math problem solving proportion

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How to Solve Proportions

Last Updated: March 26, 2024

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 97,501 times.

What is the "vertical" way to solve a proportion?

How can I solve a proportion with the "horizontal" method?

How do I solve a proportion step by step by cross-multiplying?

How do you find the missing value in a proportion with a table of ratios?

	48			128
	x			8

• Each column in this table represents a fraction. All of the fractions in this table are equal to each other.

	48	64		128
	x	4		8

32	48	64		128
2	x	4		8

• The two answers are the same, which means your answer is correct.

How do you solve percent proportions?

How do you solve proportions algebraically?

• You can change the left hand side of the equation, as long as you do the same math to the right hand side.

• To get rid of the fraction on the left, multiply both sides by 27:

How do you solve a proportion with a variable on both sides?

• Warning : This is a difficult example. If you haven't learned about quadratic equations yet, you might want to skip this part.

• You can now solve this as a quadratic equation , using any method that you've learned.

Proportions Calculator, Practice Problems, and Answers

Community Q&A

• The algebraic method above works with any proportion. But for a specific proportion, there is often a faster way to use algebra to find the answer. As you learn more algebra, this will get easier. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ http://www.mathvillage.info/node/72
• ↑ https://www.youtube.com/watch?v=nwsDiID7UtQ
• ↑ https://www.youtube.com/watch?v=Uo8HgcyfRFI
• ↑ https://www.purplemath.com/modules/ratio2.htm

About This Article

To solve proportions, start by taking the numerator, or top number, of the fraction you know and multiplying it with the denominator, or bottom number, of the fraction you don’t know. Next, take that number and divide it by the denominator of the fraction you know. Now you can replace x with this final number. For example, to figure out “x” in the problem 3/4 = x/8, multiply 3 x 8 to get 24, then divide 24 / 4 to get 6, or the value of x. To learn how to use proportions to determine percentages, read on! Did this summary help you? Yes No

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Proportion word problems

/4/8 , /3/x , /x/8 , /3/4 , /4/8 , /3/x , /x/8 , /3/4

It is very important to notice that if the ratio on the left is a ratio of number of liters of water to number of lemons, you have to do the same ratio on the right before you set them equal. 

/Number of liters of water/Number of liters of water/3/x/w/w

More interesting proportion word problems

/Length of shadow/Length of shadow/7/14/900/300/3/x/x/300/3/900/900/300/Time it takes/Time it takes/2/10/2/T/900/300 , /3/x , /x/300

Check this site if you want to solve more proportion word problems.

Ratio word problems

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Proportions

Proportion says that two ratios (or fractions) are equal.

We see that 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Example: Rope

A rope's length and weight are in proportion.

When 20m of rope weighs 1kg , then:

• 40m of that rope weighs 2kg
• 200m of that rope weighs 10kg

20 1 = 40 2

When shapes are "in proportion" their relative sizes are the same.

Here we see that the ratios of head length to body length are the same in both drawings. So they are . Making the head too long or short would look bad!

Example: International paper sizes (like A3, A4, A5, etc) all have the same proportions:

So any artwork or document can be resized to fit on any sheet. Very neat.

Working With Proportions

NOW, how do we use this?

Example: you want to draw the dog's head ... how long should it be?

Let us write the proportion with the help of the 10/20 ratio from above:

? 42 = 10 20

Now we solve it using a special method:

Multiply across the known corners, then divide by the third number

And we get this:

? = (42 × 10) / 20 = 420 / 20 = 21

So you should draw the head 21 long.

Using Proportions to Solve Percents

A percent is actually a ratio! Saying "25%" is actually saying "25 per 100":

25% = 25 100

We can use proportions to solve questions involving percents.

The trick is to put what we know into this form:

Part Whole = Percent 100

Example: what is 25% of 160 ?

The percent is 25, the whole is 160, and we want to find the "part":

Part 160 = 25 100

Multiply across the known corners, then divide by the third number:

Part = (160 × 25) / 100 = 4000 / 100 = 40

Answer: 25% of 160 is 40.

Note: we could have also solved this by doing the divide first, like this:

Part = 160 × (25 / 100) = 160 × 0.25 = 40

Either method works fine.

We can also find a Percent:

Example: what is $12 as a percent of $80 ?

Fill in what we know:

$12 $80 = Percent 100

Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:

Percent = ($12 × 100) / $80 = 1200 / 80 = 15%

Answer: $12 is 15% of $80

Or find the Whole:

Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?

