Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula. We will provide examples of how to find interest earned, calculate the rate of interest, and how to find the principal given a rate and the interest earned.
Calculating Interest Earned
We will start by solving a simple interest application to find the interest.
example
Nathalie deposited [latex]\text{\$12,500}[/latex] in her bank account where it will earn [latex]\text{4%}[/latex] interest. How much interest will Nathalie earn in [latex]5[/latex] years?
Solution
We are asked to find the Interest, [latex]I[/latex].
Organize the given information in a list.
Check your answer. Is [latex]\text{\$2,500}[/latex] a reasonable interest on [latex]\text{\$12,500}[/latex] over [latex]5[/latex] years?
At [latex]4\text{%}[/latex] interest per year, in [latex]5[/latex] years the interest would be [latex]20\text{%}[/latex] of the principal. Is [latex]20\text{%}[/latex] of [latex]\text{\$12,500}[/latex] equal to [latex]\text{\$2,500}[/latex]? Yes.
Write a complete sentence that answers the question.
The interest is [latex]\text{\$2,500}[/latex].
try it
Calculating Rate
There may be times when you know the amount of interest earned on a given principal over a certain length of time, but you don’t know the rate. For instance, this might happen when family members lend or borrow money among themselves instead of dealing with a bank. In the next example, we’ll show how to solve for the rate.
example
Loren lent his brother [latex]\text{\$3,000}[/latex] to help him buy a car. In [latex]\text{4 years}[/latex] his brother paid him back the [latex]\text{\$3,000}[/latex] plus [latex]\text{\$660}[/latex] in interest. What was the rate of interest?
Show Solution
Solution
We are asked to find the rate of interest, [latex]r[/latex].
Organize the given information.
Write a complete sentence that answers the question.
The rate of interest was [latex]5.5\text{%}[/latex].
try it
In the next video we use the simple interest formula to find the rate of interest given an amount of money borrowed and the amount if interest paid.
Calculating Principal
There may be times when you take a loan for a large purchase and the amount of the principal is not clear. This might happen, for instance, in making a car purchase when the dealer adds the cost of a warranty to the price of the car. In the next example, we will solve a simple interest application for the principal.
example
Eduardo noticed that his new car loan papers stated that with an interest rate of [latex]\text{7.5%}[/latex], he would pay [latex]\text{\$6,596.25}[/latex] in interest over [latex]5[/latex] years. How much did he borrow to pay for his car?
Show Solution
Solution
We are asked to find the principal, [latex]P[/latex].
Organize the given information.
[latex]\begin{array}{ccc}\hfill I& =& 6,596.25\hfill \\ \hfill P& =& ?\hfill \\ \hfill r& =& \text{7.5%}\hfill \\ \hfill t& =& \text{5 years}\hfill \end{array}[/latex]
Write a complete sentence that answers the question.
The amount borrowed was [latex]\text{\$17,590}[/latex].
TRY IT
In the simple interest formula, the rate of interest is given as an annual rate, the rate for one year. So the units of time must be in years. If the time is given in months, we convert it to years.
example
Caroline got [latex]\text{\$900}[/latex] as graduation gifts and invested it in a [latex]\text{10-month}[/latex] certificate of deposit that earned [latex]\text{2.1%}[/latex] interest. How much interest did this investment earn?
Show Solution
Solution
We are asked to find the interest, [latex]I[/latex].
Organize the given information.
[latex]\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$900}\hfill \\ \hfill r& =& \text{2.1%}\hfill \\ \hfill t& =& \text{10 months}\hfill \end{array}[/latex]
Write the formula.
[latex]I=Prt[/latex]
Substitute the given information, converting 10 months to [latex]\dfrac{10}{12}[/latex] of a year.
Check your answer. Is [latex]\text{\$15.75}[/latex] a reasonable amount of interest?
If Caroline had invested the [latex]\text{\$900}[/latex] for a full year at [latex]2\text{%}[/latex] interest, the amount of interest would have been [latex]\text{\$18}[/latex]. Yes, [latex]\text{\$15.75}[/latex] is reasonable.
Write a complete sentence that answers the question.
The interest earned was [latex]\text{\$15.75}[/latex].
try it
Example
A friend asks to borrow [latex]$240[/latex], offering to repay you [latex]$250[/latex] in 1 month. What annual interest rate is this equivalent to?
Show Solution
Identify the information given in the problem. Here your friend is paying back $10 more than he borrowed, so that is the interest paid.
This is equivalent to a [latex]50\%[/latex] annual interest rate.
The example video that follows shows how to determine the annual simple interest rate.
Another application of interest rate is calculating Treasury Notes.
Example
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a [latex]$1,000[/latex] T-note with a [latex]4\%[/latex] annual rate, with a maturity in [latex]2[/latex] years. How much interest will you earn?
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