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Ratio Questions And Practice Problems (KS3 & KS4): Harder GCSE Exam Style Questions Included

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Ratio questions appear throughout KS3 and KS4 building on what students have learnt in primary school. Here we provide a range of ratio questions and practice problems of varying complexity to use with your own students in class or as inspiration for creating your own. 

KS3 ratio questions

Ks4 ratio questions.

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What is ratio?

Ratio is used to compare the size of different parts of a whole. For example, in a whole class of 30 students there are 10 girls and 20 boys. The ratio of girls:boys is 10:20 or 1:2. For every one girl there are two boys. 

Uses of ratio

You might see ratios written on maps to show the scale of the map or telling you the currency exchange rate if you are going on holiday.

Ratio will be seen as a topic in its own right as well as appearing within other topics. An example of this might be the area of two shapes being in a given ratio or the angles of a shape being in a given ratio.

Ratio in KS3 and KS4

In KS3, ratio questions will involve writing and simplifying ratios, using equivalent ratios, dividing quantities into a given ratio and will begin to look at solving problems involving ratio. In KS4 these skills are recapped and the focus will be more on problem solving questions using your knowledge of ratio.

Download this 15 Ratio Questions And Practice Problems (KS3 & KS4) Worksheet

Help your students prepare for their Maths GSCE with this free Ratio worksheet of 15 multiple choice questions and answers.

Proportion and ratio

Ratio often appears alongside proportion and the two topics are related. Whereas ratio compares the size of different parts of a whole, proportion compares the size of one part with the whole. Given a ratio, we can find a proportion and vice versa.

Take the example of a box containing 7 counters; 3 red counters and 4 blue counters:

The ratio of red counters:blue counters is 3:4.

For every 3 red counters there are four blue counters.

The proportion of red counters is \frac{3}{7} and the proportion of blue counters is \frac{4}{7}

3 out of every 7 counters are red and 4 out of every 7 counters are blue.

Direct proportion and inverse proportion

In KS4, we learn about direct proportion and inverse proportion. When two things are directly proportional to each other, one can be written as a multiple of the other and therefore they increase at a fixed ratio. 

How to solve a ratio problem

When looking at a ratio problem, the key pieces of information that you need are what the ratio is, whether you have been given the whole amount or a part of the whole and what you are trying to work out. 

If you have been given the whole amount you can follow these steps to answer the question:

• Add together the parts of the ratio to find the total number of shares
• Divide the total amount by the total number of shares
• Multiply by the number of shares required

If you have been been given a part of the whole you can follow these steps:

• Identify which part you have been given and how many shares it is worth
• Use equivalent ratios to find the other parts
• Use the values you have to answer your problem

Ratio tables are another technique for solving ratio problems.

How to solve a proportion problem

As we have seen, ratio and proportion are strongly linked. If we are asked to find what proportion something is of a total, we need to identify the amount in question and the total amount. We can then write this as a fraction:

Proportion problems can often be solved using scaling. To do this you can follow these steps:

• Identify the values that you have been given which are proportional to each other
• Use division to find an equivalent relationship
• Use multiplication to find the required relationship

Real life ratio problems and proportion problems

Ratio is all around us. Let’s look at some examples of where we may see ratio and proportion:

Cooking ratio question

When making yoghurt, the ratio of starter yogurt to milk should be 1:9. I want to make 1000ml of yoghurt. How much milk should I use?

Here we know the full amount – 1000ml.

The ratio is 1:9 and we want to find the amount of milk.

• Total number of shares = 1 + 9 = 10
• Value of each share: 1000 ÷ 10 = 100
• The milk is 9 shares so 9 × 100 = 900

I need to use 900ml of milk.

Maps ratio question

The scale on a map is 1:10000. What distance would 3.5cm on the map represent in real life?

Here we know one part is 3.5. We can use equivalent ratios to find the other part.

The distance in real life would be 35000cm or 350m.

Speed proportion question

I travelled 60 miles in 2 hours. Assuming my speed doesn’t change, how far will I travel in 3 hours?

This is a proportion question.

