Simple Interest

Practice (12)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of years. Then, calculate the loan payoff amount.

   1. Principal \(P\) = $4,000, annual interest rate \(r\) = 5.5%, and number of years \(t\) = 4
   2. Principal \(P\) = $14,800, annual interest rate \(r\) = 7.9%, and number of years \(t\) = 7
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   1. Substitute the principal \(P\) = $4,000, the decimal form of the annual interest rate \(r\) = 0.055, and number of years \(t\) = 4 into the formula for simple interest, and calculate.
      \(I=P\times r\times t=4,000\times 0.055\times 4=880\).
      The simple interest, or cost of the loan, to be paid on the loan is $880.
      The loan payoff amount, or the total to be repaid, is \(T=P+I=4,000+880=4,880\), or $4,880.00.
   2. Substitute the principal \(P\) = $14,800, the decimal form of the annual interest rate \(r\) = 0.079, and number of years \(t\) = 7 into the formula for simple interest, and calculate.
      \(I=P\times r\times t=14,800\times 0.079\times 7=8,184.4\).
      The simple interest, or cost to borrow, to be paid on the loan is $8,184.40.
      The loan payoff amount, or the total to be repaid, is \(T=P+I=14,800+8,184.4=22,984.4\), or $22,984.40.

2. Riley runs an auto repair shop, and needs to purchase a new brake lathe, which costs $11,995. She takes out a two-year, simple interest loan at an annual interest rate of 14.9%. How much interest will she pay and how much total will she repay on the loan?

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   Step 1. Determine the variables, or parts of the formula. The principal \(P\) is the cost of the brake lathe, so \(P\) = $11,995. The interest rate Riley pays is 14.9%, or \(r\) = 0.149 in decimal form. The length of the loan is two years, so \(t\) = 2. We are first asked to find \(I\), the interest Riley will pay.

   Step 2. Substitute the known variables into the formula for simple interest \(I=P\times r\times t\) and solve for \(I\).

   From Step 1 we have \(I=P\times r\times t=11995\times 0.149\times 2=3,574.51\).

   This tells us that the simple interest, or cost to borrow, to be paid on the loan is $3,574.51.

   Step 3. Use the formula \(T=P+I\) to determine the total amount Riley will repay, \(T\).

   The total to be repaid is \(T=P+I=11,995+3,574.51=15,569.51\), or $15,569.51.

3. Abeje needs a loan to purchase equipment for the gym she is going to open. She visits the bank and secures a 4-month loan of $20,000. Her annual percentage rate is 6.75%. How much interest will Abeje pay and what is her loan payoff amount?

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   Abeje’s loan is for $20,000, so her principal is \(P\) = 20,000. The interest rate Abeje will pay is 6.75%, or \(r\) = 0.0675 in decimal form. The length of the loan is 4 months, so \(t=\frac{4}{12}\). Substituting these in the formula for simple interest, we find her interest to be \(I=P\times r\times t=20,000\times 0.0675\times 4/12=450\)

   The simple interest, or cost to borrow, to be paid on the loan is $450.00.

   The payoff is \(T=P+I=20,000+450=20,450\), or $20,450.00.

4. David plans to move his family from Raleigh, North Carolina to Tempe, Arizona. His company will reimburse (pay after the move) David for the move. David does research and determines that movers will cost $5,600 to move his family’s belongings to Tempe. He takes out a simple interest, 45-day loan at 11.75% interest to pay this cost. How much interest will be paid on this 45-day loan, and what is David’s loan payoff amount?

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   This loan is in terms of days, so we will use the formula \(I=P\times \frac{r}{365}\times t\), where t is the number of days and \(r\) is the annual interest rate.

   The principal for the loan is the moving cost, or \(P\) = 5,600. The annual interest rate that David will pay is 11.75%, which in decimal is 0.1175. The length of time for the loan is 45 days, so \(t\) = 45.

   Substituting these values into the formula and calculating, we find that the interest to be paid is \(I=5,600\times \frac{0.1175}{365}\times 45=81.13\), or $81.13 (remember, interest is almost always rounded up to the next cent).

   The payoff for the loan is $5,681.13.

5. In the following, determine how much interest was earned on the investment and the future value of the investment, if the investment yields simple interest.

