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GCSE Edexcel Maths › Operations with Fractions (GCSE Maths)

Operations with Fractions (GCSE Maths)

Skill Check

Q1
Question 1

Calculate $\frac{13}{15} - \frac{4}{15}$.

Give your answer in its simplest form.

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SOLUTION

Step 1  ·  Subtract the numerators

Subtract the numerators:

$$13 - 4 = 9$$

The fraction is:

$$\frac{9}{15}$$

Step 2  ·  Simplify the fraction

Simplify by dividing the top and bottom by $3$:

$$\frac{3}{5}$$

Final Answer

$\frac{3}{5}$

Q2
Question 2

Calculate $\frac{8}{9} - \frac{2}{9}$.

Give your answer in its simplest form.

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SOLUTION

Step 1  ·  Subtract the numerators

Subtract the numerators:

$$8 - 2 = 6$$

The fraction is:

$$\frac{6}{9}$$

Step 2  ·  Simplify the fraction

Simplify by dividing the top and bottom by $3$:

$$\frac{6}{9} = \frac{2}{3}$$

Final Answer

$\frac{2}{3}$

Q3
Question 3

Calculate $\frac{7}{12} + \frac{4}{12}$.

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SOLUTION

Step 1  ·  Add the numerators

$$7 + 4 = 11$$

Step 2  ·  Form the fraction

The denominator remains $12$.

Final Answer

$\dfrac{11}{12}$

Q4
Question 4

Calculate $\frac{7}{10} - \frac{3}{10}$.

Give your answer in its simplest form.

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SOLUTION

Step 1  ·  Subtract the numerators

Subtract the numerators:

$$7 - 3 = 4$$

The fraction is:

$$\frac{4}{10}$$

Step 2  ·  Simplify the fraction

Simplify by dividing the top and bottom by $2$:

$$\frac{4}{10} = \frac{2}{5}$$

Final Answer

$\dfrac{2}{5}$

Q5
Question 5

Calculate $\frac{3}{8} + \frac{2}{8}$

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SOLUTION

Step 1  ·  Add the numerators

Add the numerators together while keeping the denominator the same:

$$3 + 2 = 5$$

Final Answer

$\frac{5}{8}$

Problem Solving

Q6
Question 6

Calculate $\frac{2}{5} + \frac{3}{10}$.

Give your answer in its simplest form.

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SOLUTION

Step 1  ·  Find a common denominator

Since $10$ is a multiple of $5$, use $10$ as the common denominator.

Step 2  ·  Convert the first fraction

$$\frac{2 \times 2}{5 \times 2} = \frac{4}{10}$$

Step 3  ·  Add the fractions

$$\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$$

Final Answer

$\frac{7}{10}$

Q7
Question 7

Calculate $\frac{7}{8} - \frac{1}{3}$.

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SOLUTION

Step 1  ·  Find a common denominator

To subtract the fractions, we need a common denominator for $8$ and $3$, which is $24$.

Step 2  ·  Convert the fractions

Convert each fraction to have a denominator of $24$:

$$\frac{7 \times 3}{8 \times 3} = \frac{21}{24} \quad \text{and} \quad \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$$

Step 3  ·  Subtract the fractions

Now, subtract the converted fractions:

$$\frac{21}{24} - \frac{8}{24} = \frac{13}{24}$$

Final Answer

$\frac{13}{24}$

Q8
Question 8

Calculate $\frac{3}{4} + \frac{1}{6}$.

Give your answer in its simplest form.

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SOLUTION

Step 1  ·  Find a common denominator

Find the lowest common denominator for $4$ and $6$, which is $12$.

Step 2  ·  Convert the fractions

Convert both fractions so they have a denominator of $12$:

$$\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$
$$\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$$

Step 3  ·  Add the fractions

Add the numerators together while keeping the denominator the same:

$$\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$$

Final Answer

$\frac{11}{12}$

Q9
Question 9

Calculate $\frac{4}{5} - \frac{1}{2}$.

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SOLUTION

Step 1  ·  Find a common denominator

Find a common denominator for $5$ and $2$, which is $10$.

Step 2  ·  Convert the fractions

Convert the fractions:

$$\frac{4 \times 2}{5 \times 2} = \frac{8}{10}$$
$$\frac{1 \times 5}{2 \times 5} = \frac{5}{10}$$

Step 3  ·  Subtract the fractions

Subtract:

$$\frac{8}{10} - \frac{5}{10} = \frac{3}{10}$$

Final Answer

$\frac{3}{10}$

Q10
Question 10

Calculate $\frac{1}{3} + \frac{1}{4}$.

