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Module 7 · L3 of 12 ~25 min MS12-5 ⚡ +50 XP available

Comparing Interest Rates

Two banks offer savings accounts: Bank A advertises 6.2% p.a. compounded monthly. Bank B advertises 6.25% p.a. compounded semi-annually. Which pays more? At first glance, 6.25% beats 6.2%. But because Bank A compounds more frequently, its effective rate is actually higher. Banks know most customers won't do this calculation — they rely on nominal rates sounding impressive. In this lesson you will learn to strip away the marketing and compare financial products on equal terms using effective annual rates.

Today's hook — Bank X: 5.8% p.a. compounded quarterly. Bank Y: 5.9% p.a. compounded annually. Which is better for a saver? Predict before reading on.
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Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall — your gut answer first
+5 XP warm-up

Bank X: 5.8% p.a. compounded quarterly. Bank Y: 5.9% p.a. compounded annually. Which is better for a saver?

Before reading on — write your gut feeling. We will revisit this at the end of the lesson.

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Key ideas for this lesson
reference

Comparing financial products requires converting all rates to a common measure — the effective annual rate.

Nominal rate: the advertised annual rate before any compounding adjustments. What banks print on posters.

Effective annual rate: the true annual rate after accounting for compounding frequency. Use this to compare products fairly.

Flat rate: simple interest calculated on the original principal throughout the loan — makes true cost much higher than advertised.

EFFECTIVE ANNUAL RATE r_eff = (1 + r/k)^k - 1 r = nominal rate (decimal) k = compounding periods per year Rule of 72: years to double ≈ 72 ÷ rate%
Always convert to effective annual rate before comparing products with different compounding frequencies.
More frequent = higher effective
Monthly compounding gives a higher effective rate than quarterly, which beats semi-annual, which beats annual — even at the same nominal rate.
Flat rate trap
Flat rate loans charge interest on the original principal even as you repay it. The true (effective) rate can be nearly double the advertised flat rate.
Rule of 72
A quick mental tool: divide 72 by the interest rate (%) to estimate the number of years for an investment to double in value.
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What you will master
Know

Key facts

  • Nominal vs effective annual rate
  • Flat rate definition
  • Rule of 72
Understand

Concepts

  • Why compounding frequency affects the true rate
  • How banks use nominal rates to attract customers
  • Why flat rate loans cost more than they appear
Can do

Skills

  • Calculate effective annual rates
  • Compare different financial products
  • Estimate doubling time using Rule of 72
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Key terms
Nominal rateThe advertised annual interest rate before accounting for compounding frequency.
Effective annual rateThe true annual rate after compounding, used to compare products fairly. Also called the annual equivalent rate (AER).
Compounding frequency (k)How many times per year interest is calculated and added: monthly k=12, quarterly k=4, semi-annual k=2, annual k=1.
Flat rateA simple interest rate charged on the original principal for the full loan term, even as the balance reduces.
Rule of 72An approximation: years to double ≈ 72 ÷ interest rate (%). Useful for quick mental calculations.
Reducing balanceInterest calculated only on the outstanding balance — the standard for most mortgages and personal loans.
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Effective vs nominal rates
core concept

The nominal rate is the advertised rate. The effective annual rate is what you actually earn or pay after accounting for how often interest compounds.

Effective annual rate formula:

$$r_{\text{eff}} = \left(1 + \frac{r}{k}\right)^k - 1$$

where $r$ is the nominal rate (as a decimal) and $k$ is the number of compounding periods per year.

Steps to compare two products:

  1. Write down the nominal rate $r$ and compounding frequency $k$ for each product.
  2. Calculate $r_{\text{eff}} = (1 + r/k)^k - 1$ for each.
  3. Compare the effective rates directly — higher is better for savers, lower is better for borrowers.
Worked example: Bank A: 6.2% compounded monthly ($k=12$). Bank B: 6.25% compounded semi-annually ($k=2$).
Bank A: $(1 + 0.062/12)^{12} - 1 = 6.38\%$ effective
Bank B: $(1 + 0.0625/2)^{2} - 1 = 6.35\%$ effective
Bank A is slightly better for a saver despite the lower nominal rate!
Key insight: The same nominal rate compounds to different effective rates depending on frequency. Monthly > quarterly > semi-annual > annual for the same nominal rate. This is why banks often advertise nominal rates — they sound lower for loan products.
What to write in your book
  • Effective annual rate: $r_{\text{eff}} = (1 + r/k)^k - 1$ where $r$ = nominal rate, $k$ = periods per year.
  • More frequent compounding = higher effective rate for the same nominal rate.
  • Always convert to effective rate before comparing financial products.