• I travelled 60 miles in 2 hours.
• Dividing by 2, I travelled 30 miles in one hour
• Multiplying by 3, I would travel 90 miles in 3 hours

KS2 ratio questions

Ratio is introduced in KS2. Writing and simplifying ratios is explored and the idea of dividing quantities in a given ratio is introduced using word problems such as the question below, before being linked with the mathematical notation of ratio.

Example KS2 worded question

Richard has a bag of 30 sweets. Richard shares the sweets with a friend. For every 3 sweets Richard eats, he gives his friend 2 sweets. How many sweets do they each eat?

In KS3 ratio questions ask you to write and simplify a ratio, to divide quantities into a given ratio and to solve problems using equivalent ratios.

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Ratio questions year 7

1. In Lucy’s class there are 12 boys and 18 girls. Write the ratio of girls:boys in its simplest form.

The question asks for the ratio girls:boys so girls must be first and boys second. It also asks for the answer in its simplest form.

2. Gertie has two grandchildren, Jasmine, aged 2, and Holly, aged 4. Gertie divides £30 between them in the ratio of their ages. How much do they each get?

Jasmine £15, Holly £15

Jasmine £15, Holly £7.50

Jasmine £10, Holly £20

Jasmine £2, Holly £4

£30 is the whole amount.

Gertie divides £30 in the ratio 2:4.

The total number of shares is 2 + 4 = 6.

Each share is worth £30 ÷ 6 = £5.

Jasmine gets 2 shares, 2 x £5 = £10.

Holly gets 4 shares, 4 x £5 = £20.

Ratio questions year 8

3. The ratio of men:women working in a company is 3:5. What proportion of the employees are women?

In this company, the ratio of men:women is 3:5 so for every 3 men there are 5 women.

This means that for every 8 employees, 5 of them are women.

Therefore \frac{5}{8} of the employees are women.

4. The ratio of cups of flour:cups of water in a pizza dough recipe is 9:4. A pizza restaurant makes a large quantity of dough, using 36 cups of flour. How much water should they use?

The ratio of cups of flour:cups of water is 9:4. We have been given one part so we can work this out using equivalent ratios.

Ratio questions year 9

5. The angles in a triangle are in the ratio 3:4:5. Work out the size of each angle.

30^{\circ} , 40^{\circ} and 50^{\circ}

22.5^{\circ},  30^{\circ} and 37.5^{\circ}

60^{\circ} , 60^{\circ} and 60^{\circ}

45^{\circ} , 60^{\circ} and 75^{\circ}

The angles in a triangle add up to 180 ^{\circ} . Therefore 180 ^{\circ} is the whole and we need to divide 180 ^{\circ} in the ratio 3:4:5.

The total number of shares is 3 + 4 + 5 = 12.

Each share is worth 180 ÷ 12 = 15 ^{\circ} .

3 shares is 3 x 15 = 45 ^{\circ} .

4 shares is 4 x 15 = 60 ^{\circ} .

5 shares is 5 x 15 = 75 ^{\circ} .

6. Paint Pro makes pink paint by mixing red paint and white paint in the ratio 3:4.

Colour Co makes pink paint by mixing red paint and white paint in the ratio 5:7. 

Which company uses a higher proportion of red paint in their mixture?

They are the same

It is impossible to tell

The proportion of red paint for Paint Pro is \frac{3}{7}

The proportion of red paint for Colour Co is \frac{5}{12}

We can compare fractions by putting them over a common denominator using equivalent fractions

\frac{3}{7} = \frac{36}{84} \hspace{3cm} \frac{5}{12}=\frac{35}{84}

\frac{3}{7} is a bigger fraction so Paint Pro uses a higher proportion of red paint. 

In KS4 we apply the knowledge that we have of ratios to solve different problems. Ratio is an important topic in all exam boards, including Edexcel, AQA and OCR. Ratio questions can be linked with many different topics, for example similar shapes and probability, as well as appearing as problems in their own right.

Read more: Question Level Analysis Of Edexcel Maths Past Papers (Foundation)

Ratio GCSE exam questions foundation

7. The students in Ellie’s class walk, cycle or drive to school in the ratio 2:1:4. If 8 students walk, how many students are there in Ellie’s class altogether?

We have been given one part so we can work this out using equivalent ratios.

The total number of students is 8 + 4 + 16 = 28

8. A bag contains counters. 40% of the counters are red and the rest are yellow.

Write down the ratio of red counters:yellow counters. Give your answer in the form 1:n.

