The following examples will demonstrate how we can use the formulas introduced in the previous section.
Example 1
Courtney borrowed $1200 from her bank at 7.5% simple interest for 40 weeks. Determine the amount due at the end of the 40 weeks (rounded to twodecimal places).
Solution
Since we want to determine the final amount due, weuse the formula: \(S=P(1+rt)\)
Next, we can identify the information given in the problem:
\(P=1200,\ r=\frac{7.5}{100}=0.075,\ t=\frac{40}{52}=\frac{10}{13}\)
*Note: The \(t\) value is calculated as \(\frac{40}{52}\) since the given interest rate is the annual/yearly rate, but we we're given the time in weeks. So to convert from weeks to years, we divide by \(52\) (the number of weeks in a year).
Now that we have the appropriate values for the principal, interest rate, and time, we can substitute them into the formula and evaluate the final amount due:
\(S=1200(1+0.075(\frac{10}{13}))=1269.23\)
So, the amount due is \($1269.23\).
Example 2
Example: Jamie lends a friend $500 for fifteen days. When his friend pays it back, he gets an extra $13 as a token of appreciation. What is the annual rate of simple interest that Jamie earned (rounded to two decimal places)?
Solution
First, we can identify the information that is given in the problem:
\(P=500,\ t=\frac{15}{365},\ S=500+13=513\)
*Note: The \(t\) value is calculated as \(\frac{15}{365}\) since the interest rate we want to find is the annual rate, but we we'regiven the time in days. So to convert from days to years, we divide by \(365\) (the number of days in a year).
Since we want to determine the annual rate of simple interest, \(r\), and we're given \(P\), \(t\), and \(S\), we use the formula: \(S=P(1+rt)\) and rearrange it to find \(r\):
\(513=500(1+r(\frac{15}{365}))\)
\(\frac{513}{500}=1+r(\frac{15}{365})\)
\(\frac{513}{500}-1=r(\frac{15}{365})\)
\(\frac{\frac{513}{500}-1}{\frac{15}{365}}=r\)
\(r=0.6327\)
So, Jamie earned \(63.27\)% per annum.
Tip:This example demonstrates that we can rearrange the general formula \(S=P(1+rt)\) into \(r=\frac{\frac{S}{P}-1}{t}=\frac{S-P}{Pt}\), then use this rearranged formula to solve for \(r\).
Example 3
Example: Samantha's insurance company gives her thechoice to collect $20,000 now or $20,750 seven months from now. If money is worth 4.5% simple interest, which option is better for Samanthaand by how much (in terms of today'sdollar)?
Solution
See the below video for the solution.
