Question

Suppose we deposit $1117 in a savings account earning 2.7% per year, compounded annually. After how many years did this account value $2690.78? Round your answer to the nearest whole number.

Answer 1

The account with an initial deposit of $1117 at a 2.7% annual interest rate, compounded annually, will grow to $2690.78 in approximately 31 years when rounded to the nearest whole number.

Step-by-step explanation:

To determine after how many years an account with an initial deposit of $1117 will grow to $2690.78 at an annual interest rate of 2.7%, compounded annually, we can use the formula for compound interest:

A = P(1 + r/n)nt

Where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (decimal).
• n is the number of times that interest is compounded per year.
• t is the number of years the money is invested or borrowed for.

Given that the interest is compounded annually, n is equal to 1. We can now set up the equation with the given values and solve for t:

2690.78 = 1117(1 + 0.027/1)1t

To solve for t, we take the following steps:

1. Divide both sides by 1117 to isolate the compound interest factor:
2. 2690.78 / 1117 = (1 + 0.027)t
3. Calculate the compound interest factor by performing the division:
4. (1 + 0.027)t = 2.409
5. Take the natural logarithm of both sides to eliminate the exponent:
6. ln(2.409) = t * ln(1.027)
7. Solve for t:
8. t = ln(2.409) / ln(1.027)
9. Calculate t using a calculator:
10. t ≈ 30.97
11. Since t must be a whole number, we round to the nearest whole number:
12. t = 31 years

Therefore, it would take approximately 31 years for the account value to grow to $2690.78.