6.1 Simple Interest and Discount
Why does this section matter? Simple interest shows up in everyday financial decisions: short-term loans, overdue credit card balances, bank notes, and cash discounts. If you understand how money grows, how much interest you are actually paying, and how much cash you really receive from a discounted loan, you can make smarter borrowing decisions.
- Find simple interest.
- Find present value.
- Find discounts and proceeds.
Simple interest is one of the most important starting points in finance because it is predictable. The interest grows in a straight-line way: double the time, and you double the interest.
SIMPLE INTEREST
It costs money to borrow money. The rent you pay for the use of money is called interest. The amount of money that is borrowed or loaned is called the principal or present value. With simple interest, interest is paid only on the original amount borrowed. The borrower pays a fixed rate of interest on the principal for the time the money is used.
Although the interest rate is often stated per year, it may also be stated per week, per month, or per quarter. For example, credit card companies often list charges as monthly rates. A finance charge of \(1.5\%\) per month is not unusual.
The principal, or present value, is the original amount of money borrowed or invested.
Simple interest is interest computed only on the original principal.
Simple Interest Formulas
If an amount \(P\) is borrowed for a time \(t\) at an interest rate of \(r\) per time period, then the simple interest is
$$ I = Prt $$The total amount \(A\), also called the accumulated value or future value, is
$$ A = P + I = P + Prt $$or equivalently,
$$ A = P(1 + rt) $$where the interest rate \(r\) is written as a decimal.
These formulas let you answer two very practical questions: “How much extra will I pay?” and “How much will I owe altogether?” That is exactly what borrowers, lenders, and anyone using a credit card need to know.
Finding the Interest and Future Value
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.
Example 6.1.1 Solution
The interest is computed by multiplying the principal, the interest rate, and the time.
$$ I = Prt $$ $$ I = 600(0.15)\left(\frac{5}{12}\right) = 37.50 $$So the interest is \$37.50.
The total amount is
$$ A = P + I = 600 + 37.50 = 637.50 $$Incidentally, the total amount can also be computed directly:
$$ \begin{aligned} A &= P(1 + rt) \\ &= 600\left[1 + (0.15)\left(\frac{5}{12}\right)\right] \\ &= 600(1 + 0.0625) \\ &= 637.50 \end{aligned} $$Find the simple interest and future value if \$1800 is borrowed for 8 months at \(9\%\) per year.
Try It Now 6.1.1 Solution
Use \(I = Prt\) with \(P = 1800\), \(r = 0.09\), and \(t = \frac{8}{12}\).
$$ I = 1800(0.09)\left(\frac{8}{12}\right) = 108 $$Then
$$ A = P + I = 1800 + 108 = 1908 $$So the interest is \$108 and the future value is \$1908.
Jose deposited \$2500 in an account that pays \(6\%\) simple interest. How much money will he have at the end of 3 years?
Example 6.1.2 Solution
The total amount, or future value, is given by \(A = P(1 + rt)\).
$$ \begin{aligned} A &= 2500[1 + (0.06)(3)] \\ &= 2500(1.18) \\ &= 2950 \end{aligned} $$Jose will have \$2950 at the end of 3 years.
Notice how simple interest adds the same dollar amount each year because the interest is always based on the original principal, not on a growing balance.
Finding Present Value
Sometimes you know the total amount owed and want to work backward to find how much was originally borrowed. In that case, you solve the formula
$$ A = P(1 + rt) $$for \(P\).
This is useful when you see a final bill or payoff amount and want to know how much of it came from the original loan versus the interest charge.
Darnel owes a total of \$3060, which includes \(12\%\) interest for the three years he borrowed the money. How much did he originally borrow?
Example 6.1.3 Solution
This time we are asked to compute the principal \(P\).
$$ \begin{aligned} 3060 &= P[1 + (0.12)(3)] \\ 3060 &= P(1.36) \\ P &= \frac{3060}{1.36} \\ P &= 2250 \end{aligned} $$Darnel originally borrowed \$2250.
A total payoff of \$4240 includes simple interest at \(6\%\) for 2 years. Find the original principal.
Try It Now 6.1.2 Solution
Use \(A = P(1 + rt)\).
$$ 4240 = P[1 + (0.06)(2)] = P(1.12) $$ $$ P = \frac{4240}{1.12} = 3785.71 $$The original principal was \$3785.71.
