An initial investment of \8,000 was deposited into an account earning an annual interest r… — GMAT

An initial investment of $\$8,000$ was deposited into an account earning an annual interest rate of $r\%$ , compounded annually. If no additional deposits or withdrawals were made and the total balance in the account after $2$ years was $\$11,520$ , what is the value of $r$ ?

Answer: 20

Answer

The value of

r

is 20.

The compound interest formula for annual compounding is

A = P\left(1 + \frac{r}{100}\right)^t

. Substituting

P = 8,000

A = 11,520

, and

t = 2

gives

11,520 = 8,000\left(1 + \frac{r}{100}\right)^2

. Dividing both sides by

8,000

yields

\left(1 + \frac{r}{100}\right)^2 = 1.44

. Taking the square root gives

1 + \frac{r}{100} = 1.2

, so

\frac{r}{100} = 0.20

, which corresponds to

r = 20

Step-by-Step Solution

Set up the compound interest equation using given principal and accumulated balance.

11,520 = 8,000 \left(1 + \frac{r}{100}\right)^2

Interest compounded annually over

2

years scales the initial principal by a factor of

\left(1 + \frac{r}{100}\right)^2

Divide both sides by

8,000

to isolate the annual growth factor squared.

\left(1 + \frac{r}{100}\right)^2 = \frac{11,520}{8,000} = 1.44

Finding the ratio of final balance to initial principal gives the overall multiplier over the two-year period.

Take the square root of both sides to obtain the single-year multiplier.

1 + \frac{r}{100} = \sqrt{1.44} = 1.2

Since the interest rate

r > 0

, taking the positive square root gives the multiplier for one year.

Solve for

r

r = (1.2 - 1) \times 100 = 20

Subtracting

1

yields the annual decimal interest rate of

0.20

, which corresponds to

20\%

Key Concept

Annual Compound Interest Rate Calculation

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