Simple Interest: A Yearly Percent of the Principal | Grade 7 Math | LevelKite

When you borrow money from a bank, you usually pay a fee for using it. When you keep money in a savings account, the bank can pay you a fee for using yours. We call that fee interest.

The amount you borrow or save at the start is the principal. In this lesson the principal is always a positive amount of money.

You already find sales tax and tips by taking a percent of a stated whole, such as a price or a bill. Interest works the same way. A percent is a rate per 100 of a stated positive whole, and for interest that whole is the principal. The new idea is time: the interest also depends on how long the money stays borrowed or saved.

Two more quantities set the size of the interest.

The interest rate is the percent of the principal that you pay or earn for one year. A rate of 3% per year means 3 dollars of interest for every 100 dollars of principal, each year.
The time is how long the money stays borrowed or saved. Measure it in years, because the rate is for one year.

With simple interest, every year's interest is the same percent of the original principal. The whole stays the original principal; earlier interest never changes it.

Suppose you save 200 dollars at 3% simple interest per year and add no other money. Write 3% as 0.03, as you do for sales tax. One year of interest is $0.03 \times 200 = 6$ dollars. The table shows what happens over 4 years.

200 dollars saved at 3% simple interest per year

Year	Interest for that year (dollars)	Total interest so far (dollars)	Same total as a product
1	$0.03 \times 200 = 6$	$6$	$6 \times 1 = 6$
2	$0.03 \times 200 = 6$	$6 + 6 = 12$	$6 \times 2 = 12$
3	$0.03 \times 200 = 6$	$12 + 6 = 18$	$6 \times 3 = 18$
4	$0.03 \times 200 = 6$	$18 + 6 = 24$	$6 \times 4 = 24$

Each year adds the same 6 dollars, because each year's interest is 3% of the same 200 dollars. So you can multiply instead of adding again and again. For 4 years:

$0.03 \times 200 \times 4 = 6 \times 4 = 24$

So for simple interest, you multiply the principal by the yearly rate, written as a decimal, and then by the number of years.

When the positive principal and the yearly rate stay the same, the interest is proportional to the time. That means the interest is always the same number times the time. In the table it is 6 dollars times the number of years, so twice the time gives twice the interest and half the time gives half.

Key concept

Simple interest: principal times rate times time

Let $I$ be the interest in dollars, $P$ the principal in dollars, $r$ the interest rate for one year written as a decimal, and $t$ the time in years.

One year of simple interest is $Pr$ , the yearly percent of the principal. Multiply by the number of years to get $I = Prt$ . A part of a year earns that same part of one year's interest.

Use this rule when all of these hold:

$P$ is positive, and it is the whole that you take the percent of.
The rate is a percent of the original principal for one year, so 5% becomes $r = 0.05$ .
$t$ is in years and is not negative.
The interest is simple: earlier interest never changes the whole.

I = Prt

Worked example

Find the interest for whole years

Maya saves 500 dollars in an account that pays 4% simple interest per year. She adds no other money. How much interest does the account earn in 3 years?

Name each quantity with its unit. The principal P is the whole you take the percent of: 500 dollars, a positive amount. The time t is already in years.
$P = 500, \quad t = 3$
Write the yearly rate r as a decimal. A percent is a rate per 100, so 4% of the principal means 4 hundredths of it.
$r = \frac{4}{100} = 0.04$
Find one year of interest by taking 4% of the 500-dollar principal.
$0.04 \times 500 = 20$
Multiply by the 3 years. Simple interest pays the same 20 dollars each year, because you always take the percent of the original 500 dollars.
$I = 500 \times 0.04 \times 3 = 20 \times 3 = 60$
Check another way. Each year's 4% is a percent of the same 500 dollars, so 3 years is 12% of it in all, and 12% of 500 is also 60.
$0.12 \times 500 = 60$

Answer

The account earns 60 dollars of simple interest in 3 years.

Many problems ask for more than the interest. How much interest asks for $I$ . How much in all or how much to repay asks for the total.

Let $A$ be the total amount in dollars. When no other money goes in or comes out, the total is the positive principal plus the simple interest on it:

A = P + I

Worked example

Find the total to repay when time is in months

Carlos borrows 1200 dollars at 6% simple interest per year. He repays the whole loan after 18 months and makes no payments before then. How much does he repay in all?

Name the quantities with units. The principal is the whole: 1200 dollars, a positive amount. The rate is 6% of the principal for one year, and 6 per 100 is 0.06.
$P = 1200, \quad r = \frac{6}{100} = 0.06$
Change the time to years, because the rate is for one year. One year has 12 months, so divide the 18 months by 12.
$t = \frac{18}{12} = 1.5$
Find the simple interest. The principal times the yearly rate gives one year of interest, 72 dollars. For 1.5 years, add one year's 72 dollars and half a year's 36 dollars.
$I = 1200 \times 0.06 \times 1.5 = 72 \times 1.5 = 72 + 36 = 108$
The question asks for the total to repay. He makes no earlier payments, so the total is the principal he borrowed plus the interest.
$A = 1200 + 108 = 1308$
Check the size of the interest. One year costs 72 dollars and two years cost 144 dollars. The principal and the rate are positive, so 1.5 years must cost an amount between them.
$72 < 108 < 144$

Answer

Carlos repays 1308 dollars in all: the 1200 dollars he borrowed plus 108 dollars of simple interest.

Worked example

Find the yearly rate from the interest

Aisha saves 400 dollars in an account that pays simple interest. She adds no other money, and after 5 years the account has earned 70 dollars of interest. What is the interest rate per year?

Name the known quantities with units: the principal is 400 dollars, the time is 5 years and the interest is 70 dollars. Let r be the unknown rate for one year, written as a decimal.
$P = 400, \quad t = 5, \quad I = 70$
Put the known values into the simple interest rule: interest equals principal times rate times time. The rule applies because the interest is simple, the principal is positive and the time is in years.
$I = Prt \quad \Rightarrow \quad 70 = 400 \times r \times 5$
Divide both sides by 5. Dividing both sides of an equation by the same number that is not zero keeps the two sides equal. Now the left side is one year of interest, 14 dollars.
$\frac{70}{5} = \frac{400 \times r \times 5}{5} \quad \Rightarrow \quad 14 = 400 \times r$
Divide both sides by 400, the principal. It is positive, so it is not zero and the two sides stay equal. The result is the part of the principal that one year of interest makes up. Divide 14 and 400 each by 4 to see it as 3.5 per 100.
$r = \frac{14}{400} = \frac{3.5}{100} = 0.035$
Write the rate as a percent. A percent is a rate per 100 of the principal, and 3.5 per 100 is 3.5%.
$r = 0.035 = 3.5\%$
Check by using the rate. 3.5% of the 400-dollar principal is 14 dollars for one year, and 5 years of that is 70 dollars. This matches the interest in the problem.
$400 \times 0.035 \times 5 = 14 \times 5 = 70$

Answer

The interest rate is 3.5% per year.

Simple interest is a yearly percent of the original principal, multiplied by the time in years. When the principal is positive, you write the rate as a decimal and you measure the time in years, the interest is $I = Prt$ . When no other money goes in or comes out, the total is $A = P + I$ .

Before you compute, change the time to years and decide whether the question asks for the interest or the total.

Next, you will compare discounts and coupons to find the better deal.