A Visa credit card company charges a \(1.5\%\) 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?
Example 6.1.4 Solution
In this problem, the finance charge is given per month, not per year. So we use \(r = 0.015\) and \(t = 3\) months.
The total amount Martha owes is the previous unpaid balance plus the finance charge.
$$ A = 2350 + 2350(0.015)(3) = 2350 + 105.75 = 2455.75 $$Alternatively, we can compute the amount directly using \(A = P(1 + rt)\):
$$ A = 2350[1 + (0.015)(3)] = 2350(1.045) = 2455.75 $$So Martha now owes \$2455.75.
DISCOUNTS AND PROCEEDS
Banks often deduct the simple interest from the loan amount at the time the loan is made. When this happens, the loan is said to be discounted. The interest deducted up front is called the discount, and the actual amount given to the borrower is called the proceeds. The amount the borrower must repay at the end of the loan is called the maturity value.
A discounted loan can feel a little surprising: you agree to repay one amount, but you do not actually receive that full amount in cash. That is why it is important to distinguish between the maturity value and the proceeds.
Discounted Loan Terminology
The interest deducted at the beginning of the loan is called the discount.
The amount the borrower actually receives is called the proceeds.
The total amount the borrower must repay at the end of the loan is called the maturity value.
Discount and Proceeds Formulas
If an amount \(M\) is borrowed for a time \(t\) at a discount rate of \(r\) per year, then the discount \(D\) is
$$ D = Mrt $$The proceeds \(P\), the actual amount the borrower gets, are given by
$$ P = M - D $$ $$ P = M - Mrt $$or
$$ P = M(1 - rt) $$where the interest rate \(r\) is written as a decimal.
In a simple-interest loan, you receive the principal first and pay the interest later. In a discounted loan, the interest is taken out first, so the cash you receive is smaller than the amount you must eventually repay.
Francisco borrows \$1200 for 10 months at a simple interest rate of \(15\%\) per year. Determine the discount and the proceeds.
Example 6.1.5 Solution
The discount \(D\) is the interest on the loan that the bank deducts from the loan amount.
$$ D = Mrt $$ $$ D = 1200(0.15)\left(\frac{10}{12}\right) = 150 $$Therefore, the bank deducts \$150 from the maturity value of \$1200 and gives Francisco \$1050. Francisco is obligated to repay the bank \$1200.
In this case, the discount is \$150, and the proceeds are
$$ P = 1200 - 150 = 1050 $$An amount of \$3000 is borrowed at a discount rate of \(8\%\) for 9 months. Find the discount and the proceeds.
Try It Now 6.1.3 Solution
Use \(D = Mrt\) with \(M = 3000\), \(r = 0.08\), and \(t = \frac{9}{12}\).
$$ D = 3000(0.08)\left(\frac{9}{12}\right) = 180 $$Then
$$ P = M - D = 3000 - 180 = 2820 $$So the discount is \$180 and the proceeds are \$2820.
If Francisco wants to receive \$1200 for 10 months at a simple interest rate of \(15\%\) per year, what amount of loan should he apply for?
Example 6.1.6 Solution
In this problem, we are given the proceeds \(P\) and are being asked to find the maturity value \(M\).
We have \(P = 1200\), \(r = 0.15\), and \(t = \frac{10}{12}\). We need to find \(M\).
We know
$$ P = M - D $$but also
$$ D = Mrt $$therefore,
$$ P = M - Mrt = M(1 - rt) $$Substitute the known values:
$$ 1200 = M\left[1 - (0.15)\left(\frac{10}{12}\right)\right] $$Now simplify and solve for \(M\).
$$ 1200 = M(1 - 0.125) $$ $$ 1200 = M(0.875) $$ $$ M = \frac{1200}{0.875} $$ $$ M = 1371.43 $$Therefore, Francisco should ask for a loan of \$1371.43.
The bank will discount \$171.43, and Francisco will receive \$1200.
Suppose you want proceeds of \$5000 from a discounted loan for 1 year at a discount rate of \(10\%\). What maturity value should you request?
Try It Now 6.1.4 Solution
Use \(P = M(1 - rt)\) with \(P = 5000\), \(r = 0.10\), and \(t = 1\).
$$ 5000 = M(1 - 0.10)(1) = 0.90M $$ $$ M = \frac{5000}{0.90} = 5555.56 $$The maturity value should be \$5555.56.
