12. Muaz will need \( \$ 130000 \) in 4 years to buy a piece of property. He plans to save money by making equal quarterly deposits into an account earning \( 6.1 \% \) per year compounded quarterly. a) What is Muaz's quarterly deposit? b) How much interest will Muaz earn?

To determine Muaz's quarterly deposit, we can use the formula for the future value of an ordinary annuity. The formula is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) = Future Value (\$130,000) - \( P \) = Quarterly deposit - \( r \) = Quarterly interest rate (annual rate / 4) - \( n \) = Total number of deposits (number of years × 4) Given an annual interest rate of \( 6.1\% \), the quarterly rate is: \[ r = \frac{6.1\%}{4} = 0.01525 \] For \( 4 \) years, there are \( n = 4 \times 4 = 16 \) deposits. Rearranging the formula to solve for \( P \): \[ P = \frac{FV \cdot r}{(1 + r)^n - 1} \] Substituting the values: \[ P = \frac{130,000 \cdot 0.01525}{(1 + 0.01525)^{16} - 1} \] Calculating \( (1 + 0.01525)^{16} \) gives approximately \( 1.27232 \). Thus, \[ P = \frac{130,000 \cdot 0.01525}{1.27232 - 1} = \frac{1,982.5}{0.27232} \approx 7,281.39 \] **Answer for (a)**: Muaz's quarterly deposit will be approximately \( \$7,281.39 \). To find out how much interest Muaz will earn, we first calculate the total amount he will deposit over the duration of 4 years: \[ \text{Total Deposits} = P \times n = 7,281.39 \times 16 \approx 116,500.24 \] Now, the interest earned is calculated as: \[ \text{Interest} = FV - \text{Total Deposits} = 130,000 - 116,500.24 \approx 13,499.76 \] **Answer for (b)**: Muaz will earn approximately \( \$13,499.76 \) in interest.