If 40% of the counters are red, 60% must be yellow and therefore the ratio of red counters:yellow counters is 40:60. Dividing both sides by 40 to get one on the left gives us

Since the question has asked for the ratio in the form 1:n, it is fine to have decimals in the ratio.

9. Rosie and Jim share some sweets in the ratio 5:7. If Jim gets 12 sweets more than Rosie, work out the number of sweets that Rosie gets.

Jim receives 2 shares more than Rosie, so 2 shares is equal to 12.

Therefore 1 share is equal to 6. Rosie receives 5 shares: 5 × 6 = 30.

10. Rahim is saving for a new bike which will cost £480. Rahim earns £1500 per month. Rahim spends his money on bills, food and extras in the ratio 8:3:4. Of the money he spends on extras, he spends 80% and puts 20% into his savings account.

How long will it take Rahim to save for his new bike?

Rahim’s earnings of £1500 are divided in the ratio of 8:3:4.

The total number of shares is 8 + 3 + 4 = 15.

Each share is worth £1500 ÷ 15 = £100 .

Rahim spends 4 shares on extras so 4 × £100 = £400 .

20% of £400 is £80.

The number of months it will take Rahim is £480 ÷ £80 = 6

Ratio GCSE exam questions higher

11. The ratio of milk chocolates:white chocolates in a box is 5:2. The ratio of milk chocolates:dark chocolates in the same box is 4:1.

If I choose one chocolate at random, what is the probability that that chocolate will be a milk chocolate?

To find the probability, we need to find the fraction of chocolates that are milk chocolates. We can look at this using equivalent ratios.

To make the ratios comparable, we need to make the number of shares of milk chocolate the same in both ratios. Since 20 is the LCM of 4 and 5 we will make them both into 20 parts.

We can now say that milk:white:dark is 20:8:5. The proportion of milk chocolates is \frac{20}{33} so the probability of choosing a milk chocolate is \frac{20}{33} .

12. In a school the ratio of girls:boys is 2:3. 

25% of the girls have school dinners.

30% of the boys have school dinners.

What  is the total percentage of students at the school who have school dinners?

In this question you are not given the number of students so it is best to think about it using percentages, starting with 100%.

100% in the ratio 2:3 is 40%:60% so 40% of the students are girls and 60% are boys.

25% of 40% is 10%.

30% of 60% is 18%.

The total percentage of students who have school dinners is 10 + 18 = 28%.

13. For the cuboid below, a:b = 3:1 and a:c = 1:2.

Find an expression for the volume of the cuboid in terms of a.

If a:b = 3:1 then b=\frac{1}{3}a

If a:c = 1:2 then c=2a.

Difficult ratio GCSE questions

14. Bill and Ben win some money in their local lottery. They share the money in the ratio 3:4. Ben decides to give £40 to his sister. The amount that Bill and Ben have is now in the ratio 6:7.

Calculate the total amount of money won by Bill and Ben.

Initially the ratio was 3:4 so Bill got £3a and Ben got £4a. Ben then gave away £40 so he had £(4a-40).

The new ratio is 3a:4a-40 and this is equal to the ratio 6:7.

Since 3a:4a-40 is equivalent to 6:7, 7 lots of 3a must be equal to 6 lots of 4a-40.

The initial amounts were 3a:4a. a is 80 so Bill received £240 and Ben received £320.

The total amount won was £560.

15. On a farm the ratio of pigs:goats is 4:1. The ratio of pigs:piglets is 1:6 and the ratio of gots:kids is 1:2.

What fraction of the animals on the farm are babies?

The easiest way to solve this is to think about fractions.

\\ \frac{4}{5} of the animals are pigs, \frac{1}{5} of the animals are goats.

