Simplifying and Using Ratios

A ratio is a way of comparing how much of one quantity there is compared with another. Ratios run quietly through almost every topic in OCR GCSE Mathematics (J560): recipes, mixing paint, map scales, sharing money and even gradients all rest on ratio reasoning. This lesson covers how to write a ratio, simplify it to its lowest terms, find equivalent ratios, link a ratio to fractions, and rewrite a ratio in the useful $1:n$ form. These skills appear on the non-calculator Paper 1 and the calculator Papers 2 and 3, on both tiers.

This lesson is mostly AO1 fluency — writing and simplifying ratios accurately — with AO2 reasoning when you justify which fraction of a whole a part represents, and AO3 problem-solving when a ratio is buried inside a wordier context.

Key Vocabulary

Term	Meaning
Ratio	A comparison of two or more quantities written with a colon, e.g. $3:2$
Part	Each number in a ratio counts a number of equal "parts"
Simplest form	A ratio with no common factor left, e.g. $4:6$ simplifies to $2:3$
Equivalent ratios	Ratios that describe the same comparison, e.g. $1:3 = 5:15$
Highest common factor (HCF)	The largest number that divides exactly into every part
Unit ratio	A ratio written as $1:n$ (or $n:1$ ) so one side is exactly $1$
Order	The sequence of the numbers; $3:2$ does not mean the same as $2:3$

Writing and Simplifying Ratios

A ratio has no units of its own — it only records relative size. If you are told there are $3$ teachers for every $2$ classes, the ratio of teachers to classes is $3:2$ , and that single statement is true whether the school has $30$ teachers or $300$ . Two things matter when you write a ratio. First, the two quantities must be in the same unit before you compare them; comparing $40$ centimetres with $2$ metres directly would be meaningless until both are in centimetres. Second, the order must match the wording: "teachers to classes" is $3:2$ , while "classes to teachers" would be $2:3$ . Getting the order wrong is one of the most common ways to lose a mark, so always read the question carefully and write the quantities in the order they are mentioned.

Once a ratio is written, you simplify it exactly as you simplify a fraction: divide every part by the highest common factor (HCF) of all the parts. A ratio is in its simplest form when the only number that divides into every part is $1$ . The reason this works is that multiplying or dividing every part by the same number does not change the comparison — it is the same idea as cancelling a fraction down. Throughout this lesson, get into the habit of checking, at the end of every simplification, that no further common factor remains.

Worked Example 1

Write down the ratio of $20$ minutes to $1$ hour in its simplest form.

First put both times in the same unit. $1$ hour $= 60$ minutes, so the ratio is $20:60$ .

The HCF of $20$ and $60$ is $20$ , so divide both parts by $20$ :

$20:60 = 1:3.$

Answer: $1:3$ .

Common error: writing $20:1$ by leaving the hour as " $1$ ". Always convert to a common unit first, or the comparison is meaningless.

Worked Example 2

Simplify the ratio $24:36$ .

The HCF of $24$ and $36$ is $12$ . Dividing both parts by $12$ :

$24:36 = 2:3.$

Answer: $2:3$ .

A quick check: $2:3$ has no common factor other than $1$ , so it is fully simplified.

Worked Example 3

Simplify the three-part ratio $15:25:40$ .

Find a factor common to all three parts. The HCF of $15$ , $25$ and $40$ is $5$ :

$15:25:40 = 3:5:8.$

Answer: $3:5:8$ .

Common error: dividing each part by a different number. You must divide every part by the same factor or you change the comparison.

Simplifying Ratios with Decimals or Fractions

If a ratio contains decimals, multiply every part by a power of $10$ to clear them; if it contains fractions, multiply every part by the lowest common denominator. Then simplify as usual.

Worked Example 4

Write the ratio $0.8:1.2$ in its simplest form.

Multiply both parts by $10$ to clear the decimals: $8:12$ . The HCF is $4$ :

$8:12 = 2:3.$

Answer: $2:3$ .

Worked Example 5

Write the ratio $\dfrac{1}{2}:\dfrac{2}{3}$ in its simplest form.

The lowest common denominator of $2$ and $3$ is $6$ . Multiply both parts by $6$ :

$\dfrac{1}{2} \times 6 = 3, \qquad \dfrac{2}{3} \times 6 = 4.$

So the ratio becomes $3:4$ , which has no common factor.

