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Maths IA · Modelling · topic 73 of 97

Home loans: where does each payment go? · IB Maths IA idea: guide, sample graph and step-by-step plan

same payment, changing split: that's your IA

Pay the bank the same amount every month for 25 years, and in year one most of it never touches the loan. You're going to find the month that flips, and what a rate rise does to it.

Difficulty●●○ Medium
KitA spreadsheet, one real home-loan rate or quote, your GDC
TimeAbout 12 to 15 hours
FitsAA SL3 of 3AA HL3 of 3AI SL3 of 3AI HL3 of 3
  • ✓ your family has talked about a mortgage or a flat loan
  • ✓ you like a formula you can check line by line against a spreadsheet
  • ✓ you want an answer that matters outside school

The big idea

An amortised loan has a fixed monthly payment, but what that payment does changes every month. Early on, most of it is interest on a large balance; later, most of it pays the loan down. Write the balance as a geometric series and the interest part of payment k becomes a formula in k. From there you can find the crossover month exactly, the total interest as a function of the rate, and how sensitive that total is to a small rate change. The formula is the routine part. The marks come from what you do with it: compare terms, test a rate rise both ways (a higher payment or a longer loan), and check every result against a month-by-month table.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Find the crossover month

Solve for the payment where the interest part and the principal part are equal.

→ An exponential equation in k, solved with logarithms. Check it against your table and explain why it lands so late in the loan.
Raise the rate halfway

The rate rises by one percentage point after year 5. Keep the payment and extend the loan, or keep the end date and raise the payment.

→ Two new models from one formula. Compare the extra interest each way and decide which a borrower should choose.
Total interest as a function of r

Write the total interest I(r) = nP(r) − L and study its shape for several terms.

→ Differentiate to get the cost of each extra 0.1 percentage point. At HL, show I(r) is increasing and concave up and say what that means.
Pay a lump sum

Put an extra amount in at year 3 and recompute how long the loan lasts.

→ A new starting balance and a log equation for the months saved. Compare months saved per dollar at different times.
Shorter or longer loan

Compare 15, 20, 25 and 30-year terms for the same loan and rate.

→ A family of curves. Find where extra years stop lowering the payment much, which is a limit as n grows.
Check a real schedule

Use a lender's published repayment table or calculator output as your data.

→ Your formula against their numbers. Any gap (rounding, day count, payment timing) becomes part of your evaluation.

Your Maths IA research question: run the check

which loan? which number?

Write your own question first. Then make sure it ticks every box:

  • Names the loan: amount, annual rate, term and how often it is paid, real or assumed with a reason.
  • Names the quantity you will find: the crossover month, the total interest, or the cost of a rate change.
  • Says what the formula is tested against: a month-by-month table or a lender's own schedule.
  • Goes beyond a loan calculator: there is a formula to derive and a decision to compare.
  • States what you leave out: fees, insurance, rate changes, rounding to the cent.
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The model

One payment a month, the same size every time. Split each payment into interest and principal and draw the split for your own loan.

160120180240300P/2payment number k (25-year loan, 300 payments)interest: r·Bk−1principal: the rest of Psame payment P every month......but the split changescrossover: interest = principal.find this k exactly

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why this loan, where the rate comes from, and what you expect before calculating: when does the split flip?
The loan as a sequence~2 pages
Every symbol defined. The balance month by month, the geometric series spotted and summed, the payment formula derived.
Splitting each payment~1.5 pages
The interest part and the principal part as functions of k, checked against a spreadsheet for a few months.
The crossover month~1.5 pages
The equation, the logarithm step and the answer, then how the crossover moves as the rate changes.
A rate rise, two ways~3 pages
Higher payment against longer term: two models, the extra interest in each, one graph comparing them.
Model against a real schedule~1.5 pages
Your formula beside a lender's table. A number for the difference and a reason for it.
Conclusion~1 page
Back to your aim: when the split flips, and which response to a rate rise costs less, with numbers.
Evaluation~2 pages, the biggest part!
Fixed rates, ignored fees, rounding, what a real borrower would also weigh. Say which assumption matters most.
Bibliographynot counted
Where every rate, rule and schedule came from.

What your Maths IA graph could look like

mark the crossover!

