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Mathematics LibreTexts

5.5: Ratios and Proportions

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A bar graph titled Facebook Dominates the Social Media Landscape displays monthly active users of selected social networks and messaging services. Numbers represent million. The x-axis ranges from 0 to 2000, in increments of 400.The following social media apps are displayed: Facebook (2,006), WhatsApp (1,300), Messenger (1,200), WeChat (938), Instagram (700), Qzone (632), Weibo (340), Twitter (328), Pinterest (175), Snapchat (166), Vkontakte (95).

Learning Objectives

After completing this section, you should be able to:

  • Construct ratios to express comparison of two quantities.
  • Use and apply proportional relationships to solve problems.
  • Determine and apply a constant of proportionality.
  • Use proportions to solve scaling problems.

Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio , which is a comparison of two numbers that can be written as: /**/(\$ 1 =0.82\,{€})/**/, how many dollars should you receive? Round to the nearest cent if necessary.

Example 5.31

Solving a proportion involving weights on different planets.

A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 pounds) weigh on Mars? Round your answer to the nearest tenth.

Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.

170 64 = 1,000 x 170 64 = 1,000 x

Step 2: Cross multiply, and then divide to solve.

170 x = 1,000 ( 64 ) 170 x = 64,000 170 x 170 = 64,000 170 x = 376.5 170 x = 1,000 ( 64 ) 170 x = 64,000 170 x 170 = 64,000 170 x = 376.5

So the 1,000-pound horse would weigh about 376.5 pounds on Mars.

Your Turn 5.31

Example 5.32, solving a proportion involving baking.

A cookie recipe needs /**/1{\text{ inch}} =/**/ how many miles). Then use that scale to determine the approximate lengths of the other borders of the state of Wyoming.

Example 5.38

Solving a scaling problem involving model cars.

Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 inches long, how long is a real NASCAR automobile? Write your answer in feet.

The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.

1 24 = 9 x 24 ( 9 ) = x 216 = x 1 24 = 9 x 24 ( 9 ) = x 216 = x

This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:

216 12 = 18 216 12 = 18

The NASCAR automobile is 18 feet long.

Your Turn 5.38

Check your understanding, section 5.4 exercises.

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Ratio Problem Solving

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.

What is ratio problem solving?

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.

What is ratio problem solving?

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. 

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.

  • Dividing ratios

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

Ratios and fractions (proportion problems)

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.

Simplifying and equivalent ratios

  • Simplifying ratios

Equivalent ratios

Units and conversions ratio questions

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.

How to do ratio problem solving

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.

Explain how to do ratio problem solving

Explain how to do ratio problem solving

Ratio problem solving worksheet

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

Related lessons on ratio

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:

  • How to work out ratio  
  • Ratio to fraction
  • Ratio scale
  • Ratio to percentage

Ratio problem solving examples

Example 1: part:part ratio.

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?

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

Ratio problem solving example 1 step 1

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.

Ratio problem solving example 1 step 2

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.

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

There are 651 students that have a packed lunch.

Example 2: unit conversions

The table below shows the currency conversions on one day.

Ratio problem solving example 2

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

Ratio problem solving example 2

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

Ratio problem solving example 2 step 1 image 2

We know that we have £520.

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

Ratio problem solving example 2 step 2

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.

Example 3: writing a ratio 1:n

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.

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.

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

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 ).

Ratio problem solving example 3 step 3

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

Example 4: forming and solving an equation

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?

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

Ratio problem solving example 4 step 1

We also know that Luke receives £8.

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

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.

Ratio problem solving example 4 step 3

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

Example 5: simplifying ratios

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

Ratio problem solving example 5

Express this data as a ratio in its simplest form.

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

Ratio problem solving example 5 step 1

We need to simplify this ratio.

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

Ratio problem solving example 5 step 3

Our solution is 6:8:5 .

Example 6: combining two ratios

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.

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

Ratio problem solving example 6 step 1

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

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

Ratio problem solving example 6 step 3 image 1

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.

Ratio problem solving example 6 step 3 image 2

Example 7: using bar modelling

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?

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

Ratio problem solving example 7 step 1

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

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

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

Ratio problem solving example 7 step 3 image 1

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

Ratio problem solving example 7 step 3 image 2

We can fill in each share to be 25g.

