Finding deposits, rates, terms and accumulated amounts | Year 12

01 / LEARN

Understand the idea.

Choose the unknown before entering numbers

Start with the balance equation and identify the unknown. If the target is $F V$ and initial savings are $P$ , those savings contribute $P {(1 + i)}^{n}$ at the target date. Only the remaining gap must be supplied by the regular deposits.

For end-of-period deposits, divide that gap by the savings-annuity factor to find $R$ . A negative result means the initial savings alone exceed the target under these assumptions; it does not describe a required positive deposit.

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

Whole periods and numerical solutions

A solver may return a fractional number of periods, but deposits occur at specified dates. To find the first whole period reaching a target, test the next integer and the preceding integer. Rounding to the nearest integer can leave a shortfall.

An unknown rate appears both inside the power and in the annuity denominator. Solve the full equation numerically over a stated valid interval, then substitute the result back. Keep the rate, deposit timing and initial balance consistent throughout the calculation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build the model

What equal year-end deposit reaches AUD 10,000 after 8 years at 4% annually, starting from zero?

Worked solution

1
Model eight equal year-end deposits from zero.
$10000 = R \frac{{(1.04)}^{8} - 1}{0.04}$
The eighth deposit is made at the target date and receives no interest.
2
Isolate the regular deposit.
$\begin{aligned} R & = \frac{10000 \times 0.04}{{(1.04)}^{8} - 1} \\ \approx 1085.278320 \end{aligned}$
Dividing by the accumulated value of one unit per year gives the required deposit.
3
State the deposit to cents.
$R \approx AUD 1085.28$
The exact model value is rounded only after solving the equation.

Answer

AUD 1,085.28

EXAMPLE 02

Use the model to make a decision

Starting from zero, deposit AUD 500 at each year end at 5%. Find the first whole year when the balance reaches AUD 4,000.

Worked solution

1
Express the target as an inequality.
$500 \frac{{(1.05)}^{n} - 1}{0.05} \geq 4000$
The first qualifying year must reach or exceed the target after its deposit.
2
Solve the boundary equation.
$\begin{aligned} {(1.05)}^{n} & = 1.4 \\ n & = \frac{ln (1.4)}{ln (1.05)} \approx 6.8963 \end{aligned}$
Taking logarithms finds the continuous threshold; actual deposits occur only at whole years.
3
Check the adjacent whole years.
$\begin{aligned} F V_{6} & = 500 \frac{{(1.05)}^{6} - 1}{0.05} \approx 3400.96 \\ F V_{7} & = 500 \frac{{(1.05)}^{7} - 1}{0.05} \approx 4071.00 \end{aligned}$
Six deposits fall short, while seven deposits exceed AUD 4,000.
4
Choose the first qualifying deposit date.
$n = 7 years$
With positive deposits and interest, these balances increase each year, so no earlier year reaches the target.

Answer

7 years

03 / YOUR TURN

Try it yourself.

Check your understanding

With no interest, what monthly deposit reaches AUD 2,400 after 16 months?

Show answer & reasoning

AUD 150.00

With no interest, the 16 deposits must sum to AUD 2,400. Thus $16 R = 2400$ and $R = \frac{2400}{16} = 150$ , giving AUD 150.00 per month.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

AUD 3,000 is already saved. What year-end deposit reaches AUD 12,000 in 5 years at 4%?
3 marks
With no interest, what monthly deposit reaches AUD 2,400 after 16 months?
3 marks
A person can deposit AUD 250 at each year end for 6 years. At 3%, how much initial savings are needed to reach AUD 8,000?
3 marks
Find the annual rate $i$ between 0 and 10% for which five end-of-year deposits of AUD 1,000 accumulate to AUD 5,500.
3 marks
A solver gives $n = 9.18$ annual deposits to reach a target. How many whole deposits are needed, and what check should be made?
3 marks
At 4% annually for 10 years, compare the end-of-year and beginning-of-year deposits needed to reach AUD 20,000.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

AUD 3,000 is already saved. What year-end deposit reaches AUD 12,000 in 5 years at 4%?
3 marks
Grow the initial AUD 3,000 for 5 years.
The deposits must supply $12000 - 3000 {(1.04)}^{5}$ .
Divide this by the five-period annuity factor.
AUD 1,541.64
With no interest, what monthly deposit reaches AUD 2,400 after 16 months?
3 marks
At zero interest, $FV = n R$ .
$R = \frac{2400}{16} = 150$ .
AUD 150.00
A person can deposit AUD 250 at each year end for 6 years. At 3%, how much initial savings are needed to reach AUD 8,000?
3 marks
The deposits contribute $\frac{250 ({(1.03)}^{6} - 1)}{0.03}$ .
Subtract that from 8000, then discount the remainder by ${1.03}^{6}$ .
AUD 5,345.58
Find the annual rate $i$ between 0 and 10% for which five end-of-year deposits of AUD 1,000 accumulate to AUD 5,500.
3 marks
Solve $\frac{1000 ({(1 + i)}^{5} - 1)}{i} = 5500$ numerically, using the zero-rate limit at $i = 0$ .
A root search gives $i = 0.047673$ .
Substitution gives a final balance of AUD 5,500.
4.7673%
A solver gives $n = 9.18$ annual deposits to reach a target. How many whole deposits are needed, and what check should be made?
3 marks
The goal is not achieved after only 9 complete deposits.
Round upwards and verify the crossing, rather than rounding to the nearest integer.
10 deposits; calculate balances after 9 and 10 deposits.
At 4% annually for 10 years, compare the end-of-year and beginning-of-year deposits needed to reach AUD 20,000.
3 marks
The beginning-deposit annuity factor is 1.04 times the ordinary factor.
Divide the same target by each factor.
Beginning deposits can be smaller because they earn longer.
End: AUD 1,665.82; beginning: AUD 1,601.75.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

In this lesson

Understand the idea.

Choose the unknown before entering numbers

Whole periods and numerical solutions

See the method in action.

Build the model

Model eight equal year-end deposits from zero.

Isolate the regular deposit.

State the deposit to cents.

Use the model to make a decision

Express the target as an inequality.

Solve the boundary equation.

Check the adjacent whole years.

Choose the first qualifying deposit date.

Try it yourself.

Check your understanding

Keep practising