\frac{1}{7} of the pigs are adult pigs, so  \frac{1}{7}   of  \frac{4}{5} is  \frac{1}{7} \times \frac{4}{5} = \frac{4}{35}

\frac{6}{7} of the pigs are piglets, so \frac{6}{7} of \frac{4}{5} is \frac{6}{7} \times \frac{4}{5} = \frac{24}{35}

\frac{1}{3}   of the goats are adult goats, so \frac{1}{3} of \frac{1}{5} is \frac{1}{3} \times \frac{1}{5} = \frac{1}{15}

\frac{2}{3}   of the goats are kids, so \frac{2}{3} of \frac{1}{5} is \frac{2}{3} \times \frac{1}{5} = \frac{2}{15}

The total fraction of baby animals is \frac{24}{35} + \frac{2}{15} = \frac{72}{105} +\frac{14}{105} = \frac{86}{105}

Looking for more ratio questions and resources?

Third Space Learning’s free GCSE maths resource library contains detailed lessons with step-by-step instructions on how to solve ratio problems, as well as worksheets with ratio and proportion practice questions and more GCSE exam questions.

Take a look at the Ratio lessons today – more are added every week.

If you’re also looking for other topics, try our GCSE maths questions blog!

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What Is a Ratio?

Different ways of presenting ratios, scaling a ratio, reducing a ratio, finding unknown quantities from existing equivalent ratios, common mistakes and things to look out for when solving ratio problems, frequently asked questions, how to solve questions about ratios (2023 guide).

Updated June 8, 2023

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A ratio is a mathematical term that is used to compare the size of one number to the size of another number. It is commonly used in both mathematics and in professional environments.

Here are some everyday examples of times when you could use ratios:

• When you convert your Pounds to Dollars or Euros when you go on holiday
• When you calculate your winnings on a bet
• When you work out how many bottles of beer you need for a party
• When you share a packet of sweets fairly among your friends
• When you calculate how much tax you must pay on your income

Ratios are usually used to compare two numbers, though they can also be used to compare multiple quantities.

Ratios are often included in numerical reasoning tests, where they can be presented in a number of different ways. It is therefore important that you are able to recognize and manipulate ratios however they are presented.

Ratios are usually shown as two or more numbers separated with a colon, for example, 8:5 or 1:4 or 3:2:1.

However, they can also be shown in a number of other ways; the three examples below are all different expressions of the same ratio.

One of the reasons that ratios are useful is that they enable us to scale amounts. This means increasing or decreasing the amount of something. This is particularly handy for things like scale models or maps, where really big numbers can be converted to much smaller representations that are still accurate.

Scaling is also helpful for increasing or decreasing the amount of ingredients in a recipe or chemical reaction. Ratios can be scaled up or down by multiplying both parts of the ratio by the same number.

For example:

There are six practice questions below. Should you need further practice afterwards, we recommend the numerical reasoning packages available from JobTestPrep . These tests include ratio questions, with full explanations for all answers.

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Question 1: Scaling Ratios

Olivia wants to make pancakes for nine friends, but her recipe only makes enough pancakes for three friends (the recipe is shown below).

What quantity of the ingredients will she need to use?

Pancake recipe (serves 3)

• 2 large eggs

Sometimes ratios are not presented in their simplest form, which makes them harder to manipulate.

For example, if a farmer has seven chickens and together they lay 56 eggs every day, this would be represented by the ratio 7:56 (or presented as a fraction this would be shown as 7/56).

Reducing a ratio means converting the ratio to its simplest form, and this makes it easier to use. This is done by dividing both numbers in of the ratio by the largest number that they can both be divided by (this is just the same as reducing a fraction to its simplest form).

So, for example:

Therefore, the reduced ratio is 1:8, which tells us that each chicken lays eight eggs. This is far easier to use. For example, if the farmer needed 96 eggs per day, he could use the ratio to work out how many chickens he would need for this to be possible.

We know from the ratio that each chicken lays 8 eggs, therefore, if the farmer wants 96 eggs he must divide 96 by 8 to calculate the number of chickens required:

96/8 = 12 chickens

Similarly, if one of the farmer’s chickens stopped laying eggs, he could use the ratio to work out how much that would reduce his total number of eggs to; for example, 56 – 8 = 48.

Question 2: Reducing Ratios

Ella has 18 pigeons and they eat 54kg of grain per week. Jayden has 22 pigeons and they eat 88kg or grain per week. Who has the greediest pigeons?

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One of the ways that ratios are particularly useful is that they enable us to work out new and unknown quantities based on an existing (known) ratio. There are a couple of ways of solving this type of problem. The first is to use cross-multiplication .