Ratio problem solving example 7 step 3 image 3

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.

Common misconceptions

  • 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} ).

Ratio problem solving common misconceptions

  • 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.

Practice ratio problem solving questions

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?

GCSE Quiz True

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?

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.

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

5. A clothing company has a sale on tops, dresses and shoes. \frac{1}{3} of sales were for tops, \frac{1}{5} 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.

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.

Ratio problem solving GCSE questions

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.

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.

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

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.

Learning checklist

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

The next lessons are

  • Compound measures
  • Best buy maths

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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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A ratio compares values .

A ratio says how much of one thing there is compared to another thing.

Ratios can be shown in different ways:

A ratio can be scaled up:

Try it Yourself

Using ratios.

The trick with ratios is to always multiply or divide the numbers by the same value .

Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk.

So the ratio of flour to milk is 3 : 2

To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:

3 ×4 : 2 ×4 = 12 : 8

In other words, 12 cups of flour and 8 cups of milk .

The ratio is still the same, so the pancakes should be just as yummy.

"Part-to-Part" and "Part-to-Whole" Ratios

The examples so far have been "part-to-part" (comparing one part to another part).

But a ratio can also show a part compared to the whole lot .

Example: There are 5 pups, 2 are boys, and 3 are girls

Part-to-Part:

The ratio of boys to girls is 2:3 or 2 / 3

The ratio of girls to boys is 3:2 or 3 / 2

Part-to-Whole:

The ratio of boys to all pups is 2:5 or 2 / 5

The ratio of girls to all pups is 3:5 or 3 / 5

Try It Yourself

We can use ratios to scale drawings up or down (by multiplying or dividing).

Example: To draw a horse at 1/10th normal size, multiply all sizes by 1/10th

This horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is

1500 : 2000

What is that ratio when we draw it at 1/10th normal size?

We can make any reduction/enlargement we want that way.

"I must have big feet, my foot is nearly as long as my Mom's!"

But then she thought to measure heights, and found she is 133cm tall, and her Mom is 152cm tall.

In a table this is:

The "foot-to-height" ratio in fraction style is:

We can simplify those fractions like this:

And we get this (please check that the calcs are correct):

"Oh!" she said, "the Ratios are the same".

"So my foot is only as big as it should be for my height, and is not really too big."

You can practice your ratio skills by Making Some Chocolate Crispies

Ratios and Proportions

Malcolm McKinsey

Ratios and proportional relationships

Ratios and proportions are similar figures and concepts that are as easily confused as toads and frogs (all toads are frogs, but not all frogs are toads). Ratios compare values, while proportions compare ratios.

What are ratios?

Ratios  compare values. You can compare the number of brown-haired boys to the number of blond-haired boys, or to the number of pencils in the classroom, or to the number of brown-haired girls, or … well, you get the idea. Ratios compare values of the same things or things that are different.

Ratios & Proportions

Say you have 10 brown-haired girls in a class, and 6 blonde-haired girls in the same class. You can set up  six   different ratios:

10 16 \frac{10}{16} 16 10 ​ : Brown-haired girls to all girls

6 10 \frac{6}{10} 10 6 ​ : Blonde-haired girls to brown-haired girls

6 16 \frac{6}{16} 16 6 ​ : Blonde-haired girls to all girls

10 6 \frac{10}{6} 6 10 ​ : Brown-haired girls to blonde-haired girls

16 10 \frac{16}{10} 10 16 ​ : All girls to brown-haired girls

16 6 \frac{16}{6} 6 16 ​ : All girls to blonde-haired girls

Ratios & Proportions proper or improper fractions

Three of those ratios are improper fractions; that is okay! Ratios can be written as proper or improper fractions. They can also be written with a semicolon, like this:

What are proportions?

