Question

Simple Interest You invest \(\$ 2500\) at \(7 \%\) simple interest. How many years will it take for the investment to earn \(\$ 1000\) in interest?

Short answer

It will take approximately 5.71 years for the investment to earn $1000 in interest.

Step-by-step solution

Understand Simple Interest Formula

In order to solve this problem, first understand the simple interest formula. The formula for Simple Interest is \(I = PRT\), where \(I\) is the interest, \(P\) is the principal amount (initial investment), \(R\) is the rate of interest (expressed in decimal form), and \(T\) is the time in years.

Identify the Values

Identify the values from the problem that align with the variables in the formula. The interest \(I\) is \$1000, the principal \(P\) is \$2500, and the rate \(R\) is 7% or 0.07. The time \(T\) is what we're trying to find.

Substitute the Values Into the Formula

Next, substitute the known values into the formula and solve for \(T\). \(1000 = 2500 * 0.07 * T\)

Solve for T

Finally, solve the equation for \(T\). First, simplify the right hand side: \(1000 = 175T\). Then, divide both sides of the equation by 175 to isolate \(T\): \(T = 1000 / 175 = 5.71\) years.

Key concepts

Simple Interest Formula

Understanding simple interest is crucial when dealing with loans and investments. It is a quick way to calculate the interest you'll earn or pay on a principal sum over a certain period. The formula for simple interest is given by \( I = PRT \), where \( I \) represents the interest earned or paid, \( P \) is the principal amount, \( R \) is the interest rate, and \( T \) is the time period in years.

The simplicity of this formula lies in its direct proportionality; your interest grows linearly with time and the rate. This means that for each year, the interest will increase by the same amount, making the calculation straightforward and predictable. To correctly apply the formula, however, ensure that the interest rate is in decimal form, not as a percentage.

Principal Amount

The principal amount \( P \) is the initial sum of money put into an investment or the original amount of a loan on which interest is calculated. In our example, the principal amount is \$2500\. It's important to note that this amount does not change when dealing with simple interest; it remains constant throughout the investment period.

When you come across a problem that includes compound interest, this is where the principal can start to shift, as interest is calculated on the accumulating amount. However, for simple interest, the simplicity remains with the principal amount being your base line for all calculations.

Interest Rate

The interest rate \( R \) is the percentage that determines how much interest you'll earn on your principal over a period of time. In the exercise, the given rate is 7%, or 0.07 when converted to a decimal. This conversion is crucial for calculations, as the simple interest formula requires the rate in decimal form.

It's worth noting that interest rates can be quoted per year (annually), per month, or any other time frame. For annual calculations, no adjustments are needed, but for other time frames, you'd typically divide the annual rate by the number of periods in a year, representing the interest accrued in those smaller spans.

Time Calculation

The final variable in the simple interest equation is the time \( T \) during which the principal earns interest. It's usually expressed in years. If you're given a different time frame, like months or days, conversion to years is necessary for consistency with the formula.

Depending on the simplicity of the numbers involved, solving for \( T \) might involve basic algebra as shown in the exercise. Dividing the interest by the product of the principal and rate, you arrive at the time it takes for the investment to earn a specific amount of interest. In our case, it would take approximately 5.71 years for a \$2500\