For example, there are four rings and seven bracelets in the jewellery box. If this ratio were maintained, how many necklaces would there be if there were 28 bracelets?

Another method is to calculate the factor (size) of the increase in number of bracelets (28/7 = 4) and to then multiply the number of rings by that same number (4 x 4 = 16).

Question 3: Finding Unknown Quantities From Existing Equivalent Ratios

Gabriel and Mandeep are getting married. They have calculated that they will need 35 bottles of wine for their 70 guests. Suddenly they find out that another 20 guests are planning to attend.

How much wine do they need in total?

Make sure you are reading the ratio the right way . For example, a ratio of 2 pink to 6 red should be expressed 2:6 not 6:2. The first item in the sentence comes first.

Be careful with reading the wording . For example, people often make mistakes with questions such as “Bob has eight pigs and four cows. Calculate the ratio of pigs to animals.” It can be tempting to say the ratio is 8:4, but that would be incorrect because the question asks for the ratio of pigs to animals . Therefore you need to calculate the total number of animals (8 + 4 = 12), so the correct ratio is, therefore, 8:12 (or 2:3).

Don’t be put off by units or decimals . The principles remain the same whether they apply to whole numbers, fractions, £ or m2, but do ensure you note the units in your calculations, and where possible convert them to the same units. For example, if you have a ratio of 500g to 0.75kg, then convert both sides to either grams or kilos.

Question 4: Scaling Ratios

Otto knows that his car requires 27.5 litres of fuel to drive 110 miles. He wants to drive from London to Edinburgh, a distance of 405 miles.

How much fuel will he need?

Question 5: Reducing Ratios

Hugo and Claudia share 600 g chocolate using the ratio 4:2.

How much chocolate does Claudia get?

Question 6: Scaling Ratios

Caleb wants to paint his house. He checks the instructions on the paint tin and finds that one tin of paint will cover 5m 2 . The total wall area Caleb needs to paint is 40m 2 . How many tins of paint will he need to buy?

If you would like to practise more ratio-based questions, WikiJob has a psychometric tests app , available in both iTunes and Google Play. All tests include a timer and worked solutions at the end. Give it a try!

What is an easy way to solve ratio questions?

To easily solve a ratio question first add all the parts of a ratio together to find the total number.

For example, if you have to split 20 sweets into a ratio of 3:1, you add the ratio figures up to find the total.

In this case, that’s 3 + 1 = 4.

The second step is to divide the number of sweets you are working with by the total you found. So 20 sweets divided by 4 equals 5.

That 5 is then used in the third and final step to create the answer.

The first number in the ratio is 3 so you multiply the 5 by 3. This gives 15. The second number in the ratio is 1 so you multiply the 5 by 1 to get 5. That gives you a final ratio of 15:5 or 20 sweets split into a pile of 15 and a pile of 5.

What are the most common mistakes in ratio questions?

A common mistake people make in ratio questions is getting the numbers the wrong way around. For example, when talking about there being 5 weekdays to 2 weekend days, you would express that as 5:2. The first item being mentioned comes first in the ratio, regardless of which number is bigger.

Another common mistake in ratio questions is misreading the information, so always take care to read thoroughly before rushing to answer.

A third common mistake in ratio questions is being thrown off by units or decimals. The principles of ratios remain the same regardless – but do convert km to meters, for example, so that you’re working in the same size unit for both sides of the ratio.

How do I solve ratio questions and get an answer in minutes?

There are a few different types of ratio questions you can get. The first thing you want to do is to assess which type of question it is – for example, it could be a scaling question, a reduction of ratios or finding an unknown quantity from an existing ratio.

Once you work out which type of question it is you can get an answer in minutes.

A scaling question almost always just involves multiplying or dividing the original ratio numbers. Reducing a ratio requires working in fractions and might just require dividing the original ratio by half or another number. The last type of ratio question, which requires finding an unknown quantity, can require cross-multiplication or simple algebra.

Are there any tricks to solve ratio questions in bank/any other competitive exams?

The best way to prepare for ratio questions for bank or other competitive exams is to practice them and to refresh all your basic mathematical principles. This will build your confidence and speed.

In terms of tricks, it helps to remember that the main property of ratios is that they need to exist between like quantities (for example, the same units and type, so that you’re comparing kg to kg or lbs to lbs).