When you compare two ratios, you use  proportions . You are asking if the first ratio is the same, less than, or more than the second ratio. Compare the ratios of brown-to-all girls and blonde-to-all girls:

Ratios & Proportions - Not Equal

You can see these two   ratios are  not  equal, so they are  not  proportional :

How to solve ratios and proportions

What would proportional fractions look like? Let's add eight class pets to the classroom: 5 hamsters and 3 frogs . The  ratios  you can create are:

How to Solve Ratios & Proportions

5:3  (hamsters to frogs)

3:5  (frogs to hamsters)

5:8  (hamsters to all pets)

3:8  (frogs to all pets)

8:5  (all pets to hamsters)

8:3  (all pets to frogs)

Proportions can tell us if two ratios are equal or not. Compare the ratio of hamsters to all pets and the ratio of brown-haired girls to all girls:

You can check these fractions in a few ways, such as simplifying  10 16 \frac{10}{16} 16 10 ​  to  10 16 \frac{10}{16} 16 10 ​ , or by cross-multiplying and dividing: 5 × 16 10 = 8 \frac{5\times 16}{10}=8 10 5 × 16 ​ = 8

These two ratios  are  proportional to each other. The ratio of hamsters to all class pets is the same as the ratio of brown-haired girls to all girls in the class.

Ratios and proportions word problems

Cooking, comparing prices, driving, engineering, construction and finance are just some areas where ratios and proportions work every day.

Here is a recipe for hamster food to feed one hamster:

20g of five-cereal blend

10g small seed blend

10g rolled oats

10g dried vegetables

5g dried fruit

One hamster gets 60 grams of hamster chow. How much should you mix for five hamsters?

Ratios & Proportions - Finding the proper ratio

Whatever you multiply 1 times to get 5 , multiply 60 times the same number. You need 300 grams.

How much of each ingredient should you mix?

For every 60 grams of hamster chow for one hamster, 20 grams is five-cereal blend, a ratio of  20:60 or  1:3 .

If you want to feed five hamsters, you have to mix more of everything in the right proportions. How many grams of five-cereal blend will you need?

Say you did your calculations and mixed the five-cereal blend at a ratio of  50:300 . Is that correct? Check: Is  50 300 \frac{50}{300} 300 50 ​  proportional to  20 60 \frac{20}{60} 60 20 ​ or  20 60 \frac{20}{60} 60 20 ​ ?

You can cross-multiply and divide to check:  50 × 3 300 \frac{50\times 3}{300} 300 50 × 3 ​ .

You see that  150 300 = 1 2 \frac{150}{300}=\frac{1}{2} 300 150 ​ = 2 1 ​ , not 1 . So your mix is  not  in the right proportion because 50 is not one third of 300 .

You needed 100 g of five-cereal blend to maintain the right proportions.

Ratios and proportions examples

Perhaps you have a part-time job in a grocery store, assembling gift baskets of fruit. Your manager tells you to maintain a ratio of  2:3 of pears to apples for every size of basket. A small basket gets 2 pears and 3 apples . An extra-large basket must have the same ratio,  2:3 , but be five times larger.

Ratios & Proportions - Apples and Pears Example

The ratio of  pears:apples  is  2:3 , so multiply both parts of the ratio times 5 to get the new ratio:  10:15.

Your extra-large gift basket needs 10 pears and 15 apples .

Ratios and proportions practice

The class of 10 brown-haired and 6 blonde-haired girls also has boys in it. Of the 12 boys in the class, 4 have blond hair and 8 have brown hair.

Write three ratios using this new information.

Many ratios can be written from the information. See if you can figure out what these ratios describe:

Did you get these answers?

Ratios & Proportions - Classroom Example

4:12 (Blond-haired boys to all boys)

8:12  (Brown-haired boys to all boys)

4:28 (Blond-haired boys to all students)

8:10 (Brown-haired boys to brown-haired girls)

28:8 (All students to brown-haired boys)

Lesson summary

You have learned that ratios compare values, while proportions compare ratios. Proportions are most often used to ensure ratios are equal when they increase or decrease. You can write ratios as either a fraction or with a colon between them, like this:  10 16 \frac{10}{16} 16 10 ​ or  10 : 16 10:16 10 : 16 . Ratios can compare like and unlike things. Both ratios and proportions are useful in many aspects of everyday life.

How to solve proportions

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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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How do we solve ratio problems?

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ACT Math: How to Solve Ratio Problems

There are a few key concepts to get down in order to ace ACT Math ratios. Let’s go right into how the ACT will test you on ratios and break it down for you.

Ratio Basics

A ratio tells you the proportional quantity of one thing relative to another.

