problem solving involving ratio with solution

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Use the generator to make customized ratio worksheets. Experiment with the options to see what their effect is.

Primary Grade Challenge Math by Edward Zaccaro

A good book on problem solving with very varied word problems and strategies on how to solve problems. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Each chapter’s questions are broken down into four levels: easy, somewhat challenging, challenging, and very challenging.

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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.

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

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

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.

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.

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

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.

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

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 ).

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

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.

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.

Express this data as a ratio in its simplest form.

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

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

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

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.

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.

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

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.

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.

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} ).

• 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?

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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Solving Ratio Problems

• We add the parts of the ratio to find the total number of parts.
• There are 2 + 3 = 5 parts in the ratio in total.
• To find the value of one part we divide the total amount by the total number of parts.
• 50 ÷ 5 = 10.
• We multiply the ratio by the value of each part.
• 2:3 multiplied by 10 gives us 20:30.
• The 50 counters are shared into 20 counters to 30 counters.

• 2 + 3 = 5 and so there are 5 parts in the ratio in total.
• We divide by this total number of parts to find the value of each part.
• We multiply the original ratio by the value of each part.
• We have 20:30.

• Sharing in a Ratio: Part 1

Ratio Problems: Worksheets and Answers

How to Solve Ratio Problems

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Word Problems Involving Rates and Ratios

The ratio is to compare two numbers. Rate is one type of ratio and is used to measure the variety of one thing or quantity in comparison to other. Word problems involving comparing rates deal with distances, time, rates, wind or water current, money, and age.

A step-by-step guide to solving rates and ratios word problems

To solve the word problems involving rates and ratios, follow these steps: Step 1: Find the known ratio and the unknown ratio. Step 2: Write the proportion. Step 3: Use cross-multiply and solve. Step 4: Plug the result into the unknown ratio to check the answers.

Word Problems Involving Rates and Ratios – Examples 1

If 11 apple pies cost $88, what will 8 apple pies cost? Solution: Write as a rate. \(\frac{88÷11}{11÷11}=\frac{8}{1}\) Write a proportion to know the cost of 8 apple pies. \(\frac{8}{1}=\frac{x}{6}→8×6=1×x→x=48\)

Word Problems Involving Rates and Ratios – Examples 2

If 6 cookbooks cost $120, how much would a dozen cookbooks cost? Solution: Write as a rate. \(\frac{120÷6}{6÷6}=\frac{20}{1}\) Write a proportion to know the cost of 12 cookbooks. \(\frac{20}{1}=\frac{x}{12}→20×12=1×x→x=240\)

by: Effortless Math Team about 10 months ago (category: Articles )

Effortless Math Team

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Ratio and Proportion Examples With Answers

We may have mastered comparing objects descriptively, but how do we mathematically compare objects?

The answer is through ratio.

The ratio is the mathematical tool we use to compare quantities using division. The concept of ratios leads to the concept of proportions, which has a lot of applications in our daily lives. For example, when we convert currencies, estimate the volume of gasoline required for a car to cover a certain distance, calculate the cost of items bought, and so on.

This article features ratio and proportion examples with answers to help you effectively learn these concepts and how to apply them in your daily life.

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A ratio shows how an object’s quantity is related to another object’s quantity . For example, if there are 15 male and 23 female students in a classroom, we can compare these quantities using a ratio, particularly 15 : 23.

We usually write it in the format below to express two quantities being compared as a ratio. Note that we use a colon (:) to express a ratio.

<first number> : <second number>

Example : Aling Bela has 4 chickens and 8 pigs on her small farm. What is the ratio of her chickens to her pigs?

Solution : We can express the ratio of chickens to pigs that Aling Bela owns as 4 : 8

Ratio as a Fraction

We can write a ratio into its equivalent fractional form . We write the first number as the numerator, then the second as the denominator.

For instance, using our example above about Aling Nena’s chickens and pigs, we can express the ratio of her chickens to her pigs (i.e., 4 : 8 ) as 4⁄8.

Example 1 : For every 4 burgers you buy, you must pay ₱128 . What is the ratio of the number of burgers bought to the price you must pay? Express the ratio in fractional form.

Solution : We can express the ratio of the number of burgers bought to the price you must pay as 4 : 128. In fractional form, we can write this as 4⁄128.

