To find the size of monthly payments required to meet Sara's goal, we can use the formula for the future value of an ordinary annuity:
FV= P X (1+r)^n−1/r
where:
FV = Future Value (the goal amount Sara wants to save, which is Br. 200,000)
P = Monthly payment
r = Monthly interest rate (12% or 0.12 as a decimal)
n = Number of months (2 years, so n = 2 * 12 = 24 months)
A. Size of Monthly Payments Required:
Let's plug the values into the formula:
200,000 = P \times \{(1 + 0.12)^{24} - 1} divided by {0.12} \]
Now, solve for P:
\[ P = \ {200,000}{\ {(1 + 0.12)^{24} - 1} divided by {0.12}} \]
\[ P = \ {200,000}divided by {7.1091} \]
\[ P ≈ 28,150.59 \]
Sara would need to make monthly payments of approximately Br. 28,150.59 to meet her goal of saving Br. 200,000 in two years.
B. Total Amount Deposited by Sara:
To find the total amount deposited by Sara, we can multiply the monthly payment (P) by the number of months (n):
\[ Total Amount Deposited = P \times n \]
\[ Total Amount Deposited = 28,150.59 \times 24 \]
\[ Total Amount Deposited ≈ 675,614.16 \]
Sara will deposit approximately Br. 675,614.16 in total over the two years.
C. Amount of Interest:
The amount of interest can be calculated by subtracting the total amount deposited from the future value:
\[ Interest = FV - Total Amount Deposited \]
\[ Interest = 200,000 - 675,614.16 \]
\[ Interest ≈ -475,614.16 \]
The negative value for interest indicates that Sara will actually pay approximately Br. 475,614.16 in interest over the two years due to the compounding effect of the 12% interest rate. This is because the interest earned on the deposits will not be enough to cover the future value of Br. 200,000, leading to additional interest being paid.