Simple Interest Formula and Calculations - Seventh Grade Mathematics

The simple interest formula models the fundamental economic cost of borrowing money and the financial reward of investing capital over time.

The Simple Interest Formula: I = Prt

When money is borrowed from a financial institution or deposited into a savings bond, a fee is charged or earned for the use of those funds over time. This fee is known as interest. Simple interest is calculated solely on the original starting balance, known as the principal.

               THE CANONICAL SIMPLE INTEREST FORMULA
                             I = P * r * t
   I = Interest earned or paid (in currency dollars)
   P = Principal (initial sum borrowed or invested)
   r = Annual interest rate (expressed as a decimal)
   t = Time (measured strictly in YEARS)

Every variable in this formula must adhere to strict mathematical definitions. The interest rate r is an annual rate; a five percent rate must be entered as zero point zero five. Time t must always be expressed in years.

Converting Time Units to Fractional Years

Financial transactions frequently span durations other than whole years. When the time is given in months or days, it must be converted into a fractional or decimal year before substitution into I = Prt.

+-----------------------+-----------------------------+------------------------+
| Given Time Duration   | Conversion to Years (t)     | Decimal Equivalent     |
+-----------------------+-----------------------------+------------------------+
| 6 months              | 6 / 12 year                 | 0.5 year               |
| 3 months              | 3 / 12 = 1 / 4 year         | 0.25 year              |
| 18 months             | 18 / 12 = 3 / 2 years       | 1.5 years              |
| 9 months              | 9 / 12 = 3 / 4 year         | 0.75 year              |
| 73 days (ordinary)    | 73 / 365 year               | 0.2 year               |
+-----------------------+-----------------------------+------------------------+

Substituting eighteen directly into t instead of one point five would result in twelve times too much interest being calculated.

Total Accrued Balance: A = P + I

When a loan matures or an investment term ends, the total monetary value, designated by the balance A, equals the initial principal plus the accumulated interest.

               TOTAL BALANCE FORMULAS
                      A = P + I
                      A = P + (P * r * t) = P * (1 + r * t)

Consider depositing two thousand dollars into a certificate of deposit earning four percent simple annual interest for three years. Step 1: Compute interest: I equals two thousand times zero point zero four times three. Two thousand times zero point zero four equals eighty; eighty times three equals two hundred forty dollars in interest. Step 2: Compute total balance: A equals two thousand plus two hundred forty equals two thousand two hundred forty dollars.

Total Accrued: $2,000 + $240 = $2,240.00

At the end of three years, the account holds two thousand two hundred forty dollars.

Solving for Unknown Variables in the Interest Formula

Algebraic manipulation allows you to solve for any missing variable in the simple interest formula when the other three variables are known.

+-----------------------+-----------------------------+------------------------+
| Desired Unknown       | Algebraic Formula           | Operation Required     |
+-----------------------+-----------------------------+------------------------+
| Principal (P)         | P = I / (r * t)             | Divide I by (r * t)    |
| Interest Rate (r)     | r = I / (P * t)             | Divide I by (P * t)    |
| Time in Years (t)     | t = I / (P * r)             | Divide I by (P * r)    |
+-----------------------+-----------------------------+------------------------+

Suppose an investor earns one hundred fifty dollars in simple interest on an investment of one thousand dollars over three years. What was the annual interest rate? Use the rate formula: r equals I divided by the product of P and t. Substitute: r equals one hundred fifty divided by the quantity one thousand times three. One thousand times three equals three thousand. One hundred fifty divided by three thousand equals zero point zero five. Convert to a percent: five percent annual interest rate.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Calculate the simple interest earned on an investment of five thousand dollars at an annual interest rate of six percent over a period of four years.

Exercise 2. A consumer borrows one thousand two hundred dollars on a short-term personal loan with a simple annual interest rate of nine percent for eight months. Calculate the interest owed and the total amount required to pay off the loan.

Exercise 3. An entrepreneur invests three thousand dollars into a business note that pays simple interest. After two years, the note produces two hundred forty dollars in interest. What was the annual interest rate of the note?

Exercise 4. How many years will it take for a principal deposit of four thousand dollars to earn eight hundred dollars in simple interest at an annual interest rate of five percent?

Solutions

Solution to Exercise 1. Identify the given values: principal P equals five thousand dollars, annual rate r equals six percent (0.06), and time t equals four years. Substitute into I equals P times r times t: I equals five thousand times zero point zero six times four. Five thousand times zero point zero six equals three hundred; three hundred times four equals one thousand two hundred dollars. The simple interest earned is one thousand two hundred dollars.

Solution to Exercise 2. Principal P is one thousand two hundred dollars, and rate r is nine percent (0.09). Convert eight months into years: t equals eight-twelfths of a year, which simplifies to two-thirds of a year. Substitute into I equals P times r times t: I equals one thousand two hundred times zero point zero nine times two-thirds. One thousand two hundred times zero point zero nine equals one hundred eight. One hundred eight times two-thirds equals seventy-two dollars in interest. To find the total payoff balance, add principal and interest: one thousand two hundred plus seventy-two equals one thousand two hundred seventy-two dollars ($1,272.00).

Solution to Exercise 3. The unknown is the interest rate r. Use the formula r equals I divided by the product of P and t. Substitute interest I equals two hundred forty, principal P equals three thousand, and time t equals two: r equals two hundred forty divided by the quantity three thousand times two. Three thousand times two equals six thousand. Divide two hundred forty by six thousand: two hundred forty divided by six thousand equals zero point zero four. Multiply by one hundred: the annual interest rate was four percent (4%).

Solution to Exercise 4. The unknown is time t. Use the formula t equals I divided by the product of P and r. Substitute interest I equals eight hundred, principal P equals four thousand, and rate r equals five percent (0.05): t equals eight hundred divided by the quantity four thousand times zero point zero five. Four thousand times zero point zero five equals two hundred. Divide eight hundred by two hundred: eight hundred divided by two hundred equals four. It will take four years to accumulate eight hundred dollars in interest.