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NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

Lesson 10: Simple Interest

138

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Lesson 10: Simple Interest

Student Outcomes

Students solve simple interest problems using the formula ๐ผ = ๐‘ƒ๐‘Ÿ๐‘ก, where ๐ผ represents interest, ๐‘ƒ represents

principal, ๐‘Ÿ represents interest rate, and ๐‘ก represents time.

When using the formula ๐ผ = ๐‘ƒ๐‘Ÿ๐‘ก, students recognize that units for both interest rate and time must be

compatible; students convert the units when necessary.

Classwork

Fluency Exercise (10 minutes): Fractional Percents

Students complete a two-round Sprint provided at the end of this lesson (Fractional Percents) to practice finding the

percent, including fractional percents, of a number. Provide one minute for each round of the Sprint. Refer to the

Sprints and Sprint Delivery Script sections in the Module 2 Module Overview for directions to administer a Sprint. Be

sure to provide any answers not completed by the students. Sprints and answer keys are provided at the end of the

lesson.

Example 1 (7 minutes): Can Money Grow? A Look at Simple Interest

Students solve a simple interest problem to find the new balance of a savings account that

earns interest. Students model the interest earned over time (in years) by constructing a

table and graph to show that a proportional relationship exists between ๐‘ก, number of

years, and ๐ผ, interest.

Begin class discussion by displaying and reading the following problem to the whole class.

Allow students time to process the information presented. Small group discussion should

be encouraged before soliciting individual feedback.

Larry invests $100 in a savings plan. The plan pays 412

% interest each year on

his $100 account balance. The following chart shows the balance on his account

after each year for the next 5 years. He did not make any deposits or

withdrawals during this time.

Time (in years) Balance (in dollars)

1 104.50

2 109.00

3 113.50

4 118.00

5 122.50

Scaffolding:

Allow one calculator per

group (or student) to aid

with discovering the

mathematical pattern

from the table.

Also, consider using a

simpler percent value,

such as 2%.

MP.1

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

Lesson 10: Simple Interest

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Possible discussion questions:

What is simple interest?

How is it calculated?

What pattern(s) do you notice from the table?

Can you create a formula to represent the pattern(s) from the table?

Display the interest formula to the class, and explain each variable.

Model for the class how to substitute the given information into the interest formula to find the amount of interest

earned.

Example 1: Can Money Grow? A Look at Simple Interest

Larry invests $๐Ÿ๐ŸŽ๐ŸŽ in a savings plan. The plan pays ๐Ÿ’๐Ÿ๐Ÿ

% interest each year on his $๐Ÿ๐ŸŽ๐ŸŽ account balance.

a. How much money will Larry earn in interest after ๐Ÿ‘ years? After ๐Ÿ“ years?

๐Ÿ‘ years:

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = ๐Ÿ๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ’๐Ÿ“)(๐Ÿ‘)

๐‘ฐ = ๐Ÿ๐Ÿ‘. ๐Ÿ“๐ŸŽ

Larry will earn $๐Ÿ๐Ÿ‘. ๐Ÿ“๐ŸŽ in interest after ๐Ÿ‘ years.

๐Ÿ“ years:

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = ๐Ÿ๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ’๐Ÿ“)(๐Ÿ“)

๐‘ฐ = ๐Ÿ๐Ÿ. ๐Ÿ“๐ŸŽ

Larry will earn $๐Ÿ๐Ÿ. ๐Ÿ“๐ŸŽ in interest after ๐Ÿ“ years.

b. How can you find the balance of Larryโ€™s account at the end of ๐Ÿ“ years?

You would add the interest earned after ๐Ÿ“ years to the beginning balance. $๐Ÿ๐Ÿ. ๐Ÿ“๐ŸŽ + $๐Ÿ๐ŸŽ๐ŸŽ = $๐Ÿ๐Ÿ๐Ÿ. ๐Ÿ“๐ŸŽ.

๐ˆ๐ง๐ญ๐ž๐ซ๐ž๐ฌ๐ญ = ๐๐ซ๐ข๐ง๐œ๐ข๐ฉ๐š๐ฅ ร— ๐‘๐š๐ญ๐ž ร— ๐“๐ข๐ฆ๐ž

๐‘ฐ = ๐‘ท ร— ๐’“ ร— ๐’•

๐‘ฐ = ๐‘ท๐’“๐’•

To find the simple interest, use the following formula:

๐’“ is the percent of the principal that is paid over a period of time (usually per year).

๐’• is the time.

๐’“ and ๐’• must be compatible. For example, if ๐’“ is an annual interst rate, then ๐’• must be written in years.

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Lesson 10: Simple Interest

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Show the class that the relationship between the amount of interest earned each year can be represented in a table or

graph by posing the question, โ€œThe interest earned can be found using an equation. How else can we represent the

amount of interest earned other than an equation?โ€

Draw a table, and call on students to help you complete the table. Start with finding the amount of interest

earned after 1 year.