Example 2: There are 15 science teachers in a public high school. In that same high school, there were 10 math teachers. What is the ratio of science teachers to math teachers in that public high school? Express the ratio in fractional form .

Solution : We can express the ratio of science teachers to math teachers in that public high school as 15 : 10. In fractional form, we write it as 15⁄10.

Using Ratio to Compare a Part to a Whole

We have already defined what ratios are. However, the ratios that we tackled in our previous sections pertain to comparing an object’s quantity to a different object’s. 

This time, let us use the ratio to compare a part of a whole to the whole itself.

Suppose you and your friends bought a pizza and sliced it into 8 equal parts. Suppose that you’re able to take 2 slices from it. What is the ratio of the slices of pizza you have (a portion of the whole pizza) to the total number of slices (the whole pizza)?

The given situation above might ring a bell to you. Yes, we can use fractions to show that comparison. In particular, fraction 2⁄8 can be expressed as 2 : 8.

This means that to use the ratio to compare a part of a whole to the whole itself, we can use this format:

<portion of the whole> : <total number portions of the whole>

Example : In a classroom, 15 students are male while 20 are female. What is the ratio of female students to the total number of students in the classroom?

Solution : There are 20 female students in the classroom. Meanwhile, the total number of students in the classroom is the sum of the number of male students and the number of female students. There are 15 + 20 = 35 students in that classroom.

Therefore, the ratio of female students to the total number of students in that classroom can be expressed as 20 : 35

A proportion indicates that the two ratios are equal. In other words, proportions are equivalent ratios . Hence, if we say that ratios are proportional, we mean that those ratios are equal in value.

Let’s say we have two ratios, 1 : 2 and 2 : 4. We illustrate these ratios as shown below:

It’s seen that the ratios represent the same parts. It implies that these ratios are equivalent.

Hence, 1 : 2 = 2 : 4 is a proportion.

You might have realized that proportions are similar to the concept of equivalent fractions . Indeed, proportions indicate equivalent fractions since we can write ratios in fractional form.

How to Know if Two Ratios Are Proportional

Two ratios are proportional if they are equal . One way to determine if two ratios are equal is by converting them into fractional form and using the cross-multiplication method we discussed in the Fractions and Decimals review .

For instance, let us use the cross-multiplication method to determine if 1 : 2 = 2 : 4 is proportional.

Since the products are equal (both equal to 4), the ratios are equal. Hence, the ratios are proportional.

Moreover, you might also notice that if we multiply the numbers in a ratio, we can obtain another ratio that is proportional to that ratio. 

For example, if we multiply each number in 5 : 2 by the same number, let’s say 2, we have 10 : 4. Using the cross-multiplication method, you can verify that 5 : 2 = 10 : 4

Example: Give a ratio that is equivalent or proportional to 2 : 9

Solution : We can determine a ratio equivalent to or proportional to 2 : 9 by multiplying each number in 2 : 9 by the same number.

Let us try to multiply the numbers in 2 : 9 by 5.

(2 x 5) : (9 x 5) = 10 : 45

Hence, 2 : 9 = 10 : 45. 

Note : The number you can use to find a proportional ratio to 2 : 9 is arbitrary.  If we multiply the numbers in 2 : 9 by the same number, we will come up with a ratio that is proportional to 2 : 9. In this example, I just arbitrarily used 5. You may use any number and multiply it by the numbers in 2 : 9, and you will come up with a ratio that is proportional to it.  For example, I can multiply the numbers of 2 : 9 by 7 and obtain 14 : 63. 14 : 63 is also proportional to 2 : 9

Parts of a Proportion: Extremes and Means

Suppose a proportion a : b = c : d where a, b, c, and d represent real numbers .

The proportion’s first and last terms (i.e., a and d) are called the extremes. Meanwhile, the second and third terms (i.e., b and c ) are called the means .

Example : Determine the extremes and the means of the proportion 5 : 10 = 20 : 40

Solution : The extremes are the first and last terms of the proportion, which are 5 and 40, respectively. Meanwhile, the means are the second and third terms of the proportion, which are 10 and 20, respectively.

Properties of Proportion

Using the fact that proportions are equivalent ratios, we can mathematically derive their properties. These properties are beneficial when solving problems involving ratios and proportions.

