5.1: Simple Interest

Learning Objectives

In this section, you will learn to:

• Use the simple interest formula to find an account balance, principal, rate, or time.

Prerequisite Skills

Before you get started, take this prerequisite quiz.

1. Solve \(5=2(x+9)\).

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\(x=-\dfrac{13}{2}\)

If you missed this problem, review Section 1.1. (Note that this will open in a new window.)

2. Solve \(x=-3(7-10)\).

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\(x=9\)

If you missed this problem, review Section 1.1. (Note that this will open in a new window.)

3. Solve \(-12=x(3+8)\).

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\(x=-\dfrac{12}{11}\)

If you missed this problem, review Section 1.1. (Note that this will open in a new window.)

4. Solve \(12=4(2-x)\).

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\(x=-1\)

If you missed this problem, review Section 1.1. (Note that this will open in a new window.)

Simple Interest

It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal, also called the present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is most commonly specified for a year, it may be specified for a week, a month, or a quarter, etc. Credit card companies often list their charges as monthly rates, sometimes it is as high as 1.5% a month.

Definition: Simple Interest

If an amount \(P\) is borrowed for a time \(t\) at an interest rate of \(r\) per time period, then the simple interest is given by

\[ I = P \cdot r \cdot t\]

Definition: Accumulated Value

The total amount \(A\), also called the accumulated value or the future value, is given by

\[ \begin{align*} A &= P + I \\[4pt] &= P + Prt \end{align*}\]

or

\[ A = P(1+rt) \label{simple3}\]

where interest rate \(r\) is expressed in decimals.

Example \(\PageIndex{1}\)

Ursula borrows $600 for 5 months at a simple interest rate of 15% per year. Find the interest, and the total amount she is obligated to pay?

Solution

The interest is computed by multiplying the principal with the interest rate and the time.

\[\begin{align*} \mathrm{I} &=\operatorname{Prt} \\[4pt] &=\$ 600(0.15) \frac{5}{12} \\[4pt] &=\$ 37.50 \end{align*} \]

The total amount is

\[\begin{align*} \mathrm{A} &=\mathrm{P}+\mathrm{I} \\[4pt] &=\$ 600+\$ 37.50 \\[4pt] &=\$ 637.50 \end{align*} \]

Incidentally, the total amount can be computed directly via Equation \ref{simple3} as

\[\begin{align*} A &=P(1+r t) \\[4pt] &=\$ 600[1+(0.15)(5 / 12)] \\[4pt] &=\$ 600(1+0.0625) \\[4pt] &=\$ 637.50 \end{align*} \]

Example \(\PageIndex{2}\)

Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?

Solution

The total amount or the future value is given by Equation \ref{simple3}.

\[\begin{align*} A &= P(1 + rt) \\[4pt] &=\$ 2500[1+(.06)(3)] \\[4pt] \mathrm{A} &=\$ 2950 \end{align*}\]

Example \(\PageIndex{3}\)

Darnel owes a total of $3060 which includes 12% simple interest for the three years he borrowed the money. How much did he originally borrow?

Solution

This time we are asked to compute the principal \(P\) via Equation \ref{simple3}.

\[\begin{align*} \$ 3060 &=\mathrm{P}[1+(0.12)(3)] \\[4pt] \$ 3060 &=\mathrm{P}(1.36) \\[4pt] \dfrac{\$ 3060}{1.36}&=\mathrm{P} \\[4pt] \$ 2250 &=\mathrm{P} \quad \text { Darnel originally borrowed \$2250. } \end{align*}\]

Example \(\PageIndex{4}\)

A Visa credit card company charges a 1.5% simple interest finance charge each month on the unpaid balance. If Martha owed $2350 and has not paid her bill for three months, how much does she owe now?

Solution

Before we attempt the problem, the reader should note that in this problem the rate of finance charge is given per month and not per year.

The total amount Martha owes is the previous unpaid balance plus the finance charge.

\[ A=\$ 2350+\$ 2350(.015)(3)=\$ 2350+\$ 105.75=\$ 2455.75 \nonumber\]

Alternatively, again, we can compute the amount directly by using formula \(A = P(1 + rt)\)

\[A=\$ 2350[1+(.015)(3)]=\$ 2350(1.045)=\$ 2455.75 \nonumber\]

Summary

Below is a summary of the formulas we developed for calculations involving simple interest:

Simple interest

If an amount \(P\) is borrowed for a time \(t\) at an interest rate of \(r\) per time period, then the simple interest is given by

\[ I = P \cdot r \cdot t \nonumber\]

The total amount \(A\), also called the accumulated value or the future value, is given by

\[ A = P + I = P + Prt \nonumber\]

or

\[ A = P(1+rt) \nonumber\]

where interest rate \(r\) is expressed in decimals.