๐’• (in years) ๐‘ฐ (interest earned after ๐’• years, in dollars)

1 ๐ผ = (100)(0.045)(1) = 4.50

2 ๐ผ = (100)(0.045)(2) = 9.00

3 ๐ผ = (100)(0.045)(3) = 13.50 4 ๐ผ = (100)(0.045)(4) = 18.00 5 ๐ผ = (100)(0.045)(5) = 22.50

Possible discussion questions:

Using your calculator, what do you observe when you divide the ๐ผ by ๐‘ก for each year?

The ratio is 4.5.

What is the constant of proportionality in this situation? What does it mean? What evidence from the table

supports your answer?

The constant of proportionality is 4.5. This is the principal times the interest rate because

(100)(0.045) = 4.5. This means that for every year, the interest earned on the savings account

increases by $4.50. The table shows that the principal and interest rate are not changing; they are

constant.

What other representation could we use to show the relationship between time

and the amount of interest earned is proportional?

We could use a graph.

Display to the class a graph of the relationship.

What are some characteristics of the graph?

It has a title.

The axes are labeled.

The scale for the ๐‘ฅ-axis is 1 year.

The scale for the ๐‘ฆ-axis is 5 dollars.

By looking at the graph of the line, can you draw a conclusion about the relationship between time and the

amount of interest earned?

All pairs from the table are plotted, and a straight line passes through those points and the origin. This

means that the relationship is proportional.

Scaffolding:

Use questioning strategies to

review graphing data in the

coordinate plane for all

learners. Emphasize the

importance of an accurate

scale and making sure variables

are graphed along the correct

axes.

The amount of interest earned increases by the same amount each year, $4.50. Therefore, the ratios in the table

are equivalent. This means that the relationship between time and the interest earned is proportional.

Increase of $4.50

Increase of $4.50

Increase of $4.50

Increase of $4.50

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

Lesson 10: Simple Interest

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What does the point (4, 18) mean in terms of the situation?

It means that at the end of four years, Larry would have earned $18 in

interest.

What does the point (0, 0) mean?

It means that when Larry opens the account, no interest is earned.

What does the point (1, 4.50) mean?

It means that at the end of the first year, Larryโ€™s account earned $4.50.

4.5 is also the constant of proportionality.

What equation would represent the amount of interest earned at the end of a

given year in this situation?

๐ผ = 4.5๐‘ก

Exercise 1 (3 minutes)

Students practice using the interest formula independently, with or without technology. Review answers as a whole

class.

Exercise 1

Find the balance of a savings account at the end of ๐Ÿ๐ŸŽ years if the interest earned each year is ๐Ÿ•. ๐Ÿ“%. The principal is

$๐Ÿ“๐ŸŽ๐ŸŽ.

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = $๐Ÿ“๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ•๐Ÿ“)(๐Ÿ๐ŸŽ)

๐‘ฐ = $๐Ÿ‘๐Ÿ•๐Ÿ“

The interest earned after ๐Ÿ๐ŸŽ years is $๐Ÿ‘๐Ÿ•๐Ÿ“. So, the balance at the end of ๐Ÿ๐ŸŽ years is $๐Ÿ‘๐Ÿ•๐Ÿ“ + $๐Ÿ“๐ŸŽ๐ŸŽ = $๐Ÿ–๐Ÿ•๐Ÿ“.

0

5

10

15

20

25

0 1 2 3 4 5 6

Amount of Interest Earned (in dollars)

Am

ou

nt

of

Inte

rest

Ear

ned

(in

do

llars

)

Time (years) Scaffolding:

Provide a numbered

coordinate plane to help

build confidence for

students who struggle

with creating graphs by

hand.

If time permits, allow

advanced learners to

practice graphing the

interest formula using the

Y= editor in a graphing

calculator and scrolling the

table to see how much

interest is earned for ๐‘ฅ

number of years.

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

Lesson 10: Simple Interest

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๐‘ท ๐‘ฐ

๐’“

Example 2 (5 minutes): Time Other Than One Year

In this example, students learn to recognize that units for both the interest rate and time

must be compatible. If not, they must convert the units when necessary.

Remind the class how to perform a unit conversion from months to years. Because

1 year = 12 months, the number of months given can be divided by 12 to get the

equivalent year.

Example 2: Time Other Than One Year

A $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ savings bond earns simple interest at the rate of ๐Ÿ‘% each year. The interest is paid at the end of every month.

How much interest will the bond have earned after ๐Ÿ‘ months?

Step 1: Convert ๐Ÿ‘ months to a year.

๐Ÿ๐Ÿ months = ๐Ÿ year. So, divide both sides by ๐Ÿ’ to get ๐Ÿ‘ months =๐Ÿ๐Ÿ’

year.

Step 2: Use the interest formula to find the answer.

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = ($๐Ÿ๐ŸŽ๐ŸŽ๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ‘)(๐ŸŽ. ๐Ÿ๐Ÿ“)

๐‘ฐ = $๐Ÿ•. ๐Ÿ“๐ŸŽ

The interest earned after ๐Ÿ‘ months is $๐Ÿ•. ๐Ÿ“๐ŸŽ.