Here are the properties of proportion:

Property #1: The Product of the Means Is Equal to the Product of the Extremes

For every proportion a : b = c : d, then a x d = b x c

This property tells us that we will obtain the same number if we multiply the means and the extremes of a proportion.

For example, suppose the proportion 4 : 3 = 12 : 9. 

If we multiply the means: 3 x 12 = 36

If we multiply the extremes: 4 x 9 = 36

Note that the products of the means and the extremes equal 36.

Example 1: What must be N so that N : 8 = 2 : 16 is a proportion?

Solution : Let us use the fact that the product of the means of a proportion equals the product of the extremes.

Multiplying the means, we have: 8 x 2 = 16

Multiplying the extremes, we have 16 x N 

Now, by the first property, 16 x N = 16. What must be multiplied by 16 so that it will be 16? That number should be 1.

Hence, N = 1.

Therefore, the proportion should be 1 : 8 = 2 : 16.

Example 2: Four kilos of chicken cost ₱640. How many kilos of chicken can you buy with ₱ 3 200?

Solution : The ratio of the kilos of chicken that can be bought to the cost is 4 : 640. Now, let’s use N to represent the number of kilos of chicken that can be bought with ₱3200. Thus, we have the ratio N : 3200.

4 : 640 = N : 3200

Let us apply that the product of the means is equal to the product of extremes so we can determine N.

Multiplying the means of the ratio: 640 x N 

Multiplying the extremes of the ratio: 4 x 3200 = 12800

Since the product of the means is equal to the product of the extremes:

640 x N = 12800

What must be multiplied by 640 to obtain 12800? We determine that number by dividing 12800 by 640.

N = 12800 ÷ 640 = 20

Therefore, you can buy 20 kilos of chicken for ₱3200.

Property #2: The Reciprocals of the Ratios in a Proportion Are Equal

Recall that the reciprocal of a fraction is its multiplicative inverse or simply the same fraction but with the positions of the numerator and the denominator reversed .

For example, the reciprocal of ⅖ is 5⁄2.

Given a proportion, say a : b = c : d , we can express it in fractional form as a⁄b = c⁄d

If we get the reciprocal of both fractions in a⁄b = c⁄d, we have:

We can express  b⁄a = d⁄c in ratio as b : a = d : c

This property states that if we take the reciprocal of each ratio in proportion, the ratios are still proportional. In symbols:

a : b = c : d → b : a = d : c

Example: If 5 : 4 = 35 : 28, what should be N so that 4 : 5 = 28 : N

Solution : Since the ratios in the proportion are reciprocated, we can use the second property of proportions. Using the second property, N = 35.

Property #3: Switching the Means or the Extremes in a Proportion Will Result in a Proportion

Suppose the proportion 1 : 7 = 3 : 21. If we try to switch the positions of the means of this proportion, we have 1 : 3 = 7 : 21 . You can verify using cross-multiplication that 1 : 3 = 7 : 21 is true (that is, 1 : 3 and 7 : 21 are equivalent ratios or proportional ).

Now, let us try switching the extremes of 1 : 7 = 3 : 21. We obtain 21 : 7 = 3 : 1 . Again, you can verify using cross-multiplication that 21 : 7 = 3 : 1 is true.

Hence, for every proportion a : b = c : d , switching the means or the extremes will still result in a proportion.

a : b = c : d → a : c = b : d and d : b = c : a

Example: When A is divided by 5, the result will equal the result when you divide B by 2. What is the result if you divide A by B?

Solution : The problem sounds tricky since we have no idea what the values of A and B are. However, we can determine the result by dividing A by B using the third property of proportion.

A divided by five can be written as A⁄5, which can then be expressed into a ratio of A : 5 .

Meanwhile, B divided by 2 can be written as B⁄2, which can then be expressed into a ratio of B : 2.

Since the problem states that if A is divided by 5, the result will be equal to the result if B is divided by 2, then 

A : 5 = B : 2

We want to know what will be the result when we divide A by B or A⁄B or, as a ratio, A : B

So, from A : 5 = B : 2, how can we obtain A : B?

We can apply the property that if we switch the means of a proportion, the result is still a proportion.

Let us now switch the means of A : 5 = B : 2

We obtain A : B = 5 : 2. Expressing into a fractional form:

Therefore, if A is divided by B, the result is 5⁄2 or 2.5.