$150 Whole = 80 100

Whole = ($150 × 100) / 80 = 15000 / 80 = 187.50

Answer: the phone's normal price was $187.50

Using Proportions to Solve Triangles

We can use proportions to solve similar triangles.

Example: How tall is the Tree?

Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.

But then Sam has a clever idea ... similar triangles!

Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:

Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles:

Height: Shadow Length:     h 2.9 m = 2.4 m 1.3 m

h = (2.9 × 2.4) / 1.3 = 6.96 / 1.3 = 5.4 m (to nearest 0.1)

Answer: the tree is 5.4 m tall.

And he didn't even need a ladder!

The "Height" could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:

Let us try the ratio of "Shadow Length to Height":

Shadow Length: Height:     2.9 m h = 1.3 m 2.4 m

It is the same calculation as before.

A "Concrete" Example

Ratios can have more than two numbers !

For example concrete is made by mixing cement, sand, stones and water.

A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6 .

We can multiply all values by the same amount and still have the same ratio.

10:20:60 is the same as 1:2:6

So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones.

Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make a 1:2:6 mix?

Let us lay it out in a table to make it clearer:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:			12

You have 12 buckets of stones but the ratio says 6.

That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.

Here is the solution:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:	2	4	12

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....)

Why are they the same ratio? Well, the 1:2:6 ratio says to have :

• twice as much Sand as Cement ( 1 : 2 :6)
• 6 times as much Stones as Cement ( 1 :2: 6 )

In our mix we have:

• twice as much Sand as Cement ( 2 : 4 :12)
• 6 times as much Stones as Cement ( 2 :4: 12 )

So it should be just right!

That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same.

Direct and Inverse Proportion Practice Questions

Click here for questions, click here for answers .

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Home / United States / Math Classes / 7th Grade Math / Writing and Solving Proportions

Writing and Solving Proportions

A proportion is an equation that states that two ratios are equivalent. We can perform operations on proportions, just l ike we do with normal equations. Here we will learn to perform operations on proportions and the steps involved in solving them. ...Read More Read Less

About Proportions in Math

What are Proportions?

Writing proportions as fractions, using operations to solve proportions, solved examples.

• Frequently Asked Questions

Proportions are mathematical equations that are used to relate equivalent ratios. Two ratios having different antecedents and consequents can have the same value. This relation can be expressed with the help of proportions. Let us consider the ratios \( 2:5 \)  and \( 8:20 \) . When we simplify the ratio \( 8:20 \) , we obtain \( 2:5 \) . This implies that \( 2:5 \)  and \( 8:20 \)  are equivalent ratios. 

In other words, these ratios are in proportion. If \( a:b \)  and \( c:d \) are equivalent ratios, we can express the relation as \( a:b::c:d \) , where the ‘ \( ~::~ \) ’ sign is used to express proportion. So, we can state the proportion in the example that was just observed as, \( 2:5 :: 8:20 \) .

We have learned that a ratio is a comparison of two quantities having the same unit. We also compare two quantities using fractions. Hence, a ratio can also be written as a fraction. For example, we can write the ratio \( a:b \)  as \( \frac{a}{b} \) and \( c:d \)  as \( \frac{c}{d} \) .

Similarly, we can write a proportion as a fraction. Instead of writing \( 2:5::8:20 \) , we can write the proportion as \( \frac{2}{5} = \frac{8}{20} \)   , and this makes it easier to perform operations on proportions to solve for unknown values.

Since a proportion is basically an equation, we can perform operations on them to find unknown values. We can use operations like addition, subtraction, multiplication, and division to solve a proportion. In most cases, we only need to use multiplication and division. Let’s consider a proportion in which one of the values is unknown. 

For example, \( \frac{5}{8} = \frac{x}{40} \)

Use basic math operations to solve this equation. Begin by removing the denominator from both sides.

\( \frac{5 \times 8}{8} = \frac{x \times 8}{40} \)            [Multiply both sides by \( 8 \)]

\( 5 = \frac{x}{5} \)                    [Simplify]

\( 5 \times 5 = \frac{x\times 5}{5} \)          [Multiply both sides by \( 5 \)]

\( 25 = x \)                   [ Simplify]      

Hence, the value of \( x \) is \( 25 \). 

Similarly, we can use a combination of mathematical operations to solve proportions.

Example 1: Use math operations to find the value of \( x \) in the expression, \( \frac{3}{7} = \frac{x}{28} \) .

Solution:  

To find the value of \( x \) , simplify the equation.