SECTION 6.1 SUMMARY
Below is a summary of the formulas developed in this section for calculations involving simple interest.
Simple Interest Summary
If an amount \(P\) is borrowed for a time \(t\) at an interest rate \(r\) per time period, then the simple interest is
$$ I = Prt $$The total amount \(A\), also called the accumulated value or future value, is
$$ A = P + I = P + Prt $$ $$ A = P(1 + rt) $$where the interest rate \(r\) is expressed in decimals.
Discount and Proceeds Summary
If an amount \(M\) is borrowed for a time \(t\) at a discount rate \(r\) per year, then the discount is
$$ D = Mrt $$The proceeds \(P\), the actual amount the borrower receives at the time the money is borrowed, are given by
$$ P = M - D $$ $$ P = M - Mrt $$ $$ P = M(1 - rt) $$where the interest rate \(r\) is expressed in decimals.
At the end of the loan's term, the borrower repays the entire maturity value \(M\).
The big idea of this section is that time, rate, and principal work together in predictable ways. Once you know which quantity is missing, you can choose the correct formula and solve directly.
SECTION 6.1 PROBLEM SET: SIMPLE INTEREST AND DISCOUNT
These problems let you practice the three main skills from this section: finding simple interest, finding total amount owed, and working with discounted loans. As you solve, keep asking yourself: “Am I finding interest, present value, discount, or proceeds?”
Do the following simple interest problems.
1) If an amount of \$2,000 is borrowed at a simple interest rate of 10% for 3 years, how much is the interest?
Problem 1 Solution
Step 1: Use the simple interest formula.
$$ I = Prt $$Step 2: Substitute \(P = 2000\), \(r = 0.10\), and \(t = 3\).
$$ I = 2000(0.10)(3) $$Step 3: Compute the interest.
$$ I = 600 $$Verification:
$$ A = P + I = 2000 + 600 = 2600 $$So the interest amount is consistent.
Answer: \(\$600\)
2) You borrow \$4,500 for six months at a simple interest rate of 8%. How much is the interest?
Problem 2 Solution
Step 1: Use the simple interest formula.
$$ I = Prt $$Step 2: Convert 6 months to years.
$$ t = \frac{6}{12} = \frac{1}{2} $$Step 3: Substitute \(P = 4500\), \(r = 0.08\), and \(t = \frac{1}{2}\).
$$ I = 4500(0.08)\left(\frac{1}{2}\right) $$Step 4: Compute.
$$ I = 180 $$Verification:
$$ A = 4500 + 180 = 4680 $$Answer: \(\$180\)
3) John borrows \$2400 for 3 years at 9% simple interest. How much will he owe at the end of 3 years?
Problem 3 Solution
Step 1: Find the interest.
$$ I = Prt = 2400(0.09)(3) $$ $$ I = 648 $$Step 2: Add the interest to the principal.
$$ A = P + I = 2400 + 648 = 3048 $$Verification: Use \(A = P(1+rt)\).
$$ A = 2400[1 + (0.09)(3)] = 2400(1.27) = 3048 $$Answer: \(\$3048\)
4) Jessica takes a loan of \$800 for 4 months at 12% simple interest. How much does she owe at the end of the 4-month period?
Problem 4 Solution
Step 1: Convert 4 months to years.
$$ t = \frac{4}{12} = \frac{1}{3} $$Step 2: Find the interest.
$$ I = Prt = 800(0.12)\left(\frac{1}{3}\right) $$ $$ I = 32 $$Step 3: Find the total amount owed.
$$ A = P + I = 800 + 32 = 832 $$Verification:
$$ A = 800\left[1 + (0.12)\left(\frac{1}{3}\right)\right] = 800(1.04) = 832 $$Answer: \(\$832\)
5) If an amount of \$2,160, which includes a 10% simple interest for 2 years, is paid back, how much was borrowed 2 years earlier?
Problem 5 Solution
Step 1: Use the future value formula.
$$ A = P(1 + rt) $$Step 2: Substitute \(A = 2160\), \(r = 0.10\), and \(t = 2\).
$$ 2160 = P[1 + (0.10)(2)] $$ $$ 2160 = 1.2P $$Step 3: Solve for \(P\).
$$ P = \frac{2160}{1.2} = 1800 $$Verification:
$$ A = 1800[1 + (0.10)(2)] = 1800(1.2) = 2160 $$Answer: \(\$1800\)
6) Jamie just paid off a loan of \$2,544, the principal and simple interest. If he took out the loan six months ago at 12% simple interest, what was the amount borrowed?
