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## Solving problems with percentages

- Price difference I
- Price difference II
- How many students?

To solve problems with percent we use the percent proportion shown in "Proportions and percent".

$$\frac{a}{b}=\frac{x}{100}$$

$$\frac{a}{{\color{red} {b}}}\cdot {\color{red} {b}}=\frac{x}{100}\cdot b$$

$$a=\frac{x}{100}\cdot b$$

x/100 is called the rate.

$$a=r\cdot b\Rightarrow Percent=Rate\cdot Base$$

Where the base is the original value and the percentage is the new value.

47% of the students in a class of 34 students has glasses or contacts. How many students in the class have either glasses or contacts?

$$a=r\cdot b$$

$$47\%=0.47a$$

$$=0.47\cdot 34$$

$$a=15.98\approx 16$$

16 of the students wear either glasses or contacts.

We often get reports about how much something has increased or decreased as a percent of change. The percent of change tells us how much something has changed in comparison to the original number. There are two different methods that we can use to find the percent of change.

The Mathplanet school has increased its student body from 150 students to 240 from last year. How big is the increase in percent?

We begin by subtracting the smaller number (the old value) from the greater number (the new value) to find the amount of change.

$$240-150=90$$

Then we find out how many percent this change corresponds to when compared to the original number of students

$$90=r\cdot 150$$

$$\frac{90}{150}=r$$

$$0.6=r= 60\%$$

We begin by finding the ratio between the old value (the original value) and the new value

$$percent\:of\:change=\frac{new\:value}{old\:value}=\frac{240}{150}=1.6$$

As you might remember 100% = 1. Since we have a percent of change that is bigger than 1 we know that we have an increase. To find out how big of an increase we've got we subtract 1 from 1.6.

$$1.6-1=0.6$$

$$0.6=60\%$$

As you can see both methods gave us the same answer which is that the student body has increased by 60%

## Video lessons

A skirt cost $35 regulary in a shop. At a sale the price of the skirtreduces with 30%. How much will the skirt cost after the discount?

Solve "54 is 25% of what number?"

- Pre-Algebra
- The mean, the median and the mode
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## Ratio and Percent Word Problems

Ratio word problems, using table with multiplication.

## Using Double Number Lines

## Using Tape Diagram

## Using Graphs

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## Solving ratio problems

The ratio of horses to donkeys at an animal sanctuary is 5 : 2. What fraction of the animals are donkeys?

Add the ratio parts (5 and 2) to find the denominator of the fraction. 5 + 2 = 7. The denominator is 7

The numerator of the fraction is the ratio part that is the focus of the question (donkeys). The numerator is 2. The fraction of the animals that are donkeys is 2⁄7

To make pink paint, 2⁄5 red paint is mixed with white paint. How much white paint is needed? What is the ratio of red to white paint?

The numerator of the fraction of red paint is 2. This is the ratio part for red paint. To find the ratio part for the white paint, subtract the numerator from the denominator. 5 – 2 = 3. The ratio part for white paint is 3. 3⁄5 of the paint is white. The ratio of red to white paint is 2 : 3

A farm has sheep and goats in the ratio 7 : 5 (sheep : goats). What fraction of the animals are sheep?

## Show answer Hide answer

Add the ratio parts (7 and 5) to find the denominator of the fraction.

7 + 5 = 12. The denominator is 12

The numerator of the fraction is the ratio part that is the focus of the question (sheep). The numerator is 7

The fraction of animals that are sheep is \( \frac{7}{12} \)

A box contains toffees and truffles in the ratio 7 : 6. There are 24 truffles. How many toffees are there?

Draw a bar model and label it to illustrate the problem. The ratio of toffees to truffles is 7 : 6. Draw 7 parts for the toffees and 6 for the truffles. Label the truffle bar with 24. To work out the total number of toffees, the value of one part needs to be found.

To work the value of one part, divide the number of truffles (24) by the number of parts given (6). This gives the value of one part. 24 ÷ 6 = 4

Multiply the value of one part (4) by the number of parts asked for: the toffees (7). 4 × 7 = 28. There are 28 toffees in the box.

A library stocks books for children and books for adults in the ratio 3 : 5. There are 450 books for children. How many books are in the library?

Draw a bar model and label it to illustrate the problem. The ratio of books for children to books for adults is 3 : 5. Draw 3 parts for books for children and 5 parts for books for adults. Label the books for children bar with 450. To work out the total number of books, the value of one part needs to be found.

To find the value of one part, divide the amount of books for children (450) by the number of parts given (3). 450 ÷ 3 = 150. The value of one part is 150

Multiply the value of one part (150) by the number of parts asked for: the number of books in the library. This is the total of books for children parts and the books for adults parts. (3 + 5 = 8). The calculation is 150 × 8. The total number of books in the library is 1200.

