5.1 The nominal interest rate charged is 8% per annum, compounded monthly - NSC Technical Mathematics - Question 5 - 2023 - Paper 1

We can use the formula for compound interest:

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

Where:

• $P = 25000$
• $i = 0.0996$ (interest rate)
• $m = 4$ (compounding quarterly)
• $t = 7$ years

Calculating:

1. Calculate $n = mt = 4 \times 7 = 28$.
2. Substitute into the formula:

$A = 25000 \left(1 + \frac{0.0996}{4}\right)^{28}$

Calculating further,

\approx 25000 \times 1.99955 \approx R 48656.72$$

Using the formula for reducing balance:

$A = P(1 - r/100)^{t}$

For 80 °C: $80 = P(1 - r/100)^{6}$

For 50 °C (after 2 more minutes): $50 = P(1 - r/100)^{8}$

Dividing these equations:

$\frac{80}{50} = \frac{(1 - r/100)^{6}}{(1 - r/100)^{8}} \implies 1.6 = (1 - r/100)^{-2}$ Taking the reciprocal gives:

\implies 1 - r/100 = \sqrt{0.625} \\ \implies 1 - r/100 = 0.79$$ Thus, $$r = 21$$

Using the reducing balance formula:

$P(1 - r/100)^{t}$

Now substituting $r = 21$:

80 = P(0.79)^{6}$$ This requires first finding P: $$P = \frac{80}{(0.79)^{6}} \\ \approx \frac{80}{0.3025} \approx 264.3 °C$$