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Simplifying ratiosDividing a quantity into a ratio

Ratios to fractionsScale ratios

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Here we will learn about ratio problem solving, including how to set up and solve problems. We will also look at real life ratio problems.

There are also ratio problem solving worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Ratio problem solving** is a collection of **word problems** that link together aspects of **ratio and proportion** into more** real life** questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.

A ratio is a **relationship between two or more quantities**. They are usually written in the form a:b where a and b are two quantities. When **problem solving** with a ratio, the key facts that you need to know are,

- What is the
**ratio**involved? - What
**order**are the quantities in the ratio? - What is the
**total amount**/ what is the**part of the total amount**known? - What are you trying to
**calculate**?

As with all problem solving, there is not one unique method to solve a problem. However, this does not mean that there aren’t **similarities** between different problems that we can use to help us find an answer.

The key to any problem solving is being able to **draw from prior knowledge** and use the correct piece of information to allow you to get to the next step and then the solution.

Let’s look at a couple of methods we can use when given certain pieces of information.

When solving ratio problems it is very important that you are able to use ratios. This includes being able to use ratio notation.

For example, Charlie and David share some sweets in the ratio of 3:5. This means that for every 3 sweets Charlie gets, David receives 5 sweets.

Charlie and David share 40 sweets, how many sweets do they each get?

We use the ratio to divide 40 sweets into 8 equal parts.

3:5

3 + 5 = 8

40 ÷ 8 = 5

Then we multiply each part of the ratio by 5.

3 x 5:5 x 5 = 15:25

This means that Charlie will get 15 sweets and David will get 25 sweets.

**Step-by-step guide:** Dividing ratios (coming soon)

You have been given
And you want to | Step 1: Add the parts of the ratio together. Step 2: Divide the quantity by the sum of the parts. Step 3: Multiply the share value by each part in the ratio. | For example
Share £100 in the ratio 4:1 . (£80:£20) |
---|---|---|

You have been given
And you want to find | Step 1: Identify which part of the ratio has been given. Step 2: Calculate the individual share value. Step 3: Multiply the other quantities in the ratio by the share value. | For example
A bag of sweets is shared between boys and girls in the ratio of 5:6. Each person receives the same number of sweets. If there are 15 boys, how many girls are there? (18) |

We also need to consider problems involving fractions. These are usually proportion questions where we are stating the proportion of the total amount as a fraction.

**Step-by-step guide:** Ratios and fractions (coming soon)

You have been given
And you want to find a fraction of the total | Step 1: Add the parts of the ratio for the denominator. Step 2: State the required part of the ratio as the numerator. | For example
The ratio of red to green counters is 3:5. What fraction of the counters are green? (\frac{5}{8}) |
---|---|---|

You have been given
And you want to find | Step 1: Subtract the numerator from the denominator of the fraction. Step 2: State the parts of the ratio in the correct order. | For example
if \frac{9}{10} students are right handed, write the ratio of right handed students to left handed students. (9:1) |

**Simplifying ratios**

You have been given
And you want to find | Step 1: Calculate the highest common factor of the parts of the ratio. Step 2: Divide each part of the ratio by the highest common factor. | For example
Simplify the ratio 10:15. (2:3) |
---|---|---|

**Equivalent ratios**

You have been given
And you want to find form 1:n | Step 1: Identify which part of the ratio is to equal 1. Step 2: Divide all parts of the ratio by this value. | For example
Write the ratio 4:15 in the form 1:n. (1:3.75) |
---|---|---|

You have been given
And you want to find | Step 1: Multiply all parts of the ratio by the same amount. | For example
A map uses the scale 1:500. How many centimetres in real life is 3cm on the map? (1:500 = 3:1500, so 1500 cm) |

Units and conversions are usually **equivalent ratio problems** (see above).

- If £1:\$1.37 and we wanted to convert £10 into dollars, we would multiply both sides of the ratio by 10 to get £10 is equivalent to \$13.70.
- The scale on a map is 1:25,000. I measure 12cm on the map. How far is this in real life, in kilometres? After multiplying both parts of the ratio by 12 you
**must**then convert 12 \times 25000=300000 \ cm to km by dividing the solution by 100 \ 000 to get 3km.

