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When taking out a loan or securing a line of credit, it's very common to owe interest on the money you've borrowed. Similarly, it's common to receive interest payments on money you've placed in a bank account. To learn one way this interest can be calculated, let's take a closer look at simple interest.

Simple interest refers to the amount of money paid or earned on the principal over a set period of time (term). For instance, if you place money in the bank (principal) and earn a fixed amount on that money over the term, you will be earning simple interest. Similarly, if you borrow a loan for a specific principal amount and pay a fixed amount on that principal over the term (ex. $50 interest payment each year for 3 years), you will be paying simple interest.

This differs from compound interest, which is interest paid or earned on the initial principal as well as accumulated interest from previous periods (ex. $50 interest payment in year 1, $52.50 payment in year 2, and $55.13 payment in year 3).

The formula for simple interest is fairly straightforward.

Simple Interest: $I=prt$

In this formula, I represents the interest earned or paid, p is the principal (money invested or borrowed), r is the annual interest rate written in decimal form, and t is the time in periods for which the interest is paid or earned. So for example, if the interest was calculated monthly for 6 months $t=6$ , if it was yearly but only for 9 months then $t=\frac{9}{12}=\frac{3}{4}$ .

**Example #1**

You decide to deposit $600 at 6% simple interest over a period of 4 years. How much interest have you earned at the end of the term?

Start with the formula for simple interest.

$I=prt$

Substitute the principal (p), the annual rate of interest (r), time for which the interest is paid (t) and then multiply.

$I=600\left(0.06\right)\left(4\right)=144$

The interest earned at the end of the term is $144.

**Example #2**

You borrowed a loan for $7,300 at 4% simple interest over a term of 5 years. How much interest did you pay on the loan?

$I=7300\left(0.4\right)\left(5\right)=1460$

The interest you paid on the loan is $1,460.

**Example #3**

You borrowed $15,000 at 5% simple interest and pay it off 3 and a half years later. What was the total amount you paid?

Note: Since the time is measured annually, three and a half years will be 3.5.

$I=15000\left(0.05\right)\left(3.5\right)=2625$

Keep in mind that the total amount paid will include the interest you accrued over 3.5 years as well as the principal you borrowed. In this case, the total paid will be $\$17625=\$15000+\$2625$ .

You've learned how to find the interest accrued on money borrowed or saved, but what if you want to find the principal, the annual rate of interest, or the time for which interest is paid or earned?

Suppose Jacob borrowed a principal amount at 3% simple interest. It must be repaid in 6 years along with $1,512 in interest. What was the principal amount Jacob borrowed?

Start with the simple interest formula.

$I=prt$

Fill in 1,512 for the interest accrued (I), 0.03 for the annual rate of interest (r), and 6 for the time in years for which the interest will be paid (t).

$\mathrm{1512}=p\left(0.03\right)\left(6\right)$

Multiply r and t.

$\mathrm{1512}=0.18p$

Divide each side by 0.18.

$p=8400$

The principal amount Jacob borrowed is $8,400.

If Clara placed $3,750 in a bank account for 4 years and earned $1,050 in interest, what was the annual simple interest rate she received?

This time, you want to solve for r. Fill in the principal, time, and interest earned where appropriate.

$1050=3750\left(r\right)\left(4\right)$

Multiply p and t.

$1050=15000r$

Divide each side by 15,000.

$r=0.07$

Convert the decimal to a percentage.

Clara received a simple interest rate of 7%.

If Sarah invests $6,000 into an account with 5% annual simple interest, how many years will it take for her to earn $900 in interest?

In order to solve for t, you'll need to fill in the principal, interest rate, and interest earned where appropriate.

$900=6000\left(0.05\right)t$

Multiply p and r.

$900=300t$

Divide each side by 300.

$t=3$

It will take Sarah 3 years to earn $900 in interest.

a. If Meagan invests $12,000 at 3% simple interest for 2 years, how much interest will she earn over that period?