The ratio of desserts to pizzas in a supermarket freezer is 4 : 3 There are a total of 620 desserts. How many pizzas are in the freezer?

Draw a bar model to illustrate the problem. The ratio of desserts to pizzas is 4 : 3 . Draw 4 parts for desserts and 3 parts for pizzas.

Label it to illustrate the problem. Label the dessert bar with 620. To find the number of pizzas in the freezer, work out the value of one part.

To find the value of one part, divide the share for desserts (620) by the number of parts (4). 620 ÷ 4 = 155. The value of one part is 155

Multiply the value of one part (155) by the number of parts asked for: pizzas (3). 155 x 3 = 465

The number of pizzas in the freezer is 465

Everyone at a fancy dress party is dressed up as either a vampire or a wizard. The ratio of people dressed as vampires to wizards is 5 : 2. If there are 6 more vampires than wizards, how many people are at the party?

Draw a bar model to illustrate the problem. The ratio of vampires to wizards is 5 : 2. There are 5 parts for the vampires and 2 parts for the wizards.

Label the given information. There are 6 more vampires than wizards. The diagram shows the comparison between the vampires bar and the wizards bar. To work out the total number of people at the party, the value of one part needs to be found.

To find the value of one part, divide the difference value (6) by the number of parts that make up the difference (3). 6 ÷ 3 = 2. The value of one part is 2

Multiply the value of one part (2) by the number of parts asked for (all the people so all the parts, 7). 2 × 7 = 14. The total number of people at the party is 14

The ratio of the number of tulips to daffodils in a flower display is 3 : 7. There are 96 fewer tulips than daffodils. Find the number of daffodils in the display.

Draw a bar model to illustrate the problem. The ratio of tulips to daffodils is 3 : 7. There are 3 parts for tulips and 7 parts for daffodils.

Label the given information. There are 96 fewer tulips than daffodils. The diagram shoes the comparison between the tulips bar and the daffodils bar. To work out the number of daffodils in the display, the value of one part needs to be found.

To find the value of one part, divide the comparison value (96) by the number of parts that make up the difference (4). 96 ÷ 4 = 24. The value of one part is 24

Multiply the value of one part (24) by the number of parts asked for (all the daffodils, 7 parts). 24 × 7 = 168. The number of daffodils in the display is 168

The ratio of the number of robins to sparrows to blackbirds in a survey of garden birds is 1 : 3 : 8 (robins : sparrows : blackbirds)

There were 70 fewer robins than sparrows. How many birds were observed in this survey?

Draw a bar model to illustrate the problem. The ratio of robins to sparrows to blackbirds is 1 : 3 : 8. There is 1 part for robins, 3 for sparrows and 8 for blackbirds.

Label the given information. There are 70 fewer robins than sparrows. This is the difference between the robins bar and the sparrows bar. To find the total number of birds in the survey, the value of one part needs to be found.

To find the value of one part, divide the difference value (70) by the number of parts that make up the difference (2). 70 ÷ 2 = 35. The value of one part is 35

Multiply the value of one part (35) by the number of parts asked for (all the parts, 12). 35 x 12 = 420

A market stall has 60 T-shirts. The ratio of small, medium and large T-shirts being sold on the stall is 2 : 3 : 1. The stallholder sells 26 medium and 2 large T-shirts. What is the ratio of small, medium and large T-shirts now?

Draw a bar model to illustrate the starting information. Draw 2 parts for small T-shirts, 3 for medium and 1 for large. The total number of T-shirts is 60

Divide 60 in the ratio 2 : 3 : 1. To find the value of one part, divide the total number (60) by the sum of the parts (2 + 3 + 1 = 6). 60 ÷ 6 = 10. The value of one part is 10

2 parts are small, so 2 × 10 = 20 small T-shirts. 3 parts are medium, so 3 × 10 = 30 medium T-shirts. 1 part is large, so 1 × 10 = 10 large T-shirts.

Adjust the shared amounts according to the given information in the question. At the start there were 20 small, 30 medium and 10 large T-shirts. 26 medium and 2 large T-shirts are sold. The remaining T-shirts are 20 small, 4 medium and 8 large T-shirts.

The ratio of small to medium to large T-shirts is now 20 : 4 : 8. This ratio can be simplified.

To simplify the ratio, divide each part by their highest common factor (HCF). The HCF of 20, 4 and 8 is 4. Divide each part by 4. The new simplified ratio of small to medium to large T-shirts is 5 : 1 : 2

There are 55 big cats in a safari park. The ratio of lions to tigers is 3 : 2 12 lion cubs and 5 tiger cubs are born. What is the ratio of lions to tigers now?

