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Maths IA · Modelling · topic 68 of 97

Credit cards: how long does the minimum payment take? · IB Maths IA idea: guide, sample graph and step-by-step plan

the minimum is built to last

Owe 3,000 on a card at 26% a year, pay only the minimum, and the debt takes nearly 18 years to clear. You're going to find out why it takes so long, and what a slightly bigger payment does to it.

Difficulty●●○ Medium
KitA spreadsheet, one real card's APR and minimum-payment rule, your GDC
TimeAbout 12 to 15 hours
FitsAA SL2 of 3AA HL3 of 3AI SL3 of 3AI HL3 of 3
  • ✓ you have seen a card statement and wondered why the balance barely moves
  • ✓ you like a model with two phases that you have to stitch together
  • ✓ you want to show what a rule of thumb really costs

The big idea

Each month the balance grows by interest, then the payment comes off. If the minimum is a percentage of the balance, the balance shrinks by a fixed factor every month: a geometric sequence, and it never quite reaches zero. Cards add a floor, a fixed minimum in dollars, so the rule switches partway down. That makes a piecewise model, two formulas stitched at the month where the floor takes over. Logarithms solve each phase for its number of months. The setting-up is routine. The marks come from comparing payment plans, explaining why the total interest changes so sharply, and testing the model against a real statement.

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 →

Solve the two phases separately

Find the month where the floor takes over, then the number of months the floor phase needs.

→ Two log equations joined by a switching point. Check the total against a month-by-month table.
Fixed payment instead

Pay the first month's minimum every month, never lowering it as the balance falls.

→ The model becomes one annuity-style formula. Compare the months saved and the interest saved with minimum-only.
The payment you can never beat

Find the smallest fixed payment that ever clears the debt.

→ A threshold: below monthly interest the balance grows forever. Explain it from the formula and show it on a graph.
Change the rate or the percentage

Vary the APR and the minimum percentage p, one at a time.

→ Study the months to clear as a function of each input and say which one matters more.
Daily interest

Charge interest daily, as many cards do, instead of once a month.

→ A different compounding rule. Measure how far the monthly model is off, with a percentage error.
A continuous model

Replace the monthly steps by a differential equation dB/dt = (ρ − p)B and solve it.

→ A smooth curve to compare with your table. HL work: separate variables, then judge where the approximation breaks.

Your Maths IA research question: run the check

which card? which number?

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

  • Names the debt: starting balance, annual rate and how the card computes interest, real or assumed with a reason.
  • States the minimum-payment rule exactly: the percentage, the floor and what is included.
  • Names the quantity you will find: months to clear, total interest, or the cost of a change.
  • Says what the model is tested against: a month-by-month table or a real statement.
  • Goes beyond a card calculator: there is a formula to derive and a decision to compare.
  • States what you leave out: new spending, fees, changing rates, late payments.
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The model

Interest is added, then the payment comes off. Draw the rule for your own card and decide where the floor takes over.

Each month: B → B + rB − payment, payment = max(pB, m)r = APR ÷ 12, p = minimum percentage, m = minimum floorm/p015003000balance B (dollars)255075dollars per monthpayment: max(pB, m)interest rBfloor takes overbelow this balancethis gap is all thatrepays the loan

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 card, where the rate and the minimum rule come from, and what you expect before calculating.
The balance as a sequence~2 pages
Every symbol defined. The monthly step, the geometric phase and why it never reaches zero.
The floor and the switching month~1.5 pages
Where the minimum changes rule, the formula for that month and a check against a table.
Months to clear and total interest~2 pages
The log equations for both phases, the total interest, and the same numbers from a spreadsheet.
Other payment plans~3 pages
Fixed payments of different sizes: months, interest and the threshold payment, on one graph.
Model against a real statement~1.5 pages
Your formula beside a real or published statement. A number for the gap and a reason for it.
Conclusion~1 page
Back to your aim: how long the minimum takes, what it costs, and which plan you would advise.
Evaluation~2 pages, the biggest part!
Daily interest, new spending, fees, changing rates. Say which assumption matters most and test it.
Bibliographynot counted
Where every rate, rule and statement came from.

What your Maths IA graph could look like

mark the switch month!