To find the interest earned, use the simple interest formula: $I=prt$

$I=\text{Interest}$

$P=\text{Principal}\phantom{\rule{2pt}{0ex}}\left(\mathrm{\$12000}\right)$

$r=\text{Interest rate}\phantom{\rule{2pt}{0ex}}\left(\mathrm{3\%}=0.03\right)$

$t=\text{Time}\phantom{\rule{2pt}{0ex}}\left(2\mathrm{years}\right)$

$I=\$12000\times 0.03\times 2=\$720$

b. Suppose Bill borrows $25,000 that must be repaid over a period of 8 years at 4% simple interest? What is the total interest amount he will pay on his loan?

$I=prt$

$I=\text{Interest}$

$P=\text{Principal}\phantom{\rule{2pt}{0ex}}\left(\mathrm{\$25000}\right)$

$r=\text{Interest rate}\phantom{\rule{2pt}{0ex}}\left(\mathrm{4\%}=0.04\right)$

$t=\text{Time}\phantom{\rule{2pt}{0ex}}\left(8\mathrm{years}\right)$

$I=\$25000\times 0.04\times 8=\$8000$

c. Kelly deposited $4,000 into a bank account then chose to withdraw her funds and close the account 5.5 years later. If she earned an annual simple interest rate of 5%, how much total (principal + interest) did she receive once she closed the account?

$I=prt$

$I=\text{Interest}$

$P=\text{Principal}\phantom{\rule{2pt}{0ex}}\left(\mathrm{\$4000}\right)$

$r=\text{Interest rate}\phantom{\rule{2pt}{0ex}}\left(\mathrm{5\%}=0.05\right)$

$t=\text{Time}\phantom{\rule{2pt}{0ex}}\left(5.5\mathrm{years}\right)$

$I=\$4000\times 0.05\times 5.5=\$1100$

$\mathrm{Total\; received}=\mathrm{Principal}+\mathrm{Interest}=\mathrm{\$4,000}+\mathrm{\$1,100}=\mathrm{\$5,100}$

d. If Jeff places $500 into an account and wants to earn $52.50 in interest, how many years will it take to do so with a 3% annual simple interest rate?

Rearrange the simple interest formula to find time: $t=\frac{I}{(P\times r)}$

$I=\text{Interest}\phantom{\rule{2pt}{0ex}}\left(\$52.50\right)$

$p=\text{Principal}\phantom{\rule{2pt}{0ex}}\left(\$500\right)$

$r=\text{Interest rate}\phantom{\rule{2pt}{0ex}}\left(\mathrm{3\%}=0.03\right)$

$t=\frac{52.50}{(500\times 0.03)}=3.5\mathrm{years}$

e. What is the simple interest rate Tasha has received for a $1,750 loan with a 3-year term if she ends up paying $315 in interest?

Rearrange the simple interest formula to find the interest rate: $r=\frac{I}{(P\times t)}$

$I=\text{Interest}\phantom{\rule{2pt}{0ex}}\left(\$315\right)$

$p=\text{Principal}\phantom{\rule{2pt}{0ex}}\left(\$1750\right)$

$t=\text{Time}\phantom{\rule{2pt}{0ex}}\left(3\mathrm{years}\right)$

$r=\frac{315}{1750\times 3}=0.06$

Convert the decimal to a percentage: $0.06\times 100=6\%$

f. How much money will Clark have to deposit in the bank if he'd like to earn $1,000 in interest in 5 years after being granted a 4% simple interest rate?

Rearrange the simple interest formula to find the principal: $P=\frac{I}{(r\times t)}$

$I=\text{Interest}\phantom{\rule{2pt}{0ex}}\left(\$1000\right)$

$r=\text{Interest rate}\phantom{\rule{2pt}{0ex}}\left(\mathrm{4\%}=0.04\right)$

$t=\text{Time}\phantom{\rule{2pt}{0ex}}\left(5\mathrm{years}\right)$

$P=\frac{\$1000}{(0.04\times 5)}=\$5000$

Partioning a Segment in a given Ratio

College Algebra Diagnostic Tests

Calculating simple interest often involves reading and understanding word problems, which can make solving problems tough. Whether your student is trying to solve for I, p, r, or t, they might need help remembering the steps involved, especially if they're working on an assignment or preparing for a test. Fortunately, there are plenty of great tutors available to help your student with these and other math problems. Contact the Educational Directors at Varsity Tutors, and we will find a qualified tutor who's got the right math skills to work with your student.

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