Example 3 (5 minutes): Solving for ๐‘ท, ๐’“, or ๐’•

Students practice working backward to find the interest rate, principal, or time by dividing the interest earned by the

product of the other two values given.

The teacher could have students annotate the word problem by writing the corresponding variable above each given

quantity. Have students look for keywords to identify the appropriate variable. For example, the words investment,

deposit, and loan refer to principal. Students will notice that time is not given; therefore, they must solve for ๐‘ก.

Example 3: Solving for ๐‘ท, ๐’“, or ๐’•

Mrs. Williams wants to know how long it will take an investment of $๐Ÿ’๐Ÿ“๐ŸŽ to earn $๐Ÿ๐ŸŽ๐ŸŽ in interest if the yearly interest

rate is ๐Ÿ”. ๐Ÿ“%, paid at the end of each year.

๐‘ฐ = ๐‘ท๐’“๐’•

$๐Ÿ๐ŸŽ๐ŸŽ = ($๐Ÿ’๐Ÿ“๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ”๐Ÿ“)๐’•

$๐Ÿ๐ŸŽ๐ŸŽ = $๐Ÿ๐Ÿ—. ๐Ÿ๐Ÿ“๐’•

$๐Ÿ๐ŸŽ๐ŸŽ (๐Ÿ

$๐Ÿ๐Ÿ—. ๐Ÿ๐Ÿ“) = (

๐Ÿ

$๐Ÿ๐Ÿ—. ๐Ÿ๐Ÿ“) $๐Ÿ๐Ÿ—. ๐Ÿ๐Ÿ“๐’•

๐Ÿ”. ๐Ÿ–๐Ÿ‘๐Ÿ•๐Ÿ” = ๐’•

Six years is not enough time to earn $๐Ÿ๐ŸŽ๐ŸŽ. At the end of seven years, the interest will be over $๐Ÿ๐ŸŽ๐ŸŽ. It will take seven

years since the interest is paid at the end of each year.

MP.1 Scaffolding:

Provide a poster with the terms

semi, quarterly, and annual.

Write an example next to each

word, showing an example of a

conversion.

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Exercises 2โ€“3 (7 minutes)

Students complete the following exercises independently, or in groups of two, using the simple interest formula.

Exercise 2

Write an equation to find the amount of simple interest, ๐‘จ, earned on a $๐Ÿ”๐ŸŽ๐ŸŽ investment after ๐Ÿ๐Ÿ๐Ÿ

years if the semi-

annual (๐Ÿ”-month) interest rate is ๐Ÿ%.

๐Ÿ๐Ÿ๐Ÿ

years is the same as

๐Ÿ” months ๐Ÿ” months ๐Ÿ” months

๐ˆ๐ง๐ญ๐ž๐ซ๐ž๐ฌ๐ญ = ๐๐ซ๐ข๐ง๐œ๐ข๐ฉ๐š๐ฅ ร— ๐‘๐š๐ญ๐ž ร— ๐“๐ข๐ฆ๐ž

๐‘จ = ๐Ÿ”๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ)(๐Ÿ‘) ๐Ÿ. ๐Ÿ“ years is ๐Ÿ year and ๐Ÿ” months, so ๐’• = ๐Ÿ‘.

๐‘จ = ๐Ÿ‘๐Ÿ” The amount of interest earned is $๐Ÿ‘๐Ÿ”.

Exercise 3

A $๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ loan has an annual interest rate of ๐Ÿ’๐Ÿ๐Ÿ’

% on the amount borrowed. How much time has elapsed if the interest

is now $๐Ÿ๐Ÿ๐Ÿ•. ๐Ÿ“๐ŸŽ?

๐ˆ๐ง๐ญ๐ž๐ซ๐ž๐ฌ๐ญ = ๐๐ซ๐ข๐ง๐œ๐ข๐ฉ๐š๐ฅ ร— ๐‘๐š๐ญ๐ž ร— ๐“๐ข๐ฆ๐ž

Let ๐’• be time in years.

๐Ÿ๐Ÿ๐Ÿ•. ๐Ÿ“๐ŸŽ = (๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ’๐Ÿ๐Ÿ“)๐’•

๐Ÿ๐Ÿ๐Ÿ•. ๐Ÿ“๐ŸŽ = ๐Ÿ”๐Ÿ‘. ๐Ÿ•๐Ÿ“๐’•

(๐Ÿ๐Ÿ๐Ÿ•. ๐Ÿ“๐ŸŽ) (๐Ÿ

๐Ÿ”๐Ÿ‘. ๐Ÿ•๐Ÿ“) = (

๐Ÿ

๐Ÿ”๐Ÿ‘. ๐Ÿ•๐Ÿ“) (๐Ÿ”๐Ÿ‘. ๐Ÿ•๐Ÿ“)๐’•

๐Ÿ = ๐’•

Two years have elapsed.

Closing (2 minutes)

Explain each variable of the simple interest formula.

๐ผ is the amount of interest earned or owed.