Quick check: Bank A offers 4.8% p.a. compounded monthly. Bank B offers 4.9% p.a. compounded annually. Which has the higher effective annual rate?

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Flat rate loans — when simple interest disguises itself
core concept

Some loans advertise a flat rate of simple interest, then spread repayments evenly across the term. This makes the true interest cost much higher than it appears.

How flat rate loans work:

  • Total interest = Principal × flat rate × years
  • Total repayment = Principal + Total interest
  • Monthly repayment = Total repayment ÷ number of months

The problem: You repay principal throughout the loan, but interest is still charged on the original amount. By the halfway point, you've repaid half the principal — but you're still paying interest as if you owed the full amount.

Worked example: $10,000 loan at 8% flat rate over 4 years.
Total interest = $10{,}000 \times 0.08 \times 4 = \$3{,}200$
Total repayment = $13,200. Monthly = $13,200 ÷ 48 = $275.
The true effective rate is approximately 14–15% — nearly double the advertised flat rate.
Real-world trap: Car dealerships and furniture stores frequently use flat rates. A "0% interest" deal with a large "establishment fee" can have a significant hidden true rate. Always calculate the total cost and compare.
What to write in your book
  • Flat rate: interest on original principal for full term, even as balance reduces.
  • Total interest = P × r_flat × n. Total = P + interest.
  • True effective rate on a flat rate loan is approximately double the advertised flat rate.

True or false: A loan with a 6% flat rate has a true (effective) interest rate of approximately 6% p.a.

Worked examples · reveal each step

PROBLEM 1 · COMPARING THREE PRODUCTS

Compare: (A) 5.5% compounded monthly, (B) 5.6% compounded quarterly, (C) 5.7% compounded annually. Find the best effective rate for a saver.

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Product A: $r = 0.055$, $k = 12$
Identify nominal rate and compounding frequency for each product
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PROBLEM 2 · FLAT RATE LOAN

A car loan advertises 7% flat rate over 5 years on $25,000. Find the total repayment and monthly repayment.

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Total interest = $25{,}000 \times 0.07 \times 5 = \$8{,}750$
Flat rate: interest on full principal for all 5 years
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Rule of 72 — quick mental estimation
core concept

The Rule of 72 estimates how long an investment takes to double in value:

$$\text{Doubling time} \approx \frac{72}{\text{interest rate (\%)}}$$

Examples:

  • At 6% p.a.: doubles in approx. $72 \div 6 = 12$ years. Exact: $\ln(2)/\ln(1.06) = 11.9$ years.
  • At 4% p.a.: doubles in approx. $72 \div 4 = 18$ years.
  • At 8% p.a.: doubles in approx. $72 \div 8 = 9$ years.
  • At 12% p.a.: doubles in approx. $72 \div 12 = 6$ years.
Why it works: The Rule of 72 is a clever approximation of the exact formula $t = \ln(2)/\ln(1+r)$. Because $\ln(2) \approx 0.693$ and for small $r$, $\ln(1+r) \approx r$, we get $t \approx 0.693/r \approx 70/r$. Using 72 instead of 69.3 gives slightly better accuracy for typical interest rates of 4–12%.
HSC application: Use the Rule of 72 for quick sanity checks and estimates. For exact answers, use the compound interest formula or logarithms.
What to write in your book
  • Rule of 72: doubling time (years) ≈ 72 ÷ interest rate (%). Quick estimation only.
  • Exact doubling time: $t = \ln(2) / \ln(1+r)$.
  • At 6%: ≈ 12 years. At 8%: ≈ 9 years. At 12%: ≈ 6 years.

Fill the gap: Using the Rule of 72, an investment at 9% p.a. will double in approximately years.

Common errors · the 3 traps that cost marks

Trap 01
Comparing nominal rates directly
Never rank financial products by nominal rate alone. 6.2% compounded monthly beats 6.25% compounded semi-annually for a saver. Always calculate effective annual rates first.
Trap 02
Forgetting to convert r to a decimal
If the nominal rate is 6.2%, you must use $r = 0.062$ in the formula $(1 + r/k)^k - 1$. Using $r = 6.2$ gives a nonsensical answer many thousands of times too large.
Trap 03
Thinking flat rate = effective rate
A flat rate loan at 8% does not have an 8% true interest cost. Because you repay principal but still pay interest on the original amount, the effective rate is roughly double the flat rate.
What to write in your book
  • Always convert to effective annual rate before comparing products.
  • Use $r$ as a decimal in formulas (6.2% → 0.062).
  • Flat rate true cost ≈ double the advertised rate.