A made-up loan to show the form of a strong graph, not the answer: 400,000 over 25 years at 4% a year. Yours will look different. example only

05010015020025030004008001200160020002400Payment number k (months)Part of the payment / dollarsinterest partprincipal partyour spreadsheet values (made-up)the two parts cross near k = 93:solve for k with logarithmsILLUSTRATIVE · MADE-UP DATA
  • Both parts of the payment on one set of axes, so the crossover is visible at a glance.
  • Axis labels give units: payment number in months, amounts in dollars.
  • Calculate the crossover, then mark it. Reading it off the graph is not enough.
  • Plot a few rows of your spreadsheet as points: agreement with the curve is your evidence the formula is right.
  • Another graph to consider: total interest against the annual rate, one curve for each loan term.

The maths in action

worked through, not sketched!

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

Two panels. Left: stacked areas of interest and principal per monthly payment over 300 payments, with crossover lines at payments 93 and 162. Right: total interest rising with the interest rate for 15, 20, 25 and 30-year terms, with a tangent line at 4 percent.
(a) The interest and principal parts of each payment on a 400,000 loan over 25 years: at 4% the parts cross at payment 93, at 6% not until payment 162. (b) Total interest against the annual rate for four terms, with the tangent at 4% for 25 years.Source: Computed from the amortisation formula (no real borrower data).

How much interest does the term cost?

Rank the terms on one number: total interest paid. Work it out for your own loan, and explain why the gaps between terms are not equal.

15 years133k20 years182k25 years233k30 years287klonger term, lower payment, much more interestTotal interest on 400,000 at 4% a year, in thousandsILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictFind out how a fixed-payment loan works. Before any calculation, guess in which year the principal part first beats the interest part.
  2. Choose the loanPick an amount, rate and term you can justify: a real quote, a published rate or a typical loan where you live.
  3. Derive the paymentWrite the balance after each month, spot the geometric series and derive the payment formula. Define every symbol.
  4. Trial run (1 lesson)Build a 12-month spreadsheet and compare it with your formula. Fix any mismatch now: monthly rate, payment timing, rounding.
  5. Split the paymentsFind the interest and principal parts of payment k as formulas, and check three of them against the table.
  6. Solve for the crossoverSet the two parts equal, solve with logarithms, and repeat for three or four rates. Describe how the crossover moves.
  7. Test a rate riseModel the same rise twice: a higher payment, or a longer loan. Compare the extra interest each way on one graph.
  8. Go further, with a reasonDifferentiate the total interest with respect to the rate, or add a lump-sum payment. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it means for a real borrower and what it assumes. These lines become your evaluation.
  10. Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Geometric series – Write the balance after payments as a geometric series and derive the fixed payment from .
  • Exponential equations and logarithms – Solve for the crossover payment, and solve for the new term after a rate rise.
  • Functions of the rate – Treat total interest as a function and compare its graph for several terms.
  • Differentiation – Find with the quotient and chain rules to price each extra 0.1 percentage point; check concavity with .
  • Technology as a check – A month-by-month spreadsheet that every formula result is tested against.

Key equations

Syllabus links

  • 1.3Geometric sequences and series
  • 1.4Compound interest and depreciation
  • 1.7Rational exponents and laws of logarithms
  • 2.9Exponential and logarithmic functions
  • 5.6Standard derivatives; chain, product and quotient rules
  • 5.7Second derivative and graph behaviour

Read up first

Data and sources to start from

A published home-loan rate (bank, housing board or national average) and, if possible, a lender's repayment schedule to test the formula against.

References to cite

  1. Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
  2. Federal Reserve Bank of St. Louis. (2026). 30-year fixed rate mortgage average in the United States [MORTGAGE30US]. FRED.
  3. Housing & Development Board. (2026). Housing loans. https://www.hdb.gov.sg/

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

monthly rate is not the annual rate
  • Using the annual rate as if it were monthly. Divide by 12 only for a nominal rate compounded monthly, and say which yours is.
  • Assuming the interest part falls in a straight line. It does not, and its shape is part of your answer.
  • Quoting a bank calculator's output without deriving the formula behind it.
  • Rounding the payment to the cent and not explaining why the last balance is not exactly zero.
  • Comparing a rate rise only by the new payment, not by the total extra interest.
  • Calling a 30-year loan cheaper because the payment is lower, without counting the extra years of interest.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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