4. Percents

4.01 Proportional reasoning with percents

4.02 Percent increase and decrease

Investigation: Heart rate

4.03 Discounts, fees and markups

4.04 Sales tax, commissions and tips

4.05 Simple interest

Lesson

Practice

Topic Review: Percents

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United States of America

Grade 7

4.05 Simple interest

Lesson

Practice

Introduction

It costs money to borrow money. The extra money that banks and other lenders charge us to borrow money is called interest. Interest may also refer to the additional money that is earned from investing money, such as into a savings accounts. There are different types of interest and in this lesson we are going to talk about simple interest.

Ideas

Simple interest
Simple interest calculations for time periods other than years

Simple interest

Simple interest, or flat rate interest, describes a method of calculating interest where the interest amount is fixed, which means it doesn't change. The interest charge is always based on the original amount borrowed or invested.

Many financial institutions give their interest rates per year. For example, an interest rate might be given as 3\% per year.

To calculate simple interest, three quantities are involved: the principal amount P that is borrowed or invested, the number of time periods t, and the interest rate of r per time period. Simple interest is then calculated using the formula:I=Prt

Note that sometimes n is used instead of t to represent the number of time periods.

To find the total value of the investment or loan after a given time period, we add the interest to the principal amount P.

Examples

Example 1

Calculate the simple interest on a loan of \$8580 at 2\% per year for 10 years.

Calculate the simple interest.

Worked Solution

Create a strategy

Use the simple interest formula.

Apply the idea

We are given: P=\$8580, { } r=2\% and t=10

\displaystyle I	\displaystyle =	\displaystyle 8580\times 2\%\times 10	Substitute the given values to the formula
	\displaystyle =	\displaystyle 8580\times 0.02\times 10	Convert the percentage to a decimal
	\displaystyle =	\displaystyle \$1716.00	Evaluate

What is the total value of the loan after 10 years?

Worked Solution

Create a strategy

The total value is the original loan plus the interest that has accrued over 10 years.

Apply the idea

\displaystyle \text{Total value}	\displaystyle =	\displaystyle 1716+8580	Add the interest and the loan
	\displaystyle =	\displaystyle \$10\,296.00	Evaluate

Example 2

The interest on an investment of \$3600 over 10 years is \$2520.00. If the annual interest rate is r, find r as a percentage.

Worked Solution

Create a strategy

Use the simple interest formula.

Apply the idea

We are given: I=\$2520, { } P=\$3600 and t=10

\displaystyle 2520	\displaystyle =	\displaystyle 3600\times r\times 10	Substitute the given values to the formula
\displaystyle 2520	\displaystyle =	\displaystyle 36000r	Evaluate the multiplication
\displaystyle r	\displaystyle =	\displaystyle \dfrac{2520}{36000}	Divide both sides of equation by 36000
	\displaystyle =	\displaystyle 0.07	Convert the fraction to decimal
	\displaystyle =	\displaystyle 7\%	Convert the decimal to percentage

Example 3

For a simple interest rate of 6\% per year , calculate the number of years t needed for an investment of \$1957 to earn \$1174.20 in interest.

Give your answer as a whole number of years.

Worked Solution

Create a strategy

Use the simple interest formula.

Apply the idea

We are given: I=\$1174.20, { } P=\$1957 and r=6\%

\displaystyle 1174.2	\displaystyle =	\displaystyle 1957\times 6\% \times t	Substitute the given values to the formula
\displaystyle 1174.2	\displaystyle =	\displaystyle 1957\times 0.06\times t	Convert the percentage to decimal
\displaystyle 1174.2	\displaystyle =	\displaystyle 117.42t	Evaluate the multiplication
\displaystyle t	\displaystyle =	\displaystyle \dfrac{1174.2}{117.42}	Divide both sides of equation by 117.42
	\displaystyle =	\displaystyle 10	Evaluate

Idea summary

Simple interest is calculated asI=Prt

where P is the principal amount invested (or borrowed), r is the interest rate per time period, and t is the number of time periods.

The total amount or value A, earned after t interest periods, is then calculated asA=P+I

Simple interest calculations for time periods other than years

When calculating simple interest for time periods that are not years, such as months, weeks or days, we need to make sure the interest rate and the time periods are expressed using the same period. For example, if the rate r is given per year then the value for t needs to be expressed in years too.

Examples

Example 4

Calculate the simple interest generated on a loan of \$3860 at a rate of 9\% per year for 13 months.

Worked Solution

Create a strategy

Convert the given t in years and use the simple interest formula.

Apply the idea

There are 12 months in 1 year.

\displaystyle t	\displaystyle =	\displaystyle 13 \times \dfrac{1}{12}	Multiply 13 by \dfrac{1}{12}
	\displaystyle =	\displaystyle \dfrac{13}{12}	Evaluate
\displaystyle I	\displaystyle =	\displaystyle 3860\times 9\% \times \frac{13}{12}	Substitute the given values to the formula
	\displaystyle =	\displaystyle 3860\times 0.09 \times \frac{13}{12}	Convert the percentage to decimal
	\displaystyle =	\displaystyle \$ 376.35	Evaluate

Example 5

Calculate the simple interest earned on an investment of \$5000 at 1.7\% per quarter for 9 years.

Worked Solution

Create a strategy

Convert the given t in quarters and use the simple interest formula.

Apply the idea

There are 4 quarters in a year.

\displaystyle t	\displaystyle =	\displaystyle 9 \times 4	Multiply 9 by 4
	\displaystyle =	\displaystyle 36	Evaluate
\displaystyle I	\displaystyle =	\displaystyle 5000\times 1.7\% \times 36	Substitute the given values to the formula
	\displaystyle =	\displaystyle 5000\times 0.017 \times 36	Convert the percentage to decimal
	\displaystyle =	\displaystyle \$ 3060.00	Evaluate

Example 6

Calculate the simple interest earned on an investment of \$5320 at 6\% per year for 95 weeks.

Assume that a year has 52 weeks.

Worked Solution

Create a strategy

Convert the given t in years and use the simple interest formula.

Apply the idea

\displaystyle t	\displaystyle =	\displaystyle 95 \times \dfrac{1}{52}	Multiply 95 by \dfrac{1}{52}
	\displaystyle =	\displaystyle \dfrac{95}{52}	Evaluate
\displaystyle I	\displaystyle =	\displaystyle 5320\times 6\% \times \frac{95}{52}	Substitute the given values to the formula
	\displaystyle =	\displaystyle 5320\times 0.06 \times \frac{95}{52}	Convert the percentage to decimal
	\displaystyle =	\displaystyle \$ 583.15	Evaluate

Idea summary

We need to make sure the interest rate and the time periods are expressed using the same period when computing for the amount of interest.

Remember the equivalent time periods of 1 year:

\displaystyle 1 \text{ year}	\displaystyle =	\displaystyle 12 \text{ months}
	\displaystyle =	\displaystyle 52 \text{ weeks}
	\displaystyle =	\displaystyle 4 \text{ quarters} \text{ (of } 3 \text{ months each)}
	\displaystyle =	\displaystyle 365 \text{ days}

4.05 Simple interest

Introduction

Ideas

Simple interest

Examples

Example 1

Example 2

Example 3

Simple interest calculations for time periods other than years

Examples

Example 4

Example 5

Example 6

Outcomes

7.RP.A.3

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