Simple Interest: Principal, Rate, Time

Simple Interest

In a simple interest environment, you calculate interest solely on the amount of money at the beginning of the transaction (amount borrowed or lent).

Assume $1,000 is placed into an account with 12% simple interest for a period of 12 months. For the entire term of this transaction, the amount of money in the account always equals $1,000. During this period, interest accrues at a rate of 12%, but the interest is never placed into the account. When the transaction ends after 12 months, the $120 of interest and the initial $1,000 are then combined to total $1,120.

A loan or investment always involves two parties—one giving and one receiving. No matter which party you are in the transaction, the amount of interest remains unchanged. The only difference lies in whether you are earning or paying the interest.

The Formula

[latex]\colorbox{LightGray}{Formula 8.1}\; \color{BlueViolet}{\text{Simple Interest:}\; I=Prt}[/latex]

where,

I is Interest Amount. The interest amount is the dollar amount of interest that is paid or received.

P is Present Value or Principal. The present value is the amount borrowed or invested at the beginning of a period.

r is Simple Interest Rate. The interest rate is the rate of interest that is charged or earned during a specified time period. It is expressed as a percent.

t is Time Period. The time period or term is the length of the financial transaction for which interest is charged or earned.

Example 8.1.1: How Much Interest is Owed? 

Julio borrowed $1,100 from Maria five months ago. When he first borrowed the money, they agreed that he would pay Maria 5% simple interest. If Julio pays her back today, how much interest does he owe her?

Solution:

Step 1: Given information:

P = $1,100; r = 5%; per year; t = 5 months

Step 2: The rate is annual, and the time is in months. Convert the time period from months to years; [latex]t=\frac{5}{12}[/latex]

 

Step 3: Solve for the amount of interest, I.

[latex]\begin{align} I&=Prt\\ &=\$1,\!100 \times 5\% \times \frac{5}{12}\\ &=\$1,\!100 \times 0.05 \times 0.41\overline{6}\\ &=\$22.92 \end{align}[/latex]

For Julio to pay back Maria, he must reimburse her for the $1,100 principal borrowed plus an additional $22.92 of simple interest as per their agreement.

Solving for P, r or t

Four variables are involved in the simple interest formula, which means that any three can be known, requiring you to solve for the fourth missing variable. To reduce formula clutter, the triangle technique illustrated in the video below will help you remember how to rearrange the simple interest formula as needed.

Example 8.1.2: What did You Start With?

What amount of money invested at 6% annual simple interest for 11 months earns $2,035 of interest?

Solution:

Step 1: Given information:

r = 6%; t = 11 months; I = $2,035

Step 2: Convert the time from months to an annual basis to match the interest rate; [latex]t=\frac{11}{12}[/latex]

 

Step 3: Solve for the present value, P.

[latex]\begin{align} P&=\frac{I}{rt}\\ &=\frac{\$2,\!035}{6\% \times \frac{11}{12}}\\ &=\frac{\$2,\!035}{0.06 \times 0.91\overline{6}}\\ &=\$37,\!000 \end{align}[/latex]

To generate $2,035 of simple interest at 6% over a time frame of 11 months, $37,000 must be invested.

Example 8.1.3: How long?

For how many months must $95,000 be invested to earn $1,187.50 of simple interest at an interest rate of 5%?

Solution:

Step 1: Given information:

P = $95,000; I = $1,187.50; r = 5%; per year

Step 2: Convert the interest rate to a “per month” format; [latex]r=\frac{0.05}{12}[/latex]

 

Step 3: Solve for the time period, t.

[latex]\begin{align} t&=\frac{I}{Pr}\\ &=\frac{\$1,\!187.50}{\$95,\!000 \times \frac{0.05}{12}}\\ &=\frac{\$1,\!187.50}{\$95,\!000 \times 0.0041\overline{6}}\\ &=3 \; \text {months} \end{align}[/latex]

For $95,000 to earn $1,187.50 at 5% simple interest, it must be invested for a three-month period.

In the examples of simple interest so far, the time period was given in months. While this is convenient in many situations, financial institutions and organizations calculate interest based on the exact number of days in the transaction, which changes the interest amount.
To illustrate this, assume you had money saved for the entire months of July and August, where [latex]t = \frac{2}{12}[/latex] or [latex]t = 0.16666...=0.1\overline{6}[/latex] of a year. However, if you use the exact number of days, the 31 days in July and 31 days in August total 62 days. In a 365-day year that is [latex]t =\frac{62}{365}[/latex] or t = 0.169863 of a year. Notice a difference of 0.003196 (0.169863 – 0.16) occurs. Therefore, to be precise in performing simple interest calculations, you must calculate the exact number of days involved in the transaction.

Using The BA 2+ Plus Date Function to Calculate the Exact Number of Days

In the video below we’ll demonstrate how to use the BA2+ Date Function.

Example 8.1.4: Time Using Dates

On September 13, 2011, Aladdin decided to pay back the Genie on his loan of $15,000 at 9% simple interest. If he paid the Genie the principal plus $1,283.42 of interest, on what day did he borrow the money from the Genie?

Solution:

Step 1: Given variables:
P = $15,000; I = $1,283.42; r = 9% per year; End Date = September 13, 2011

Step 2: The time is in days, but the rate is annual. Convert the rate to a daily rate; [latex]r=\frac{9\%}{365}[/latex]

Step 3: Solve for the time, t

[latex]\begin{align} t&=\frac{I}{Pr}\\ &=\frac{\$1,\!283.42}{\$15,\!000 \times \frac{0.09}{365}}\\ &= 346.998741=347\; \text{days} \end{align}[/latex]

Step 4: Use the DATE function to calculate the start date (DT1). Use the time in days.

Calculator Instructions

If Aladdin owed the Genie $1,283.42 of simple interest at 9% on a principal of $15,000, he must have borrowed the money 347 days earlier, which is October 1, 2010.

Image Descriptions

Figure 8.1.1: Timeline showing PV =$1,100 at the Start with an arrow pointing to the end (right) where I = ? when t=5 months. r = 5% annually [Back to Figure 8.1.1]

Figure 8.1.2: Timeline showing PV = ? at the Start with an arrow pointing to the end (right) where I = $2,035 when t=11 months. r = 6% annually [Back to Figure 8.1.2]

Figure 8.1.3: Timeline showing PV =$95,000 at the Start with an arrow pointing to the end (right) where I = $1,187.50 when t=?. r = 5% annually [Back to Figure 8.1.3]

Figure 8.1.4: Calculator Instructions to use the Date Function. Instructions are: 2ND DATE, Down Arrow, DT2 = 9-13-2011, Down Arrow, DBD = 347, Up Arrow, Up Arrow, CPT DT1, DT1 = 10-01-2010 [Back to Figure 8.1.4]