   1. Principal is $1,000, annual interest rate is 2.01%, and time is 5 years
   2. Principal is $5,000, annual interest rate is 1.85%, and time is 30 years
   3. Principal is $10,000, annual interest rate is 1.25%, and time is 18 months
   4. Principal is $7,000, annual interest rate is 3.26%, and time is 100 days
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   1. The principal is \(P\) = $1,000, the annual interest rate, in decimal form, is 0.0201, and the term is 5 years, or \(t\) = 5. Since the term is an integer number of years, the interest earned on the investment is \(I=P\times r\times t=1,000\times 0.0201\times 5=100.5\), or the interest earned was $100.50.
      To find the future value, we use the formula \(FV=P+I\). Substituting the values and calculating, we find the future value of the investment to be \(FV=P+I=1,000+100.5=1100.5\). The future value of the investment at the end of 5 years is $1,100.50.
      Notice that the future value could have been calculated directly with \(FV=P+P\times r\times t\)
   2. The principal is \(P\) = $5,000, the annual interest rate, in decimal form, is 0.0185, and the term is 30 years, or \(t\) = 30. Since the term is an integer number of years, the interest earned on the investment is \(I=P\times r\times t=5,000\times 0.0185\times 30=2,775\), or the interest earned was $2,775.00. To find the future value, we use the formula \(FV=P+I\). Substituting the values and calculating, we find the future value of the investment to be \(FV=P+I=5,000+2,775=7,775\). The future value of the investment at the end of 30 years is $7,775.00.
   3. The principal is \(P\) = $10,000, the annual interest rate, in decimal form, is 0.0125, and the term is 18 months. Since the term is in months, we have to write the months in terms of years. For 18 months, we use 18/12 as \(t\). The interest earned on the investment is \(I=P\times r\times t=10,000\times 0.0125\times \frac{18}{12}=187.5\), or the interest earned was $187.50. To find the future value, we use the formula \(FV=P+I\). Substituting the values and calculating, we find the future value of the investment to be \(FV=P+I=10,000+187.5=10,187.5\). The future value of the investment at the end of 18 months is $10,187.50.
   4. Principal is $7,000, annual interest rate is 3.26%, and time is 100 days. The principal is \(P\) = $7,000, the annual interest rate, in decimal form, is 0.0326, and the term is 100 days. Since the term is in days, we have to write the time using actual/365, or \(t\) = 100/365. The interest earned on the investment is \(I=P\times r\times t=7,000\times 0.0326\times \frac{100}{365}=62.52\), or the interest earned was $62.52. To find the future value, we use the formula \(FV=P+I\). Substituting the values and calculating, we find the future value of the investment to be \(FV=P+I=7,000+62.52=7,062.52\). The future value of the investment at the end of 100 days is $7,062.52.

6. Jonas deposits $2,500 in a CD bearing 3.25% simple interest for a term of 3 years. When he redeems his CD at the end of the 3 years, how much will he receive?

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   This is a future value example. We know that \(P\) = $2,500 is the amount deposited. The annual simple interest rate in decimal form is \(r\) = 0.0325. The term of the investment is \(t\) = 3 years.

   Substituting those values into the future value formula, we have \(FV=P+P\times r\times t=2,500+2,500\times 0.0325\times 3=2,500+243.75=2,743.75\).

   When the CD is redeemed, Jonas will receive $2,743.75.