๐‘ƒ is the principal, or the amount invested or borrowed.

๐‘Ÿ is the interest rate for a given time period (yearly, quarterly, monthly).

๐‘ก is time.

What would be the value of the time for a two-year period for a quarterly interest rate? Explain.

๐‘ก would be written as 8 because a quarter means every 3 months, and there are four quarters in one

year. So, 2 ร— 4 = 8.

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Lesson 10: Simple Interest

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Exit Ticket (6 minutes)

๐ˆ๐ง๐ญ๐ž๐ซ๐ž๐ฌ๐ญ = ๐๐ซ๐ข๐ง๐œ๐ข๐ฉ๐š๐ฅ ร— ๐‘๐š๐ญ๐ž ร— ๐“๐ข๐ฆ๐ž

๐‘ฐ = ๐‘ท ร— ๐’“ ร— ๐’•

๐‘ฐ = ๐‘ท๐’“๐’•

Lesson Summary

Interest earned over time can be represented by a proportional relationship between time, in years,

and interest.

The simple interest formula is

๐’“ is the percent of the principal that is paid over a period of time (usually per year).

๐’• is the time.

The rate, ๐’“, and time, ๐’•, must be compatible. If ๐’“ is the annual interest rate, then ๐’• must be written in

years.

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Name Date

Lesson 10: Simple Interest

Exit Ticket

1. Ericaโ€™s parents gave her $500 for her high school graduation. She put the money into a savings account that earned

7.5% annual interest. She left the money in the account for nine months before she withdrew it. How much

interest did the account earn if interest is paid monthly?

2. If she would have left the money in the account for another nine months before withdrawing, how much interest

would the account have earned?

3. About how many years and months would she have to leave the money in the account if she wants to reach her goal

of saving $750?

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Lesson 10: Simple Interest

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Exit Ticket Sample Solutions

1. Ericaโ€™s parents gave her $๐Ÿ“๐ŸŽ๐ŸŽ for her high school graduation. She put the money into a savings account that earned

๐Ÿ•. ๐Ÿ“% annual interest. She left the money in the account for nine months before she withdrew it. How much

interest did the account earn if interest is paid monthly?

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = (๐Ÿ“๐ŸŽ๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ•๐Ÿ“) (๐Ÿ—

๐Ÿ๐Ÿ)

๐‘ฐ = ๐Ÿ๐Ÿ–. ๐Ÿ๐Ÿ๐Ÿ“

The interest earned is $๐Ÿ๐Ÿ–. ๐Ÿ๐Ÿ‘.

2. If she would have left the money in the account for another nine months before withdrawing, how much interest

would the account have earned?

๐‘ฐ = ๐‘ท๐’“๐’•

๐‘ฐ = (๐Ÿ“๐ŸŽ๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ•๐Ÿ“) (๐Ÿ๐Ÿ–

๐Ÿ๐Ÿ)

๐‘ฐ = ๐Ÿ“๐Ÿ”. ๐Ÿ๐Ÿ“

The account would have earned $๐Ÿ“๐Ÿ”. ๐Ÿ๐Ÿ“.

3. About how many years and months would she have to leave the money in the account if she wants to reach her goal

of saving $๐Ÿ•๐Ÿ“๐ŸŽ?

๐Ÿ•๐Ÿ“๐ŸŽ โˆ’ ๐Ÿ“๐ŸŽ๐ŸŽ = ๐Ÿ๐Ÿ“๐ŸŽ She would need to earn $๐Ÿ๐Ÿ“๐ŸŽ in interest.

๐‘ฐ = ๐‘ท๐’“๐’•

๐Ÿ๐Ÿ“๐ŸŽ = (๐Ÿ“๐ŸŽ๐ŸŽ)(๐ŸŽ. ๐ŸŽ๐Ÿ•๐Ÿ“)๐’•

๐Ÿ๐Ÿ“๐ŸŽ = ๐Ÿ‘๐Ÿ•. ๐Ÿ“๐’•

๐Ÿ๐Ÿ“๐ŸŽ (๐Ÿ

๐Ÿ‘๐Ÿ•. ๐Ÿ“) = (

๐Ÿ

๐Ÿ‘๐Ÿ•. ๐Ÿ“) (๐Ÿ‘๐Ÿ•. ๐Ÿ“)๐’•

๐Ÿ”๐Ÿ

๐Ÿ‘= ๐’•

It would take her ๐Ÿ” years and ๐Ÿ– months to reach her goal because ๐Ÿ๐Ÿ‘

ร— ๐Ÿ๐Ÿ months is ๐Ÿ– months.

Problem Set Sample Solutions

1. Enrique takes out a student loan to pay for his college tuition this year. Find the interest on the loan if he borrowed

$๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ at an annual interest rate of ๐Ÿ”% for ๐Ÿ๐Ÿ“ years.

๐‘ฐ = ๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ”)(๐Ÿ๐Ÿ“)

๐‘ฐ = ๐Ÿ, ๐Ÿ๐Ÿ“๐ŸŽ

Enrique would have to pay $๐Ÿ, ๐Ÿ๐Ÿ“๐ŸŽ in interest.