A total amount and ratio is given. Draw a bar model to illustrate this starting information. Draw 3 parts for lions and 2 parts for tigers, with a total of 55

Divide the total number of big cats (55) in the ratio 3 : 2. To find the value of one part, divide the amount (55) by the total number of parts (5). 55 ÷ 5 = 11. The value of one part is 11. Multiply one part by the number of parts for each big cat. There are 11 × 3 = 33 lions. There are 11 × 2 = 22 tigers.

Adjust the shared amounts according to given information in the question. After the cubs have been born there are 33 + 12 = 45 lions and 22 + 5 = 27 tigers.

Write the new shares as a ratio and simplify. The new ratio of lions to tigers is 45 : 27. This ratio can be simplified by finding the HCF of 45 and 27. This is 9. Divide each ratio share by 9. The ratio simplifies to 5 : 3

The ratio of lions to tigers is now 5 : 3

## Real-world maths

## More on Ratio

Find out more by working through a topic

Scale drawings

- count 4 of 5

Map scales and ratio

- count 5 of 5

Equivalent ratios and simplifying ratios

- count 1 of 5

Division in a given ratio

- count 2 of 5

## Ratio Worksheet

FREE DOWNLOAD

Help your students prepare for their Maths GCSE with this free ratio worksheet of 44 questions and answers

- Section 1 of the ratio worksheet contains 36 skills-based ratio questions, in 3 groups to support differentiation
- Section 2 contains 4 applied ratio questions with a mix of worded problems and deeper problem solving questions
- Section 3 contains 4 foundation and higher level GCSE exam style ratio questions
- Answers and a mark scheme for all ratio questions are provided
- Questions follow variation theory with plenty of opportunities for students to work independently at their own level
- All questions created by fully qualified expert secondary maths teachers
- Suitable for GCSE maths revision for AQA, OCR and Edexcel exam boards

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Raise maths attainment across your school with hundreds of flexible and easy to use GCSE maths worksheets and lessons designed by teachers for teachers.

## Ratio at a glance

A ratio compares how much there is of one thing in relation to another. For example if the ratio of the number of dogs to the number of cats is 2:3 then for every 2 dogs there are 3 cats.

Ratios can often be simplified by dividing each part by a common factor. Writing ratios in their simplest form makes it easier to visualise the relationship between the quantities in the ratios. It also makes it easier to use ratios in other contexts.

Ratio word problems may ask us to write a ratio, simplify a ratio, divide a quantity into a ratio or use a ratio to find quantities. We can use a bar model to represent a given ratio and this can help us visualise the ratio problem more easily.

Looking forward, students can then progress to additional ratio and proportion worksheets , for example a speed distance time worksheet or a direct proportion worksheet.

For more teaching and learning support on Ratio and Proportion our GCSE maths lessons provide step by step support for all GCSE maths concepts.

## Related worksheets

## Proportion Diagnostic Questions

Ratio and Proportion

## Ratio Diagnostic Questions

## Measurements & scales Diagnostic Questions

## Scale drawing worksheet

Popular for gcse.

## Do you have KS4 students who need more focused attention to succeed at GCSE?

There will be students in your class who require individual attention to help them succeed in their maths GCSEs. In a class of 30, it’s not always easy to provide.

Help your students feel confident with exam-style questions and the strategies they’ll need to answer them correctly with our dedicated GCSE maths revision programme.

Lessons are selected to provide support where each student needs it most, and specially-trained GCSE maths tutors adapt the pitch and pace of each lesson. This ensures a personalised revision programme that raises grades and boosts confidence.

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## VIDEO

## COMMENTS

Advice • Read each question carefully before you start to answer it. • Keep an eye on the time. • Try to answer every question. • Check your answers if you have time at the end mathsgenie.co.uk 1 The ratio of dogs to cats is 5:3 The ratio of fish to dogs is 6:1 Find the ratio of cats to fish. Give your answer in its simplest form.

Finding a Percentage of an Amount Watch on Finding a Percentage of an Amount without a calculator Percent means out of 100 When we work out percentages without a calculator we usually start by working out 50%, 10% or 1%. 50% is the same as 50 100 We can simplify 50 100 by dividing the top and bottom by 50 50 100 = 1 2

Maths revision video on the topic solving problems relating to direct and inverse proportion relationships.

Join this channel to get access to perks:https://www.youtube.com/channel/UCStPzCGyt5tlwdpDXffobxA/joinA video revising the techniques and strategies for solv...

The trick to ratio problems made easy. Essential skills for GCSE maths. Become a maths genius, get the basics of ratio and proportion correct.Designed by www...