The other main property of ratios to remember is that all parts of it must be multiplied or divided by the same quantity. So if you are doubling a recipe, for example, all parts of it must be increased by a factor of two.

What is an example of a ratio question?

A simple ratio question might be working out the ratio of boys and girls in a class. If there are 10 boys and 12 girls you could divide each number by a common denominator (in this case 2). So you get the answer that there's a 5:6 ratio of boys to girls.

This is an example of a ratio reduction. An example of a scaling question could be: “If 10 liters of fuel are needed to drive 50 miles, how much is needed to drive 200 miles?” In this case, you can see that the mileage has increased 4 times, therefore the fuel amount also needs to be multiplied by 4, – giving an answer of 40 liters.

Do ratios perform a proportion?

In terms of definitions, a ratio is an ordered pair of numbers, such as a and b, often written a:b. So to answer the question of whether ratios perform a proportion, you could say that they do show the proportions of a versus b.

For example, if we are talking about how many smokers there are in an office you could say that the ratio is 1:10. That would mean there is one smoker to every 10 non-smokers – and this ratio is the most simplified way to look at the numbers.

As a proportion, that same concept would be expressed as 1/10. If the office was bigger, you might say that the proportion is 10/100. Each of these two fractions (1/10 and 10/100) is equivalent to the same proportion.

How are ratios used in everyday life?

Ratios are used in everyday life in various ways. For example, you might be on a diet 6 days a week and have 1 cheat day. That would be expressed as a 6:1 ratio.

Ratios are also used when traveling. For example, if you needed to convert British Sterling from US Dollars it might help you to know that the ratio is 1:1.4 when shopping. That way you can work out what you’re spending in your home currency versus just the travel currency.

Another everyday use of ratios can happen if you are planning a party and you predict most people will drink 3 beverages. You could then use ratios to work out how much to buy. If 15 people are coming, for example, you would then use ratios and multiply by 3 to buy 45 beverages.

What are the rules of ratio?

Ratios are typically presented with colons . They can also be seen in using fractions or the word ‘to’.

For example, a 1:2 ratio could be expressed as 1/2 or with the words '1 to 2'.

In terms of basic rules, you need to treat each side of the ratio the same way. For example, if you’re tripling one side of a ratio, you would do the same to the other side.

Another rule is that ratios should exist between quantities of the same kind, such as mm to mm or kg to kg. You can always adjust the units to match – for example, if you were given two items, one in grams and one in kg you can multiply the latter by 1,000 to turn both items into gram measurements.

How does ratio and proportion help me as a student?

If you are studying mathematics , any related scientific disciplines or even a subject like Geography then having a grasp of ratios and proportions will be helpful.

For example, you might be asked to work out the ratio of white sand beaches to black sand beaches in the world, so being able to manipulate data in these simple ways is a useful skill to have.

Also, when it comes to applying for jobs or internships as a student, you will need to prepare for assessment tests. These often include numerical reasoning tests, where ratios and proportions can be presented in several different ways.

As a student, it is therefore important that you can recognize and work with data to make the best of the next steps in your career.

What prior knowledge is needed for ratios?

Not much prior knowledge is needed to solve ratios , but it helps if you are comfortable with numbers and basic mathematical concepts.

If you have previously studied fractions or algebra then the concept of ratios is easier to grasp. Even basic human constructs such as sharing food fairly between people can be helpful to give ratios a context that is relatable.

For example, the knowledge of how to share a box of chocolates between five people can be useful to understand ratios. If there are 30 chocolates in the box you can describe the distribution of them in a ratio of 6:1 – so 6 chocolates are given per person.

How do you teach students ratios?

It helps to make the examples easy to grasp when teaching students ratios. For example, you can talk about a family that has one cat and two dogs. The ratio of cats to dogs is therefore 1:2.

Giving as many examples as possible from real life will make it more straightforward to teach a student about ratios. Simple examples will make it easier for them to understand ratios and you can also relate ratios to the concept of fairness – as students often care a lot about this.

For example, you could say that in the school next door the teacher-to-student ratio is 1:10. You could compare this to the school you are in and ask them whether it’s fair that the ratio is different in one learning environment versus another.

