An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly. At the end of the 4 years, the total amount in the account was R13 080. Determine the value of r. A businesswoman deposited R9 000 into an account at the end of January 2014. She continued to make monthly deposits of R9 000 at the end of each month up to the end of December 2018. The account earned interest at a rate of 7,5% p.a., compounded monthly. 7.2.1 Calculate how much money was in the account immediately after 60 deposits had been made. 7.2.2 The businesswoman left the amount calculated in QUESTION 7.2.1 for a further n months in the account. The interest rate remained unchanged and no further payments were made. The total interest earned over the entire investment period was R190 214,14. Determine the value of n.

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An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly - NSC Mathematics - Question 7 - 2021 - Paper 1

Question 7

An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly. At the end of the 4 years, the total amount in the account was R13 ... show full transcript

Worked Solution & Example Answer:An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly - NSC Mathematics - Question 7 - 2021 - Paper 1

Step 1

Determine the value of r

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Answer

To determine the interest rate r, we can use the compound interest formula:

$A = P \left(1 + \frac{i}{n}\right)^{nt}$

Where:

A = R13 080
P = R10 000
n = 4 (quarterly compounding)
t = 4 years

Substituting values into the formula:

$13 080 = 10 000 \left(1 + \frac{r}{4}\right)^{4 \cdot 4}$

This simplifies to: $\left(1 + \frac{r}{4}\right)^{16} = \frac{13 080}{10 000}$

Calculating the right side gives: $\left(1 + \frac{r}{4}\right)^{16} = 1.308$

Taking the 16th root yields: $1 + \frac{r}{4} = 1.308^{\frac{1}{16}} \\ \frac{r}{4} = 1.308^{\frac{1}{16}} - 1$

Now calculate and find: $r = 4 \left(1.0677 - 1\right) = 0.271\text{ or } 6.77\%$

Step 2

7.2.1 Calculate how much money was in the account immediately after 60 deposits had been made

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Answer

Using the future value formula for an annuity, we have:

$F = P \left(\frac{(1 + i)^{nt} - 1}{i}\right)$

Where:

P = R9 000
i = \frac{0.075}{12} = 0.00625
n = 60 (60 deposits made over 5 years)

Now substituting values:

$F = 9 000 \left(\frac{(1 + 0.00625)^{60} - 1}{0.00625}\right)$

Calculating this gives: $F \approx R652 743.95$

Step 3

7.2.2 Determine the value of n

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Answer

After 60 deposits, the total amount in the account is R652 743.95. This amount is left to earn interest for an additional n months, yielding a total interest of R190 214,14 over the entire period.

We use the formula:

$A = P(1 + i)^{n}$

Thus, $R190 214.14 = 652 743.95 \left(1 + 0.00625\right)^{n} - 540 000$

Which simplifies to: $730 214.14 = 652 743.95 \left(1 + 0.00625\right)^{n}$

Then, isolate the power: $1.1186 = (1.00625)^{n}$

Taking the logarithm, $n = \frac{\log(1.1186)}{\log(1.00625)} \approx 18 \text{ months}$

An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly - NSC Mathematics - Question 7 - 2021 - Paper 1

Worked Solution & Example Answer:An amount of R10 000 was invested for 4 years, earning interest at r% p.a., compounded quarterly - NSC Mathematics - Question 7 - 2021 - Paper 1

Determine the value of r

7.2.1 Calculate how much money was in the account immediately after 60 deposits had been made

7.2.2 Determine the value of n

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