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B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

Financial Mathematics: Compound Interest and Annuities: Unit Review

B3.1 B3.2 B3.3 B3.4 B3.5 B3.6 B3.7About 40 minutes
Learning goals
  • I can compare simple and compound interest and explain why compound interest is exponential growth.
  • I can find the amount, the principal (present value), the rate and the number of periods for compound interest.
  • I can find the future value, present value and regular payment of an ordinary simple annuity, and the total interest.
  • I can investigate how changing conditions affects an annuity and make a justified financial decision.

Words to know

compound interest
Interest on the principal and on interest already earned: A = P(1 + i)ⁿ.
present value
The value now of a future amount: P = A(1 + i)⁻ⁿ.
ordinary simple annuity
Equal payments at the end of each period, with payment interval = compounding period.
future value of an annuity
FV = R[(1 + i)ⁿ − 1] ÷ i.
present value of an annuity
PV = R[1 − (1 + i)⁻ⁿ] ÷ i.
TVM Solver
A tool with N, I%, PV, PMT, FV, P/Y and C/Y that solves for any one missing value.

Let's review!

In this unit you compared simple and compound interest and saw that compound interest is exponential growth. You used A = P(1 + i)ⁿ to find future values and P = A(1 + i)⁻ⁿ to find present values, and solved for the rate and the time with rational exponents, tables and the TVM Solver. Then you met annuities: you found their future value, present value and regular payment, built an amortization table, investigated how changing the conditions changes the result, and used real, dated Canadian rates and rules to make decisions.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For every question, first write down what is known (P, A, R, i, n) and what is unknown, and decide: lump sum or annuity? Now or later? Find i and n carefully from the words. Keep full calculator precision and round money to the cent only at the end. For every mistake, write down why (wrong formula? wrong i or n? bracket error? sign error in the TVM Solver?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Which tool? Ask: one amount or a stream? Is the unknown now or later?Lump sum, find later valueA = P(1 + i)ⁿLessons 1, 2, 4Lump sum, find value nowP = A(1 + i)⁻ⁿLesson 3Lump sum, find i or ni = (A ÷ P)^(1/n) − 1; table or TVM for nLessons 5, 6Annuity, find value at the endFV = R[(1 + i)ⁿ − 1] ÷ iLessons 7, 8Annuity, find value now (loan or fund)PV = R[1 − (1 + i)⁻ⁿ] ÷ iLesson 9Annuity, find the paymentsolve FV or PV formula for R; amortizationLessons 10, 11, 12
A map of the unit. Decide whether there is one amount (lump sum) or a stream of equal payments (annuity), and whether the unknown is now or later.
Diego's car loan (Lesson 10)N=60I%=7.2PV=15000PMT=-298.44FV=0P/Y=12C/Y=12PMT=END BEGINPV = +15 000He receives the loan: positive.PMT = −298.44He pays each month: negative.FV = 0Nothing owed at the end.
TVM Solver sign check: money you receive is positive and money you pay out is negative. For a savings plan the signs flip: deposits (PMT) are negative and the future value is positive.
Just for fun

Review tip from compound interest itself: a little studying every day earns interest on yesterday's studying. Cramming the night before is more like simple interest, with a late fee.

Simple Interest and Compound Interest: Tables and Graphs

1

Find the amount after 4 years on $900 at 5%: (a) simple interest, (b) compounded annually.

2

Values $700, $742, $786.52, $833.71: simple or compound? Rate?

3

On a graph, which kind of interest gives a straight line?

The Compound Interest Formula: Finding the Amount (Future Value)

1

Find i and n for 5.4% compounded monthly for 3 years.

2

Find the amount: $2 200 at 5.4% compounded monthly for 3 years. Interest?

3

$1 000 at 4% compounded quarterly for 2 years, then reinvested at 6% compounded semi-annually for 1 year. Final amount?

Present Value: How Much Do I Need to Invest Now?

1

How much must be invested now at 4.5% compounded semi-annually to have $6 000 in 5 years?

2

Is the present value of $1 000 due in 3 years more or less than $1 000 (positive rate)?

3

$5 000 now or $5 600 in 3 years, at 3% compounded annually? Which is worth more today?

Compound Interest Is Exponential Growth

1

A(n) = 3200(1.0075)ⁿ, n in months. Principal? Annual rate?

2

Rule of 72: about how long to double at 6% compounded annually?