A made-up card to show the form of a strong graph, not the answer: 3,000 owed at 26% a year, minimum 3% or 25, whichever is larger. Yours will look different. example only

0204060801001201401601802002200400800120016002000240028003200Months since the balance was 3,000, nBalance owed B / dollarsminimum only: 3% phaseminimum only: floor phasefixed 90 a monthyour statement balances (made-up)floor takes over nearmonth 154: where exactly?ILLUSTRATIVE · MADE-UP DATA
  • Balance in dollars against months, with both payment plans on one set of axes.
  • The kink where the minimum rule switches is a result: calculate it, then mark it.
  • The fixed payment ends years earlier. Say where the curve meets zero and how you found it.
  • Plot a few real statement balances as points: agreement with the curve is your evidence the model works.
  • Another graph to consider: months to clear against the fixed monthly payment, with the threshold marked.

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: balance against months for four plans, each ending where it reaches zero, with the minimum-only curve far longer than the rest. Right: total interest against fixed monthly payment, rising steeply as the payment falls towards a vertical threshold, with the minimum-only plan marked.
(a) A 3,000-dollar balance at 26% APR: minimum-only (3% or 25 dollars) takes 213 months (17.8 years); paying a fixed 90, 150 or 300 dollars takes 60, 27 or 12 months. (b) Total interest against a fixed payment, with the threshold at 65 dollars.Source: Computed from the model (no real cardholder data); assumed card terms.

How much interest does each plan cost?

Rank the plans on one number: total interest paid. Compute it for your own card, and explain why a small rise in the payment saves so much.

Minimum only6294Fixed 90 a month2378Fixed 150 a month975Fixed 300 a month418the minimum-only plan costs more than the debt itselfTotal interest paid on 3,000 at 26% a year, in dollarsILLUSTRATIVE · 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 card's minimum payment and interest work. Before any calculation, guess how many years minimum-only takes.
  2. Choose the cardPick a balance, an APR and a minimum rule you can justify: a real card's terms, a published rate or a typical case where you live.
  3. Write the monthly stepDefine every symbol and write the balance after one month. Decide how interest is charged and say so.
  4. Trial run (1 lesson)Build a 12-month spreadsheet and compare it with your formula. Fix any mismatch now: rate per month, order of interest and payment.
  5. Solve the geometric phaseSpot the geometric sequence, find where the floor takes over and use logarithms for the number of months.
  6. Solve the floor phaseTreat the fixed payment as an annuity and solve for the months left. Add the two phases and check against the table.
  7. Compare payment plansRun fixed payments of several sizes. Find the threshold payment and the months and interest for each plan.
  8. Go further, with a reasonTest daily interest or build the continuous model. 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 cardholder 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 sequences – Show that with a percentage-only minimum the balance after months is , and explain why it never reaches zero.
  • Logarithms – Solve for the switching month, and solve the fixed-payment phase for its number of months.
  • Piecewise models – Join the geometric phase and the floor phase at the month where first holds, and check the join is continuous.
  • Annuity formula – Treat the floor phase as a loan repaid by a fixed payment, using the sum of a geometric series, to find months and total interest.
  • Threshold analysis – Show that a fixed payment clears the debt only if , and graph months to clear as approaches that value.
  • Differential equation (HL) – Solve by separation of variables and compare with the monthly recurrence.

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
  • 5.18First-order differential equations; Euler's methodHL

Read up first

Data and sources to start from

Take a real card's APR and minimum-payment rule from its published terms, and check the model against a real or published statement.

References to cite

  1. Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
  2. Credit Card Accountability Responsibility and Disclosure Act of 2009, Pub. L. No. 111-24, 123 Stat. 1734.
  3. Board of Governors of the Federal Reserve System. (2026). Consumer credit: G.19. https://www.federalreserve.gov/releases/g19/current/

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

Traps to avoid in this Maths IA

interest first, then payment
  • Using the annual rate as if it were monthly. Divide by 12 only for a nominal rate charged monthly, and say which yours is.
  • Forgetting that a percentage-only minimum never reaches zero. The floor is what finishes the loan, and your model needs it.
  • Rounding the number of months down. Say why you round up, and what the last payment looks like.
  • Ignoring the order of events: interest on the balance before the payment, or after it, gives different numbers.
  • Quoting a card website's calculator without deriving the formula behind it.
  • Treating the rule as universal. Minimum rules differ between countries and cards; cite the one you use.

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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