how to solve multi step ratio problems

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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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Ratio & Percent Problems (Grade 7)

Suggested learning targets.

• I can calculate percentage of increase and percentage of decrease.
• I can calculate simple interest, tax, markups and markdowns.
• I can calculate percent increase and decrease.
• I can calculate gratuities, commissions, and fees.
• I can calculate percent error.
• I can solve multi-step ratio and percent problems using proportional relationships.
• I can justify multi-step ratio and percent in real life situations.
• I can iIdentify or describe errors to given multi-step problems and present corrected solutions.

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Paine in the Math

Mr. Paine's Online Course Resources

Lesson 14: Multistep Ratio Problems

Essential Question: How do we solve multi-step ratio problems including fractional markdowns, markups, commissions, fees, etc?

[expand title=”Warm up”]

3. $40 – (Answer to #1)

4. $33 – (Answer to #2)

[expand title=”Lesson”]

S.58 Example 2: Big Al’s Used Cars (Partner Work)

• Can you write an equation for determining the commission for any car sold?

S.59 Example 3: Tax time

S.59 Example 4: Born to Ride

[expand title=”Summary”]

[expand title=”Problems”]

[expand title=”Exit”]

Curriculum  /  Math  /  7th Grade  /  Unit 1: Proportional Relationships  /  Lesson 17

Proportional Relationships

Lesson 17 of 18

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Solve multi-step ratio and rate problems using proportional reasoning, including fractional price increase and decrease, commissions, and fees.

Common Core Standards

Core standards.

The core standards covered in this lesson

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Foundational Standards

The foundational standards covered in this lesson

Number and Operations—Fractions

5.NF.B.6 — Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Find a fractional amount of a quantity and reason about what to do with the value using the context of the problem.
• Find an original value when given either a discount or a markup and a fractional rate. 
• Use different strategies to solve price increase, decrease, commission, and fee problems, including tape diagrams, proportions, equations, and other proportional reasoning.

Suggestions for teachers to help them teach this lesson

• In terms of pacing, this lesson may be extended over more than one day depending on the amount of practice and/or exposure to different variations of problems students need. The Target Task includes 2 problems which could be split over 2 days. 
• The Anchor Problems specifically cover the topics of price increase and price decrease. The other topics of commissions and fees should be included in the problem set.
• Percent problems are not included in this lesson; all problems involve fractional amounts rather than percentages. In Unit 5, students will revisit this topic, but with percentages. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

Peter’s Pants Palace advertises the following sale: 

a.   If a pair of shoes originally costs $40, what is the sale price?

b.   A pair of pants usually sells for $33. What is the sale price of the pants?

Guiding Questions

Grade 7 Mathematics > Module 1 > Topic C > Lesson 14 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

A phone case was on sale for $${{1\over3}}$$  off the original price. Terry purchased the phone case at this discounted price for $24. What was the original price of the phone case?

An autographed baseball card of Pedro Martinez was originally worth $240. In 2015, when the retired player was inducted into the Baseball Hall of Fame, the value of the card increased in value by $$\frac{1}{5}$$  its original value. What is the new value of the card?

A motorcycle dealer paid a certain price for a motorcycle and marked it up by $${{1\over5}}$$  of the price he paid. Later, he sold it for $14,000. What is the original price? 

Solve this using two different strategies.

A set of suggested resources or problem types that teachers can turn into a problem set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

A furniture store offers a deal to new customers. On your first purchase, you can use a $$\frac{1}{4}$$ discount on the price of any table.

Brian is a new customer at the store and wants to buy a table that originally costs $$$96$$ . What discounted price would Brian pay for the table?

A washing machine is on sale, and its price is marked down by $$\frac{1}{5}$$ of its original price. The sale price of the washing machine is $$$304$$ . What is the original price of the washing machine before the discount?

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems similar to each anchor problem, asking students to find both the prices after the markup or markdown, and the prices before the markup or markdown.
• Include problems about commission.
• Include problems that involve fractional fees, such as late fees, registration fees, tax fees, etc.
• EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic C > Lesson 14 — Examples 2-3, Problem Set

Topic A: Representing Proportional Relationships in Tables, Equations, and Graphs

Solve ratio and rate problems using double number lines, tables, and unit rate.