2. Your family plans to start a small business in your neighborhood. Your father borrows $๐Ÿ๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ from the bank at an

annual interest rate of ๐Ÿ–% rate for ๐Ÿ‘๐Ÿ” months. What is the amount of interest he will pay on this loan?

๐‘ฐ = ๐Ÿ๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ–)(๐Ÿ‘)

๐‘ฐ = ๐Ÿ, ๐Ÿ’๐ŸŽ๐ŸŽ

He will pay $๐Ÿ, ๐Ÿ’๐ŸŽ๐ŸŽ in interest.

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3. Mr. Rodriguez invests $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ in a savings plan. The savings account pays an annual interest rate of ๐Ÿ“. ๐Ÿ•๐Ÿ“% on the

amount he put in at the end of each year.

a. How much will Mr. Rodriguez earn if he leaves his money in the savings plan for ๐Ÿ๐ŸŽ years?

๐‘ฐ = ๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ(๐ŸŽ. ๐ŸŽ๐Ÿ“๐Ÿ•๐Ÿ“)(๐Ÿ๐ŸŽ)

๐‘ฐ = ๐Ÿ, ๐Ÿ๐Ÿ“๐ŸŽ

He will earn $๐Ÿ, ๐Ÿ๐Ÿ“๐ŸŽ.

b. How much money will be in his savings plan at the end of ๐Ÿ๐ŸŽ years?

At the end of ๐Ÿ๐ŸŽ years, he will have $๐Ÿ‘, ๐Ÿ๐Ÿ“๐ŸŽ because $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ + $๐Ÿ, ๐Ÿ๐Ÿ“๐ŸŽ = $๐Ÿ‘, ๐Ÿ๐Ÿ“๐ŸŽ.

c. Create (and label) a graph in the coordinate plane to show the relationship between time and the amount of

interest earned for ๐Ÿ๐ŸŽ years. Is the relationship proportional? Why or why not? If so, what is the constant of

proportionality?

Yes, the relationship is proportional because the graph shows a straight line touching the origin. The constant

of proportionality is ๐Ÿ๐Ÿ๐Ÿ“ because the amount of interest earned increases by $๐Ÿ๐Ÿ๐Ÿ“ for every one year.

d. Explain what the points (๐ŸŽ, ๐ŸŽ) and (๐Ÿ, ๐Ÿ๐Ÿ๐Ÿ“) mean on the graph.

(๐ŸŽ, ๐ŸŽ) means that no time has elapsed and no interest has been earned.

(๐Ÿ, ๐Ÿ๐Ÿ๐Ÿ“) means that after ๐Ÿ year, the savings plan would have earned $๐Ÿ๐Ÿ๐Ÿ“. ๐Ÿ๐Ÿ๐Ÿ“ is also the constant of

proportionality.

e. Using the graph, find the balance of the savings plan at the end of seven years.

From the table, the point (๐Ÿ•, ๐Ÿ–๐ŸŽ๐Ÿ“) means that the balance would be $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ + $๐Ÿ–๐ŸŽ๐Ÿ“ = $๐Ÿ, ๐Ÿ–๐ŸŽ๐Ÿ“.

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12

Amount of Interest Mr. Rodriguez Earns (in dollars)

Time (in years)

Inte

rest

Ear

ne

d (

in d

olla

rs)

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

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f. After how many years will Mr. Rodriguez have increased his original investment by more than ๐Ÿ“๐ŸŽ%? Show

your work to support your answer.

๐๐ฎ๐š๐ง๐ญ๐ข๐ญ๐ฒ = ๐๐ž๐ซ๐œ๐ž๐ง๐ญ ร— ๐–๐ก๐จ๐ฅ๐ž

Let ๐‘ธ be the account balance that is ๐Ÿ“๐ŸŽ% more than the original investment.

๐‘ธ > (๐Ÿ + ๐ŸŽ. ๐Ÿ“๐ŸŽ)(๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ)

๐‘ธ > ๐Ÿ‘, ๐ŸŽ๐ŸŽ๐ŸŽ

The balance will be greater than $๐Ÿ‘, ๐ŸŽ๐ŸŽ๐ŸŽ beginning between ๐Ÿ– and ๐Ÿ— years because the graph shows (๐Ÿ–, ๐Ÿ—๐Ÿ๐ŸŽ)

and (๐Ÿ—, ๐Ÿ๐ŸŽ๐Ÿ‘๐Ÿ“), so $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ + $๐Ÿ—๐Ÿ๐ŸŽ = $๐Ÿ, ๐Ÿ—๐Ÿ๐ŸŽ < $๐Ÿ‘, ๐ŸŽ๐ŸŽ๐ŸŽ, and $๐Ÿ, ๐ŸŽ๐ŸŽ๐ŸŽ + $๐Ÿ, ๐ŸŽ๐Ÿ‘๐Ÿ“ = $๐Ÿ‘, ๐ŸŽ๐Ÿ‘๐Ÿ“ > $๐Ÿ‘, ๐ŸŽ๐ŸŽ๐ŸŽ.