Notice that for all three of these examples, the **units** are important. For example if we write the mapping example as the ratio 4cm:1km, this means that 4cm on the map is 1km in real life.

**Top tip: **if you are converting units, always write the units in your ratio.

Usually with ratio problem solving questions, the problems are quite **wordy**. They can involve **missing values**, **calculating ratios**, **graphs**, **equivalent fractions**, **negative numbers**, **decimals** and **percentages**.

**Highlight** the important pieces of information from the question, know what you are trying to find or **calculate**, and use the **steps** above to help you start practising how to solve problems involving ratios.

In order to solve problems including ratios:

**Identify key information within the question.****Know what you are trying to calculate.****Use prior knowledge to structure a solution.**

Get your free ratio problem solving worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free ratio problem solving worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON**Ratio problem solving** is part of our series of lessons to support revision on **ratio**. You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Within a school, the number of students who have school dinners to packed lunches is 5:7. If 465 students have a school dinner, how many students have a packed lunch?

**Identify key information within the question.**

Within a school, the number of students who have **school dinners to packed lunches **is \bf{5:7.} If \bf{465} **students have a school dinner**, how many students have a **packed lunch**?

Here we can see that the ratio is 5:7 where the first part of the ratio represents school dinners (S) and the second part of the ratio represents packed lunches (P).

We could write this as

Where the letter above each part of the ratio links to the question.

We know that 465 students have school dinner.

2**Know what you are trying to calculate.**

From the question, we need to calculate the number of students that have a packed lunch, so we can now write a ratio below the ratio 5:7 that shows that we have 465 students who have school dinners, and p students who have a packed lunch.

We need to find the value of p.

3**Use prior knowledge to structure a solution.**

We are looking for an equivalent ratio to 5:7. So we need to calculate the multiplier. We do this by dividing the known values on the same side of the ratio by each other.

465\div 5 = 93So the value of p is equal to 7 \times 93=651.

There are 651 students that have a packed lunch.

The table below shows the currency conversions on one day.

Use the table above to convert £520 (GBP) to Euros € (EUR).

**Identify key information within the question.**

The table below shows the currency conversions on one day.

Use the table above to **convert ** \bf{£520} ** (GBP)** to** Euros ** \bf{€} ** (EUR).**

The two values in the table that are important are GBP and EUR. Writing this as a ratio, we can state

We know that we have £520.

**Know what you are trying to calculate.**

We need to convert GBP to EUR and so we are looking for an equivalent ratio with GBP = £520 and EUR = E.

**Use prior knowledge to structure a solution.**

To get from 1 to 520, we multiply by 520 and so to calculate the number of Euros for £520, we need to multiply 1.17 by 520.

1.17 \times 520=608.4

So £520 = €608.40.

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the 500ml of concentrated plant food must be diluted into 2l of water. Express the ratio of plant food to water respectively in the ratio 1:n.

**Identify key information within the question.**

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the** ** \bf{500ml} ** of concentrated plant food** must be diluted into \bf{2l} ** of water**. Express the ratio of **plant food to water **respectively as a ratio in the form 1:n.

Using the information in the question, we can now state the ratio of plant food to water as 500ml:2l. As we can convert litres into millilitres, we could convert 2l into millilitres by multiplying it by 1000.

2l = 2000ml

So we can also express the ratio as 500:2000 which will help us in later steps.

**Know what you are trying to calculate.**

We want to simplify the ratio 500:2000 into the form 1:n.

**Use prior knowledge to structure a solution.**

We need to find an equivalent ratio where the first part of the ratio is equal to 1. We can only do this by dividing both parts of the ratio by 500 (as 500 \div 500=1 ).

So the ratio of plant food to water in the form 1:n is 1:4.

Three siblings, Josh, Kieran and Luke, receive pocket money per week proportional to their age. Kieran is 3 years older than Josh. Luke is twice Josh’s age. If Josh receives £8 pocket money, how much money do the three siblings receive in total?

