Writing and Simplifying Ratio

What you'll learn

Write ratios in the correct order from words, units, and values.
Simplify ratios by dividing all parts by the same number.
Turn fraction information, such as $38\frac{3}{8}$ , into a ratio.
Work with forms like $n : 1$ , $1 : n$ , and simple three-part ratios.

1. What a ratio means

A ratio is a way to compare amounts. It tells you how much of one quantity there is compared with another quantity.

Definition

Ratio

A ratio compares quantities by showing how many parts of one thing match how many parts of another. The ratio of $A$ to $B$ is written as $A : B$ .

A part is one equal share in the comparison. For example, the ratio 2 : 5 means “2 parts of the first thing for every 5 parts of the second thing”.

The order matters. “Red to blue” is not the same as “blue to red”.

Key Idea

Order matters

Always write the ratio in the same order as the words in the question. If the question says “apples to oranges”, the apple amount must come first.

Example

Writing and simplifying a length ratio

A ribbon is 420 cm long and a cord is 30 cm long. Write the ratio of ribbon length to cord length in simplest form.

Check the units. Both quantities are in cm, so you can compare them directly.
Write the ratio in the order asked: ribbon to cord.
$420 : 30$
Divide both parts by 30:
$420 : 30 = 14 : 1$
The ratio of ribbon length to cord length is 14 : 1.

2. Simplifying a two-part ratio

To simplify a ratio, divide every part by the same whole number.

A common factor is a whole number that divides into each part exactly. The highest common factor, or HCF, is the biggest common factor.

Definition

Simplest form

A ratio is in simplest form when there is no whole number bigger than 1 that divides exactly into every part of the ratio.

Example

Simplifying a ratio

Simplify 84 : 36.

Look for a common factor of 84 and 36. Both numbers divide by 12.
Divide both parts by 12:
$84 : 36 = 7 : 3$
Since 7 and 3 have no common factor bigger than 1, the simplest form is 7 : 3.

Common Mistake

Dividing only one side

If you divide one part of a ratio, you must divide every part by the same number. For example, 84 : 36 cannot become 7 : 36.

3. Ratios from values, not just counts

Sometimes the question gives objects, but asks for the ratio of their values. You must find the total value of each group first.

For coins, this means multiplying the number of coins by the value of each coin.

Example

Ratio of coin values

Sam has four 20p coins and three 50p coins. Write the ratio of the value of the 20p coins to the value of the 50p coins.

Find the total value of the 20p coins. Four 20p coins are worth 80p.
Find the total value of the 50p coins. Three 50p coins are worth 150p.
Write the ratio in the order asked: value of 20p coins to value of 50p coins.
$80 : 150$
Divide both parts by 10:
$80 : 150 = 8 : 15$
The ratio is 8 : 15.

Common Mistake

Counting objects instead of values

If a question asks for the ratio of values, do not compare the number of coins. Four 20p coins to three 50p coins would be 4 : 3, but that is not the ratio of their values.

4. Turning a fraction into a ratio

A fraction tells you how many equal parts are in one group.

In a fraction such as $38\frac{3}{8}$ , the numerator is the top number, 3, and the denominator is the bottom number, 8. The denominator tells you the total number of equal parts.

The rest means everything not included in the named group. The bar below shows why a fraction can turn into a ratio: if 3 out of 8 parts are one type, then the remaining 5 parts are the other type.

Bar model showing 3 out of 8 parts becoming the ratio 3 to 5

Example

Fraction of a group to ratio

In a class, $29\frac{2}{9}$ of the students are left-handed. Write the ratio of left-handed students to right-handed students.

The denominator is 9, so imagine the whole class split into 9 equal parts.
The numerator is 2, so 2 parts are left-handed.
The rest are right-handed. Work out the remaining parts:
$9 - 2 = 7$
Write the ratio in the order asked: left-handed to right-handed.
$2 : 7$

5. Writing ratios in the form $n : 1$ or $1 : n$

Sometimes the question asks for a ratio in a special form.

$n : 1$ means the second part must be 1.
$1 : n$ means the first part must be 1.

To do this, divide both parts by the part you want to become 1.

Tip

Which part becomes 1?

For $n : 1$ , divide by the second part. For $1 : n$ , divide by the first part.

