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Mathematics LibreTexts

2.2: Simple Interest

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Learning Objectives

In this section, you will learn to:

  • Use the simple interest formula to calculate the future value of a lump sum

So that they can analyze financial scenarios to make informed decisions.

Discussing interest starts with the principal, or the amount your account starts with. This could be a starting investment or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05)=$5. The total amount you would repay would be $105, the original principal plus the interest.

Simple One-time Interest

I=P0r

A=P0+I=P0+P0r=P0(1+r)

where

  • I is the interest
  • A is the end amount: principal plus interest
  • P0 is the principal (starting amount)
  • r is the interest rate (in decimal form. Example: 5%=0.05)

Example  2.2.1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?

Solution

P0=$300the principal r=0.033% rateI=$300(0.03)=$9.You will earn $9 interest.

You try it 2.2.1

One-time simple interest is only common for extremely short-term loans. For longer-term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bondholder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.

Example  2.2.2

Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?

Solution

Each year, you would earn 5% interest: $1000(0.05)=$50 in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.

We can generalize this idea of simple interest over time.

Simple Interest over Time

I=P0rt

A=P0+I=P0+P0rt=P0(1+rt)

where

  • I is the interest
  • A is the end amount: principal plus interest
  • P0 is the principal (starting amount)
  • r is the interest rate in decimal form
  • t is time

The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.

APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.
A 4% annual rate paid quarterly would be divided into four 1% payments.

Example  2.2.3: Treasury Notes

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

Solution

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.

P0=$1000the principal r=0.022% ratet=84 years = 8 half-yearsI=$1000(0.02)(8)=$160.You will earn $160 interest total over the four years.

 

You try it 2.2.2

This page titled 2.2: Simple Interest is shared under a CC BY-SA license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) .

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