Question

find the interest rate for a $7500 deposit accumulating to $12,036, compounded annually for 6 years.

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the number of years. We are given $P=\$7500$, $A = \$12036$, and $t = 6$. Substitute these values into the formula: $12036=7500(1 + r)^6$.

Step2: Isolate $(1 + r)^6$

Divide both sides of the equation by $7500$: $\frac{12036}{7500}=(1 + r)^6$. Simplify $\frac{12036}{7500}$ to get $1.6048=(1 + r)^6$.

Step3: Solve for $1 + r$

Take the sixth - root of both sides. $1 + r=1.6048^{\frac{1}{6}}$. Using a calculator, $1.6048^{\frac{1}{6}}\approx1.08$.

Step4: Solve for $r$

Subtract $1$ from both sides: $r=1.08 - 1=0.08$.

Answer:

The interest rate is $8\%$.