7.RP.A.1 7.RP.A.2

Represent proportional relationships in tables, and define the constant of proportionality.

7.RP.A.2 7.RP.A.2.B

Determine the constant of proportionality in tables, and use it to find missing values.

7.RP.A.2.A 7.RP.A.2.B

Write equations for proportional relationships presented in tables.

7.RP.A.2.B 7.RP.A.2.C

Write equations for proportional relationships from word problems.

7.RP.A.2 7.RP.A.2.C

Represent proportional relationships in graphs.

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.D

Interpret proportional relationships represented in graphs.

7.RP.A.2 7.RP.A.2.D

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Topic B: Non-Proportional Relationships

Compare proportional and non-proportional relationships.

Determine if relationships are proportional or non-proportional.

Topic C: Connecting Everything Together

Make connections between the four representations of proportional relationships (Part 1).

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.B 7.RP.A.2.C 7.RP.A.2.D

Make connections between the four representations of proportional relationships (Part 2).

Use different strategies to represent and recognize proportional relationships.

Topic D: Solving Ratio & Rate Problems with Fractions

Find the unit rate of ratios involving fractions.

Find the unit rate and use it to solve problems.

7.RP.A.1 7.RP.A.3

Solve ratio and rate problems by setting up a proportion.

Solve ratio and rate problems by setting up a proportion, including part-part-whole problems.

Use proportional reasoning to solve real-world, multi-step problems.

7.RP.A.1 7.RP.A.2 7.RP.A.3

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Related Topics

Other topics in analyze proportional relationships and use them to solve real-world and mathematical problems. :.

• Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
• Recognize and represent proportional relationships between quantities.
• Terms of Use

Engage your students with effective distance learning resources. ACCESS RESOURCES>>

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

• Inspiration

7.2.2B Problem Solving with Proportions

Standard 7.2.2.

Solve multi-step problems involving proportional relationships in numerous contexts.

For example : Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Another example : How many kilometers are there in 26.2 miles?

Use knowledge of proportions to assess the reasonableness of solutions.

For example : Recognize that it would be unreasonable for a cashier to request $200 if you purchase a $225 item at 25% off.

Standard 7.2.2 Essential Understandings

Students have had prior experience with situations involving a change in one quantity effecting a corresponding change in another.  This previous experience has included graphical, tabular, and function rule representations of these relationships. This standard extends the prior understanding to proportional situations. Proportional relationships are a specific linear relationship. When these proportional relationships are graphed, the representation is a line passing through the origin. In other representations of a proportional relationship (tabular, verbal, symbols, or equations), the idea may not be initially evident, but in each representation a constant rate of change can be determined. From translation between these representations, students explore this constant rate of change to determine a unit rate (constant of proportionality or slope). The more connections students can make between these multiple forms of representation, the deeper their understanding of the relationship. With this understanding, students will be able to move to other linear relationships that are not proportional (with a graph that is a line that does not pass through the origin). 

All Standard Benchmarks

7.2.2.1 Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations. 7.2.2.2 Solve multi-step problems involving proportional relationships in numerous contexts. 7.2.2.3 Use knowledge of proportions to assess the reasonableness of solutions. 7.2.2.4 Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

Benchmark Group B - Proportional Problem Solving

7.2.2.2  Solve multi-step problems involving proportional relationships in numerous contexts.   For example: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Another example: How many kilometers are there in 26.2 miles?

7.2.2.3 Use knowledge of proportions to assess the reasonableness of solutions. For example : Recognize that it would be unreasonable for a cashier to request $200 if you purchase a $225 item at 25% off.

What students should know and be able to do [at a mastery level] related to these benchmarks

• students should be able to solve multiple-problem types, including missing values, numerical comparison, qualitative comparison, and qualitative prediction;
• assess reasonableness of solutions in the context of the problem.

Work from previous grades that supports this new learning includes: 

• Students can fluently translate from percent to decimal;
• Students can assess reasonableness of solutions;
• Students can use proportions (or other strategies) to do measurement conversions;
• Students know the basic metric and standard measurement equivalencies;
• students can solve proportions in a variety of ways (unit rate, factor of change, tabular, graphical, and fraction).