7. 1. A simple interest loan for $6,500 is taken out at 12.6% annual percentage rate. A partial payment is made 45 days into the loan period. How much of the partial payment will be for interest?
   2. A simple interest loan for $13,700 is taken out at 6.55% annual interest rate. A partial payment is to be made after 60 days. How much of the partial payment will be for interest?
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   1. To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the origination of the loan and the payment day, or 45 days.
      The principal is $6,500. The annual interest rate, in decimal form, is 0.126.
      The interest paid for 45 days is found by substituting the values for principal \(P\), rate \(r\), and time \(t\) into the formula \(I=P\times \frac{r}{365}\times t\).
      Calculating, we have \(I=P\times \frac{r}{365}\times t=6,500\times \frac{0.126}{365}\times 45=100.9726\). Rounding up, the portion of the partial payment that will be paid for interest is $100.98.
   2. To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the origination of the loan and the payment day, or 60 days.
      The principal is $13,700. The annual interest rate, in decimal form, is 0.0655.
      The interest paid for those 60 days is found by substituting those values into the formula \(I=P\times \frac{r}{365}\times t\). Calculating, we have \(I=P\times \frac{r}{365}\times t=13,700\times \frac{0.0655}{365}\times 60=147.5096\). Rounding up, the portion of the partial payment that will be paid for interest is $147.51.
8. 1. A simple interest loan for $45,500 is taken out at 11.8% annual percentage rate. A partial payment of $20,000 is made 50 days into the loan period. After this payment, what will the remaining balance of the loan be?
   2. A simple interest loan for $150,000 is taken out at 5.85% annual percentage rate. A partial payment of $50,000 is made 70 days into the loan period. After this payment, what will the remaining balance of the loan be?
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   1. The principal is $45,500, which will be treated as the balance, \(B\), of the loan. The annual simple interest rate, in decimal form, is 0.118. The time is \(t\) = 50 days.
      Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values above into the formula \(I=P\times \frac{r}{365}\times t\) and calculate. Calculating, the amount of the payment that is applied to interest is \(I=45,500\times \frac{0.118}{365}\times 50=735.4795\). Rounding up, we have \(I\) = $735.48.
      Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the amount of the partial payment that is applied to the interest. The payment is $2,000. The amount that is applied to the balance is \(P-I=\text{\$}20,000-\text{\$}735.48=\text{\$}19,264.52\).
      Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous balance, or \(B-(P-I)=\text{\$}45,500-\text{\$}19,264.52=\text{\$}26,235.48\).
      The remining balance after the partial payment is $26,235.48.
   2. The principal is $150,000, which will be treated as the balance, \(B\), of the loan. The annual simple interest rate, in decimal form, is 0.0585. The time is \(t\) = 70 days.
      Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values above into the formula \(I=P\times \frac{r}{365}\times t\) and calculate. Calculating, the amount of the payment that is applied to interest is \(I=150,000\times \frac{0.0585}{365}\times 70=1,682.8767\). Rounding up, we have \(I\) = $1,682.88.
      Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the amount of the partial payment that is applied to the interest. The payment is $50,000. The amount that is applied to the balance is \(P-I=\text{\$}50,000-\text{\$}1,682.88=\text{\$}48,317.12\).
      Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous balance, or \(B-(P-I)=\text{\$}150,000-\text{\$}48,317.12=\text{\$}101,682.88\).
      The remining balance after the partial payment is $101,682.88.

9. Laura takes out an $18,400 loan for 120 days at 17.9% simple interest. She makes a partial payment of $7,500 after 45 days. What is her payoff amount at the end of the loan?

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   The initial balance, or principal, of her loan is $18,400. The interest rate in decimal form is 0.179. Her partial payment of $7,500 is made after 45 days. Using these values, we can determine how much of the partial payment is applied to the balance. From there, we can determine her final loan payoff after 120 days.

   Step 1: Determine the remaining balance after the partial payment. Using the partial payment process outlined in the previous example, we first find that the amount of the partial payment that is applied to the balance. Their interest paid in the partial payment is \(I=P\times \frac{r}{365}\times t=18,400\times \frac{0.179}{365}\times 45=406.0603\), or $406.07 (remember to round up!). Using this and that the loan amount was for $18,400, the remaining balance on the loan after the partial payment is \(B-(P-I)=\text{\$}18,400-(\text{\$}7,500-\text{\$}406.07)=\text{\$}11,306.07\).

   Step 2: The number of days between the partial payment and the date that the loan is to be paid off is 120 – 45 = 75. This means that the time between the partial payment and the final payment is 75 days.

   Step 3: To calculate the payoff amount, use \(\text{payoff}=P+P\times \frac{r}{365}\times t\), with \(P\) = $11,306.07 (the remaining balance), \(t\) = 75 (from Step 2) and \(r\) = 0.179. The payoff amount, then, is \(\text{payoff}=11,306.07+11,306.07\times \frac{0.179}{365}\times 75=11,721.9165\). Rounding up, the payoff amount is $11,721.92.

10. Desiree buys a new car, by taking a loan out from her credit union. The balance of her loan is $27,845.00. The annual interest rate that Desiree will pay is 7.3%. She plans to pay this off over 4 years. How much will Desiree’s monthly payment be?

