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. $\newline$ ( $1$ ) Click the icon to view some finance formulas. $\newline$ a. The future value is $\$$ \square. $\newline$ (Round to the nearest cent 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

(

1

) Click the icon to view some finance formulas.

\newline

a. The future value is

\$

\square.

\newline

(Round to the nearest cent as needed.)

Compound Interest Formula: To find the future value of the deposit after one year with quarterly compounding interest, we use the compound interest formula: $\newline$ Future Value = Principal $\times$ $\left(1 + \frac{Interest\ Rate}{Number\ of\ Compounding\ Periods}\right)^{Number\ of\ Compounding\ Periods \times Time}$ $\newline$ Here, Principal = $\$10,000$ , Interest Rate = $1.43\%$ , Number of Compounding Periods = $4$ (quarterly), Time = $1$ year.
Convert to Decimal: First, convert the interest rate from a percentage to a decimal by dividing by $100$ : $\text{Interest Rate (decimal)} = \frac{1.43}{100} = 0.0143$
Plug Values into Formula: Now, plug the values into the compound interest formula: $\newline$ Future Value = $10,000 \times (1 + (0.0143 / 4))^{(4 \times 1)}$
Calculate Inside Parentheses: Calculate the term inside the parentheses: $\newline$ $(1 + (0.0143 / 4)) = 1 + 0.003575 = 1.003575$
Exponentiation Calculation: Raise this term to the power of $4$ (since the interest is compounded quarterly for $1$ year): $(1.003575)^4$
Find Future Value: Perform the exponentiation: $(1.003575)^4 \approx 1.0143$
Calculate Simple Interest: Multiply this result by the principal to find the future value: $\newline$ Future Value $\approx 10,000 \times 1.0143 \approx \$(10,143)$
Rearrange Formula for Rate: For part b, we need to find the effective annual yield using the simple interest formula. The effective annual yield is the equivalent interest rate when interest is compounded more frequently than once a year. $\newline$ The formula for simple interest is: $\newline$ Simple Interest $= Principal \times Rate \times Time$ $\newline$ We will use the future value we found and solve for the $Rate$ to find the effective annual yield.
Calculate Simple Interest: We know the future value is approximately \$$10$,$143$, the principal is \$$10$,$000$, and the time is $1$ year. We can rearrange the simple interest formula to solve for the Rate:$\newline$Rate = $\frac{\text{Simple Interest}}{\text{Principal} \times \text{Time}}$
Find Effective Annual Yield: First, calculate the simple interest by subtracting the principal from the future value:$\newline$Simple Interest = Future Value - Principal = $10,143 - 10,000 = $$\$143$
Find Effective Annual Yield: First, calculate the simple interest by subtracting the principal from the future value:$\newline$Simple Interest = Future Value - Principal = $10,143 - 10,000 = $$\$143$Now, plug the values into the rearranged formula to find the Rate:$\newline$Rate = $\frac{143}{(10,000 \times 1)} = 0.0143$ or $1.43\%$
Find Effective Annual Yield: First, calculate the simple interest by subtracting the principal from the future value:$\newline$Simple Interest = Future Value - Principal = $10,143 - 10,000 = $$\$143$Now, plug the values into the rearranged formula to find the Rate:$\newline$Rate = $\frac{143}{(10,000 \times 1)} = 0.0143$ or $1.43\%$The effective annual yield is the same as the nominal interest rate of $1.43\%$ because the simple interest formula does not take compounding into account. Therefore, the effective annual yield is $1.43\%$.