Challenge Problem:

4. George went on a game show and won $๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ. He wanted to invest it and found two funds that he liked. Fund

250 earns ๐Ÿ๐Ÿ“% interest annually, and Fund 100 earns ๐Ÿ–% interest annually. George does not want to earn more

than $๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ in interest income this year. He made the table below to show how he could invest the money.

๐‘ฐ ๐‘ท ๐’“ ๐’•

Fund 100 ๐ŸŽ. ๐ŸŽ๐Ÿ–๐’™ ๐’™ ๐ŸŽ. ๐ŸŽ๐Ÿ– ๐Ÿ

Fund 250 ๐ŸŽ. ๐Ÿ๐Ÿ“(๐Ÿ”๐ŸŽ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐’™) ๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐’™ ๐ŸŽ. ๐Ÿ๐Ÿ“ ๐Ÿ

Total ๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ ๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ

a. Explain what value ๐’™ is in this situation.

๐’™ is the principal, in dollars, that George could invest in Fund ๐Ÿ๐ŸŽ๐ŸŽ.

b. Explain what the expression ๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐’™ represents in this situation.

๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐’™ is the principal, in dollars, that George could invest in Fund 250. It is the money he would have

left over once he invests in Fund 100.

c. Using the simple interest formula, complete the table for the amount of interest earned.

See the table above.

d. Write an inequality to show the total amount of interest earned from both funds.

๐ŸŽ. ๐ŸŽ๐Ÿ–๐’™ + ๐ŸŽ. ๐Ÿ๐Ÿ“(๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐’™) โ‰ค ๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ

e. Use algebraic properties to solve for ๐’™ and the principal, in dollars, George could invest in Fund 100. Show

your work.

๐ŸŽ. ๐ŸŽ๐Ÿ–๐’™ + ๐Ÿ—, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐ŸŽ. ๐Ÿ๐Ÿ“๐’™ โ‰ค ๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ

๐Ÿ—, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐ŸŽ. ๐ŸŽ๐Ÿ•๐’™ โ‰ค ๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ

๐Ÿ—, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐Ÿ—, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐ŸŽ. ๐ŸŽ๐Ÿ•๐’™ โ‰ค ๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ โˆ’ ๐Ÿ—, ๐ŸŽ๐ŸŽ๐ŸŽ

โˆ’๐ŸŽ. ๐ŸŽ๐Ÿ•๐’™ โ‰ค โˆ’๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ

(๐Ÿ

โˆ’๐ŸŽ. ๐ŸŽ๐Ÿ•) (โˆ’๐ŸŽ. ๐ŸŽ๐Ÿ•๐’™) โ‰ค (

๐Ÿ

โˆ’๐ŸŽ. ๐ŸŽ๐Ÿ•) (โˆ’๐Ÿ, ๐Ÿ“๐ŸŽ๐ŸŽ)

๐’™ โ‰ฅ ๐Ÿ๐Ÿ, ๐Ÿ’๐Ÿ๐Ÿ–. ๐Ÿ“๐Ÿ•

๐’™ approximately equals $๐Ÿ๐Ÿ, ๐Ÿ’๐Ÿ๐Ÿ–. ๐Ÿ“๐Ÿ•. George could invest $๐Ÿ๐Ÿ, ๐Ÿ’๐Ÿ๐Ÿ–. ๐Ÿ“๐Ÿ• or more in Fund 100.

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f. Use your answer from part (e) to determine how much George could invest in Fund 250.

He could invest $๐Ÿ‘๐Ÿ–, ๐Ÿ“๐Ÿ•๐Ÿ. ๐Ÿ’๐Ÿ‘ or less in Fund 250 because ๐Ÿ”๐ŸŽ, ๐ŸŽ๐ŸŽ๐ŸŽ โˆ’ ๐Ÿ๐Ÿ, ๐Ÿ’๐Ÿ๐Ÿ–. ๐Ÿ“๐Ÿ• = ๐Ÿ‘๐Ÿ–, ๐Ÿ“๐Ÿ•๐Ÿ. ๐Ÿ’๐Ÿ‘.

g. Using your answers to parts (e) and (f), how much interest would George earn from each fund?

Fund 100: ๐ŸŽ. ๐ŸŽ๐Ÿ– ร— ๐Ÿ๐Ÿ, ๐Ÿ’๐Ÿ๐Ÿ–. ๐Ÿ“๐Ÿ• ร— ๐Ÿ approximately equals $๐Ÿ, ๐Ÿ•๐Ÿ๐Ÿ’. ๐Ÿ๐Ÿ—.

