If you want to earn $1,000 of simple interest at a rate of 5% in a span of five months, how much money must you invest?
The correct answer and explanation is:
To find out how much money must be invested to earn $1,000 of simple interest at a rate of 5% in five months, we use the formula for simple interest:
I=P×r×tI = P \times r \times t
Where:
- II is the interest earned ($1,000)
- PP is the principal (the amount invested, which we need to find)
- rr is the annual interest rate (5% or 0.05 as a decimal)
- tt is the time the money is invested, expressed in years
Since the time given is five months, convert it to years:
t=512 yearst = \frac{5}{12} \text{ years}
Now substitute the known values into the formula:
1,000=P×0.05×5121,000 = P \times 0.05 \times \frac{5}{12}
Simplify the right side:
1,000=P×0.05×512=P×0.2512=P×0.02083331,000 = P \times \frac{0.05 \times 5}{12} = P \times \frac{0.25}{12} = P \times 0.0208333
To solve for PP:
P=1,0000.0208333≈48,000P = \frac{1,000}{0.0208333} \approx 48,000
So, the amount to invest is approximately $48,000.
Explanation:
Simple interest is calculated on the original principal for the entire duration of the investment. The formula relates the principal, the interest rate, and the time period to the interest earned. Since the interest rate is annual, the time must be expressed in years to match the rate’s time frame.
In this case, the interest rate is 5% per year. Because the time period is five months, it must be converted into a fraction of a year by dividing by 12. This ensures the time and rate units are consistent.
Once time is converted, plugging the values into the formula gives a straightforward way to find the principal. Solving the equation shows that an investment of about $48,000 is required to earn $1,000 of interest in five months at 5%.
This method is very useful for planning investments or loans where interest is calculated simply without compounding. It helps understand how principal, rate, and time interact to determine total interest earned.