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Zaki invests \( \$ 500 \) into a money market account that pays him \( 5.5 \% \) interest on his investment annually. After 2 years, Zaki puts an additional \( \$ 750 \) in the account. 3 years after that, he deposits another \( \$ 1,300 \). He then lets the account sit for 5 years. a. How much money does Zaki have after the first 2 years (before the deposit)? b. What is his new balance after depositing \( \$ 750 \) ? c. How much money does Zaki have in the account at the end of the next 3 years (before the next deposit)? d. What is his new balance after depositing \( \$ 1,300 \) ? e. How much money does Zaki have in the account at the end of the 10 year period? e. \( 50.51+6.055)^{2}=556.51 \) How much did Zaki earn in interest?

Ask by Erickson Craig. in the United States
Jan 23,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

After 10 years, Zaki has \$3,702.61 in the account and earned \$1,152.61 in interest.

Solution

Let's tackle each part of Zaki's investment step by step. ### **Given:** - **Initial Investment:** \$500 - **Annual Interest Rate:** 5.5% - **Additional Deposits:** - **After 2 years:** \$750 - **After 5 years (3 years after the second deposit):** \$1,300 - **Total Investment Period:** 10 years Assuming the interest is **compounded annually**, let's proceed with each part. --- ### **a. Amount After the First 2 Years (Before the \$750 Deposit)** **Calculation:** \[ A = P(1 + r)^n \] Where: - \( P = \$500 \) - \( r = 5.5\% = 0.055 \) - \( n = 2 \) years \[ A = 500 \times (1 + 0.055)^2 = 500 \times 1.113025 = \$556.51 \] **Answer:** After the first 2 years, Zaki has **\$556.51** in the account. --- ### **b. New Balance After Depositing \$750** **Calculation:** \[ \text{New Balance} = \text{Previous Amount} + \text{Deposit} = 556.51 + 750 = \$1,306.51 \] **Answer:** After depositing \$750, the new balance is **\$1,306.51**. --- ### **c. Amount After the Next 3 Years (Before the \$1,300 Deposit)** **Calculation:** \[ A = P(1 + r)^n \] Where: - \( P = \$1,306.51 \) - \( r = 5.5\% = 0.055 \) - \( n = 3 \) years \[ A = 1,306.51 \times (1.055)^3 \approx 1,306.51 \times 1.17424 \approx \$1,534.14 \] **Answer:** At the end of the next 3 years, Zaki has **\$1,534.14** in the account. --- ### **d. New Balance After Depositing \$1,300** **Calculation:** \[ \text{New Balance} = \text{Previous Amount} + \text{Deposit} = 1,534.14 + 1,300 = \$2,834.14 \] **Answer:** After depositing \$1,300, the new balance is **\$2,834.14**. --- ### **e. Amount at the End of the 10-Year Period** **Calculation:** \[ A = P(1 + r)^n \] Where: - \( P = \$2,834.14 \) - \( r = 5.5\% = 0.055 \) - \( n = 5 \) years First, calculate \( (1.055)^5 \): \[ (1.055)^5 \approx 1.3064 \] Then, \[ A = 2,834.14 \times 1.3064 \approx \$3,702.61 \] **Answer:** At the end of 10 years, Zaki has **\$3,702.61** in the account. --- ### **f. Total Interest Earned** **Calculation:** \[ \text{Total Deposits} = 500 + 750 + 1,300 = \$2,550 \] \[ \text{Final Amount} = \$3,702.61 \] \[ \text{Interest Earned} = \text{Final Amount} - \text{Total Deposits} = 3,702.61 - 2,550 = \$1,152.61 \] **Answer:** Zaki earned **\$1,152.61** in interest over the 10-year period.

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To calculate Zaki's account balance and interest earned, let's break it down step by step. ### a. After the first 2 years (before new deposit): Zaki invests \$500 at an annual interest rate of 5.5%. The formula for calculating the future value with compound interest is: \[ A = P(1 + r)^t \] Where: - \( P \) = principal amount (\$500) - \( r \) = annual interest rate (5.5% or 0.055) - \( t \) = number of years (2) \[ A = 500(1 + 0.055)^2 = 500(1.113025) \approx 556.51 \] So, after 2 years, Zaki has approximately **\$556.51** in his account. ### b. After depositing \$750: Now, he deposits another \$750, so the new balance is: \[ 556.51 + 750 = 1306.51 \] Zaki's balance after this deposit is **\$1306.51**. ### c. At the end of the next 3 years (before new deposit): Using the same formula, we calculate the future value for the new balance over the next 3 years: \[ A = 1306.51(1 + 0.055)^3 \approx 1306.51(1.174617) \approx 1535.87 \] Thus, after the next 3 years, Zaki has approximately **\$1535.87** in the account. ### d. After depositing \$1,300: Now he deposits \$1,300, so the new balance is: \[ 1535.87 + 1300 = 2835.87 \] Now, Zaki's balance is **\$2835.87** after this deposit. ### e. At the end of the 5-year period: Finally, we calculate the total amount after letting it sit for 5 years: \[ A = 2835.87(1 + 0.055)^5 \approx 2835.87(1.302169) \approx 3690.52 \] Thus, after the full 10 years, Zaki has approximately **\$3690.52** in the account. ### f. Total interest earned: To find out how much Zaki earned in interest, we can use the formula: \[ \text{Interest} = \text{Final Amount} - \text{Total Deposits} \] The total deposits made by Zaki are: \[ 500 + 750 + 1300 = 2550 \] So, we calculate interest earned: \[ \text{Interest} = 3690.52 - 2550 = 1140.52 \] Zaki earned approximately **\$1140.52** in interest over the 10 years.

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