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?

검증된 단계별 안내

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: $20000 = 16000 \left(1 + \frac{r}{4}\right)^{7.25 \times 4}$.

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 $r$: $r = 4 \left( \left(\frac{20000}{16000}\right)^{\frac{1}{29}} - 1 \right)$. Finally, convert $r$ to a percentage by multiplying by 100 and round to the nearest hundredth of a percent.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Compound Interest Formula

The compound interest formula calculates the amount of money accumulated over time with interest added periodically. The formula A = P(1 + r/n)^(nt) is used when interest is compounded a specific number of times per year, where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years.

Solving for the Interest Rate

To find the interest rate when other variables are known, rearrange the compound interest formula to isolate r. This involves algebraic manipulation and taking roots or logarithms to solve for r, especially when interest is compounded multiple times per year.

Compounding Frequency and Time

Compounding frequency (n) affects how often interest is added to the principal, impacting growth. Time (t) is the duration the money is invested. Both values influence the exponent in the formula, so understanding their roles is essential for accurate calculations.

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?