You deposit $\$12,000$ in an account that pays $1.49\%$ interest compounded quarterly. Find the future value after one year. Use the future value formula for simple interest to determine the effective annual yield. ( $1$ ) Click the icon to view some finance formulas. The future value is $\$$ \square. (Round to the nearest cent as needed.)

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Math Problems

Grade 8

Compound interest

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Q. You deposit

\$12,000

in an account that pays

1.49\%

interest compounded quarterly. Find the future value after one year. Use the future value formula for simple interest to determine the effective annual yield. (

1

) Click the icon to view some finance formulas. The future value is

\$

\square. (Round to the nearest cent as needed.)

Calculate Future Value: First, let's calculate the future value of the $\$12,000$ deposit with an interest rate of $1.49\%$ compounded quarterly after one year. $\newline$ The formula for compound interest is: $\newline$ $A = P(1 + \frac{r}{n})^{nt}$ $\newline$ Where: $\newline$ $A$ = the future value of the investment/loan, including interest $\newline$ $P$ = the principal investment amount ( $\$12,000$ ) $\newline$ $r$ = the annual interest rate (decimal) ( $1.49\%$ or $0.0149$ ) $\newline$ $n$ = the number of times that interest is compounded per year (quarterly, so $4$ ) $\newline$ $1.49\%$ $0$ = the time the money is invested for, in years ( $1.49\%$ $1$ year)
Plug in Values: Now, let's plug in the values into the formula and calculate the future value. $\newline$ $A = 12000(1 + 0.0149/4)^(4*1)$ $\newline$ $A = 12000(1 + 0.003725)^(4)$ $\newline$ $A = 12000(1.003725)^(4)$ $\newline$ $A = 12000 \times (1.003725)^4$ $\newline$ $A = 12000 \times 1.014963$ $\newline$ $A = 12179.56$ $\newline$ The future value after one year is approximately $\$12,179.56$ .
Calculate Effective Yield: Next, we will calculate the effective annual yield using the simple interest formula. $\newline$ The formula for simple interest is: $\newline$ $A = P(1 + rt)$ $\newline$ However, to find the effective annual yield, we need to rearrange the formula to solve for $r$ : $\newline$ $r = \frac{A/P - 1}{t}$ $\newline$ We will use the future value we found from the compound interest formula as $A$ , and compare it to the principal to find the effective rate.
Calculate Effective Yield: Next, we will calculate the effective annual yield using the simple interest formula. $\newline$ The formula for simple interest is: $\newline$ $A = P(1 + rt)$ $\newline$ However, to find the effective annual yield, we need to rearrange the formula to solve for $r$ : $\newline$ $r = \frac{A/P - 1}{t}$ $\newline$ We will use the future value we found from the compound interest formula as $A$ , and compare it to the principal to find the effective rate.Now, let's plug in the values into the rearranged simple interest formula to find the effective annual yield. $\newline$ $r = \frac{12179.56/12000 - 1}{1}$ $\newline$ $r = \frac{1.014963 - 1}{1}$ $\newline$ $r = 0.014963$ $\newline$ To express this as a percentage, we multiply by $100$ : $\newline$ Effective annual yield = $0.014963 \times 100$ $\newline$ Effective annual yield = $1.4963\%$ $\newline$ The effective annual yield is approximately $1.4963\%$ .