Question

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)tn and A = Pert At what interest rate, to the nearest hundredth of a percent, will $$16,000 grow to 20$$,000 if invested for 7.25 yr and interest is compounded quarterly?

Accepted Answer

Identify the given values: the principal amount $$P = 16000$$, the amount after time $$A = 20000$$, the time period $$t = 7.25$$ years, and the number of compounding periods per year $$n = 4 ($$quarterly compounding). Write down the compound interest formula for interest compounded quarterly: $$A = P \left(1 + \frac{r}{n}\$$right)^{tn}$$, where r$ is the annual interest rate (in decimal form) we need to find. Substitute the known values into the formula: 20000 = 16000$$ \left(1 + \frac{r}{4}\$$right)^{7.25 	imes 4}. $$Divide both sides of the equation by 16000 to isolate the compound factor: $$\frac{20000}{16000} = \left(1 + \frac{r}{4}\$$right)^{29}$$. Take the 29th root of both sides to solve for 1$$ + \frac{r}{4}$$, then subtract 1 and multiply by 4 to solve for r$: r = 4$$ \left( \left(\frac{20000}{16000}\$$right)^{\frac{1}$${29}} - 1 ight)$$. Finally, convert r$ to a percentage by multiplying by 100 and round to the nearest hundredth of a percent.

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt At what interest rate, to the nearest hundredth of a percent, will \$16,000 grow to \$20,000 if invested for 7.25 yr and interest is compounded quarterly?

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Key Concepts

Compound Interest Formula

Solving for the Interest Rate

Compounding Frequency and Time

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)tn and A = Pert At what interest rate, to the nearest hundredth of a percent, will \$16,000 grow to \$20,000 if invested for 7.25 yr and interest is compounded quarterly?