Advice Read each question carefully before you start to answer it. Keep an eye on the time. Try to answer every question. Check your answers if you have time at the end. 1. A piece of wood is of length 45 cm. The length is divided in the ratio 7 : 2 Work out the length of each part. ...................... cm, ..................... cm (3 marks) 2.

Ratio, Fractions, Decimals and Percentages Back to Scheme Overview Stage 1 Stage 2 In addition to the topics in stage 1, students should be fluent in the following topics. Stage 3 In addition to the topics in stage 1 and 2, students should be fluent in the following topics. Maths Genie is a free GCSE and A Level revision site.

Part A The magic carpet is made with a total of 150 meters of yarn. How much silver yarn is in the magic carpet? meters Part B Jasmine buys the magic carpet on sale for $ 374 . Jasmine saved $ 66 off the regular price. What percent was the price of the magic carpet discounted? % Problem 2: Soccer Joel is training for soccer season.

Instructions Use black ink or ball-point pen. Answer all questions. Answer the questions in the spaces provided there may be more space than you need. Diagrams are NOT accurately drawn, unless otherwise indicated. You must show all your working out. Information The marks for each question are shown in brackets

Advice • Read each question carefully before you start to answer it. • Keep an eye on the time. • Try to answer every question. • Check your answers if you have time at the end mathsgenie.co.uk 1 In a bag there are blue sweets and red sweets. The ratio of blue sweets to red sweets is 5:3 What fraction of the sweets are blue?

To solve problems with percent we use the percent proportion shown in "Proportions and percent". a b = x 100 a b = x 100. a b ⋅b = x 100 ⋅ b a b ⋅ b = x 100 ⋅ b. a = x 100 ⋅ b a = x 100 ⋅ b. x/100 is called the rate. a = r ⋅ b ⇒ Percent = Rate ⋅ Base a = r ⋅ b ⇒ P e r c e n t = R a t e ⋅ B a s e. Where the base is the ...

Grade 7 math (FL B.E.S.T.) 8 units · 90 skills. Unit 1 Equivalent forms of numbers. Unit 2 Operations with rational numbers. Unit 3 Ratios and percentages. Unit 4 Proportional relationships. Unit 5 Expressions, equations, & inequalities. Unit 6 Area and volume. Unit 7 Scale copies and scale drawings.

Pre-algebra 15 units · 179 skills. Unit 1 Factors and multiples. Unit 2 Patterns. Unit 3 Ratios and rates. Unit 4 Percentages. Unit 5 Exponents intro and order of operations. Unit 6 Variables & expressions. Unit 7 Equations & inequalities introduction. Unit 8 Percent & rational number word problems.

To solve percent problems, you can use the equation, Percent ⋅ Base = Amount , and solve for the unknown numbers. Or, you can set up the proportion, Percent = amount base , where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion. Percents are a ratio of a number and 100, so they are ...

Common Core Connection for Grades 6 and 7. Understand the concept of ratio and describe the relationship between two quantities. Use ratio and rate reasoning to solve real-world and mathematical problems. Recognize and represent proportional relationships between quantities. Use proportional relationships to solve multistep ratio and percent ...

Ratio: Problem Solving Textbook Exercise - Corbettmaths. October 7, 2019 corbettmaths.

In order to convert a ratio to a percentage: Add the parts of the ratio for the denominator of the fractions. Convert each part of the ratio to a fraction. Convert the fractions to percentages. Explain how to convert a ratio to a percentage Ratio to percentage worksheet Related lessons on ratio

Ratio problem solving GCSE questions. 1. One mole of water weighs 18 18 grams and contains 6.02 \times 10^ {23} 6.02 × 1023 water molecules. Write this in the form 1gram:n 1gram: n where n n represents the number of water molecules in standard form. 2.

A ratio is a comparison of any two quantities. It can be written as a to b, a: b or a/b. Percent is a ratio. Percent should be viewed as a part-to-whole ratio that compares a number to a whole divided into 100 equal parts. In these lessons, we will learn how to solve ratio word problems and how to use ratios to help us solve percent word problems.

To find the value of one part, divide the difference value (6) by the number of parts that make up the difference (3). 6 ÷ 3 = 2. The value of one part is 2. Image caption, Multiply the value of ...

Click here for Questions Click here for Answers Percentages (calculator) Click here for Questions Click here for Answers Practice Questions Previous Foundation Solving Quadratics Next Ratio Videos The Corbettmaths Practice Questions on finding a percentage of an amount.

The Corbettmaths Practice Questions on Ratio. Videos, worksheets, 5-a-day and much more

Help your students prepare for their Maths GCSE with this free ratio worksheet of 44 questions and answers Section 1 of the ratio worksheet contains 36 skills-based ratio questions, in 3 groups to support differentiation Section 2 contains 4 applied ratio questions with a mix of worded problems and deeper problem solving questions