show the common denominator in addition and subtraction.
Give answers in their simplest form, as a mixed number if the question uses mixed numbers.
In reverse problems, show the fraction that corresponds to the known amount.

Fraction rules

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

Add with a common denominator.

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

Multiply straight across.

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

Divide by multiplying by the reciprocal.

Watch out for these

Adding tops and bottoms:

\frac{1}{2} + \frac{1}{3} = \frac{2}{5}

Use a common denominator:

\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Multiplying mixed numbers part by part:

2 \frac{1}{2} \times 1 \frac{1}{2} = 2 \frac{1}{4}

Convert first:

\frac{5}{2} \times \frac{3}{2} = \frac{15}{4} = 3 \frac{3}{4}

Flipping the first fraction when dividing.

Keep the first, flip the second.

In a reverse problem, multiplying the known amount by the fraction.

Divide by the numerator, then multiply by the denominator.

Stretch yourself

Sam spends $\frac{1}{4}$ of his money on a game and $\frac{2}{5}$ of what is left on a book. He has £27 remaining. How much did he start with?

Hint — The book fraction is of the remainder, not of the original amount. Work out what fraction of the original is left at the end.

Questions students ask

Do I always need the lowest common denominator?

Any common denominator works, but the lowest keeps numbers small and reduces simplifying at the end.

Why does dividing by a fraction less than 1 make the answer bigger?

You are counting how many small pieces fit in, and more small pieces fit than large ones.

Should I give answers as improper fractions or mixed numbers?

Either is usually accepted if simplified, but match the form used in the question when it gives a clue.

Key takeaways

Equivalent fractions let you simplify, compare and find common denominators.
Add and subtract only with a common denominator.
Multiply across; divide by multiplying by the reciprocal.
Convert mixed numbers to improper fractions before multiplying or dividing.
Reverse problems: find one part, then the whole.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Work out $4 \frac{2}{5} - 2 \frac{2}{3}$ . Give your answer as a mixed number.[2]

Question 2Grade 43 marks

Show that $1 \frac{3}{4} \div 2 \frac{1}{3} = \frac{3}{4}$ .[3]

Question 3Grade 63 marks

Tom spends $\frac{1}{3}$ of his wages on rent and $\frac{1}{4}$ of what is left on food. He has £390 left.

Work out Tom's wages.[3]

Unlimited practice

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Fractions

Equivalent fractions and mixed numbers

Adding and subtracting

Multiplying and dividing

Fractions of amounts and reverse problems

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Fractions

Equivalent fractions and mixed numbers

Adding and subtracting

Multiplying and dividing

Fractions of amounts and reverse problems

Practise it the way it is examined

Skill drill

Practise this topic

Related topics