**Identify key information within the question.**

Three siblings, Josh, Kieran and Luke, receive pocket money per week proportional to their ages.** Kieran is ** \bf{3} ** years older than Josh**. **Luke is twice Josh’s age.** If **Luke receives ** \bf{£8} pocket money, how much money do the three siblings receive in** total**?

We can represent the ages of the three siblings as a ratio. Taking Josh as x years old, Kieran would therefore be x+3 years old, and Luke would be 2x years old. As a ratio, we have

We also know that Luke receives £8.

**Know what you are trying to calculate.**

We want to calculate the total amount of pocket money for the three siblings.

**Use prior knowledge to structure a solution.**

We need to find the value of x first. As Luke receives £8, we can state the equation 2x=8 and so x=4.

Now we know the value of x, we can substitute this value into the other parts of the ratio to obtain how much money the siblings each receive.

The total amount of pocket money is therefore 4+7+8=£19.

Below is a bar chart showing the results for the colours of counters in a bag.

Express this data as a ratio in its simplest form.

**Identify key information within the question.**

From the bar chart, we can read the frequencies to create the ratio.

**Know what you are trying to calculate.**

We need to simplify this ratio.

**Use prior knowledge to structure a solution.**

To simplify a ratio, we need to find the highest common factor of all the parts of the ratio. By listing the factors of each number, you can quickly see that the highest common factor is 2.

\begin{aligned}
&12 = 1, {\color{red} 2}, 3, 4, 6, 12 \\\\
&16 = 1, {\color{red} 2}, 4, 8, 16 \\\\
&10 = 1, {\color{red} 2}, 5, 10
\end{aligned}

HCF (12,16,10) = 2

Dividing all the parts of the ratio by 2 , we get

Our solution is 6:8:5 .

Glass is made from silica, lime and soda. The ratio of silica to lime is 15:2. The ratio of silica to soda is 5:1. State the ratio of silica:lime:soda.

**Identify key information within the question.**

Glass is made from silica, lime and soda. The ratio of **silica to lime is ** \bf{15:2.} The ratio of **silica to soda is ** \bf{5:1.} State the **ratio of silica:lime:soda**.

We know the two ratios

**Know what you are trying to calculate.**

We are trying to find the ratio of all 3 components: silica, lime and soda.

**Use prior knowledge to structure a solution.**

Using equivalent ratios we can say that the ratio of silica:soda is equivalent to 15:3 by multiplying the ratio by 3.

We now have the same amount of silica in both ratios and so we can now combine them to get the ratio 15:2:3.

India and Beau share some popcorn in the ratio of 5:2. If India has 75g more popcorn than Beau, what was the original quantity?

**Identify key information within the question.**

India and Beau share some popcorn in the ratio of \bf{5:2.} If **India** has \bf{75g} ** more** popcorn** than Beau**, what was the **original quantity?**

We know that the initial ratio is 5:2 and that India has three more parts than Beau.

We want to find the original quantity.

Drawing a bar model of this problem, we have

Where India has 5 equal shares, and Beau has 2 equal shares.

**Know what you are trying to calculate.**

Each share is the same value and so if we can find out this value, we can then find the total quantity.

**Use prior knowledge to structure a solution.**

From the question, India’s share is 75g more than Beau’s share so we can write this on the bar model.

We can find the value of one share by working out 75 \div 3=25g.

We can fill in each share to be 25g.

Adding up each share, we get

India = 5 \times 25=125g

Beau = 2 \times 25=50g

The total amount of popcorn was 125+50=175g.

**Mixing units**

Make sure that all the units in the ratio are the same. For example, in example 6 , all the units in the ratio were in millilitres. We did not mix ml and l in the ratio.

**Ratio written in the wrong order**

For example the number of dogs to cats is given as the ratio 12:13 but the solution is written as 13:12.

**Ratios and fractions confusion**

Take care when writing ratios as fractions and vice-versa. Most ratios we come across are part:part. The ratio here of red:yellow is 1:2. So the fraction which is red is \frac{1}{3} (not \frac{1}{2} ).