How are ratios used in cooking?

When scaling up a recipe while cooking you might use ratios. For example, a common shortbread recipe uses flour, sugar and butter in a 4:2:1 ratio of cups.

If you wanted to make double the amount of shortbread you were originally making, you would then use 8 cups of flour, 4 cups of sugar and 2 cups of butter.

Therefore just doubling the initial measurements is a way of using ratios.

Using ratios this way removes the need to remember exact weights or measurements, which is helpful when you don’t have a scale to hand.

Learning commonly-used ratios like the amount of flour to liquid when making bread or fritters, can result in more powerful and intuitive cooking skills than slavishly following a recipe each time would.

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Ratios and proportions and how to solve them

Let's talk about ratios and proportions. When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.

A ratio can be written in three different ways and all are read as "the ratio of x to y"

$$x\: to\: y$$

$$\frac{x}{y}$$

A proportion on the other hand is an equation that says that two ratios are equivalent. For instance if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.

$$\frac{20}{1}=\frac{40}{2}$$

A proportion is read as "x is to y as z is to w"

$$\frac{x}{y}=\frac{z}{w} \: where\: y,w\neq 0$$

If one number in a proportion is unknown you can find that number by solving the proportion.

You know that to make 20 pancakes you have to use 2 eggs. How many eggs are needed to make 100 pancakes?

$$\frac{eggs}{pancakes}=\frac{eggs}{pancakes}\: \: or\: \: \frac{pancakes}{eggs}=\frac{pancakes}{eggs}$$

If we write the unknown number in the nominator then we can solve this as any other equation

$$\frac{x}{100}=\frac{2}{20}$$

Multiply both sides with 100

$${\color{green} {100\, \cdot }}\, \frac{x}{100}={\color{green} {100\, \cdot }}\, \frac{2}{20}$$

$$x=\frac{200}{20}$$

If the unknown number is in the denominator we can use another method that involves the cross product. The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal

If we again use the example with the cookie mix used above

$$\frac{{\color{green} {20}}}{{\color{blue} {1}}}=\frac{{\color{blue} {40}}}{{\color{green} {2}}}$$

$${\color{blue} {1}}\cdot {\color{blue} {40}}={\color{green} {2}}\cdot {\color{green} {20}}=40$$

It is said that in a proportion if

If you look at a map it always tells you in one of the corners that 1 inch of the map correspond to a much bigger distance in reality. This is called a scaling. We often use scaling in order to depict various objects. Scaling involves recreating a model of the object and sharing its proportions, but where the size differs. One may scale up (enlarge) or scale down (reduce).  For example, the scale of 1:4 represents a fourth. Thus any measurement we see in the model would be 1/4 of the real measurement. If we wish to calculate the inverse, where we have a 20ft high wall and wish to reproduce it in the scale of 1:4, we simply calculate:

$$20\cdot 1:4=20\cdot \frac{1}{4}=5$$

In a scale model of 1:X where X is a constant, all measurements become 1/X - of the real measurement. The same mathematics applies when we wish to enlarge. Depicting something in the scale of 2:1 all measurements then become twice as large as in reality. We divide by 2 when we wish to find the actual measurement.

Video lesson

$$\frac{x}{x + 20} = \frac{24}{54}$$

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Algebra: Ratio Word Problems

Related Pages Two-Term Ratio Word Problems More Ratio Word Problems Algebra Lessons

In these lessons, we will learn how to solve ratio word problems that have two-term ratios or three-term ratios.

Ratio problems are word problems that use ratios to relate the different items in the question.

The main things to be aware about for ratio problems are:

• Change the quantities to the same unit if necessary.
• Write the items in the ratio as a fraction .
• Make sure that you have the same items in the numerator and denominator.

Ratio Problems: Two-Term Ratios

Example 1: In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. If the bag contains 120 green sweets, how many red sweets are there?

Solution: Step 1: Assign variables: Let x = number of red sweets.

Step 2: Solve the equation. Cross Multiply 3 × 120 = 4 × x 360 = 4 x

Answer: There are 90 red sweets.

Example 2: John has 30 marbles, 18 of which are red and 12 of which are blue. Jane has 20 marbles, all of them either red or blue. If the ratio of the red marbles to the blue marbles is the same for both John and Jane, then John has how many more blue marbles than Jane?