Fund 250: ๐ŸŽ. ๐Ÿ๐Ÿ“ ร— ๐Ÿ‘๐Ÿ–, ๐Ÿ“๐Ÿ•๐Ÿ. ๐Ÿ’๐Ÿ‘ ร— ๐Ÿ approximately equals $๐Ÿ“, ๐Ÿ•๐Ÿ–๐Ÿ“. ๐Ÿ•๐Ÿ or $๐Ÿ•, ๐Ÿ“๐ŸŽ๐ŸŽ โˆ’ $๐Ÿ, ๐Ÿ•๐Ÿ๐Ÿ’. ๐Ÿ๐Ÿ—.

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Fractional Percentsโ€”Round 1

Directions: Find the part that corresponds with each percent.

1. 1% of 100 23. 1

4% of 100

2. 1% of 200 24. 1

4% of 200

3. 1% of 400 25. 1

4% of 400

4. 1% of 800 26. 1

4% of 800

5. 1% of 1,600 27. 1

4% of 1,600

6. 1% of 3,200 28. 1

4% of 3,200

7. 1% of 5,000 29. 1

4% of 5,000

8. 1% of 10,000 30. 1

4% of 10,000

9. 1% of 20,000 31. 1

4% of 20,000

10. 1% of 40,000 32. 1

4% of 40,000

11. 1% of 80,000 33. 1

4% of 80,000

12. 1

2% of 100 34. 1% of 1,000

13. 1

2% of 200 35.

1

2% of 1,000

14. 1

2% of 400 36.

1

4% of 1,000

15. 1

2% of 800 37. 1% of 4,000

16. 1

2% of 1,600 38.

1

2% of 4,000

17. 1

2% of 3,200 39.

1

4% of 4,000

18. 1

2% of 5,000 40. 1% of 2,000

19. 1

2% of 10,000 41.

1

2% of 2,000

20. 1

2% of 20,000 42.

1

4% of 2,000

21. 1

2% of 40,000 43.

1

2% of 6,000

22. 1

2% of 80,000 44.

1

4% of 6,000

Number Correct: ______

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

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Fractional Percentsโ€”Round 1 [KEY]

Directions: Find the part that corresponds with each percent.

1. 1% of 100 ๐Ÿ 23. 1

4% of 100

๐Ÿ

๐Ÿ’

2. 1% of 200 ๐Ÿ 24. 1

4% of 200

๐Ÿ

๐Ÿ

3. 1% of 400 ๐Ÿ’ 25. 1

4% of 400 ๐Ÿ

4. 1% of 800 ๐Ÿ– 26. 1

4% of 800 ๐Ÿ

5. 1% of 1,600 ๐Ÿ๐Ÿ” 27. 1

4% of 1,600 ๐Ÿ’

6. 1% of 3,200 ๐Ÿ‘๐Ÿ 28. 1

4% of 3,200 ๐Ÿ–

7. 1% of 5,000 ๐Ÿ“๐ŸŽ 29. 1

4% of 5,000 ๐Ÿ๐Ÿ

๐Ÿ

๐Ÿ

8. 1% of 10,000 ๐Ÿ๐ŸŽ๐ŸŽ 30. 1

4% of 10,000 ๐Ÿ๐Ÿ“

9. 1% of 20,000 ๐Ÿ๐ŸŽ๐ŸŽ 31. 1

4% of 20,000 ๐Ÿ“๐ŸŽ

10. 1% of 40,000 ๐Ÿ’๐ŸŽ๐ŸŽ 32. 1

4% of 40,000 ๐Ÿ๐ŸŽ๐ŸŽ

11. 1% of 80,000 ๐Ÿ–๐ŸŽ๐ŸŽ 33. 1

4% of 80,000 ๐Ÿ๐ŸŽ๐ŸŽ

12. 1

2% of 100

๐Ÿ

๐Ÿ 34. 1% of 1,000 ๐Ÿ๐ŸŽ

13. 1

2% of 200 ๐Ÿ 35.

1

2% of 1,000 ๐Ÿ“

14. 1

2% of 400 ๐Ÿ 36.

1

4% of 1,000 ๐Ÿ. ๐Ÿ“

15. 1

2% of 800 ๐Ÿ’ 37. 1% of 4,000 ๐Ÿ’๐ŸŽ

16. 1

2% of 1,600 ๐Ÿ– 38.

1

2% of 4,000 ๐Ÿ๐ŸŽ

17. 1

2% of 3,200 ๐Ÿ๐Ÿ” 39.

1

4% of 4,000 ๐Ÿ๐ŸŽ

18. 1

2% of 5,000 ๐Ÿ๐Ÿ“ 40. 1% of 2,000 ๐Ÿ๐ŸŽ

19. 1

2% of 10,000 ๐Ÿ“๐ŸŽ 41.

1

2% of 2,000 ๐Ÿ๐ŸŽ

20. 1

2% of 20,000 ๐Ÿ๐ŸŽ๐ŸŽ 42.

1

4% of 2,000 ๐Ÿ“

21. 1

2% of 40,000 ๐Ÿ๐ŸŽ๐ŸŽ 43.

1

2% of 6,000 ๐Ÿ‘๐ŸŽ

22. 1

2% of 80,000 ๐Ÿ’๐ŸŽ๐ŸŽ 44.