\( \frac{3}{7} = \frac{x}{28} \)                    [Write the equation]

\( \frac{3 \times 7}{7} = \frac{x \times 7}{28} \)              [Multiply both sides by \( 7 \)]

\( 3 = \frac{x}{4} \)                      [Simplify]

\( 3 \times 4 = \frac{x \times 4}{4} \)            [Multiply both sides by \( 4 \)]

\( 12 = x \)                    [Simplify]

So, the value of \( x \) is \( 12 \).

Example 2: Solve the proportion to find the unknown value: \( 15:y :: 25:55 \) .

The proportion is \( 15:y :: 25:55 \) and this expression can also be written as \( \frac{15}{y} = \frac{25}{55} \)

To find the value of \( y \) , simplify the equation.

\( \frac{15}{y} = \frac{25}{55} \)                   [Write the proportion]

\( \frac{y}{15} = \frac{55}{25} \)                    [Taking reciprocal of both sides]

\( \frac{y \times 15}{15} = \frac{55 \times 15}{25} \)           [Multiplying both sides by \( 15 \)]

\( y = \frac{11 \times 15}{55} \)                  [Simplify]

\( y = 11 \times 3 \)                [Simplify]

\( y = 33 \)                      [Multiply]

Hence, the unknown value, \( y \) is \( 33 \).

Example 3: An athlete can run \( 100 \) meters in \( 11 \) seconds. If she runs at a constant pace, how long will she take to run \( 800 \) meters?

Time taken by the athlete to cover \( 100 \) meters \( = 11 \) seconds

Let us assume the time taken by the athlete to cover \( 800 \) meters \( = x \) seconds

Since her speed is constant, the ratio of distance and time in both cases is in proportion. 

\( \frac{100}{11} = \frac{800}{x} \)                      [Write the above condition in proportion]

\( \frac{11}{100} = \frac{x}{800} \)                      [Taking reciprocal of both sides]

\( \frac{11 \times 100}{100} = \frac{x \times 100}{800} \)              [Multiplying both sides by \( 100 \)]

\( 11 = \frac{x}{8} \)                          [Simplify]

\( x = 88 \)                            [Multiplying both sides by \( 8 \)]

Hence, the athlete will take \( 88 \) seconds to run \( 800 \) meters.

Are ratios related to proportions?

Yes, a proportion is an equation that states that two ratios are equivalent. So, we can say that proportions are directly related to ratios.

Can we perform mathematical operations on proportions?

Since proportions are basically mathematical equations, we can perform all the mathematical operations on them, just like we do with a normal mathematical equation.

How do we solve a proportion?

We can solve a proportion to find the unknown value by performing mathematical operations on them. The goal is to isolate the unknown value on one side of the equation. Thus, by solving the equation, we will get the value of the unknown on the other side of the equation.

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Proportions

Ratios Proportions Proportionality Solving Word Problems Similar Figures Sun's Rays / Parts

A ratio is one thing or value compared with or related to another thing or value; it is just a statement or an expression, and can only perhaps be simplified or reduced.

On the other hand, a proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved.

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Solving Proportions

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Solving a proportion means that we have been given an equation containing two fractions which have been set equal to each other, and we are missing one part of one of the fractions; we then need to solve for that one missing value. For instance, suppose we are given the following equation:

Verifying what we already knew, we get a solution of x  = 5 .

Often times, students are asked to solve proportions before they've learned how to solve rational equations , which can be a bit of a problem. If one hasn't yet learned about rational expressions (that is, polynomial fractions), then it will be necessary to "get by" with "cross-multiplication".

To cross-multiply , we start with an equation in which two fractions are set equal to each other. Then we take each denominator and move it aCROSS the "equals" sign and then MULTIPLY it against the other fraction's numerator. The cross-multiplication solution of the above exercise looks like this:

Then we would solve the resulting linear equation by dividing through by 2 to again arrive at x  = 5 .

Note the process in the above. We multiplied the left-hand side's denominator by the right-hand side's numerator, and then divided by the right-hand side's denominator. You may see this process explicitly applied for the solving of proportions. The method of solution would then by to cross-multiply the numbers (that is, in the direction that does not involve the variable), and then divide by the remaining number. In very informal notation, the process looks like this:

The green arrow pointing northeast (that is, from bottom left to upper right) indicates the multiplication step; the purple looping arrow that ends up pointing at the variable indicates the division step.

