Let's Practice
Finding the right car or home is exciting, but an important part of making that final decision is being sure you can afford the loan. How much more would you pay with compound interest compared to simple interest on a loan? Now that you have seen how to find the compound interest on a loan or investment, it is time to calculate this type of interest on your own. The formula for compound interest is:
\( A = P\left( 1 + \frac{r}{n} \right)^{n \cdot t} \)
- \( A = \) total amount of the loan or investment you will pay (if it's a loan) or the total amount you will have earned (if it's an investment) at the end of the term (interest included)
- \( P = \) principal or initial amount
- \( r = \) interest rate (as a decimal)
- \( n = \) number of times per year that the interest is compounded
- \( t = \) term, which is the time period of the loan or investment (in years)
For each of the practice questions, determine the interest and/or total amount. Remember that the interest rate percentage must be converted to a decimal (by dividing it by 100) and the time must be in years (by dividing months by 12). Round all answers to the nearest thousandth place, and round all monetary values to the nearest penny. When ready, click each answer button to reveal the solution.
Going back to Ida's situation, her auto loan is for $20,000, to be paid back over two years, at a 4% interest rate. If this interest is compounded twice a year (compounding twice a year means semi-annual; semi means half and annual means year), what will be the total amount of interest that Ida will pay on the loan? What will the interest be if it is compounded four times a year (compounding four times a year is also known as quarterly) instead? Will Ida end up paying more on this loan if it is compounded twice a year or four times a year?
Calculate the total amount to be paid with \( P = 20000,\ r = 0.04,\ n = 2 \), and \( t = 2. \) Then find the total interest by subtracting the principal from the amount. (Amount \( − \) Principal \( = \) Interest).
Substitute these values into the formula to solve:
$20,000 loan compounded 2 times a year
\( A = 20000(1 + \frac{0.04}{2})^{2 \cdot 2} \)
\( A = 20000(1 + 0.02)^{4} \)
\( A = 20000(1.02)^{4} \)
\( A \approx 20000\left( 1.082 \right) \)
\( A = 21640.00 \)
\( A - P: \) $21,640 - $20,000 = $1,640 interest.
$20,000 compounded 4 times a year
\( A = 20000(1 + \frac{0.04}{4})^{4 \cdot 2} \)
\( A = 20000(1 + 0.01)^{8} \)
\( A = 20000(1.01)^{8} \)
\( A \approx 20000\left( 1.083 \right) \)
\( A = 21660.00 \)
\( A - P: \) $21,660 - $20,000 = $1,660 interest.
So, Ida will have to pay $20 more on this loan if it were compounded four times than if it were compounded twice a year.
Wes invests $5,000, at 7% interest, over 60 months, compounded semi-annually (twice a year). How much will Wes earn on his investment?
Here, \( P = 5000,\ r = 0.07,\ n = 2 \), and \( t = 60 \) months (convert the time to years):
\( A = 5000(1 + \frac{0.07}{2})^{2 \cdot 5} \)
\( A = 5000(1 + 0.035)^{10} \)
\( A = 5000(1.035)^{10} \)
\( A \approx 5000\left( 1.411 \right) \)
\( A = 7055 \)
So, the money Wes invested will equal $7,055 after five years, compounded semi-annually. This means, after the five years, he will have earned $7,055 \( - \) $5,000 \( = \) $2,055 on his investment.
Which investment will earn Rita more money: a $500 investment, over eight years, with an interest rate of 12%, compounded bimonthly (six times a year) or a $1,500 investment, over six years, with an interest rate of 8%, compounded monthly (12 times a year)?
Substitute the values into the formula to solve.
Then find the total interest by subtracting the principal from the amount. (Amount \( − \) Principal \( = \) Interest).
8 years compounded bimonthly
\( P = 500,\ r = 0.12,\ n = 6 \), and \( t = 8 \)
\( A = 500(1 + \frac{0.12}{6})^{6 \cdot 8} \)
\( A = 500(1 + 0.02)^{48} \)
\( A = 500(1.02)^{48} \)
\( A \approx 500\left( 2.587 \right) \)
\( A = 1293.50 \)
\( A-P: \) \( (\$ 1,293.50 - \$ 500) = \$ 793.50 \)
Rita will earn $793.50 on this investment after eight years.
6 years compounded monthly
\( P = 1500,\ r = 0.08,\ n = 12 \), and \( t = 6 \)
\( A = 1500(1 + \frac{0.08}{12})^{12 \cdot 6} \)
\( A \approx 1500(1 + 0.007)^{72} \)
\( A = 1500(1.007)^{72} \)
\( A \approx 1500\left( 1.652 \right) \)
\( A = 2478.00 \)
\( A-P: \) \( (\$ 2,478 - \$ 1,500) = \$ 978.00 \)
Rita will earn $978.00 on this investment after six years.
Since $978 > $793, a $1,500 investment, over six years, with an interest rate of 8%, compounded monthly, will earn Rita more money than a $500 investment, over eight years, with an interest rate of 12%, compounded bimonthly.
Question
How much more in interest does Rita earn with the 8% investment than with the 12% investment?
$185.
You have decided to purchase your first house and have researched two mortgages. One has interest compounded annually and the other uses simple interest. The two loans' details are:
Loan 1 will let you borrow $100,000, for 30 years, compounded annually, with an interest rate of 2%.
Loan 2 is the same but uses simple interest.
Which loan, 1 or 2, will end up costing you less in interest?
Loan 1 Compound Interest
\( P = 100000,\ r = 0.02,\ n = 1 \), and \( t = 30 \)
\( A = 100000(1 + \frac{0.02}{1})^{1 \cdot 30} \)
\( A = 100000(1 + 0.02)^{30} \)
\( A = 100000(1.02)^{30} \)
\( A \approx 100000\left( 1.811 \right) \)
\( A = 181100 \)
\( A-P: \) \( (\$ 181,100 - \$ 100,000) \) \( = \$ 81,100 \)
Loan 2 Simple Interest
\( P = 100000,\ r = 0.02, \) and \( t = 30 \)
\( I = Prt \)
\( I = 100000 \cdot 0.02 \cdot 30 \)
\( I = 60000 \)
Since $60,000 < $81,100, you will pay less interest with Loan 2 than with Loan 1.
Question
When taking out a loan, is it better to have simple interest or compound interest?
It's better to have simple interest because you pay less in total interest.