Problem 6 Solution
Step 1: Use the future value formula.
$$ A = P(1 + rt) $$Step 2: Convert 6 months to years.
$$ t = \frac{6}{12} = \frac{1}{2} $$Step 3: Substitute \(A = 2544\), \(r = 0.12\), and \(t = \frac{1}{2}\).
$$ 2544 = P\left[1 + (0.12)\left(\frac{1}{2}\right)\right] $$ $$ 2544 = 1.06P $$Step 4: Solve for \(P\).
$$ P = \frac{2544}{1.06} = 2400 $$Verification:
$$ A = 2400\left[1 + (0.12)\left(\frac{1}{2}\right)\right] = 2400(1.06) = 2544 $$Answer: \(\$2400\)
7) Shanti charged \$800 on her charge card and did not make a payment for six months. If there is a monthly charge of 1.5%, how much does she owe?
Problem 7 Solution
Step 1: Since the rate is monthly, use \(r = 0.015\) per month and \(t = 6\) months.
Step 2: Find the finance charge.
$$ I = Prt = 800(0.015)(6) $$ $$ I = 72 $$Step 3: Add the charge to the unpaid balance.
$$ A = P + I = 800 + 72 = 872 $$Verification:
$$ A = 800[1 + (0.015)(6)] = 800(1.09) = 872 $$Answer: \(\$872\)
8) A credit card company charges 18% interest on the unpaid balance. If you owed \$2000 three months ago and have been delinquent since, how much do you owe?
Problem 8 Solution
Step 1: Convert 3 months to years.
$$ t = \frac{3}{12} = \frac{1}{4} $$Step 2: Find the interest.
$$ I = Prt = 2000(0.18)\left(\frac{1}{4}\right) $$ $$ I = 90 $$Step 3: Find the total amount owed.
$$ A = P + I = 2000 + 90 = 2090 $$Verification:
$$ A = 2000\left[1 + (0.18)\left(\frac{1}{4}\right)\right] = 2000(1.045) = 2090 $$Answer: \(\$2090\)
9) An amount of \$2000 is borrowed for 3 years. At the end of the three years, \$2660 is paid back. What was the simple interest rate?
Problem 9 Solution
Step 1: Use the future value formula.
$$ A = P(1 + rt) $$Step 2: Substitute \(A = 2660\), \(P = 2000\), and \(t = 3\).
$$ 2660 = 2000(1 + 3r) $$Step 3: Divide by 2000.
$$ 1.33 = 1 + 3r $$Step 4: Solve for \(r\).
$$ 0.33 = 3r $$ $$ r = 0.11 $$Verification:
$$ A = 2000[1 + (0.11)(3)] = 2000(1.33) = 2660 $$Answer: \(11\%\)
10) Nancy borrowed \$1,800 and paid back \$1,920, four months later. What was the simple interest rate?
Problem 10 Solution
Step 1: Use the future value formula.
$$ A = P(1 + rt) $$Step 2: Convert 4 months to years.
$$ t = \frac{4}{12} = \frac{1}{3} $$Step 3: Substitute \(A = 1920\), \(P = 1800\), and \(t = \frac{1}{3}\).
$$ 1920 = 1800\left(1 + \frac{r}{3}\right) $$Step 4: Divide by 1800.
$$ \frac{1920}{1800} = 1 + \frac{r}{3} $$ $$ \frac{16}{15} = 1 + \frac{r}{3} $$Step 5: Solve for \(r\).
$$ \frac{1}{15} = \frac{r}{3} $$ $$ r = \frac{3}{15} = 0.20 $$Verification:
$$ A = 1800\left[1 + (0.20)\left(\frac{1}{3}\right)\right] = 1800\left(\frac{16}{15}\right) = 1920 $$Answer: \(20\%\)
11) Jose agrees to pay \$2,000 in one year at an interest rate of 12%. The bank subtracts the discount of 12% of \$2,000, and gives the rest to Jose. Find the amount of the discount and the proceeds to Jose.