How To Solve Problems Involving Ratio and Proportion

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In this section, let us try to solve some real-life word problems that can be solved using the concepts of ratio and proportion.

Problem 1: Suppose that on a specific date, 1 US dollar equals ₱50. How many US dollars is equivalent to ₱650?

Solution: The ratio of US dollars to the Philippine Peso can be expressed as 1 : 50. Let N be the number of dollars we can obtain from ₱650.

Hence, we have this proportion: 1 : 50 = N : 650

The product of the means is equal to the product of extremes. Thus:

What must be multiplied by 50 to obtain 650? 

N = 650 ÷ 50 = 13

Therefore, ₱650 is equal to 13 US dollars.

Problem 2 : Leonor loves animals. He has a lot of dogs and cats in his house. The ratio of dogs to his cats is 1 : 3. The total number of dogs and cats is 8. How many cats does Leonor own?

Solution: The ratio of dogs to cats is 1 : 3. This doesn’t mean Leonor has 1 dog and 3 cats. 1 : 3 is just a ratio used to compare the number of dogs to cats. To find Leonor’s actual number of dogs and cats, we need to find two numbers with a sum of 8 that, when expressed as a ratio, will be proportional to 1 : 3.

Recall that we can obtain a ratio proportional to 1 : 3 if we multiply 1 and 3 by the same number.

Let us multiply the parts of the ratio with a number a.

(1 x a) : (3 x a) 

This means that we have two numbers, 1 x a and 3 x a . 1x a represents the total number of dogs Leonor has, while 3 x a represents the total number of cats Leonor has.

Since the total number of dogs and cats that Leonor has is 8:

(1 x a) + (3 x a) = 8

We can simplify the expression above as:

(4 x a) = 8

What must be multiplied by 4 to obtain 8? Simple, that number must be 2.

Hence, a = 2.

Recall that 1 x a represents the number of dogs Leonor has; since we have computed that a = 2, Leonor has 1 x (2) = 2 dogs.

We are doing the same thing to find Leonor’s number of cats: 3 x (2) = 6.

Therefore, Leonor has six cats.

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Home / United States / Math Classes / 7th Grade Math / Solving Ratio & Proportion Problems Using Scale Drawing

Solving Ratio & Proportion Problems Using Scale Drawing

We use ratios to compare two quantities that have the same unit. Two ratios are said to be in proportion if they are equ al. Learn how to check whether two ratios are in proportion by using scale and scale factors. We will also look at some solved examples to help us understand the concept better. ...Read More Read Less

Table Of Contents

What is a Ratio?

What is a proportion, what is a scale drawing, scale and scale factor, solved examples.

• Frequently Asked Questions

A ratio is a comparison of quantities having the same unit. A ratio helps us indicate how big or how small a quantity is, when compared to another quantity.

For example, “ a : b ” compares the value of a with b .

Proportion is a concept that is closely interlinked with ratios and fractions . A proportion is an equation that states that two ratios or two fractions are equivalent. That is, two ratios are said to be proportional when they are equal.

For example, if \(\frac{5}{6} = \frac{10}{12}\) . This indicates that 5 : 6 and 10 : 12  are equivalent ratios. 

Do architects draw sketches that are as big as the structures they have in mind? This is usually not the case. Similarly, can we draw the cells of the human body according to  their actual size? It’s not always possible to draw real-life objects as per their actual dimensions. That’s where scale drawings can come in handy. 

Scale drawings make it easy to make 2-D representations of real objects on paper. Scale drawings may depict a reduced or an enlarged size of a real-life object, depending on the size of the object. The interesting thing about scale drawings is that every part of the drawing gets magnified or reduced with the same factor called scale . Maps are examples of scale drawings that we use in everyday life.

Similarly, a scale model is a proportional 3-D model of a real object.

As we have already learned, the dimensions of scale drawings and models are proportional to the dimensions of real objects. The scale gives us the ratio which compares the dimensions of the drawings or models with the size of real objects.

A ratio is usually expressed as a : b . That means it can also be expressed as \( 1:\frac{b}{a}\) . When writing the scale of a drawing or a model, we usually keep the antecedent, or the first unit, as 1.

When you change the size of the shape by making it larger or smaller, the degree to which you change the size is measured by its scale factor. A linear scale factor is simply the size of the enlargement or reduction of the shape.