Answer: $3:4$ .

Equivalent Ratios

Two ratios are equivalent if you can get from one to the other by multiplying (or dividing) every part by the same number — just like equivalent fractions. This is the engine behind scaling recipes and reading map scales.

Ratio	Operation	Equivalent ratio
$2:5$	$\times 3$	$6:15$
$3:4$	$\times 5$	$15:20$
$18:12$	$\div 6$	$3:2$

Worked Example 6

In a fruit drink, orange and mango juice are mixed in the ratio $5:2$ . A jug contains $14$ ml of mango juice. Work out how much orange juice it contains.

The mango part is $2$ , and that represents $14$ ml. So one part is $14 \div 2 = 7$ ml.

Orange juice is $5$ parts: $5 \times 7 = 35$ ml.

Answer: $35$ ml of orange juice.

Ratios and Fractions

A ratio splits a whole into parts, so each quantity is a fraction of the total. This link between ratios and fractions is tested often, and confusing the two is one of the most frequent errors at GCSE. The key idea is the difference between part-to-part and part-to-whole: a ratio compares one part with another, but a fraction compares one part with the whole. If beads are red to blue in the ratio $3:4$ , then there are $3+4 = 7$ parts in total, so $\dfrac{3}{7}$ of the beads are red and $\dfrac{4}{7}$ are blue. Notice that the denominator is $7$ (the total), not $4$ (the other colour). A good way to remember this is to picture the beads laid out in $7$ equal groups: $3$ groups are red, so the fraction red is " $3$ out of $7$ ".

Worked Example 7

In a class the ratio of left-handed to right-handed students is $2:9$ . What fraction of the class is left-handed? Give a reason for your answer.

Total parts $= 2 + 9 = 11$ . Left-handed students are $2$ of those $11$ parts.

So the fraction is $\dfrac{2}{11}$ , because the ratio divides the class into $11$ equal parts and $2$ of them are left-handed.

Answer: $\dfrac{2}{11}$ .

Common error: writing $\dfrac{2}{9}$ . That compares left-handed to right-handed, not left-handed to the whole class. The denominator of a fraction must be the total number of parts.

The $1:n$ Form

Writing a ratio as $1:n$ makes comparisons and map scales easy, because it tells you "for every $1$ on the left there are $n$ on the right". To reach $1:n$ , divide both parts by the left-hand value. The value of $n$ does not have to be a whole number — it is often a decimal or a fraction, and that is perfectly acceptable as a final answer. This form is the natural language of map scales (where $1$ cm on the map stands for $n$ cm in reality), model-making (where $1$ unit on the model stands for $n$ units on the real object), and any "rate per one" comparison. The closely related $n:1$ form, which you reach by dividing both parts by the right-hand value instead, is useful when you want to compare against a single unit on the right.

Worked Example 8

Write the ratio $4:10$ in the form $1:n$ .

Divide both parts by $4$ :

$4:10 = 1:\dfrac{10}{4} = 1:2.5.$

Answer: $1:2.5$ .

A map drawn so that $1$ cm represents $2.5$ km would be described by exactly this kind of ratio.

Extended Worked Examples

Worked Example 9: A map-scale ratio

A map has a scale of $1:50{,}000$ . Two churches are $6$ cm apart on the map. Work out the real distance between them in kilometres.

The scale $1:50{,}000$ means $1$ cm on the map is $50{,}000$ cm in real life.

Real distance $= 6 \times 50{,}000 = 300{,}000$ cm.

Convert to metres: $300{,}000 \div 100 = 3{,}000$ m. Convert to kilometres: $3{,}000 \div 1{,}000 = 3$ km.

Answer: $3$ km.

Common error: stopping at $300{,}000$ cm. Always read the units the question asks for and convert all the way.

Worked Example 10: A ratio hidden in a worded problem

A recipe for mortar mixes cement and sand in the ratio $1:5$ by mass. A builder has $9$ kg of cement. Work out the mass of sand needed to use all of the cement.

The cement part is $1$ and represents $9$ kg, so one part is $9$ kg. Sand is $5$ parts:

$5 \times 9 = 45 \text{ kg}.$

Answer: $45$ kg of sand.

Worked Example 11: Simplifying a mixed-unit ratio

Write the ratio $450$ g to $1.2$ kg in its simplest form.