NCTM Standards

Relate and compare different forms of representation for a relationship:

• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

Common Core State Standards CCSS

Ratios and Proportional Relationships 

7.RP:   Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.1   Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½ /1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2 . Recognize and represent proportional relationships between quantities.

• 7.RP.2b . Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• 7.RP.2c . Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = np.

7.RP.3   Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Misconceptions

Student Misconceptions and Common Errors

• the difference between 5 % and 105% of an item;
• increasing by 20% is different than increasing by 20;
• forgetting to check the reasonableness of a solution.

In the Classroom

Math Vignette - The Best-Buy Problem

A sixth grade class is beginning to work with ratio and proportion.  The purpose of this lesson is to use an open-ended investigation to develop the models for thinking about ratios and equivalent fractions.

The teacher gathers the class together and tells the students that he needs their help to solve a problem he encountered in his life.  The teacher tells them that he has recently gotten a kitten from an animal shelter and the kitten needs to eat a special kind of food.  This food is sold at two neighborhood stores and the teacher would like to know which store offers the best buy on cat food.  Bob's store sells 12 cans of kitten food for $15.00 and Maria's store sells 20 cans of the same food for $23.00.

After the teacher poses the problem he invites students to brainstorm their initial ideas about how they might determine which store offers the best deal.  Three students give their ideas:

• "You could think about how many more cans you are getting at Maria's store and if that makes a difference in the price, but that kind of also reflects on how much each can costs."
• "You could try and see if you can make like an equivalent fraction maybe like 3 goes into 12 and it goes into 15 and you can see if like a common fraction or something."
• "You could use the 15 and 5 goes into it so you might say 5 out of the 12 cans might be how much money."

The teacher acknowledges each comment and rephrases them without making any value judgments. Next the teacher sends the students off to work in pairs to solve this problem.  Students may use any strategy or approach they choose and must record their solution and their method on a large sheet of poster paper that will later be shared with the class.

Three pairs of students are seen working on the problem:

Helaina and Lucy:  They thought of $1 per can as the "overall price." At each store this leaves $3 extra to be divided among the total number of cans. They divided the $3 among 12 cans and $3 among 20 cans to get the price per can.

Andres and Zach (sitting across from Helaina and Lucy): They figured the price for 60 cans at each store and saw which store had the lower total for 60 cans.

Dylan and Tristan : They made a list showing the cost for various numbers of cans using the original values given in the problem as the starting point.  Their list says

After students have solved the problem and created posters showing their solutions and strategies, the teacher brings the group back together and asks some groups to share with the class.

www.ode.state.or.us/.../hooperklecknerhibbardbreakout vignette b.doc (broken link)

Teacher Notes

Students may need support in further development of previously studied concepts and skills.

• A good video lesson on finding the rate of two jets , given in word format. 
• Proportional relationships are relationships between two equal ratios. For example, oranges are sold in a bag of 5 for $2. The ratio of oranges to their cost is 5:2. If I bought 20 oranges, I could set up a proportion to determine my cost.

$\frac{5}{2}=\frac{20}{x}$

One common way of solving a proportion is to use cross-products. This would give you the equation $2\times 20 = 5\times x$ or $40 = 5\times x$. To solve this, divide both sides by 5 to see that $x = 8$.

$\frac{5}{2}=\frac{20}{8}$

You can also solve this by figuring out how many bags of 5 you need to have 20 oranges. You would need 4 bags of 5 oranges to equal 20. Therefore, you would multiply $2 times 4 to get $8.

• Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships (Common Core State Standards for Mathematics, p. 46).
• Remind students that 5% of a number is like 5 out of 100, or correlate to $0.05 of a dollar.
• Students need to be constantly reminded to ask themselves "Does this answer make sense?" after they complete a problem. 
• Shopping Mall Math   Students participate in an activity in which they develop number sense in and around the shopping mall. They solve problems involving percent and scale drawings.
• Understanding Rational Numbers and Proportions    In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving.  The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions.
• Capture-Recapture In this lesson, students experience an application of proportion that scientists actually use to solve real-life problems. Students learn how to estimate the size of a total population by taking samples and using proportions. The ratio of "tagged" items to the number of items in a sample is the same as the ratio of tagged items to the total population.

