R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly. After 24 months, the value of the investment was R13 459. Determine the value of m. On 31 January 2022, Tino deposited R1 000 in an account that paid interest at 7,5% p.a., compounded monthly. He continued depositing R1 000 on the last day of every month. He will make the last deposit on 31 December 2022. Will Tino have sufficient funds in the account on 1 January 2023 to buy a computer that costs R13 000? Justify your answer by means of an appropriate calculation. Thabo plans to buy a car that costs R250 000. He will pay a deposit of 15% and take out a loan for the balance. The interest on the loan is 13% p.a., compounded monthly. Calculate the value of the loan. The first repayment will be made 6 months after the loan has been granted. The loan will be repaid over a period of 6 years after it has been granted. Calculate the MONTHLY instalment.

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R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly - NSC Mathematics - Question 6 - 2022 - Paper 1

Question 6

R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly. After 24 months, the value of the investment was R13 459. Determine the value o... show full transcript

Worked Solution & Example Answer:R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly - NSC Mathematics - Question 6 - 2022 - Paper 1

Step 1

6.1 Determine the value of m.

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Answer

To find the interest rate m, we can use the compound interest formula:

$A = P \left(1 + \frac{m}{400}\right)^{y}$

Where:

A = R13 459
P = R12 000
y = 24 months = 2 years

Substituting the values into the formula:

$13 459 = 12 000 \left(1 + \frac{m}{400}\right)^{2}$

Now dividing both sides by R12 000:

$\frac{13 459}{12 000} = \left(1 + \frac{m}{400}\right)^{2}$

Calculating the left side:

$\frac{13 459}{12 000} \approx 1.12158$

Taking the square root:

$1 + \frac{m}{400} \approx \sqrt{1.12158}$

Calculating the square root:

$1 + \frac{m}{400} \approx 1.05807$

Now, rearranging gives:

\Rightarrow m \approx 0.05807 \times 400 \\ \Rightarrow m \approx 23.229$$ Thus, the value of m is approximately **5.78%**.

Step 2

6.2 Will Tino have sufficient funds?

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Answer

To determine if Tino has enough to buy the computer:

First, we calculate the future value F of Tino's monthly deposits using the future value of an annuity formula:

$F = P \frac{(1 + i)^{n}-1}{i}$

Where:

P = R1 000 (monthly deposit)
i = 7.5% p.a. compounded monthly = 0.075/12 = 0.00625
n = 12 months

Calculating F:

$F = 1000 \frac{(1 + 0.00625)^{12} - 1}{0.00625}$

Calculating the expression:

$F = 1000 \frac{(1.0759) - 1}{0.00625} \approx 1000 \frac{0.0759}{0.00625} \approx 12121.22$

So, the balance in the account as of 1 January 2023 is approximately R12 421.22.

Since R12 421.22 < R13 000, Tino will not be able to buy the computer.

Step 3

6.3 Calculate the value of the loan.

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Answer

To find the value of the loan, we first determine Thabo's deposit:

Deposit = 15% of R250 000 = 0.15 * R250 000 = R37 500

Thus, the loan amount will be:

Loan Amount = R250 000 - R37 500 = R212 500.

Step 4

6.3.2 Calculate the MONTHLY instalment.

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Answer

To calculate the monthly installment, we will use the loan repayment formula:

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

Where:

P = R212 500 (loan amount)
i = 13% p.a. compounded monthly = 0.13/12 = 0.0108333
n = 6 years * 12 months = 72 months

Substituting the values:

$A = 212500 \frac{0.0108333(1+0.0108333)^{72}}{(1+0.0108333)^{72}-1}$

After performing the calculations:

The monthly installment is approximately R4 724.96.

R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly - NSC Mathematics - Question 6 - 2022 - Paper 1

Worked Solution & Example Answer:R12 000 was invested in a fund that paid interest at m% p.a., compounded quarterly - NSC Mathematics - Question 6 - 2022 - Paper 1

6.1 Determine the value of m.

6.2 Will Tino have sufficient funds?

6.3 Calculate the value of the loan.

6.3.2 Calculate the MONTHLY instalment.

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