Problem 11 Solution
Step 1: Use the discount formula.
$$ D = Mrt $$Step 2: Substitute \(M = 2000\), \(r = 0.12\), and \(t = 1\).
$$ D = 2000(0.12)(1) $$ $$ D = 240 $$Step 3: Find the proceeds.
$$ P = M - D = 2000 - 240 = 1760 $$Verification:
$$ P = M(1 - rt) = 2000(1 - 0.12) = 2000(0.88) = 1760 $$Answer: Discount \(= \$240\); proceeds \(= \$1760\)
12) Tasha signs a note for a discounted loan agreeing to pay \$1200 in 8 months at an 18% discount rate. Determine the amount of the discount and the proceeds to her.
Problem 12 Solution
Step 1: Convert 8 months to years.
$$ t = \frac{8}{12} = \frac{2}{3} $$Step 2: Find the discount.
$$ D = Mrt = 1200(0.18)\left(\frac{2}{3}\right) $$ $$ D = 144 $$Step 3: Find the proceeds.
$$ P = M - D = 1200 - 144 = 1056 $$Verification:
$$ P = 1200\left[1 - (0.18)\left(\frac{2}{3}\right)\right] = 1200(0.88) = 1056 $$Answer: Discount \(= \$144\); proceeds \(= \$1056\)
13) An amount of \$8,000 is borrowed at a discount rate of 12%, find the proceeds if the length of the loan is 7 months.
Problem 13 Solution
Step 1: Convert 7 months to years.
$$ t = \frac{7}{12} $$Step 2: Find the discount.
$$ D = Mrt = 8000(0.12)\left(\frac{7}{12}\right) $$ $$ D = 560 $$Step 3: Find the proceeds.
$$ P = M - D = 8000 - 560 = 7440 $$Verification:
$$ P = 8000\left[1 - (0.12)\left(\frac{7}{12}\right)\right] = 8000(0.93) = 7440 $$Answer: \(\$7440\)
14) An amount of \$4,000 is borrowed at a discount rate of 10%, find the proceeds if the length of the loan is 180 days.
Problem 14 Solution
Step 1: Convert 180 days to years using a 360-day year.
$$ t = \frac{180}{360} = \frac{1}{2} $$Step 2: Find the discount.
$$ D = Mrt = 4000(0.10)\left(\frac{1}{2}\right) $$ $$ D = 200 $$Step 3: Find the proceeds.
$$ P = M - D = 4000 - 200 = 3800 $$Verification:
$$ P = 4000\left[1 - (0.10)\left(\frac{1}{2}\right)\right] = 4000(0.95) = 3800 $$Answer: \(\$3800\)
15) Derek needs \$2400 new equipment for his shop. He can borrow this money at a discount rate of 14% for a year. Find the amount of the loan he should ask for so that his proceeds are \$2400.
Problem 15 Solution
Step 1: Use the proceeds formula.
$$ P = M(1 - rt) $$Step 2: Substitute \(P = 2400\), \(r = 0.14\), and \(t = 1\).
$$ 2400 = M(1 - 0.14) $$ $$ 2400 = 0.86M $$Step 3: Solve for \(M\).
$$ M = \frac{2400}{0.86} = \frac{120000}{43} \approx 2790.70 $$Verification:
$$ P = \left(\frac{120000}{43}\right)(0.86) = 2400 $$Answer: He should ask for a loan of approximately \(\$2790.70\).
16) Mary owes Jim \$750, and wants to repay him. Mary decides to borrow the amount from her bank at a discount rate of 16%. If she borrows the money for 10 months, find the amount of the loan she should ask for so that her proceeds are \$750?
Problem 16 Solution
Step 1: Use the proceeds formula.
$$ P = M(1 - rt) $$Step 2: Convert 10 months to years.
$$ t = \frac{10}{12} = \frac{5}{6} $$Step 3: Substitute \(P = 750\), \(r = 0.16\), and \(t = \frac{5}{6}\).
$$ 750 = M\left[1 - (0.16)\left(\frac{5}{6}\right)\right] $$ $$ 750 = M\left(1 - \frac{2}{15}\right) $$ $$ 750 = M\left(\frac{13}{15}\right) $$Step 4: Solve for \(M\).
$$ M = 750\left(\frac{15}{13}\right) = \frac{11250}{13} \approx 865.38 $$Verification:
$$ P = \left(\frac{11250}{13}\right)\left(\frac{13}{15}\right) = 750 $$Answer: She should ask for a loan of approximately \(\$865.38\).