You deposit $\$10,000$ in an account that pays $1.43\%$ interest compounded quarterly. $\newline$ a. Find the future value after one year. $\newline$ b. Use the future value formula for simple interest to determine the effective annual yield. $\newline$ ( $3$ ) Click the icon to view some finance formulas. $\newline$ a. The future value is $\$10,143.77^{7}$ . $\newline$ (Round to the nearest cent as needed.) $\newline$ b. The effective annual yield is $\square\%$ . $\newline$ (Round to the nearest hundredth as needed.)

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Math Problems

Grade 8

Compound interest

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Q. You deposit

\$10,000

in an account that pays

1.43\%

interest compounded quarterly.

\newline

a. Find the future value after one year.

\newline

b. Use the future value formula for simple interest to determine the effective annual yield.

\newline

(

3

) Click the icon to view some finance formulas.

\newline

a. The future value is

\$10,143.77^{7}

\newline

(Round to the nearest cent as needed.)

\newline

b. The effective annual yield is

\square\%

\newline

(Round to the nearest hundredth as needed.)

Calculate Future Value: We need to calculate the future value of a $\$10,000$ deposit with an interest rate of $1.43\%$ compounded quarterly after one year. $\newline$ To do this, we use the compound interest formula: $\newline$ Future Value = Principal $\times$ $\left(1 + \frac{\text{Interest Rate}}{\text{Number of Compounding Periods}}\right)^{\text{Number of Compounding Periods} \times \text{Time in years}}$ $\newline$ Here, Principal = $\$10,000$ , Interest Rate = $1.43\%$ , Number of Compounding Periods = $4$ (quarterly), Time = $1$ year. $\newline$ First, convert the interest rate from a percentage to a decimal by dividing by $100$ . $\newline$ Interest Rate (decimal) = $\frac{1.43}{100} = 0.0143$
Convert Interest Rate: Now, plug the values into the formula: $\newline$ Future Value = $10000 \times (1 + (0.0143 / 4))^{(4 \times 1)}$ $\newline$ Calculate the term inside the parentheses: $\newline$ $(1 + (0.0143 / 4)) = 1 + 0.003575 = 1.003575$
Plug Values into Formula: Next, raise this term to the power of $4$ (since the interest is compounded quarterly for $1$ year): $\newline$ Future Value = $10000 \times (1.003575)^4$ $\newline$ Calculate the exponent: $\newline$ $(1.003575)^4 \approx 1.014377$
Calculate Term Inside Parentheses: Finally, multiply the principal by this result to find the future value: $\newline$ Future Value $\approx 10000 \times 1.014377$ $\newline$ Future Value $\approx \$10143.77$
Raise Term to Power: For part b, we need to find the effective annual yield using the simple interest formula: $\newline$ Effective Annual Yield = $(\text{Future Value} - \text{Principal}) / \text{Principal} \times 100\%$ $\newline$ Substitute the values we have: $\newline$ Effective Annual Yield = $(10143.77 - 10000) / 10000 \times 100\%$ $\newline$ Calculate the difference: $\newline$ $10143.77 - 10000 = 143.77$
Multiply Principal by Result: Now, divide this difference by the principal and multiply by $100$ to get the percentage: $\newline$ Effective Annual Yield = $\frac{143.77}{10000} \times 100\%$ $\newline$ Effective Annual Yield = $1.4377\%$