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5 Marisha invested R2 500 in a savings account. Calculate how much she will have in the account at the end of 5 years if the interest rate is 7,5% p.a. compounded quarterly for the first year, 6% p.a. compounded monthly for the second year and 8,3% p.a. compounded semi-annually thereafter. [7]​

User BlackBox
by
8.9k points

2 Answers

6 votes

Answer:

To calculate the future value of Marisha's investment, we can break it down into three periods based on the different interest rates and compounding frequencies:

**Period 1 (First Year):**

- Principal amount (initial investment) = R2,500

- Annual interest rate = 7.5%

- Compounding frequency = Quarterly (4 times a year)

We can use the formula for compound interest in this period:

\[A1 = P \left(1 + \frac{r}{n}\right)^{n*t1}\]

Where:

- A1 is the future value after the first year.

- P is the principal amount (R2,500).

- r is the annual interest rate (7.5% or 0.075 as a decimal).

- n is the number of times the interest is compounded per year (quarterly, so 4).

- t1 is the number of years in this period (1 year).

Plug in the values:

\[A1 = 2,500 \left(1 + \frac{0.075}{4}\right)^{4*1} \]

Calculate A1.

**Period 2 (Second Year):**

- Principal amount (starting with A1 from the first year) = A1 from the previous calculation.

- Annual interest rate = 6%

- Compounding frequency = Monthly (12 times a year)

Now, we'll use the same compound interest formula, but with the new interest rate and compounding frequency:

\[A2 = A1 \left(1 + \frac{r}{n}\right)^{n*t2}\]

Where:

- A2 is the future value after the second year.

- A1 is the amount from the end of the first year.

- r is the new annual interest rate (6% or 0.06 as a decimal).

- n is the new compounding frequency (monthly, so 12).

- t2 is the number of years in this period (1 year).

Plug in the values:

\[A2 = A1 \left(1 + \frac{0.06}{12}\right)^{12*1}\]

Calculate A2.

**Period 3 (Remaining 3 Years):**

- Principal amount (starting with A2 from the second year) = A2 from the previous calculation.

- Annual interest rate = 8.3%

- Compounding frequency = Semi-annually (2 times a year)

Use the compound interest formula once more:

\[A3 = A2 \left(1 + \frac{r}{n}\right)^{n*t3}\]

Where:

- A3 is the future value after the remaining 3 years.

- A2 is the amount from the end of the second year.

- r is the new annual interest rate (8.3% or 0.083 as a decimal).

- n is the new compounding frequency (semi-annually, so 2).

- t3 is the number of years in this period (3 years).

Plug in the values:

\[A3 = A2 \left(1 + \frac{0.083}{2}\right)^{2*3}\]

Calculate A3.

Now, to find the total amount Marisha will have at the end of 5 years, simply add up the amounts from each period:

\[Total Amount = A1 + A2 + A3\]

Calculate the total amount, and you will have the answer.

User Orifjon
by
8.5k points
3 votes

Answer:

R3648.87

Explanation:

F = P(1 + r/n)^(nt)

This investment takes place in 3 periods.

First period:

P = present amount = 2500

interest rate = r = 7.5% = 0.075

number of compounding periods per year = n = 4

number of years = t = 1

F = 2500(1 + 0.075/4)^(4 × 1)

F = 2692.83966

Second period of the investment:

P = present amount = 2692.83966

interest rate = r = 6% = 0.06

number of compounding periods per year = n = 12

number of years = t = 1

F = 2692.83966(1 + 0.06/12)^(12 × 1)

F = 2858.92811

Third period of the investment:

P = present amount = 2858.92811

interest rate = r = 8.3% = 0.083

number of compounding periods per year = n = 2

number of years = t = 3

F = 2858.92811(1 + 0.083/2)^(2 × 3)

F = 3648.87

Answer: R3648.87

User Caerbanog
by
8.1k points
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