Makgabo needs R45,000 for a down payment in 16 months. She can save R2,200 monthly, earning 15.4% interest…

Question

Makgabo needs R45,000 for a down payment in 16 months. She can save R2,200 monthly, earning 15.4% interest compounded monthly. How much more does she need?

ScanSolve AI Tutor · Accepted Answer

To determine the amount Makgabo still needs, we first need to calculate the future value of her savings. This is an annuity problem.

Here are the given details:

Total amount needed = R45,000
Monthly deposit ( $Pmt$ ) = R2,200
Number of months ( $n$ ) = 16
Annual interest rate = 15.4%
Interest compounded monthly, so the monthly interest rate ( $i$ ) = $\frac{0.154}{12}$

Since Makgabo "starts to save immediately" by depositing R2,200 every month, this is an annuity due. The future value of an annuity due ( $FV_{due}$ ) is given by the formula:

FV_{due} = Pmt \cdot s_{\overline{n}|i} \cdot (1+i)

where $s_{\overline{n}|i} = \frac{(1+i)^n - 1}{i}$ is the future value interest factor of an ordinary annuity.

Calculate the future value of Makgabo's savings.
Substitute the given values into the annuity due formula:
$FV_{savings} = 2200 \cdot s_{\overline{16}|\frac{0.154}{12}} \cdot \left(1 + \frac{0.154}{12}\right)$
Calculate the amount Makgabo still needs.
The amount she still needs ( $X$ ) is the total amount required minus the future value of her savings:
$X = \text{Total amount needed} - FV_{savings}$ $X = 45000 - \left[2200 \cdot s_{\overline{16}|\frac{0.154}{12}} \cdot \left(1 + \frac{0.154}{12}\right)\right]$
Comparing this expression with the given options, we can see that option d matches our derived formula, where $i$ in $(1+i)$ represents the monthly interest rate $\frac{0.154}{12}$ .

The correct option is d.
$X = 45000 - (1 + i)2200s_{\overline{16}|0.154/12}$
where $i = \frac{0.154}{12}$ .

Makgabo needs R45,000 for a down payment in 16 months. She can save R2,200 monthly, earning 15.4% interest compounded monthly. How much more does she need?

Answer

Step-by-step solution

Calculate the future value of Makgabo's savings.

Calculate the amount Makgabo still needs.

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