Example

Using $n : 1$ and $1 : n$

Work with these two ratios: write 10.5 : 3.5 in the form $n : 1$ , and write 14 : 35 in the form $1 : n$ .

For 10.5 : 3.5, make the second part 1, so divide both parts by 3.5:
$10.5 : 3.5 = 3 : 1$
For 14 : 35, make the first part 1, so divide both parts by 14:
$14 : 35 = 1 : 2.5$
So the values of $n$ are 3 and 2.5.

6. Three-part ratios

A three-part ratio compares three quantities, such as blue : red : yellow.

When a question says “twice as many” or “three times as many”, choose one group as a starting point and build the other parts from it.

Example

Building a three-part ratio

In a box, there are green, black and red pens. There are three times as many green pens as black pens. There are twice as many red pens as green pens. Write the ratio of green pens to black pens to red pens.

Start with black pens as 1 part.
Green pens are three times as many as black pens, so green pens are 3 parts.
Red pens are twice as many as green pens, so red pens are 6 parts.
Write the ratio in the order asked: green to black to red.
$3 : 1 : 6$

If you get a half part, multiply every part by 2 to make whole numbers.

Example

Making all parts whole numbers

There are blue, red and yellow sweets. The number of red sweets is three times the number of blue sweets. The number of yellow sweets is half the number of red sweets. Write the ratio blue : red : yellow using whole numbers.

Start with blue sweets as 1 part.
Red sweets are three times blue sweets, so red sweets are 3 parts.
Yellow sweets are half of red sweets, so yellow sweets are 1.5 parts.
The ratio is currently:
$1 : 3 : 1.5$
Multiply every part by 2 to remove the decimal:
$2 : 6 : 3$

7. Using totals, “the rest”, and percentages

Some ratio questions give a total number of people or objects. Use the information step by step, then subtract from the total to find “the rest”.

Example

Using a total to find a ratio

There are 96 people in a hall. Half are Year 11. The number of Year 11 people is three times the number of Year 10 people. The rest are Year 9. Find Year 9 : Year 10 in the form $n : 1$ .

Half of 96 is 48, so there are 48 Year 11 people.
Year 11 is three times Year 10, so Year 10 is 16 people.
Find the number of Year 9 people by subtracting the known groups from the total:
$96 - 48 - 16 = 32$
Write Year 9 : Year 10.
$32 : 16 = 2 : 1$
So $n = 2$ .

A percentage means “out of 100”. To find a percentage from a ratio, add the ratio parts to get the total number of parts, then find the fraction you need.

Example

Finding a percentage from ratio parts

In a bag, green counters are three times as many as orange counters. Purple counters are half as many as orange counters. Find the percentage of counters that are orange.

Choose orange counters as 2 parts, because then half of orange is a whole number.
Green counters are three times orange counters, so green counters are 6 parts.
Purple counters are half of orange counters, so purple counters are 1 part.
Add the parts to find the total number of parts:
$6 + 2 + 1 = 9$
Orange counters are 2 out of 9 parts, so the percentage is:
$29×100≈22.2%\frac{2}{9} \times 100 \approx 22.2\%$

Exam technique

In the exam

Check the order carefully: “A to B” means A comes first.
Make sure you are comparing the right quantities, such as values of coins rather than numbers of coins.
Simplify by dividing every part by the same number, and stop only when there is no common factor left.
For fraction questions, use the denominator as the total parts and subtract to find “the rest”.

Self review

Check yourself

If $37\frac{3}{7}$ of a group are red, what ratio method would you use to compare red to not red?
When writing a ratio in the form $1 : n$ , which part should you divide by?
In a three-part ratio, how do you deal with a part like 1.5?

Writing and Simplifying Ratio

What you'll learn

Write ratios in the correct order from words, units, and values.
Simplify ratios by dividing all parts by the same number.
Turn fraction information, such as $38\frac{3}{8}$ , into a ratio.
Work with forms like $n : 1$ , $1 : n$ , and simple three-part ratios.

1. What a ratio means

A ratio is a way to compare amounts. It tells you how much of one quantity there is compared with another quantity.

Definition

Ratio

A ratio compares quantities by showing how many parts of one thing match how many parts of another. The ratio of $A$ to $B$ is written as $A : B$ .

