Financial Maths, STD2 2026 HSC 38

Maya plans to go on an overseas trip in 5 years' time. Maya expects that the trip will cost $\$14\ 000$. To save for the trip, Maya began by contributing $\$750$ every 6 months into an annuity. After making a number of payments, Maya realised that the annuity would not reach $\$14\ 000$ in 5 years. Maya decides to double the contribution until the trip.

The interest rate was 2% per annum compounding half yearly.

Future values of an annuity of $1
Rate Period	0.01	0.02	0.03	0.04
2	2.0100	2.0200	2.0300	2.0400
4	4.0604	4.1216	4.1836	4.2465
6	6.1520	6.3081	6.4684	6.6330
8	8.2857	8.5830	8.8923	9.2142
10	10.4622	10.9497	11.4639	12.0061
12	12.6825	13.4121	14.1920	15.0258

What is the maximum number of $\$750$ contributions Maya could make before she will need to double her payments to reach $\$14\ 000$? (4 marks)

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$2\ \text{contributions}$

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$\text{Find annuity total of \$750 payments only:}$

$r=\dfrac{0.02}{2}=0.01,\ \ n=5\times2=10$

$\text{Table factor = 10.4622}$

$\text{Value of annuity} = 750 \times 10.4622=\$7846.65$

$\text{Extra money needed (at end of annuity)}\ = 14\ 000-7846.65=\$6153.35$

$\text{Estimated table factor where \$750 annuity has a total \$6153.35}$

$750 \times\ \text{Factor}\ = 6153.35\ \ \Rightarrow \ \ \text{Factor} = \dfrac{6153.35}{750} = 8.20 \ldots$

$\text{Using the table:}$

$\text{8 periods at $r=0.01$ is the lowest that will create an annuity > \$6153.35}$

$\text{Since there are 10 contributions, only 2 can be \$750 only.}$