For example, if the scale factor is \(3\), this means that the new shape is thrice the size of the original shape. Similarly, the scale factor of \(\frac{1}{5}\)  tells us the new shape is one-fifth the size of the original. When the units are the same, the scale can be written without units. If y = Kx, where y and x represent dimensions on two different scale drawings, \(K\) is the scale factor of the two drawings.

Example 1: The actual wingspan of an aircraft is 54 feet. Find the scale of the models if the wingspan of the small aircraft model is 1 foot and the wingspan of the bigger model is two feet.

Wingspan of actual aircraft = 54 feet

The wingspan of the smaller model = 1 foot

The scale factor of the smaller aircraft model = 1 : 54

The wingspan of the smaller aircraft model is twice as long as the wingspan of the bigger aircraft model. 

The wingspan of the larger model = 2 feet

Since the wingspan of the larger model is twice as long as the wingspan of the smaller model, the scale factor of the larger model

We can simplify this ratio by dividing both sides by 2. 

= \(\frac{2}{2}:\frac{54}{2}\)       Divide both sides of the ratio by 2

= \(1:\frac{54}{2}\)       Simplify

Example 2: A model of the Empire State Building is 14.54 inches tall.  The height of the actual building is 1454 feet tall. What does one inch in the model represent in real life? What is the scale factor of the model?

The height of the model is 14.54 inches and the height of the real tower is 1454 feet. 

14.54 inch : 1454 feet

To find what 1 inch represents, i.e. to find the scale, we need to divide both the sides by 14.54

\(\frac{14.54~\text{inch}}{14.54~\text{inch}}:\frac{1454~\text{feet}}{14.45~\text{inch}}\)

⇒ 1:100\(\frac{\text{feet}}{\text{inch}}\)

Hence, an inch of the model represents 100 feet in real life. 

To find the scale factor, the units have to be the same. That is, we need to convert feet into inches. 

1 feet = 12 inches

⇒ \(1:100\frac{\text{feet}}{\text{inch}}=1:100\times 12 = 1 : 1200\frac{\text{inch}}{\text{inch}}\)

So, the scale factor of the model is \(\frac{1}{1200} \) .

Example 3: The scale of a map of Texas is 1 cm : 100 miles . If the distance between Dallas and Austin on the map is 1.84 cm, find the actual distance between the two cities.

The distance between Austin and Dallas on the map is 1.84 cm.

The scale of the map is 1 cm : 100 miles .

Use the scale 1 cm : 100 miles and the ratio 1.84 cm : d miles to write and solve a proportion.

Therefore,\(\frac{1~\text{cm}}{100~miles} = \frac{1.84~\text{cm}}{d~\text{miles}}\) , where “ \(d\) ” is the actual distance between Dallas and Austin.

\(d = 1.84\times 100\)         Cross product

= 184 miles              Multiply

The actual distance between Austin and Dallas = 184 miles

Example 4: Howard made a scale model of the solar system whose scale is 1 inch  : 3 million miles . If the distance between the center of the Earth and the center of the Sun on the scale model is 31 inches, find the actual distance between Earth and the Sun.

The distance between Earth and the Sun on the scale model = 31 inches

Scale factor of the model = 1  inch : 3 million miles

Assume that the ratio between the distance on the scale model and the actual distance is 31 inches : d million miles , where d is the actual distance between Earth and the Sun (in million miles).

We can use the scale 1 inch  : 3 million miles and the ratio 31 inches:d million miles to form a proportion. 

\(\frac{1~\text{inch}}{100~\text{million miles}} = \frac{31~\text{inch}}{d~\text{million miles}}\)

\(d = 31\times 3\)       Cross product

\(d = 93\)               Multiply

Since d is the distance between Earth and the Sun in million miles, we got the actual distance as 93 million miles.

What is the difference between scale and scale factor?

The ratio that compares the dimensions of drawings or models with real-life objects is known as  scale . A scale factor is a number with which you multiply the dimension of a drawing or a model to get the dimension of the real-life object.

Can a model have two scale factors?

If a model has more than one scale factor, it means that the model is not proportional to the real-life object.

How can we find the dimensions of the real-life object from the scale factor?

To find the dimensions of the real-life object, we just need to multiply the dimensions of the drawing or model with the scale factor. 

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