A part is one equal share in the comparison. For example, the ratio 2 : 5 means “2 parts of the first thing for every 5 parts of the second thing”.

The order matters. “Red to blue” is not the same as “blue to red”.

Key Idea

Order matters

Always write the ratio in the same order as the words in the question. If the question says “apples to oranges”, the apple amount must come first.

Example

Writing and simplifying a length ratio

A ribbon is 420 cm long and a cord is 30 cm long. Write the ratio of ribbon length to cord length in simplest form.

Check the units. Both quantities are in cm, so you can compare them directly.
Write the ratio in the order asked: ribbon to cord.
$420 : 30$
Divide both parts by 30:
$420 : 30 = 14 : 1$
The ratio of ribbon length to cord length is 14 : 1.

2. Simplifying a two-part ratio

To simplify a ratio, divide every part by the same whole number.

A common factor is a whole number that divides into each part exactly. The highest common factor, or HCF, is the biggest common factor.

Definition

Simplest form

A ratio is in simplest form when there is no whole number bigger than 1 that divides exactly into every part of the ratio.

Example

Simplifying a ratio

Simplify 84 : 36.

Look for a common factor of 84 and 36. Both numbers divide by 12.
Divide both parts by 12:
$84 : 36 = 7 : 3$
Since 7 and 3 have no common factor bigger than 1, the simplest form is 7 : 3.

Common Mistake

Dividing only one side

If you divide one part of a ratio, you must divide every part by the same number. For example, 84 : 36 cannot become 7 : 36.

3. Ratios from values, not just counts

Sometimes the question gives objects, but asks for the ratio of their values. You must find the total value of each group first.

For coins, this means multiplying the number of coins by the value of each coin.

Example

Ratio of coin values

Sam has four 20p coins and three 50p coins. Write the ratio of the value of the 20p coins to the value of the 50p coins.

Find the total value of the 20p coins. Four 20p coins are worth 80p.
Find the total value of the 50p coins. Three 50p coins are worth 150p.
Write the ratio in the order asked: value of 20p coins to value of 50p coins.
$80 : 150$
Divide both parts by 10:
$80 : 150 = 8 : 15$
The ratio is 8 : 15.

Common Mistake

Counting objects instead of values

If a question asks for the ratio of values, do not compare the number of coins. Four 20p coins to three 50p coins would be 4 : 3, but that is not the ratio of their values.

4. Turning a fraction into a ratio

A fraction tells you how many equal parts are in one group.

In a fraction such as $38\frac{3}{8}$ , the numerator is the top number, 3, and the denominator is the bottom number, 8. The denominator tells you the total number of equal parts.

Bar model showing 3 out of 8 parts becoming the ratio 3 to 5

Example

Fraction of a group to ratio

In a class, $29\frac{2}{9}$ of the students are left-handed. Write the ratio of left-handed students to right-handed students.

The denominator is 9, so imagine the whole class split into 9 equal parts.
The numerator is 2, so 2 parts are left-handed.
The rest are right-handed. Work out the remaining parts:
$9 - 2 = 7$
Write the ratio in the order asked: left-handed to right-handed.
$2 : 7$

5. Writing ratios in the form $n : 1$ or $1 : n$

Sometimes the question asks for a ratio in a special form.

$n : 1$ means the second part must be 1.
$1 : n$ means the first part must be 1.

To do this, divide both parts by the part you want to become 1.

Tip

Which part becomes 1?

For $n : 1$ , divide by the second part. For $1 : n$ , divide by the first part.

Example

Using $n : 1$ and $1 : n$

Work with these two ratios: write 10.5 : 3.5 in the form $n : 1$ , and write 14 : 35 in the form $1 : n$ .

For 10.5 : 3.5, make the second part 1, so divide both parts by 3.5:
$10.5 : 3.5 = 3 : 1$
For 14 : 35, make the first part 1, so divide both parts by 14:
$14 : 35 = 1 : 2.5$
So the values of $n$ are 3 and 2.5.

6. Three-part ratios

A three-part ratio compares three quantities, such as blue : red : yellow.

When a question says “twice as many” or “three times as many”, choose one group as a starting point and build the other parts from it.