Convert to a common unit. $1.2$ kg $= 1{,}200$ g, so the ratio is $450:1{,}200$ .

The HCF of $450$ and $1{,}200$ is $150$ :

$450:1{,}200 = 3:8.$

Answer: $3:8$ .

Worked Example 12: Equivalent ratios with a missing value

The ratio $3:8$ is equivalent to the ratio $12:n$ . Work out the value of $n$ .

To get from $3$ to $12$ , you multiply by $4$ . Equivalent ratios are made by multiplying every part by the same number, so you must also multiply the second part by $4$ :

$n = 8 \times 4 = 32.$

Answer: $n = 32$ , so $3:8 = 12:32$ .

Common error: adding the same number to each part instead of multiplying. Adding $9$ to each part of $3:8$ gives $12:17$ , which is not equivalent — only multiplying (or dividing) every part by the same number preserves the comparison.

Worked Example 13: Combining two ratios

In an orchestra the ratio of strings to brass is $5:2$ , and the ratio of brass to woodwind is $3:4$ . Work out the ratio of strings to brass to woodwind.

Brass appears in both ratios but with different numbers ( $2$ and $3$ ). To join the ratios, make the brass figure the same in each by using the lowest common multiple of $2$ and $3$ , which is $6$ .

Scale the first ratio so brass is $6$ : multiply $5:2$ by $3$ to get $15:6$ .

Scale the second ratio so brass is $6$ : multiply $3:4$ by $2$ to get $6:8$ .

Now the brass figure matches, so combine: strings : brass : woodwind $= 15:6:8$ .

Answer: $15:6:8$ .

Worked Example 14: Using a unit ratio to compare

Two photographs are enlarged. Photo A uses the ratio (original : enlargement) of $2:7$ , and photo B uses $3:10$ . By writing each in the form $1:n$ , work out which photograph is enlarged by the greater factor.

Photo A: divide both parts by $2$ to get $1:3.5$ .

Photo B: divide both parts by $3$ to get $1:3.\overline{3}$ (that is, $1:3.33\ldots$ ).

Since $3.5 > 3.33\ldots$ , photo A has the greater enlargement factor.

Answer: photo A, because $1:3.5$ is a bigger enlargement than $1:3.33\ldots$ .

This shows why the $1:n$ form is so useful: once both ratios start with $1$ , you can compare the second numbers directly.

Answering at different grade levels

Specimen question modelled on the OCR J560 paper format: In a car park the ratio of cars to vans is $7:3$ . Write down what fraction of the vehicles are vans, and give a reason for your answer. (2 marks)

Grades 3–4 response: There are $7 + 3 = 10$ parts. Vans are $3$ parts, so the fraction is $\dfrac{3}{10}$ .

Grades 5–6 response: The ratio $7:3$ splits the vehicles into $7 + 3 = 10$ equal parts. Vans take up $3$ of those parts, so the fraction of vehicles that are vans is $\dfrac{3}{10}$ . The denominator is $10$ because that is the total number of parts.

Grades 7–9 response: Adding the parts gives a total of $10$ , so each "part" is $\dfrac{1}{10}$ of the car park. Vans are $3$ parts, hence $\dfrac{3}{10}$ of the vehicles. The reason the denominator is the total (not the other side of the ratio) is that a fraction compares a part to the whole, whereas the ratio $7:3$ compares one part to another; converting between them is just a matter of choosing the denominator.

Examiner-style commentary: the mark for the reason depends on saying the denominator is the total number of parts; the top band explicitly contrasts part-to-whole with part-to-part, which is the AO2 discriminator.

Answering at different grade levels

Specimen question modelled on the OCR J560 paper format: Write the ratio $1.5:2$ in its simplest form. (2 marks)

Grades 3–4 response: Multiply both by $2$ to get $3:4$ .

Grades 5–6 response: The decimal $1.5$ needs clearing, so multiply both parts by $2$ : $1.5 \times 2 = 3$ and $2 \times 2 = 4$ , giving $3:4$ . Since $3$ and $4$ share no common factor, this is simplest form.

Grades 7–9 response: Multiplying by $2$ clears the decimal in one step: $1.5:2 = 3:4$ . I chose $2$ rather than $10$ because $1.5$ has only one decimal place that doubling removes, keeping the numbers small. As $\gcd(3,4)=1$ , the ratio is fully simplified.

