Ratios and Proportional Relationships

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Ratios and Proportional Relationships

McGraw Hill | Ratios and Proportional Relationships

This material may be reproduced for licensed classroom use

only and may not be further reproduced or distributed.

Warm Up

Find the missing value.

1. 2.

3. 4.

5.

McGraw Hill | Ratios and Proportional Relationships

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Warm Up

Find the missing value.

1. 2.

3. 4.

5.

18

35

12

10

7

McGraw Hill | Ratios and Proportional Relationships

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MA.7.AR.3.2
Apply previous understanding of ratios to solve real-world
problems involving proportions.

Florida’s B.E.S.T. Standards for
Mathematics

McGraw Hill | Ratios and Proportional Relationships

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Florida’s Mathematical Thinking and
Reasoning Standards

MA.K12.MTR.3.1
Complete tasks with mathematical fluency.

MA.K12.MTR.4.1
Engage in discussions that reflect on the mathematical thinking
of self and others.

MA.K12.MTR.5.1
Use patterns and structure to help understand and connect
mathematical concepts.

McGraw Hill | Ratios and Proportional Relationships

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only and may not be further reproduced or distributed.

Florida’s Mathematical Thinking and
Reasoning Standards

MA.K12.MTR.7.1
Apply mathematics to real-world contexts.

McGraw Hill | Ratios and Proportional Relationships

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Content Objective
Students will write and solve proportions for unknown values.

Language Objective

Students will explain orally and in writing how to solve
proportions for unknown values using must and need to.

Lesson Objectives

McGraw Hill | Ratios and Proportional Relationships

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Learn
Ratios and Proportions

Recall that a ratiois a comparison between two quantities,
in which for every a units of one quantity, there are b units of
another quantity. A proportionis an equation stating that
two ratios are equivalent. You can use proportions to solve
problems in which one quantity is unknown.

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Kasem has a job raking leaves from neighborhood yards
to earn money to buy a new computer. He needs to rake
leaves from 30 yards to earn enough money.

If each yard takes the same amount of time, and he
can rake 4 yards in 5 hours and 30 minutes, how long
will it take him to rake 30 yards?

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Talk About It!

How could you use a table of equivalent ratios to solve
this problem?

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Step 1 Write the original ratio.

Write a ratio that compares the number of yards
raked to the number of hours.

5 hours and 30 minutes is equivalent
to 5.5 hours.

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Step 2 Write an equivalent ratio.

Write a ratio that compares the total number of yards
he needs to rake to the number of hours.

Use x for the unknown number of hours.

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Step 3 Write and solve a proportion.
Because Kasem can rake each yard in the same
amount of time, the two ratios are equivalent.

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Set the ratios equal to
each other.

Because 4 × 7.5 is 30,
multiply 5.5 by 7.5.

So, it will take Kasem 41.25 hours or 41 hours and
15 minutes to rake 30 yards.

McGraw Hill | Ratios and Proportional Relationships

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Example 1
Solve Real-World Problems Involving
Proportions

Check

Cora clears out gardens at the nearby community
garden to earn money for a new bike. She needs to
clear 10 gardens to earn enough money. If each
garden takes the same amount of time to clear, and
she can clear 4 gardens in 3 hours, how long will it
take her to clear 10 gardens?

McGraw Hill | Ratios and Proportional Relationships

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only and may not be further reproduced or distributed.

Example 1
Solve Real-World Problems Involving
Proportions

Check

Cora clears out gardens at the nearby community
garden to earn money for a new bike. She needs to
clear 10 gardens to earn enough money. If each
garden takes the same amount of time to clear, and
she can clear 4 gardens in 3 hours, how long will it
take her to clear 10 gardens?

7.5 hours or 7 hours and 30 minutes

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Last week, Johanna sold 20 hand-made greeting cards,
earning a total profit of $34.20. She plans to make 50
cards for next week’s school craft fair.

If she sells all 50 cards, how much profit can
Johanna expect to earn?

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Think About It!
What is one ratio you can write with the given
information?

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Let x represent the total profit for selling 50 cards.

Write the proportion.

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Because 20 × 2.5 is 50,
multiply $34.20 by 2.5.

So, if Johanna sells all 50 cards, she will make a profit
of $85.50.

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Talk About It!

How much profit does Johanna earn per card? Explain.

McGraw Hill | Ratios and Proportional Relationships

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Example 2
Solve Real-World Problems
Involving Proportions

Check

Last month, Carter sold 12 tie-dye T-shirts, earning a total
profit of $90. He plans to make 25 T-shirts for next month’s
school spirit shop opening. If he sells all 25 T-shirts, how
much profit can Carter expect to earn?

McGraw Hill | Ratios and Proportional Relationships

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only and may not be further reproduced or distributed.

Example 2
Solve Real-World Problems
Involving Proportions

Check

Last month, Carter sold 12 tie-dye T-shirts, earning a total
profit of $90. He plans to make 25 T-shirts for next month’s
school spirit shop opening. If he sells all 25 T-shirts, how
much profit can Carter expect to earn?

$187.50

McGraw Hill | Ratios and Proportional Relationships

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Apply
Swimming

Camila usually swims in her team’s
200-yard freestyle race, which is 8
lengths of the pool, or a total of
approximately 182.88 meters. Next
season, Camila would like to swim in
the longest individual freestyle race,
which is 20 lengths of the pool.
Approximately how many meters is this
race? How does this compare to
Camila’s normal race distance?

McGraw Hill | Ratios and Proportional Relationships

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Apply
Swimming

Talk About It!

How can writing a proportion help you determine the
number of meters if you do not know how many meters
are in a yard?

McGraw Hill | Ratios and Proportional Relationships

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Apply
Swimming

Check

The regulation dimension of a track is measured in meters
using the total length of the first lane. Terry usually runs in
his team’s 3,000-meter race, which is 7.5 laps around the
track, or a total of approximately 1.86 miles. Next season,
Terry would like to run in the longest race, which is 25 laps.
Approximately how many miles is this race? How does this
compare to Terry’s normal race distance?

McGraw Hill | Ratios and Proportional Relationships

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Apply
Swimming

Check

The regulation dimension of a track is measured in meters
using the total length of the first lane. Terry usually runs in
his team’s 3,000-meter race, which is 7.5 laps around the
track, or a total of approximately 1.86 miles. Next season,
Terry would like to run in the longest race, which is 25 laps.
Approximately how many miles is this race? How does this
compare to Terry’s normal race distance?

6.2 miles; Sample answer: This is 7,000 meters or
approximately 4.34 miles longer than his normal race distance.

McGraw Hill | Ratios and Proportional Relationships

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Pause and Reflect

You may have learned in a previous grade how to solve
ratio problems using other methods, such as a bar
diagram, double number line, a table of equivalent ratios,
and other. Reflect on your understanding of ratios and
proportions, including how you prefer to solve problems.
Do you have another method of solving problems
involving proportions? If so, explain.

McGraw Hill | Ratios and Proportional Relationships

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Exit Ticket

An artist can draw 3 illustrations in 3.5 hours. How many
hours will it take to draw 15 illustrations? Explain or show
how you found your answer.

McGraw Hill | Ratios and Proportional Relationships

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Exit Ticket

An artist can draw 3 illustrations in 3.5 hours. How many
hours will it take to draw 15 illustrations? Explain or show
how you found your answer.