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Use the Simple Interest FormulaDo you know that banks pay you to let them keep your money? The money you put in the bank is called the principal, \(P\), and the bank pays you interest, \(I\). The interest is computed as a certain percent of the principal; called the rate of interest, \(r\). The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable for time, t, represents the number of years the money is left in the account. Definition: simple interestIf an amount of money, \(P\), the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, \(I\), earned is \[I = Prt \nonumber\] where
Interest earned according to this formula is called simple interest. The formula we use to calculate simple interest is I = Prt. To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information by listing all four variables and filling in the given information. Example \(\PageIndex{1}\)Find the simple interest earned after 3 years on $500 at an interest rate of 6%. Solution Organize the given information in a list. I = ?, P = $500, r = 6%, t = 3 years We will use the simple interest formula to find the interest.
Exercise \(\PageIndex{1}\):Find the simple interest earned after 4 years on $800 at an interest rate of 5%. Answer$160 Exercise \(\PageIndex{2}\):Find the simple interest earned after 2 years on $700 at an interest rate of 4%. Answer$56 In the next example, we will use the simple interest formula to find the principal. Example \(\PageIndex{2}\):Find the principal invested if $178 interest was earned in 2 years at an interest rate of 4%. Solution Organize the given information in a list. I = $178, P = ?, r = 4%, t = 2 years We will use the simple interest formula to find the principal.
Exercise \(\PageIndex{3}\):Find the principal invested if $495 interest was earned in 3 years at an interest rate of 6%. Answer$2,750 Exercise \(\PageIndex{4}\)Find the principal invested if $1,246 interest was earned in 5 years at an interest rate of 7%. Answer$3,560 Now we will solve for the rate of interest. Example \(\PageIndex{3}\)Find the rate if a principal of $8,200 earned $3,772 interest in 4 years. Solution Organize the given information. I = $3,772, P = $8,200, r = ?, t = 4 years We will use the simple interest formula to find the rate.
Exercise \(\PageIndex{5}\):Find the rate if a principal of $5,000 earned $1,350 interest in 6 years. Answer4.5% Exercise \(\PageIndex{6}\):Find the rate if a principal of $9,000 earned $1,755 interest in 3 years. 6.5% Solve Simple Interest ApplicationsApplications 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 start by solving a simple interest application to find the interest. Example \(\PageIndex{4}\):Nathaly deposited $12,500 in her bank account where it will earn 4% interest. How much interest will Nathaly earn in 5 years? Solution We are asked to find the Interest, I. Organize the given information in a list. I = ?, P = $12,500, r = 4%, t = 5 years
Exercise \(\PageIndex{7}\):Areli invested a principal of $950 in her bank account with interest rate 3%. How much interest did she earn in 5 years? Answer$142.50 Exercise \(\PageIndex{8}\):Susana invested a principal of $36,000 in her bank account with interest rate 6.5%. How much interest did she earn in 3 years? Answer$7,020 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 \(\PageIndex{5}\):Loren lent his brother $3,000 to help him buy a car. In 4 years his brother paid him back the $3,000 plus $660 in interest. What was the rate of interest? Solution We are asked to find the rate of interest, r. Organize the given information. I = 660, P = $3,000, r = ?, t = 4 years
Exercise \(\PageIndex{9}\):Jim lent his sister $5,000 to help her buy a house. In 3 years, she paid him the $5,000, plus $900 interest. What was the rate of interest? 6% Exercise \(\PageIndex{10}\):Hang borrowed $7,500 from her parents to pay her tuition. In 5 years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of interest? Answer4% 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 \(\PageIndex{6}\):Eduardo noticed that his new car loan papers stated that with an interest rate of 7.5%, he would pay $6,596.25 in interest over 5 years. How much did he borrow to pay for his car? Solution We are asked to find the principal, P. Organize the given information. I = 6,596.25, P = ?, r = 7.5%, t = 5 years
Exercise \(\PageIndex{11}\):Sean's new car loan statement said he would pay $4,866.25 in interest from an interest rate of 8.5% over 5 years. How much did he borrow to buy his new car? Answer$11,450 Exercise \(\PageIndex{12}\):In 5 years, Gloria's bank account earned $2,400 interest at 5%. How much had she deposited in the account? Answer$9,600 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 \(\PageIndex{7}\):Caroline got $900 as graduation gifts and invested it in a 10-month certificate of deposit that earned 2.1% interest. How much interest did this investment earn? Solution We are asked to find the interest, I. Organize the given information. I = ?, P = $900, r = 2.1%, t = 10 months
Exercise \(\PageIndex{13}\):Adriana invested $4,500 for 8 months in an account that paid 1.9% interest. How much interest did she earn? Answer$57.00 Exercise \(\PageIndex{14}\):Milton invested $2,460 for 20 months in an account that paid 3.5% interest How much interest did he earn? Answer$143.50 Practice Makes PerfectUse the Simple Interest FormulaIn the following exercises, use the simple interest formula to fill in the missing information.
In the following exercises, solve the problem using the simple interest formula.
Solve Simple Interest ApplicationsIn the following exercises, solve the problem using the simple interest formula.
Everyday Math
Writing Exercises
Self Check(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. (b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? Contributors and Attributions
What is compound interest on an amount of Rs 10000 at 8% per annum for 2 years?Solution. ∴ Compound interest is ₹ 1698.58.
How much is $1000 worth at the end of 2 years if the interest rate of 6% is compounded daily?Compound interest formulas
Hence, if a two-year savings account containing $1,000 pays a 6% interest rate compounded daily, it will grow to $1,127.49 at the end of two years.
In what time will 10000 double itself at the rate of 10% simple interest?Hence, it will take 10 years for the sum of money to double itself with the rate of 10% per annum simple interest.
How long will it take money to triple itself if invested at 8% compounded annually?Answer: Approximately 13.5 years to triple.
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