Solve the proportion: katex.render("\\small{ \\bm{\\color{green}{ \\dfrac{6}{13} = \\dfrac{18}{y} }}}", typed08); 6/13 = 18/ y

The variable in this proportion is in the denominator of the right-hand side's fraction, but that's okay. I can still cross-multiply and solve.

6/13 = 18/ y

(6)( y ) = (13)(18)

(6 y )/6 = 234/6

The solution of the proportion is the value of the variable, so my answer is:

If I'd done the shorthand method (shown with the green and purple arrows above), the computations would have been:

(13 × 18) ÷ 6 = 39 = y

It's harder to "show your work" using the shorthand method, but the shorthand method is easier to plug into your calculator. Use whatever method works well for you.

Solve the proportion: katex.render("\\small{ \\bm{\\color{green}{\\dfrac{5}{b} = \\dfrac{42}{35} }}}", typed10); 5/ b = 42/25

I'll cross-multiply, and then divide:

5/ b = 42/35

(5)(35) = ( b )(42)

(175)/42 = (42 b )/42

Hm... Can proportions have fractional solutions? Yes, definitely; they can! I mustn't let the only-whole-number exercises and examples mislead me into thinking that proportions must always have whole-number answers. They don't. My fractional answer is perfectly fine.

Proportions wouldn't be of much use if you only used them for reducing fractions. A more typical use would be something like the following:

The ratio of waterfowl in a given park is 16 ducks to 9 geese. Suppose that there are 192 ducks in total. How many geese are there in total?

They've asked me to solve for an unknown value, so I'll need an equation with a variable. They've given me a ratio, so my equation will be a proportion.

I'll let " G " stand for the unknown number of geese. I'll clearly label the orientation of my ratios (so I don't confuse which number stands for what), and then I'll set up my proportional equation:

I'll cross-multiply to solve for the value of G :

16 G = 1728

(16 G )/16 = (1728)/16

I labelled things clearly at the beginning, so I know that " G " stands for "the number of geese in the park". So my answer is:

"Cross-multiplying" is standard classroom language, in that it is very commonly used by students and instructors, but it is not technically a mathematical term. You might not see "cross-multiplication" mentioned in your textbook, but you will almost certainly hear it in your class or study group.

Notice how, when I was setting up my equation at the beginning of my solution above, I prefaced my proportion by writing out my ratio in words; namely:

This is not standard notation (in the sense of your textbook being likely to use it), but it can be very helpful for setting up proportions. By clearly labelling which values are represented by the numerators and denominators, respectively, you will help yourself keep track of what each number stands for; you won't mix up which number or unit goes where.

In other words, using this method will help you set up your proportions correctly. If you do not set up the ratios consistently (for instance, if, in the above example, I'd mixed up where the values for the "ducks" and the "geese" were supposed to go in the various fractions), I'd have gotten an incorrect answer. Clarity in your set-up is crucially important when working with proportions. We will return to this subject later.

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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)

You have been given And you want to	Step 1: Add the parts of the ratio together. Step 2: Divide the quantity by the sum of the parts. Step 3: Multiply the share value by each part in the ratio.	For example Share £100 in the ratio 4:1 . (£80:£20)
You have been given And you want to find	Step 1: Identify which part of the ratio has been given. Step 2: Calculate the individual share value. Step 3: Multiply the other quantities in the ratio by the share value.	For example A bag of sweets is shared between boys and girls in the ratio of 5:6. Each person receives the same number of sweets. If there are 15 boys, how many girls are there? (18)

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.

You have been given And you want to find	Step 1: Add the parts of the ratio for the denominator. Step 2: State the required part of the ratio as the numerator.	For example The ratio of red to green counters is 3:5. What fraction of the counters are green? (\frac{5}{8})
You have been given And you want to find	Step 1: Subtract the numerator from the denominator of the fraction. Step 2: State the parts of the ratio in the correct order.	For example if \frac{9}{10} students are right handed, write the ratio of right handed students to left handed students. (9:1)

Simplifying and equivalent ratios

• Simplifying ratios

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

Equivalent ratios

You have been given And you want to find	Step 1: Identify which part of the ratio is to equal 1. Step 2: Divide all parts of the ratio by this value.	For example Write the ratio 4:15 in the form 1:n. (1:3.75)
You have been given And you want to find	Step 1: Multiply all parts of the ratio by the same amount.	For example A map uses the scale 1:500. How many centimetres in real life is 3cm on the map? (1:500 = 3:1500, so 1500 cm)

Units and conversions 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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