Classroom · Senior extension
Build a savings plan,
one period at a time.
Four payments of $100 make $400 of your own money. How much interest do they earn, and why does the date of each payment matter?
1. Predict, then investigate
| Period | Opening ($) | Interest ($) | Payment ($) | Closing ($) |
|---|
Arithmetic is carried without intermediate rounding; displayed amounts are rounded to cents. $ denotes illustrative dollars. No tax, fees, inflation or changing rates are modelled.
2. Follow the money
The starter uses 5% per period, deliberately high enough to make the pattern visible. It is not a quoted annual or monthly savings rate. A period may represent a year, month or another equal interval, but the interest rate must refer to that same interval.
With end-of-period payments, interest is calculated first and the new payment is then added. Starting from zero, the first four closing balances are $100, $205, $315.25 and $431.0125 (displayed as $431.01).
End payments: next balance = opening balance × (1 + r) + payment.
Beginning payments: next balance = (opening balance + payment) × (1 + r).
A beginning-of-period payment earns interest in the period in which it arrives. For the starter, the fourth closing balance becomes $452.563125, displayed as $452.56. The own contributions are still $400; the rest is interest.
From four rows to an interest factor
For four $1 end payments at 5% per period, the final balance is 1.053 + 1.052 + 1.05 + 1 = 4.310125. Multiply this factor by the regular payment: $100 × 4.310125 = $431.0125.
For n periods, end-payment factor = ((1 + r)n − 1) / r. At r = 0, use n instead of dividing by zero. Beginning-payment factor = end-payment factor × (1 + r). A non-zero starting balance grows separately as P(1 + r)n.
3. Build it in a spreadsheet
Download the classroom workbook (.xlsx)
Start on the student sheet: complete the first four periods before checking the worked models. The workbook includes editable inputs, 24-period beginning/end-payment models, a factor table and teacher answers. It uses visible arithmetic, rather than hiding the investigation inside an FV function.
- Calculate the first four closing balances with $200 paid at the end of each period, starting from zero at 5% per period. Predict before changing the payment.
- A student adds four payments and gives all $400 four periods of interest. Explain which payments have been given too much interest.
- Change the rate to 0%. Explain why beginning and end payments now agree.
- Use an interest factor to check a $50 end payment over four periods at 5%. Then explain how to include a separate $1,000 starting balance.
Reveal worked answers
- $200, $410, $630.50 and $862.025 (displayed $862.03). With zero starting balance, doubling every payment doubles the whole balance.
- The four end payments earn interest for 3, 2, 1 and 0 periods respectively, not four periods each. The last payment arrives only at the end.
- No interest is added. Four $100 payments give $400 in either timing model.
- $50 × 4.310125 = $215.50625, displayed $215.51. Add $1,000 × 1.054 = $1,215.50625 from the starting balance, giving $1,431.0125 in total, displayed $1,431.01.
Teaching notes and curriculum
Ask students to identify the exact time interest and payments occur before writing a formula. The first four rows should be understood before extending the spreadsheet.
Designed around NSW Mathematics Standard 2 Year 12 MST-12-S2-03, including first-four-period modelling, spreadsheets and interest-factor tables. This references the 2024 syllabus, with Year 12 teaching from Term 4 2026 and the first HSC examination in 2027. It is not labelled as 2026 HSC revision or endorsed by NESA.
For a monthly scenario, use the monthly deposits guide. A nominal annual rate compounded monthly is divided by 12; an effective annual rate is not. Keep those conventions separate from the per-period rate used here.
Return to the classroom learning sequence or practise the Excel compound interest formula.