Match each term to its correct description:

Quick-fire practice · 2 activities

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Bank A offers 5.4% p.a. compounded monthly. Bank B offers 5.5% p.a. compounded quarterly. (a) Calculate the effective annual rate for each. (b) Which is better for a saver? (c) Use the Rule of 72 to estimate the doubling time for each effective rate.

2

A store offers a $2,000 purchase with "0% interest" but charges a $200 establishment fee, repaid over 12 months. (a) What is the total cost? (b) What is the true annual interest rate? (c) Explain why this is misleading advertising.

Top 3 list: List THREE real-world situations where understanding effective annual rates would help a consumer make a better financial decision. For each, explain what they should calculate and why.

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Revisit your thinking

Bank X effective rate = $(1 + 0.058/4)^4 - 1 = 5.92\%$. Bank Y effective rate = 5.9% (annual compounding means effective = nominal). Bank X is slightly better despite the lower nominal rate because quarterly compounding boosts the effective return above Bank Y's 5.9%. This is why comparing nominal rates alone can mislead — always convert to effective rates before making financial decisions.

What has changed in your understanding? What did you get right? What surprised you?

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence — that tells the system what to drill next.

Q1. What is the effective annual rate for a nominal rate of 6% p.a. compounded monthly?

Q2. A $15,000 loan at 10% flat rate over 3 years. What is the total interest charged?

Q3. Using the Rule of 72, approximately how long does it take for $1,000 to double at 8% p.a.?

Q4. Which of the following gives the HIGHEST effective annual rate?

Q5. Why is a flat rate loan more expensive than it appears?

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Short answer
ApplyBand 42 marks

SA 1. Calculate effective annual rates for: (i) 6.4% compounded monthly, (ii) 6.5% compounded quarterly, (iii) 6.6% compounded annually. Rank them from best to worst for a saver and explain why the ranking differs from the nominal rate ranking. (2 marks)

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ApplyBand 42 marks

SA 2. A $20,000 car loan at 6% flat rate over 4 years. (a) Find the total interest. (b) Find the monthly repayment. (c) Explain why the true interest rate is much higher than 6%. (2 marks)

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AnalyseBand 53 marks

SA 3. (a) Use the Rule of 72 to estimate how long $5,000 takes to double at 9% p.a. (b) Calculate the exact doubling time using logarithms. (c) A bank advertises "Your money doubles in under 8 years!" What minimum nominal rate (compounded monthly) would make this claim true? (3 marks)

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Comprehensive answers (click to reveal)

MC 1 — C: $(1 + 0.06/12)^{12} - 1 = 6.168\% \approx 6.17\%$.

MC 2 — B: $15{,}000 \times 0.10 \times 3 = \$4{,}500$.

MC 3 — A: $72 \div 8 = 9$ years.

MC 4 — D: For the same nominal rate, more frequent compounding always gives the highest effective rate.

MC 5 — B: Interest is calculated on the original principal even as repayments reduce the balance.

SA 1 (2 marks): (i) 6.59%, (ii) 6.66%, (iii) 6.60% [1 mark]. Ranking: ii > iii > i. Quarterly compounding on 6.5% more than compensates vs annual 6.6% [1 mark].

SA 2 (2 marks): (a) $4,800 (b) $516.67/month [1 mark]. (c) By halfway, $10,000 repaid but interest still on $20,000 full balance — true rate ≈ 11–12% [1 mark].

SA 3 (3 marks): (a) 8 years [0.5]. (b) $\ln(2)/\ln(1.09) = 8.04$ years [1]. (c) $(1+r/12)^{96} \geq 2$; $r/12 \geq 2^{1/96} - 1 = 0.00724$; $r \geq 8.69\%$ [1.5].

Drill 1: A effective: 5.535%; B effective: 5.576%. B is better. Doubling: A ≈ 13.0 yrs; B ≈ 12.9 yrs.

Drill 2: Total = $2,200. True rate = $200 on $2,000 over 1 year = 10% (flat rate). Misleading because "0% interest" hides the fee which is functionally interest.

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Boss battle · The Rate Inspector
earn bronze · silver · gold

Five timed questions on effective rates, flat rate loans, and the Rule of 72. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Enter the arena
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Science Jump · platform challenge

Climb platforms by answering questions on comparing interest rates. Pool: lesson 3.

Mark lesson as complete

Tick when you've finished the practice and review.

Module overview · Maths Standard