Sure, let's go through this step-by-step to determine how long it will take for an investment of [tex]$7,000 at 7% annual simple interest to grow to $[/tex]15,820.
We can use the formula for simple interest:
[tex]\[ A = P(1 + rt) \][/tex]
Here:
- [tex]\( A \)[/tex] is the final amount.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the rate of interest per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
In this problem:
- [tex]\( A = 15,820 \)[/tex]
- [tex]\( P = 7,000 \)[/tex]
- [tex]\( r = 0.07 \)[/tex]
We need to solve for [tex]\( t \)[/tex]. Rearranging the simple interest formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{A - P}{P \times r} \][/tex]
Let's plug in the values:
[tex]\[ t = \frac{15,820 - 7,000}{7,000 \times 0.07} \][/tex]
First, calculate the numerator:
[tex]\[ 15,820 - 7,000 = 8,820 \][/tex]
Next, calculate the denominator:
[tex]\[ 7,000 \times 0.07 = 490 \][/tex]
Now, divide the numerator by the denominator:
[tex]\[ t = \frac{8,820}{490} \approx 18 \][/tex]
So, it takes approximately 18 years for the investment of [tex]$7,000 at 7% annual simple interest to grow to $[/tex]15,820 when rounded to the nearest whole number.
Therefore, the answer is:
It will take 18 years.