1

4% of 6,000 ๐Ÿ๐Ÿ“

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

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Fractional Percentsโ€”Round 2

Directions: Find the part that corresponds with each percent.

1. 10% of 30 23. 1012

% of 100

2. 10% of 60 24. 1012

% of 200

3. 10% of 90 25. 1012

% of 400

4. 10% of 120 26. 1012

% of 800

5. 10% of 150 27. 1012

% of 1,600

6. 10% of 180 28. 1012

% of 3,200

7. 10% of 210 29. 1012

% of 6,400

8. 20% of 30 30. 1014

% of 400

9. 20% of 60 31. 1014

% of 800

10. 20% of 90 32. 1014

% of 1,600

11. 20% of 120 33. 1014

% of 3,200

12. 5% of 50 34. 10% of 1,000

13. 5% of 100 35. 1012

% of 1,000

14. 5% of 200 36. 1014

% of 1,000

15. 5% of 400 37. 10% of 2,000

16. 5% of 800 38. 1012

% of 2,000

17. 5% of 1,600 39. 1014

% of 2,000

18. 5% of 3,200 40. 10% of 4,000

19. 5% of 6,400 41. 1012

% of 4,000

20. 5% of 600 42. 1014

% of 4,000

21. 10% of 600 43. 10% of 5,000

22. 20% of 600 44. 1012

% of 5,000

Number Correct: ______

Improvement: ______

NYS COMMON CORE MATHEMATICS CURRICULUM 7โ€ข4 Lesson 10

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Fractional Percentsโ€”Round 2 [KEY]

Directions: Find the part that corresponds with each percent.

1. 10% of 30 ๐Ÿ‘ 23. 1012

% of 100 ๐Ÿ๐ŸŽ. ๐Ÿ“

2. 10% of 60 ๐Ÿ” 24. 1012

% of 200 ๐Ÿ๐Ÿ

3. 10% of 90 ๐Ÿ— 25. 1012

% of 400 ๐Ÿ’๐Ÿ

4. 10% of 120 ๐Ÿ๐Ÿ 26. 1012

% of 800 ๐Ÿ–๐Ÿ’

5. 10% of 150 ๐Ÿ๐Ÿ“ 27. 1012

% of 1,600 ๐Ÿ๐Ÿ”๐Ÿ–

6. 10% of 180 ๐Ÿ๐Ÿ– 28. 1012

% of 3,200 ๐Ÿ‘๐Ÿ‘๐Ÿ”

7. 10% of 210 ๐Ÿ๐Ÿ 29. 1012

% of 6,400 ๐Ÿ”๐Ÿ•๐Ÿ

8. 20% of 30 ๐Ÿ” 30. 1014

% of 400 ๐Ÿ’๐Ÿ

9. 20% of 60 ๐Ÿ๐Ÿ 31. 1014

% of 800 ๐Ÿ–๐Ÿ

10. 20% of 90 ๐Ÿ๐Ÿ– 32. 1014

% of 1,600 ๐Ÿ๐Ÿ”๐Ÿ’

11. 20% of 120 ๐Ÿ๐Ÿ’ 33. 1014

% of 3,200 ๐Ÿ‘๐Ÿ๐Ÿ–

12. 5% of 50 ๐Ÿ. ๐Ÿ“ 34. 10% of 1,000 ๐Ÿ๐ŸŽ๐ŸŽ

13. 5% of 100 ๐Ÿ“ 35. 1012

% of 1,000 ๐Ÿ๐ŸŽ๐Ÿ“

14. 5% of 200 ๐Ÿ๐ŸŽ 36. 1014

% of 1,000 ๐Ÿ๐ŸŽ๐Ÿ. ๐Ÿ“

15. 5% of 400 ๐Ÿ๐ŸŽ 37. 10% of 2,000 ๐Ÿ๐ŸŽ๐ŸŽ

16. 5% of 800 ๐Ÿ’๐ŸŽ 38. 1012

% of 2,000 ๐Ÿ๐Ÿ๐ŸŽ

17. 5% of 1,600 ๐Ÿ–๐ŸŽ 39. 1014

% of 2,000 ๐Ÿ๐ŸŽ๐Ÿ“

18. 5% of 3,200 ๐Ÿ๐Ÿ”๐ŸŽ 40. 10% of 4,000 ๐Ÿ’๐ŸŽ๐ŸŽ

19. 5% of 6,400 ๐Ÿ‘๐Ÿ๐ŸŽ 41. 1012

% of 4,000 ๐Ÿ’๐Ÿ๐ŸŽ

20. 5% of 600 ๐Ÿ‘๐ŸŽ 42. 1014

% of 4,000 ๐Ÿ’๐Ÿ๐ŸŽ

21. 10% of 600 ๐Ÿ”๐ŸŽ 43. 10% of 5,000 ๐Ÿ“๐ŸŽ๐ŸŽ

22. 20% of 600 ๐Ÿ๐Ÿ0 44. 1012

% of 5,000 ๐Ÿ“๐Ÿ๐Ÿ“

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