Make sure not to get ratios confused with fractions. Fractions tell you the proportional quantity of something relative to its whole. Ratios expressed as fractions do not tell you the whole. One instance where you need to use the concept of ratios involves baking. If you want to make double the amount of cookies that a recipe will yield, then you need to double the quantity of each ingredient.

ACT Math ratio

ACT Math: Dealing With Ratios

You might see ACT ratios written in fraction form, colon form, or in plain English. Whatever the case may be, you can treat them all the same way. In the case of the fraction form, do not get it confused with a regular fraction! The denominator of a ratio is not necessarily equivalent to the denominator of a ratio.

For example, the ratio 12/8, 12:8, and 12 to 8 are all the same. Like fractions, you should reduce ratios down to simplest terms – in this case, it is 3/2. Keep your numbers manageable, especially when you need to look for the lowest common multiple later on in the multi-step ratio section.

On the test, ratios will be clearly spelled out for you. If you are looking at a ratio problem, you’ll know it because the test makers will make it obvious.

The important part lies in knowing how to manipulate ratios to get to your answer. The two main things you need to know are proportions and multi-step ratios.

Proportions

You’ll find that these are very common on the ACT. Thankfully, they are also easy to solve.

You will usually be given a ratio along with a hypothetical quantity of one of the things on the original ratio. The key is to set up two ratios and cross-multiply as you would two fractions to solve for the missing fourth quantity.

If you have a ratio of 3 cats to 2 dogs, how many cats do you have if you have 20 dogs? You could use mental math or set up two fractions to get 30 cats as your answer.

Multi-Step Ratios

These are a little bit more involved, but shouldn’t pose much of a threat to your ACT Math score one you learn about how to go about solving them.

Here you’ll be given two ratios and three different types of quantities: a, b, and c. The two ratios given compare a to b and b to c. You’ll then be asked to figure out the ratio of a to c.

In order to solve, you need to figure out the least common multiple of the two b’s and multiply the respective a’s accordingly. Now with your b’s equal to each other, simply take the values of your a and c and create a new ratio.

For example, if a:b is 2:3 and b:c is 6:9, then what is a:c? Here we need to multiply our first ratio by 2 in order to get 4:6. Since our b from the first and second ratio match, we can take our a and c and form a new ratio.

Our answer is 4:9.

Minh Nguyen

Minh’s passion for helping students succeed grew during his time as a career counselor at the University of California, Irvine. Now, he’s helping students all over the world by spilling SAT/ACT secrets through blog posts on Magoosh. When he’s not busy tutoring or writing, he enjoys playing guitar, traveling, and talking about himself in third-person.

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Course: 6th grade   >   Unit 1

  • Ratios with tape diagrams
  • Ratios with tape diagrams (part:whole)

Equivalent ratio word problems

  • Simplify a ratio from a tape diagram
  • Equivalent ratios with equal groups
  • Ratios and double number lines
  • Create double number lines
  • Ratios with double number lines
  • Relate double number lines and ratio tables

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Video transcript

COMMENTS

  1. Ratio Problem Solving

    40 \div 8=5 40 ÷ 8 = 5. Then you multiply each part of the ratio by 5. 5. 3\times 5:5\times 5=15 : 25 3 × 5: 5 × 5 = 15: 25. This means that Charlie will get 15 15 sweets and David will get 25 25 sweets. There can be ratio word problems involving different operations and types of numbers.

  2. Ratios and proportions

    It compares the amount of one ingredient to the sum of all ingredients. part: whole = part: sum of all parts. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed.

  3. Ratios and rates

    Do you want to learn how to compare and measure different quantities using ratios and rates? Khan Academy's pre-algebra course offers you a comprehensive introduction to these concepts, with interactive exercises and videos. You will also learn how to use proportions to solve word problems and graph proportional relationships. Join Khan Academy and start your journey to master ratios and rates!

  4. Ratio Math Problems

    Two-Term Ratio Word Problems More Ratio Word Problems Math Word Problems More Algebra Lessons. Ratio problems are word problems that use ratios to relate the different items in the question. Ratio problems: Three-term Ratios. Example 1: A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5.

  5. 1.3: Ratios

    In general, if you find a way to solve a problem that works for you, as long as you get the same numerical answer, you can use that method. We will cover proportions in the next section. In this section, we are choosing to focus on equivalent ratios as a method for problem-solving. Let's see one more example.