Additional Instructional Resources

Have the students bring in grocery ads from newspapers. Have the students figure and compare unit prices on given items at two different stores to find the best buy. (2007 Mississippi Mathematics Framework Revised Strategies, p. 37)

Teach students how to solve proportions using factor of change or unit rate. Factor of change: 20 min. n 4 mi. 12 mi. = Multiply by 3 20 x 3 = 60 min. 4 x 3 = 12 Unit Rate: 20 min. n 4 mi. 12 mi. = It takes 20 minutes to go 4 miles. 20 ÷ 4 = 5, so it takes 5 minutes to go 1 mile. 5 min. 1 mi. 12 miles  = 60 minutes (2007 Mississippi Mathematics Framework Revised Strategies, p. 37)

Consumer Math with Percent Applications . A website with a variety of links regarding consumer math using proportions, percents, etc.

Top Ten Ideas about Proportional Relationships (that students should take into High School Mathematics).

Reflection - critical questions regarding the teaching and learning of these benchmarks

• Do the students know WHY they set up the proportion the way they did?
• Are students assessing their answers to make sure they make sense in the context of the problem?
• Are students using prior knowledge and mental math to solve problems, or are they always "relying" on the algorithm of the proportion to solve the problem?
• Can students solve multi-step problems?

NCTM's book Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics : Grades 6-8

Bounds, H. M., Chapman, C., Green, T., Kaase, K., Sewell, B.H., & Thompson, M. (2007). Mississippi mathematics framework revised strategies. Jackson, MS: Mississippi Dept. of Education.

Cramer, K., & Post, T. (1993, May ). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404-407 .

Helping students gain understanding and self-confidence in algebra.  www . purplemath . com

Insights into Algebra 1: Linear functions and inequalities.  http :// www . learner . org / workshops / algebra / workshop 2/ index 2. html

National Council of Teachers of Mathematics. (n.d.). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. Reston, VA: NCTM.

Top ten ideas about proportional relationships (that students should take into high school mathematics ). http :// web . me . com / serpmedia / ToolPresentation / Tool ___ Articulation _2_ files / Top %20 ten %20 list -1. pdf   

Answer: Choice C

Massachusetts Comprehensive Assessment System Release of Spring 2009 Test Items

Answer: Choice D

Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

TAKS (Texas Assessment of Knowledge and Skills) Mathematics, Grade 8, 2010 released items.

• The price of a pair of shoes rose to $45 from $24.

What percent increase is this?

       Answer:  87.5% increase

Differentiation

Struggling Students

• Provide calculators.
• Simplify word problems to only include relevant information.
• Provide calculators and graphic organizers
• Rewrite word problems using fewer and easier words so that your ELL students can practice both their math and language skills without being stuck on what a particular phrase or word means as they read the problem. Keep the sentences short and to the point. Take out extra detail that is not needed to convey the gist of the problem.
• LESSON: A GIGABYTE OF MUSIC, HOW MUCH IS THAT?

Using Mathematical Conversions to Better Understand Numbers in the News, A very rich real world problem on converting megabytes of music to understand in more concrete way. http :// www . pbs . org / newshour / extra / teachers / lessonplans / math / download _10-2. html   Broken link

Analyze the Data

e.    To find the AVERAGE number of tagged fish, add up all 10 samples of the tagged fish and divide by 10. Do the same thing to find the AVERAGE number of total fish in your samples. (Using the AVERAGE number with 10 samples is more reliable than using any one sample's data.)

f.     Use the proportion below to estimate the total number of fish in your lake:

ESTIMATED POPULATION: _____________________ Now count the total number of fish in your lake to determine how close your estimate from the "sampling" is to the actual number of fish in the lake. ACTUAL POPULATION: _____________________ How close were you to the actual number of fish?

Parents/Admin

Administrative/Peer Classroom Observation

Parent Resources A video describing how to calculate percent change. http :// mathplayground . com / mv _ percent _ change . html

Related Frameworks

7.2.2a represent proportional relationships.

• 7.2.2.1 Represent Proportional Relationships
• 7.2.2.2 Problems involving Proportional Relationships
• 7.2.2.3 Proportions & Reasonableness

7.2.2C Represent: Equations & Inequalities

• 7.2.2.4 Represent Using Equations & Inequalities

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Unit 3: Ratios and rates

About this unit.

