Maths IA · Modelling · topic 72 of 97
Flat rate or effective rate: what does a car loan really cost? · IB Maths IA idea: guide, sample graph and step-by-step plan
the quoted rate hides the real oneA lender says 3% a year, flat, for 5 years. The real rate on what you actually owe is closer to 5.6%. You're going to find why the two rates differ, and how to turn one into the other.
- ✓ your family has compared a car or personal loan offer
- ✓ you like equations that have no neat answer and must be solved numerically
- ✓ you want to show that a quoted rate can hide the real cost
The big idea
A flat rate charges interest on the full amount borrowed for the whole term, even though each payment shrinks what you owe. So the monthly payment is easy to write down, and the cost looks small. The effective rate asks a different question: what rate on the outstanding balance gives the same payments? That rate solves an annuity equation with no algebraic solution, so you find it numerically, with a solver or Newton's method. The setting-up is routine. The marks come from the comparison: how the gap changes with the flat rate and the length of the loan, how good a shortcut formula is, and what a borrower should conclude.
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 →
Equate the flat-rate payment to the annuity formula and solve for the monthly rate.
→ An equation with no closed form. Solve it with a GDC solver, then check the answer against a month-by-month table.Code the iteration yourself and show how fast it converges from a rough starting guess.
→ A numerical method from scratch. Explain the starting value, the stopping rule and the error after each step.Test the rule that the effective rate is about 2n/(n+1) times the flat rate, where n is the number of payments.
→ An approximation and its error. Find when it is good and when it fails, as a function of the rate and the term.Convert your monthly-compounded rate into the annual effective rate (EIR).
→ Two ways of stating the same cost. Say which one a fair comparison needs, and why.Keep the flat rate fixed and change the loan from 1 to 10 years.
→ A family of results. Explain why the effective rate is nearly twice the flat rate and how the ratio moves.Include an arrangement fee, or a down payment, and recompute the effective rate.
→ A changed starting amount. Show how fees raise the true rate and say which assumption matters most.Your Maths IA research question: run the check
which loan? which rate?Write your own question first. Then make sure it ticks every box:
- Names the loan: amount, flat rate, term, how often it is paid, real or assumed with a reason.
- Names the quantity you will find: the effective rate, the gap from the flat rate, or the error of a shortcut.
- Says how the equation is solved and how you know the answer is correct.
- Says what the answer is tested against: a month-by-month table or a lender's own effective-rate figure.
- States what you leave out: fees, early repayment, insurance, rounding to the cent.
The model
One loan, one set of equal payments, two ways of quoting its cost. Draw the timeline for your own loan and label what each rate measures.
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.
Why this loan, where the flat rate comes from, and what you expect before calculating.
Every symbol defined. The payment formula, the total interest and a month-by-month table.
The annuity equation, why it has no closed form, and the solution by a solver and by Newton's method.
The gap as a function of the flat rate and the term, on one graph, with a reason for its size.
The shortcut formula, tested against exact answers, with a number for the error and where it fails.
Your formula beside a lender's quoted effective rate. A number for the difference and a reason for it.
Back to your aim: the real cost of a flat-rate loan and what a borrower should compare.
Fees, early repayment, rounding, solver accuracy. Say which assumption matters most and test it.
Where every rate, rule and offer came from.
What your Maths IA graph could look like
equal areas: why?A made-up loan to show the form of a strong graph, not the answer: 50,000 over 5 years at 3% flat. Yours will look different. example only
- Interest in dollars against the payment number, both with units.
- Two descriptions of one loan on one set of axes: their areas must be equal.
- Say why the curve starts above the flat line and ends below it.
- Plot spreadsheet rows as points: agreement with the curve is your evidence the rate is right.
- Another graph to consider: the effective rate against the flat rate, one curve per 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.
How far is the effective rate from the flat rate?
Rank the quotes on one number: the effective rate that matches each flat rate. Work it out for your own term, and say why the ratio is close to the same each time.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out how flat-rate quotes work. Before any calculation, guess the effective rate for a 3% flat loan over 5 years.
- Choose the loanPick an amount, flat rate and term you can justify: a real offer, a published rate or a typical loan where you live.
- Write the flat-rate paymentDerive the total interest and the monthly payment from the flat rate. Define every symbol.
- Trial run (1 lesson)Build a 12-month spreadsheet at a guessed effective rate and see if the balance matches. Fix any mismatch now: monthly rate, timing.
- Set up the equationEquate your payment to the annuity formula, with the monthly rate as the unknown. Say why no algebra solves it.
- Solve it numericallyUse a solver, then Newton's method or bisection. Report the answer to a stated accuracy and check it in the table.
- Compare rates and termsVary the flat rate and the term, graph the effective rate, and describe the pattern.
- Go further, with a reasonTest a shortcut formula for the effective rate or add fees. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a borrower and what it assumes. These lines become your evaluation.
- 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
- Flat-rate payment – Total interest is , so the monthly payment is for payments.
- Annuity formula (geometric series) – Derive by summing the discounted payments, where is the monthly rate.
- Numerical solution – Solve for with a GDC solver, then by bisection or Newton's method, with a stated tolerance and a check in a table.
- Approximation and error – Test against exact answers and describe the percentage error as and change.
- Nominal and effective rates – Convert to the annual effective rate and say which one a lender's disclosure uses.
Key equations
Syllabus links
- 1.3Geometric sequences and series
- 1.4Compound interest and depreciation
- 2.10Solving equations graphically and analytically
- 1.7Rational exponents and laws of logarithms
- 1.3Geometric sequences and series
- 1.4Compound interest and depreciation
- 1.6Approximation, bounds and percentage error
- 1.7Loans, amortisation and annuities
- 1.8Solving systems and polynomial equations with technology
Read up first
Data and sources to start from
Take real flat-rate loan offers (term, flat rate, quoted effective rate where given) from lenders' published pages, and check the quoted effective rate against your own.
References to cite
- Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
- Burden, R. L., & Faires, J. D. (2011). Numerical analysis (9th ed.). Brooks/Cole.
- Truth in Lending Act, 15 U.S.C. § 1601 et seq.; Regulation Z, 12 C.F.R. part 1026.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
monthly, nominal or effective?- Calling the monthly rate the annual rate. State whether yours is monthly, a nominal annual rate or an annual effective rate.
- Quoting the solver's answer without checking it in a month-by-month table.
- Starting Newton's method from a guess with no reason, and not showing the error at each step.
- Comparing a flat rate with an effective rate as if they were the same kind of number.
- Testing a shortcut formula at one rate only, then calling it accurate.
- Leaving out fees and then comparing with a lender's effective rate that includes them.
More Maths IA ideas like this
Home loans: where does each payment go?●●○
Loans and savings: when does a plan break even?●○○
Credit cards: how long does the minimum payment take?●●○
Newton's method: how fast, and when does it fail?●●○
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.