  6. 5.5: Ratios and Proportions

    Determine and apply a constant of proportionality. Use proportions to solve scaling problems. Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note ...

  7. Ratio Problem Solving

    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.

  8. Algebra: Ratio Word Problems

    Ratio Word Problems: relating different things using ratios and algebra, how to solve ratio word problems that have two-term ratios or three-term ratios, How to solve proportion word problems, questions and answers, with video lessons, examples and step-by-step solutions.

  9. Solving ratio problems with tables (video)

    The ratio 54:3 shows us that the simplified form is 18:1. This is because 54/3 = 18. Let's just call the blank part x. To find 36: x, you have to divide 36 by 18 to get 2 (so the ratio is 36:2). To find x:5, you have to multiply 18*5, which is 90 (the ratio is 90:5) Hope this helps! :) 4 comments.

  10. Ratios

    Using Ratios. The trick with ratios is to always multiply or divide the numbers by the same value. Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. So the ratio of flour to milk is 3 : 2. To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4: 3 ×4 : 2 ×4 = 12 : 8.

  11. Art of Problem Solving: Introducing Ratios

    Art of Problem Solving's Richard Rusczyk introduces ratios.Learn more about problem solving at our website: http://bit.ly/ArtofProblemSolving

  12. Ratios and Proportions

    The ratio of pears:apples is 2:3, so multiply both parts of the ratio times 5 to get the new ratio: 10:15. Your extra-large gift basket needs 10 pears and 15 apples.. Ratios and proportions practice. The class of 10 brown-haired and 6 blonde-haired girls also has boys in it. Of the 12 boys in the class, 4 have blond hair and 8 have brown hair.. Write three ratios using this new information.

  13. Ratios and proportions and how to solve them

    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 ⋅ 1: 4 = 20 ⋅ 1 4 = 5 20 ⋅ 1: 4 = 20 ⋅ 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.

  14. 100% of Math Students Need to UNDERSTAND! Rates, Ratios and ...

    How to solve math problems with rates, ratios and proportions.For more in-depth math help check out my catalog of courses. Every course includes over 275 vid...

  15. Part to whole ratio word problem using tables

    What you need to do in any word problem involving the ratios is exactly the same. Take the entire amount and divide it by the sum of the ratios. This will give you the number you need to multiply both ratios by. So the entire amount of playtimes is 30, and the sum of the ratios is 2+3, which is 5.

  16. Solving ratio problems

    Divide the total amount in the initial ratio. Find the value of one part by dividing the total amount by the sum of the parts. Multiply the value of one part by the number of parts for each share ...

  17. Ratios with tape diagrams (video)

    Tape diagrams are visual models that use rectangles to represent the parts of a ratio. Since they are a visual model, drawing them requires attention to detail in the setup. In this problem David and Jason have numbers of marbles in a ratio of 2:3. This ratio is modeled here by drawing 2 rectangles to represent David's portion, and 3 ...

  18. Solving Ratios, Practice with SAT Problems

    Definition: A comparison between quantities using division. Examples : 3:2 , 3:2:88, 3 to 2, 3 to 2 to 88. A 2 to 5 ratio can be represented as 2:5. A ration between X and Y can be written. X/Y. X:Y. X to Y. MEDIUM SAT PROBLEM#8 out of a 25 problem section. A bucket holds 4 quarts of popcorn.

  19. ACT Math: How to Solve Ratio Problems

    The two ratios given compare a to b and b to c. You'll then be asked to figure out the ratio of a to c. In order to solve, you need to figure out the least common multiple of the two b's and multiply the respective a's accordingly. Now with your b's equal to each other, simply take the values of your a and c and create a new ratio.

  20. Equivalent ratio word problems (practice)

    Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. ... Solving ratio problems with tables. Ratio tables. Equivalent ratios. Equivalent ratios: recipe ...

  21. Equivalent ratio word problems (video)

    Equivalent ratio word problems. Google Classroom. About. Transcript. This video teaches solving ratio word problems, using examples like Yoda Soda for guests, fish ratios in a tank, ice cream sundae ingredients, and dog color ratios at a park. Mastering these techniques helps students tackle real-world math challenges.