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SOLUTION

Step 1  ·  Find a common denominator

The common denominator for $3$ and $4$ is $12$.

Step 2  ·  Convert the fractions

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

Step 3  ·  Add the numerators

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

Final Answer

$\dfrac{7}{12}$

Exam-Style Questions

Q11
Question 11

A school buys $12\frac{1}{2}$ litres of paint for an art project.

Each group of students requires $1\frac{1}{4}$ litres of paint.

How many groups can be fully supplied with paint?

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SOLUTION

Step 1  ·  Set up the division

$$12\frac{1}{2} \div 1\frac{1}{4}$$

Step 2  ·  Convert both mixed numbers to improper fractions

$$\frac{25}{2} \div \frac{5}{4}$$

Step 3  ·  Multiply by the reciprocal

$$\frac{25}{2} \times \frac{4}{5} = \frac{100}{10}$$

Step 4  ·  Simplify the fraction

$$\frac{100}{10} = 10$$

Final Answer

$10$

Q12
Question 12

A bag of flour weighs $5\frac{1}{4}$ kg.

A baker uses $2\frac{2}{3}$ kg to make a batch of bread.

How much flour is left in the bag?

Give your answer as a mixed number.

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SOLUTION

Step 1  ·  Set up and convert to improper fractions

Set up the subtraction:

$$5\frac{1}{4} - 2\frac{2}{3}$$

Convert both to improper fractions:

$$\frac{21}{4} - \frac{8}{3}$$

Step 2  ·  Find a common denominator

Find a common denominator ($12$):

$$\frac{63}{12} - \frac{32}{12}$$

Step 3  ·  Subtract the numerators

Subtract the numerators:

$$\frac{31}{12}$$

Step 4  ·  Convert back to a mixed number

Convert back to a mixed number ($31 \div 12 = 2$ remainder $7$):

$$2\frac{7}{12}\text{ kg}$$

Final Answer

$2\frac{7}{12}\text{ kg}$

Q13
Question 13

A water tank is $\frac{7}{8}$ full.

After watering the garden, the tank is $\frac{1}{4}$ full.

What fraction of the tank's capacity was used?

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SOLUTION

Step 1  ·  Set up the subtraction

$$\frac{7}{8} - \frac{1}{4}$$

Step 2  ·  Find a common denominator

Find a common denominator of $8$ for both fractions:

$$\frac{7}{8} - \frac{2}{8}$$

Step 3  ·  Subtract the numerators

Subtract the numerators while keeping the denominator the same:

$$\frac{7 - 2}{8} = \frac{5}{8}$$

Final Answer

$\frac{5}{8}$

Q14
Question 14

A piece of wood is $3\frac{1}{2}$ metres long.

You need pieces of length $\frac{3}{4}$ metre for a project.

How many full pieces can you cut?

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SOLUTION

Step 1  ·  Set up the division

$$3\frac{1}{2} \div \frac{3}{4}$$

Step 2  ·  Convert to an improper fraction

$$\frac{7}{2} \div \frac{3}{4}$$

Step 3  ·  Multiply by the reciprocal

$$\frac{7}{2} \times \frac{4}{3} = \frac{28}{6}$$

Step 4  ·  Simplify the fraction

$$\frac{14}{3} = 4\frac{2}{3}$$

Step 5  ·  Interpret the result

The question asks for full pieces, which corresponds to the whole number part of $4\frac{2}{3}$.

Final Answer

$4$

Q15
Question 15

Harmeet baked $\frac{1}{3}$ of a tray of brownies in the morning for an event in St Albans, and $\frac{2}{5}$ of a tray in the afternoon.

How much of a full tray did he bake in total?

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SOLUTION

Step 1  ·  Set up the addition

$$\dfrac{1}{3} + \dfrac{2}{5}$$

Step 2  ·  Find a common denominator

Find a common denominator of $15$ to make the fractions equivalent:

$$\dfrac{5}{15} + \dfrac{6}{15}$$

Step 3  ·  Add the numerators

Add the numerators together while keeping the denominator the same:

$$\dfrac{11}{15}$$

Final Answer

$\dfrac{11}{15}$

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