**Counting the number of parts in the ratio, not the total number of shares**

For example, the ratio 5:4 has 9 shares, and 2 parts. This is because the ratio contains 2 numbers but the sum of these parts (the number of shares) is 5+4=9. You need to find the value per share, so you need to use the 9 shares in your next line of working.

**Ratios of the form**\bf{1:n}

The assumption can be incorrectly made that n must be greater than 1 , but n can be any number, including a decimal.

1. An online shop sells board games and computer games. The ratio of board games to the total number of games sold in one month is 3:8. What is the ratio of board games to computer games?

3:5

3:11

5:3

8:3

8-3=5 computer games sold for every 3 board games.

2. The volume of gas is directly proportional to the temperature (in degrees Kelvin). A balloon contains 2.75l of gas and has a temperature of 18^{\circ}K. What is the volume of gas if the temperature increases to 45^{\circ}K?

29.75l

6.875l

1.1l

0.91l

\begin{aligned}
&2.75l:18^{\circ}K \\\\
&45 \div 18=2.5 \\\\
&2.75 \times 2.5=6.875l \\\\
\end{aligned}

3. The ratio of prime numbers to non-prime numbers from 1-200 is 45:155. Express this as a ratio in the form 1:n.

1:3

1:111

3.4:n

1:3\frac{4}{9}

155\div{45}=3\frac{4}{9}

4. The angles in a triangle are written as the ratio x:2x:3x. Calculate the size of each angle.

30^{\circ}, 60^{\circ}, 90^{\circ}

36^{\circ}, 72^{\circ}, 108^{\circ}

60^{\circ}, 120^{\circ}, 180^{\circ}

10^{\circ}, 20^{\circ}, 30^{\circ}

\begin{aligned}
&x+2x+3x=180 \\\\
&6x=180 \\\\
&x=30^{\circ} \\\\
&2x=60^{\circ} \\\\
&3x=90^{\circ}
\end{aligned}

5. A clothing company has a sale on tops, dresses and shoes. \frac{1}{5} of sales were for tops, \frac{1}{3} of sales were for dresses, and the rest were for shoes. Write a ratio of tops to dresses to shoes sold in its simplest form.

10:8:6

\frac{1}{3}:\frac{1}{5}:\frac{1}{8}

5:3:7

\frac{5}{15}:\frac{3}{15}:\frac{7}{15}

\begin{aligned}
&\frac{1}{3}+\frac{1}{5}=\frac{5+3}{15}=\frac{8}{15} \\\\
&1-\frac{8}{15}=\frac{7}{15}\\\\
&5:3:7
\end{aligned}

6. During one month, the weather was recorded into 3 categories: sunshine, cloud and rain. The ratio of sunshine to cloud was 2:3 and the ratio of cloud to rain was 9:11. State the ratio that compares sunshine:cloud:rain for the month.

2:12:11

6:18:11

2:9:11

6:9:11

\begin{aligned}
&3 \times S:C=6:9 \\\\
&6:9:11
\end{aligned}

1. One mole of water weighs 18 grams and contains 6.02 \times 10^{23} water molecules.

Write this in the form 1gram:n where n represents the number of water molecules in standard form.

**(3 marks)**

Show answer

6.02 \div 18=0.334 \ (3dp)

**(1)**

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2. A plank of wood is sawn into three pieces in the ratio 3:2:5. The first piece is 36cm shorter than the third piece.

Calculate the length of the plank of wood.

**(2 marks)**

Show answer

5-3=2 \ parts = 36cm so 1 \ part = 18cm

**(1)**

**(1)**

3. (a) Jenny is x years old. Sally is 4 years older than Jenny. Kim is twice Jenny’s age. Write their ages in a ratio J:S:K.

(b) Sally is 16 years younger than Kim. Calculate the sum of their ages.

**(6 marks)**

Show answer

(a)

x+4 or 2x

**(1)**

**(1)**

(b)

x+4+16=2x

**(1)**

**(1)**

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You have now learned how to:

- Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
- Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
- Make and use connections between different parts of mathematics to solve problems

- Proportion
- Compound measures
- Best buy maths

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