Examiner-style commentary: all three reach $3:4$ , but the higher bands justify the choice of multiplier and confirm there is no remaining common factor — communication marks reward that final check.

Answering at different grade levels

Specimen question modelled on the OCR J560 paper format: A model car is built to a scale of $1:43$ . The real car is $4.3$ m long. Work out the length of the model in centimetres. (3 marks)

Grades 3–4 response: $4.3$ m $= 430$ cm. Divide by $43$ : $430 \div 43 = 10$ cm.

Grades 5–6 response: The scale $1:43$ means the real car is $43$ times the model. Converting the real length, $4.3$ m $= 430$ cm. So the model length is $430 \div 43 = 10$ cm.

Grades 7–9 response: With scale $1:43$ , model length $=$ real length $\div 43$ . Working in consistent units, $4.3$ m $= 430$ cm, so the model is $430 \div 43 = 10$ cm. The key step is converting to a single unit before dividing; leaving the real length in metres would give $0.1$ m, which is the same length but in the wrong unit for the question.

Examiner-style commentary: the discriminator is handling the unit conversion cleanly; the top band notes that $0.1$ m and $10$ cm are equal, showing the conversion is understood rather than mechanical.

Common Mistakes and Misconceptions

Forgetting to convert to a common unit before writing a ratio (e.g. mixing cm and m).
Dividing the parts of a ratio by different numbers when simplifying.
Confusing a ratio with a fraction: in $3:4$ the first quantity is $\dfrac{3}{7}$ of the whole, not $\dfrac{3}{4}$ .
Reversing the order, e.g. writing red to blue as blue to red.
Leaving a ratio unsimplified when the question asks for "simplest form".
In the $1:n$ form, dividing by the wrong part (always divide by the value you want to become $1$ ).

Going Further

Ratios in $1:n$ form are exactly how cartographers, model-makers and architects record scale, and the same idea reappears in Geometry as the scale factor of similar shapes and as the gradient of a line (a ratio of vertical to horizontal change). In Chemistry, reacting masses are read straight from ratios, and in Design and Technology, gear ratios decide how fast a wheel turns. Spotting that a wordy problem is "really" a ratio is one of the most transferable skills in the whole GCSE.

Exam Tips

AO1 (use and apply standard techniques): convert to a common unit, then divide every part by the HCF. Show the factor you divided by so the method is visible.
AO2 (reason, interpret and communicate): when asked to express a part as a fraction, state that the denominator is the total number of parts — that sentence often carries the reasoning mark.
AO3 (solve problems): in worded contexts, identify which quantity matches which part of the ratio before calculating; a quick labelled sketch or table prevents mixing them up.
Watch the OCR command words: "Write down" expects a quick answer with no working; "Work out" and "Calculate" expect method; "Give a reason for your answer" expects a short justification, not just a number.

Summary

A ratio compares quantities of the same unit and carries no units of its own.
Simplify by dividing every part by the HCF; clear decimals or fractions first.
Equivalent ratios come from multiplying or dividing every part by the same number.
A part of a ratio is a fraction of the total: in $a:b$ the first quantity is $\dfrac{a}{a+b}$ .
The $1:n$ form (divide by the left part) is the natural language of map scales and models.
Always check there is no common factor left, and that your answer is in the unit the question asks for.

This content is aligned with the OCR GCSE Mathematics (J560) specification.

Simplifying and Using Ratios

Simplifying and Using Ratios

Key Vocabulary

Writing and Simplifying Ratios

Worked Example 1

Worked Example 2

Worked Example 3

Simplifying Ratios with Decimals or Fractions

Worked Example 4

Worked Example 5

Equivalent Ratios

Worked Example 6

Ratios and Fractions

Worked Example 7

The 1:n1:n1:n Form

Worked Example 8

Extended Worked Examples

Worked Example 9: A map-scale ratio

Worked Example 10: A ratio hidden in a worded problem

Worked Example 11: Simplifying a mixed-unit ratio

Worked Example 12: Equivalent ratios with a missing value

Worked Example 13: Combining two ratios

Worked Example 14: Using a unit ratio to compare

Answering at different grade levels

Answering at different grade levels

Answering at different grade levels

Common Mistakes and Misconceptions

Going Further

Exam Tips

Summary

More in Mathematics

The $1:n$ Form