Solution: Step 1: Sentence: Jane has 20 marbles, all of them either red or blue. Assign variables: Let x = number of blue marbles for Jane 20 – x = number red marbles for Jane

Step 2: Solve the equation

Cross Multiply 3 × x = 2 × (20 – x ) 3 x = 40 – 2 x

John has 12 blue marbles. So, he has 12 – 8 = 4 more blue marbles than Jane.

Answer: John has 4 more blue marbles than Jane.

How To Solve Word Problems Using Proportions?

This is another word problem that involves ratio or proportion.

Example: A recipe uses 5 cups of flour for every 2 cups of sugar. If I want to make a recipe using 8 cups of flour. How much sugar should I use?

How To Solve Proportion Word Problems?

When solving proportion word problems remember to have like units in the numerator and denominator of each ratio in the proportion.

• Biologist tagged 900 rabbits in Bryer Lake National Park. At a later date, they found 6 tagged rabbits in a sample of 2000. Estimate the total number of rabbits in Bryer Lake National Park.
• Mel fills his gas tank up with 6 gallons of premium unleaded gas for a cost of $26.58. How much would it costs to fill an 18 gallon tank? 3 If 4 US dollars can be exchanged for 1.75 Euros, how many Euros can be obtained for 144 US dollars?

Ratio problems: Three-term Ratios

Example 1: A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. If a bag of the mixture contains 3 pounds of rice, how much corn does it contain?

Solution: Step 1: Assign variables: Let x = amount of corn

Step 2: Solve the equation Cross Multiply 2 × x = 3 × 5 2 x = 15

Answer: The mixture contains 7.5 pounds of corn.

Example 2: Clothing store A sells T-shirts in only three colors: red, blue and green. The colors are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many T-shirts does it have altogether?

Solution: Step 1: Assign variables: Let x = number of red shirts and y = number of green shirts

Step 2: Solve the equation Cross Multiply 3 × 20 = x × 4 60 = 4 x x = 15

5 × 20 = y × 4 100 = 4 y y = 25

The total number of shirts would be 15 + 25 + 20 = 60

Answer: There are 60 shirts.

Algebra And Ratios With Three Terms

Let’s study how algebra can help us think about ratios with more than two terms.

Example: There are a total of 42 computers. Each computer runs one of three operating systems: OSX, Windows, Linux. The ratio of the computers running OSX, Windows, Linux is 2:5:7. Find the number of computers that are running each of the operating systems.

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Word Problems: Ratios

A ratio is a comparison of two numbers. It can be written with a colon ( 1 : 5 ) , or using the word "to" ( 1     to     5 ) , or as a fraction: 1 5

A backyard pond has 12 sunfish and 30 rainbow shiners. Write the ratio of sunfish to rainbow shiners in simplest form .

Write the ratio as a fraction.

Now reduce the fraction .

So the ratio of sunfish to rainbow shiners is 2 : 5 .

(Note that the ratio of rainbow shiners to sunfish is the reciprocal : 5 2 or 5 : 2 .)

Read word problems carefully to check whether the ratio you're being asked for is a fraction of the total or the ratio of one part to another part .

Ms. Ekpebe's class has 32 students, of which 20 are girls. Write the ratio of girls to boys.

Careful! Don't write 20 32 ... that's the fraction of the total number of students that are girls. We want the ratio of girls to boys.

Subtract 20 from 32 to find the number of boys in the class.

32 − 20 = 12

There are 12 boys in the class. So, ratio of girls to boys is 20 : 12 .

You can reduce this ratio, the same way you reduce a fraction. Both numbers have a common fact of 4 , so divide both by 4 .

In simplest form, this ratio is 5 : 3 .

Some ratio word problems require you to solve a proportion.

A recipe calls for butter and sugar in the ratio 2 : 3 . If you're using 6 cups of butter, how many cups of sugar should you use?

The ratio 2 : 3 means that for every 2 cups of butter, you should use 3 cups of sugar.

Here you're using 6 cups of butter, or 3 times as much.

So you need to multiply the amount of sugar by 3 .

3 × 3 = 9

So, you need to use 9 cups of sugar.

You can think of this in terms of equivalent fractions :

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