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

01 / LEARN

The idea, step by step

Choose the unknown before entering numbers

A savings goal can ask for the regular payment, duration, rate or initial amount. Write the full balance equation and identify which quantities are given. For an ordinary annuity, isolate $R$ by dividing the required accumulated deposit component by the annuity factor.

When an initial balance is present, first grow it to the target date and subtract it from the target. A negative required deposit indicates that the initial savings alone exceed the target under the model.

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

Whole periods and numerical solutions

If the equation gives a non-integer number of end deposits, the first whole period reaching the target is usually the next integer. Verify both the previous and chosen periods. Do not round a period count to the nearest integer.

An unknown rate inside an annuity factor appears in several places, so use a finance solver or numerical root search. State the equation and the valid interval, then substitute the result into the original model. Any computed rate is an assumption for the problem, not a guaranteed future return.

A common trap

Rounding a required number of deposits down.

Instead: Use the next whole deposit and verify that one fewer deposit fails to reach the target.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

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

Set $10000 = R ({(1.04)}^{8} - 1) / 0.04$ .
Divide by the annuity factor to find $R = A U D$ 1,085.28.

AUD 1,085.28

Example 2

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.

Solve $n = ln (1 + 4000 \times 0.05 / 500) / ln (1.05)$ .
The non-integer result must be rounded up.
Check the balances after 6 and 7 deposits: AUD 3,400.96 and AUD 4,071.00.

7 years

Check your understanding

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

Show answer

AUD 150.00

At zero interest, $F V = n R$ . $R = 2400 / 16 = 150$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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, $F V = n R$ .
$R = 2400 / 16 = 150$ .
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 $250 ({(1.03)}^{6} - 1) / 0.03$ .
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 $1000 ({(1 + i)}^{5} - 1) / i = 5500$ 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 ↗

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