a
Method 1: Compound Interest Formula
\( A = 37,000 \times \left(1 + \frac{6.4}{100 \times 4}\right)^{2 \times 4} = 37,000 \times 1.016^8 \)
\( 1.016^8 \approx 1.135201345 \)
\( A \approx 37,000 \times 1.135201345 \approx 42,002.44976 \approx 42,002 \)
Method 2: Financial Calculator
Settings: \( N = 8 \), \( I\% = 6.4 \), \( PV = -37,000 \), \( P/Y = 4 \), \( C/Y = 4 \)
Compute: \( FV \approx 42,002 \)
Result: 42,002 AUD [3].

b
Method 1: Logarithmic Solution
\( 37,000 \times 1.016^{m/3} > 50,000 \)
\( 1.016^{m/3} > \frac{50,000}{37,000} \approx 1.351351351 \)
\( \frac{m}{3} \ln(1.016) \approx \ln(1.351351351) \)
\( m \approx 56.8854 \)
Test: \( m = 57 \) (19 quarters): \( A \approx 50,003.99 > 50,000 \)
\( m = 56 \): \( A \approx 49,716.58 < 50,000 \)
Method 2: Financial Calculator
Settings: \( PV = -37,000 \), \( FV = 50,000 \), \( I\% = 6.4 \), \( P/Y = 4 \), \( C/Y = 4 \)
Compute: \( N \approx 18.9692 \) quarters
Convert: \( 18.9692 \times 3 \approx 56.8854 \) months
Smallest integer: \( m = 57 \)
Result: 57 months [4].

c
Loan amount: \( 200,000 \times (1 – 0.25) = 150,000 \)
Result: 150,000 AUD [1].

d
(i) Total interest:
Total payments: \( 1,700 \times 12 \times 10 = 204,000 \)
Interest: \( 204,000 – 150,000 = 54,000 \)
Result: 54,000 AUD [2].
(ii) Interest rate:
\( 150,000 = 1,700 \times \frac{1 – \left(1 + \frac{r}{12}\right)^{-120}}{\frac{r}{12}} \)
Numerical solution: \( r \approx 6.46\% \)
Result: 6.46% [3].

e
Final payment after 60 months:
\( PV_{\text{remaining}} = 1,700 \times \frac{1 – \left(1 + \frac{6.46}{1200}\right)^{-60}}{\frac{6.46}{1200}} \)
\( \approx 1,700 \times 51.1614 \approx 86,974.38 \approx 86,974 \)
Result: 86,974 AUD [3].

f
Savings:
Total if continued: \( 1,700 \times 120 = 204,000 \)
Paid: \( 1,700 \times 60 + 86,974 = 188,974 \)
Savings: \( 204,000 – 188,974 = 15,026 \)
Result: 15,026 AUD [3].