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    To use the formula for monthly payments, we need the principal, the interest rate, and the number of years. The principal is $27,845. The annual rate, in decimal form, is 0.073. Dividing 0.073 by 12 gives the monthly interest rate \(0.073/12=0.00608\overset{\bar}{3}\). She takes the loan out for 4 years, which is \(t=12\times 4=48\) months. Substituting these values into the formula, \(A=P\times \frac{r\times {(1+r)}^{t}}{{(1+r)}^{t}-1}\), we calculate:

    \[\begin{array}{lll}A & = & 27,845\times \frac{0.00608\overset{\bar}{3}\times {(1+0.00608\overset{\bar}{3})}^{48}}{{(1+0.00608\overset{\bar}{3})}^{48}-1} \\ & = & 27,845\times \frac{0.00608\overset{\bar}{3}\times {(1.00608\overset{\bar}{3})}^{48}}{{(1.00608\overset{\bar}{3})}^{48}-1} \\ & = & 27,845\times \frac{0.00608\overset{\bar}{3}\times 1.337918996}{1.337918996-1} \\ & = & 27,845\times \frac{0.008139007}{0.337918996} \\ & = & 27,845\times 0.024085675 \\ & = & 670.6656\end{array}\]

    Using the formula and rounding up to the next cent, we see that Desiree’s monthly payment will be $670.67.

11. Compute the present value of the investment described. Interpret the result.

    1. \(FV\) = $10,000, \(t\) = 15 years, annual simple interest rate of 5.5%
    2. \(FV\) = $150,000, \(t\) = 20 years, annual simple interest rate of 6.25%
    3. \(FV\) = $250,000, \(t\) = 486 months, annual simple interest rate of 4.75%
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    1. The future value is \(FV\) = $10,000. The time of the investment is in years, so \(t\) = 15. The annual, simple interest rate is 5.5%, which in decimal form is 0.055. We substitute those values into the formula and calculate. \(PV=\frac{FV}{(1+rt)}=\frac{10,000}{(1+(0.055)\times (15))}=\frac{10,000}{(1+0.825)}=\frac{10,000}{(1.825)}=5,479.452\). Rounding up, we see that the present value of $10,000 invested at a simple annual interest rate of 5.5% for 15 years is $5,479.46. This means that $5,479.46 needs to be invested so that, after 15 years at 5.5% interest, the investment will be worth $10,000.
    2. The future value is \(FV\) = $150,000. The time of the investment is in years, so \(t\) = 20. The annual, simple interest rate is 6.25%, which in decimal form is 0.0625. We substitute those values into the formula and calculate. \(PV=\frac{FV}{(1+rt)}=\frac{150,000}{(1+(0.0625)\times (20))}=\frac{150,000}{(1+1.25)}=\frac{150,000}{(2.25)}=66,666.\overset{\bar}{6}\). Rounding up, we see that the present value of $150,000 invested at a simple annual interest rate of 6.25% for 20 years is $66,666.67. This means that $66,666.67 needs to be invested so that, after 20 years at 6.25% interest, the investment will be worth $150,000.
    3. The future value is \(\text{FV}\) = $250,000. The time of the investment is 486 months. This needs to be converted to years. To do so, divide the number of months by 12, giving \(years=\frac{486}{12}=40.5\), so \(t\) = 40.5 years. The annual, simple interest rate is 4.75%, which in decimal form is 0.0475. We substitute those values into the formula and calculate. \(PV=\frac{FV}{(1+rt)}=\frac{250,000}{(1+(0.0475)\times (40.5))}=\frac{150,000}{(1+1.92375)}=\frac{150,000}{(2.92375)}=129,954.5159\). Rounding up, we see that the present value of $250,000 invested at a simple annual interest rate of 4.75% for 486 months is $129,954.52. This means that $129,954.52 needs to be invested so that, after 486 months at 4.75% interest, the investment will be worth $150,000.

12. Beatriz will invest some money in a CD that yields 3.99% simple interest when invested for 30 years. How much must Beatriz invest so that after those 30 years, her CD is worth $300,000?

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    Beatriz needs to know how much to deposit now so that her CD is worth $300,000 after 30 years. This means she needs to know the present value of that $300,000. The time is 30 years and the annual simple interest rate, in decimal form, is 0.0399. Using that information and the formula for present value, we calculate the present value of that $300,000. \(PV=\frac{FV}{(1+rt)}=\frac{300,000}{(1+(0.0399)\times (30))}=\frac{300,000}{(1+1.197)}=\frac{300,000}{(2.197)}=136,549.8407\). Rounding up, Beatriz needs to invest $136,549.85 so that she has $300,000 in 30 years.

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Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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