Example

Building a three-part ratio

Start with black pens as 1 part.
Green pens are three times as many as black pens, so green pens are 3 parts.
Red pens are twice as many as green pens, so red pens are 6 parts.
Write the ratio in the order asked: green to black to red.
$3 : 1 : 6$

If you get a half part, multiply every part by 2 to make whole numbers.

Example

Making all parts whole numbers

Start with blue sweets as 1 part.
Red sweets are three times blue sweets, so red sweets are 3 parts.
Yellow sweets are half of red sweets, so yellow sweets are 1.5 parts.
The ratio is currently:
$1 : 3 : 1.5$
Multiply every part by 2 to remove the decimal:
$2 : 6 : 3$

7. Using totals, “the rest”, and percentages

Some ratio questions give a total number of people or objects. Use the information step by step, then subtract from the total to find “the rest”.

Example

Using a total to find a ratio

There are 96 people in a hall. Half are Year 11. The number of Year 11 people is three times the number of Year 10 people. The rest are Year 9. Find Year 9 : Year 10 in the form $n : 1$ .

Half of 96 is 48, so there are 48 Year 11 people.
Year 11 is three times Year 10, so Year 10 is 16 people.
Find the number of Year 9 people by subtracting the known groups from the total:
$96 - 48 - 16 = 32$
Write Year 9 : Year 10.
$32 : 16 = 2 : 1$
So $n = 2$ .

A percentage means “out of 100”. To find a percentage from a ratio, add the ratio parts to get the total number of parts, then find the fraction you need.

Example

Finding a percentage from ratio parts

In a bag, green counters are three times as many as orange counters. Purple counters are half as many as orange counters. Find the percentage of counters that are orange.

Choose orange counters as 2 parts, because then half of orange is a whole number.
Green counters are three times orange counters, so green counters are 6 parts.
Purple counters are half of orange counters, so purple counters are 1 part.
Add the parts to find the total number of parts:
$6 + 2 + 1 = 9$
Orange counters are 2 out of 9 parts, so the percentage is:
$29×100≈22.2%\frac{2}{9} \times 100 \approx 22.2\%$

Exam technique

In the exam

Check the order carefully: “A to B” means A comes first.
Make sure you are comparing the right quantities, such as values of coins rather than numbers of coins.
Simplify by dividing every part by the same number, and stop only when there is no common factor left.
For fraction questions, use the denominator as the total parts and subtract to find “the rest”.

Self review

Check yourself

If $37\frac{3}{7}$ of a group are red, what ratio method would you use to compare red to not red?
When writing a ratio in the form $1 : n$ , which part should you divide by?
In a three-part ratio, how do you deal with a part like 1.5?

Writing and Simplifying Ratio

What you'll learn

1. What a ratio means

2. Simplifying a two-part ratio

3. Ratios from values, not just counts

4. Turning a fraction into a ratio

5. Writing ratios in the form $n : 1$ or $1 : n$

6. Three-part ratios

7. Using totals, “the rest”, and percentages

Recap questions

You've reached the end

Writing and Simplifying Ratio

What you'll learn

1. What a ratio means

2. Simplifying a two-part ratio

3. Ratios from values, not just counts

4. Turning a fraction into a ratio

5. Writing ratios in the form $n : 1$ or $1 : n$

6. Three-part ratios

7. Using totals, “the rest”, and percentages

Recap questions

You've reached the end

Writing and Simplifying Ratio

What you'll learn

1. What a ratio means

2. Simplifying a two-part ratio

3. Ratios from values, not just counts

4. Turning a fraction into a ratio

5. Writing ratios in the form n:1n:1n:1 or 1:n1:n1:n

6. Three-part ratios

7. Using totals, “the rest”, and percentages

Recap questions

You've reached the end

Writing and Simplifying Ratio Revision Guide

Writing and Simplifying Ratio

What you'll learn

1. What a ratio means

2. Simplifying a two-part ratio

3. Ratios from values, not just counts

4. Turning a fraction into a ratio

5. Writing ratios in the form n:1n:1n:1 or 1:n1:n1:n

6. Three-part ratios

7. Using totals, “the rest”, and percentages

Recap questions

You've reached the end

Writing and Simplifying Ratio Revision Guide

5. Writing ratios in the form $n : 1$ or $1 : n$

5. Writing ratios in the form $n : 1$ or $1 : n$