Learn all about proportional relationships. How are they connected to ratios and rates? What do their graphs look like? What types of word problems can we solve with proportions?

Intro to ratios

• Intro to ratios (Opens a modal)
• Basic ratios (Opens a modal)
• Part:whole ratios (Opens a modal)
• Ratio review (Opens a modal)
• Basic ratios Get 5 of 7 questions to level up!

Visualize ratios

• Ratios with tape diagrams (Opens a modal)
• Equivalent ratio word problems (Opens a modal)
• Ratios and double number lines (Opens a modal)
• Ratios with tape diagrams Get 3 of 4 questions to level up!
• Equivalent ratios with equal groups Get 3 of 4 questions to level up!
• Create double number lines Get 3 of 4 questions to level up!
• Ratios with double number lines Get 3 of 4 questions to level up!
• Relate double number lines and ratio tables Get 3 of 4 questions to level up!

Equivalent ratios

• Ratio tables (Opens a modal)
• Solving ratio problems with tables (Opens a modal)
• Equivalent ratios (Opens a modal)
• Equivalent ratios: recipe (Opens a modal)
• Understanding equivalent ratios (Opens a modal)
• Ratio tables Get 3 of 4 questions to level up!
• Equivalent ratios Get 3 of 4 questions to level up!
• Equivalent ratio word problems Get 3 of 4 questions to level up!
• Equivalent ratios in the real world Get 3 of 4 questions to level up!
• Understand equivalent ratios in the real world Get 3 of 4 questions to level up!

Ratio application

• Ratios on coordinate plane (Opens a modal)
• Ratios and measurement (Opens a modal)
• Part to whole ratio word problem using tables (Opens a modal)
• Ratios on coordinate plane Get 3 of 4 questions to level up!
• Ratios and units of measurement Get 3 of 4 questions to level up!
• Part-part-whole ratios Get 3 of 4 questions to level up!

Intro to rates

• Intro to rates (Opens a modal)
• Solving unit rate problem (Opens a modal)
• Solving unit price problem (Opens a modal)
• Rate problems (Opens a modal)
• Comparing rates example (Opens a modal)
• Rate review (Opens a modal)
• Unit rates Get 5 of 7 questions to level up!
• Rate problems Get 3 of 4 questions to level up!
• Comparing rates Get 3 of 4 questions to level up!

• Rating Count
• Price (Ascending)
• Price (Descending)
• Most Recent

Multistep ratio and proportion problems

Resource type.

Multi-Step Ratio and Percent Problems Notes | Proportions & Real World Percents

Using Proportions to Solve Multistep Ratio and Percent Problems Maze 7.RP.3

Proportions and Ratio and Percent Problems - 7.RP.3

Ratios and Proportions Math Recipes Math Word Problems Ratio Unit Rate Questions

Percentages and Ratios Multi-Step Word Problems for 7th Grade Math

7.4d Ratios , Rates, Proportions , Percent & Financial Literacy Notes

Halloween Math Recipe Word Problems Ratios and Proportions Worksheets

Algebra 1 Warm-Ups - Solving Equations and Proportions (12 topics, 46 Problems )

Ratios , Rates, Proportions and Percents Unit : 7th Grade Math Curriculum

Proportions and Percents Math Stations

Percents Performance Task | Percent Proportions and Percent of Change Activity

Multi-Step Ratio and Percent Problems Task Card and Poster Set 7.RP.3

Ratios , Unit Rates and Proportions BOOM Cards

Algebra 1 - Homework / Practice Problems - Linear Equations

Ratios and Proportions Math Recipes Unit Rates Cupcakes Math Word Problems

Proportions and Percents Activities

Ratios and Proportions Rate Reasoning Grade 6 Math Grade 7 Math Word Problems

Solve Percent Problems Using Proportions

Proportions and Percents Card Sort Activity Lesson

Ratios , Rates, Proportions and Percents Digital Interactive Notebook - 7th Grade

Spectrum Word Problems Workbook Grade 8 Printable 704494-EB

Spectrum Word Problems Workbook Grade 7 Printable 704493-EB

Ratios and Proportions Recipes Metric Measurements Cup Cakes 6th Grade Math

Pancake Day Math Ratios and Proportions Metric Conversions

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