Fractions and ratios

Visual model

A ratio can be converted into fractions of the whole

3 parts2 parts

3:2\Rightarrow \frac{3}{5}\text{ and }\frac{2}{5}

whole = 5 equal parts

Gold-standard guide

20 mins

What you will learn

Connect ratios, fractions and proportions.

Use a clear step-by-step method for fractions and ratios.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier ratio, proportion and rates of change skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Fractions describe a part out of a whole

Step 1

Find the number who walk

$\frac{3}{8}$ × 40 = 15 students walk

Step 2

Find the number who do not walk

40 − 15 = 25 students do not walk

Step 3

Write the ratio

Walk : do not walk = 15:25

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers)

Core method

Fractions describe a part out of a whole

Exam check

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers)

Worked example

$\frac{3}{8}$ of a class of 40 students walk to school. Write the ratio of students who walk to students who do not walk.

Find the number who walk: $\frac{3}{8}$ × 40 = 15 students walk.

Find the number who do not walk: 40 − 15 = 25 students do not walk.

Write the ratio: Walk : do not walk = 15:25. Simplify by dividing by 5: 3:5.

Final answer

3:5

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

$\frac{3}{8}$ of a class of 40 students walk to school. Write the ratio of students who walk to students who do not walk.

3 marks4 minsfractions-and-ratios-worked

Show solution

Worked solution

1.Find the number who walk: $\frac{3}{8}$ × 40 = 15 students walk.
2.Find the number who do not walk: 40 − 15 = 25 students do not walk.
3.Write the ratio: Walk : do not walk = 15:25. Simplify by dividing by 5: 3:5.

Final answer

3:5

Mark points

M1: find the number who walk
M1: find the number who do not walk
M1: write the ratio
A1: 3:5

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Diagnosticrecall

$\frac{2}{5}$ of a bag of sweets are red. Write the ratio of red to non-red.

1 mark2 minsfractions-and-ratios-q1

Show solution

Worked solution

1.Spot the skill: Fractions describe a part out of a whole.
2.Use the find the number who walk stage first, then find the number who do not walk.
3.Keep the final answer visible: 2:3.

Final answer

2:3

Mark points

M1: use the correct fractions describe a part out of a whole. ratios compare two or more parts.to convert: if fraction = a/b, ratio of that part to rest = a:(b−a).to convert ratio a:b to a fraction of the total: each part is a/(a+b) or b/(a+b).
A1: 2:3

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Easyprocedure

In a ratio 3:7, what fraction of the total does the first part represent?

2 marks3 minsfractions-and-ratios-q2

Show solution

Worked solution

1.Spot the skill: Fractions describe a part out of a whole.
2.Use the find the number who do not walk stage first, then write the ratio.
3.Keep the final answer visible: $\frac{3}{1}0$ .

Final answer

$\frac{3}{1}0$

Mark points

M1: use the correct fractions describe a part out of a whole. ratios compare two or more parts.to convert: if fraction = a/b, ratio of that part to rest = a:(b−a).to convert ratio a:b to a fraction of the total: each part is a/(a+b) or b/(a+b).
A1: $\frac{3}{1}0$

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Mediumreasoning

A map scale is 1:25,000. Express as a fraction.

3 marks4 minsfractions-and-ratios-q3

Show solution

Worked solution

1.Spot the skill: Fractions describe a part out of a whole.
2.Use the write the ratio stage first, then find the number who walk.
3.Keep the final answer visible: $\frac{1}{2}5$ ,000.

Final answer

$\frac{1}{2}5$ ,000

Mark points

M1: use the correct fractions describe a part out of a whole. ratios compare two or more parts.to convert: if fraction = a/b, ratio of that part to rest = a:(b−a).to convert ratio a:b to a fraction of the total: each part is a/(a+b) or b/(a+b).
A1: $\frac{1}{2}5$ ,000

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Hardproblem solving

The ratio of boys to girls in a school is 7:9. What fraction of students are boys?

3 marks5 minsfractions-and-ratios-q4

Show solution

Worked solution

1.Spot the skill: Fractions describe a part out of a whole.
2.Use the find the number who walk stage first, then find the number who do not walk.
3.Keep the final answer visible: $\frac{7}{1}6$ .

Final answer

$\frac{7}{1}6$

Mark points

M1: use the correct fractions describe a part out of a whole. ratios compare two or more parts.to convert: if fraction = a/b, ratio of that part to rest = a:(b−a).to convert ratio a:b to a fraction of the total: each part is a/(a+b) or b/(a+b).
A1: $\frac{7}{1}6$

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Exam-stylemulti-step

A recipe uses flour and sugar in ratio 5:2. What fraction of the mix is sugar?

4 marks6 minsfractions-and-ratios-q5

Show solution

Worked solution

1.Spot the skill: Fractions describe a part out of a whole.
2.Use the find the number who do not walk stage first, then write the ratio.
3.Keep the final answer visible: $\frac{2}{7}$ .

Final answer

$\frac{2}{7}$

Mark points

M1: use the correct fractions describe a part out of a whole. ratios compare two or more parts.to convert: if fraction = a/b, ratio of that part to rest = a:(b−a).to convert ratio a:b to a fraction of the total: each part is a/(a+b) or b/(a+b).
A1: $\frac{2}{7}$

Watch out

Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Grade 9 stretchproblem solving

A:B = 3:5 and B:C = 2:7. Find A:B:C.

4 marks7 minsfraction-ratio-g9

Show solution

Worked solution

1.Find a common value for B.
2.Scale both ratios so B matches.
3.Combine the ratios.

Final answer

6:10:35

Mark points

M1: scale 3:5 to 6:10
M1: scale 2:7 to 10:35
A1: 6:10:35

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Fractions and ratios - 2 marks

\frac{2}{5}

of a bag of sweets are red. Write the ratio of red to non-red.Mark answer

Answer

2:3

2Ratio and sharing - 2 marksTwo people share profit in ratio 5:3. Total profit £640. Find each share.Mark answer

Answer

£400 and £240

3Direct and inverse proportion - 2 marksF is directly proportional to m × a. When m = 2 and a = 3, F = 24. Find F when m = 5 and a = 4.Mark answer

Answer

4Percentage change - 3 marksA car depreciates by 20% per year. It was worth £12,000 new. Find its value after 2 years.Mark answer

Answer

£7,680

Mastery check

I can explain the method for fractions and ratios.
I can show clear working without skipping key steps.
I can avoid this mistake: Students write the ratio as 3:8 (fraction numerator and denominator) rather than 3:5 (walkers to non-walkers).The ratio compares the two groups to each other, not the group to the whole.

Ask Nova Bot

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A ratio can be converted into fractions of the whole

What you will learn

Know the rule, then use it

Method

Find the number who walk

Find the number who do not walk

Write the ratio

Watch out

38\frac{3}{8}83​ of a class of 40 students walk to school. Write the ratio of students who walk to students who do not walk.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

$\frac{3}{8}$ of a class of 40 students walk to school. Write the ratio of students who walk to students who do not walk.