3

Domain of A(n) = 500(1.04)ⁿ for an investment of 5 years compounded annually?

Finding the Interest Rate: Rational Exponents and the TVM Solver

1

$4 000 grows to $5 000 in 5 years, compounded annually. Find the rate.

2

$1 000 grows to $1 300 in 4 years, compounded quarterly. Find the nominal annual rate.

3

In a TVM Solver, why must PV and FV have opposite signs?

Finding the Number of Compounding Periods: How Long Will It Take?

1

Smallest whole n with 1.04ⁿ ≥ 1.5?

2

A TVM Solver gives N = 30.3 with monthly compounding. When is the goal reached?

3

How long for $2 500 to reach $3 000 at 6% compounded monthly?

What Is an Annuity? Regular Payments in Tables and Graphs

1

Is $80 at the end of each month, compounded quarterly, an ordinary simple annuity?

2

$400 at the end of each year for 3 years at 5% compounded annually. Make a balance table.

3

How many years does the first payment of an 8-year ordinary annuity earn interest?

The Future Value of an Ordinary Simple Annuity

1

FV of $60 at the end of each month for 4 years at 3.6% compounded monthly?

2

How much of that is interest?

3

TVM entries for the first question?

The Present Value of an Ordinary Simple Annuity

1

How much is needed now to withdraw $400 at the end of each month for 2 years at 6% compounded monthly?

2

A loan is repaid with 36 payments of $250 at 8.4% compounded monthly. Amount borrowed? Total interest?

3

For a loan, what does the PV of the payments represent?

Finding the Regular Payment: Saving Goals, Loans and Amortization

1

Monthly deposit to reach $4 000 in 3 years at 3% compounded monthly?

2

Monthly payment on a $9 000 loan at 6% compounded monthly for 3 years? Total interest?

3

First payment of that loan: how much is interest and how much is principal?

Changing the Conditions: What Makes an Annuity Grow or Cost More?

1

A savings annuity's payment is cut in half. What happens to its FV?

2

Which costs more total interest on the same loan at the same rate: a 4-year or a 6-year term?

3

$1 200 a year for 20 years at 5%: annual deposits (compounded annually) or $100 monthly (compounded monthly)? Find both.

Making Saving, Investing and Borrowing Decisions in Canada

1

What was the Bank of Canada's policy rate in October 2026, and when was it last held?

2

Name one thing a calculation does not show when choosing between a GIC and a savings account.

3

Payment and total interest on a $2 000 credit card balance at 19.99% compounded monthly, repaid in 12 months?

Check your understanding

Quiz

1.

Which shows compound interest?

2.

$3 000 at 4% compounded quarterly for 5 years grows to about:

3.

The present value of $8 000 due in 6 years at 5% compounded annually is about:

4.

Compound interest is an example of:

5.

$2 000 grew to $2 420 in 2 years, compounded annually. The rate is:

6.

A TVM Solver gives N = 18.2 quarters. The goal is reached after:

7.

An ordinary simple annuity has payments:

8.

The FV of $500 at the end of each year for 2 years at 4% compounded annually is:

9.

A loan of $10 000 is repaid with 48 payments of $240. The total interest is:

10.

Lengthening the term of a loan (same amount and rate) gives:

For parents and teachers

Use the review to find what to revisit:

  • Mixing up simple and compound interest, or misreading tables and graphs → Lesson 1 (Simple Interest and Compound Interest: Tables and Graphs).
  • Wrong i or n, or bracket errors in A = P(1 + i)ⁿ → Lesson 2 (The Compound Interest Formula).
  • Trouble working backwards or with negative exponents → Lesson 3 (Present Value).
  • Not recognizing growth factors, domain, or doubling time → Lesson 4 (Compound Interest Is Exponential Growth).
  • Errors finding the rate (missing brackets around 1/n, forgetting × 12) → Lesson 5 (Finding the Interest Rate).
  • Rounding n down, or not converting to years and months → Lesson 6 (Finding the Number of Compounding Periods).
  • Not sure what counts as an ordinary simple annuity → Lesson 7 (What Is an Annuity?).
  • Using A = P(1 + i)ⁿ on the total of all deposits → Lesson 8 (The Future Value of an Ordinary Simple Annuity).
  • Using FV when the money is needed now, or misjudging a loan amount → Lesson 9 (The Present Value of an Ordinary Simple Annuity).
  • Choosing the wrong payment formula, or amortization table errors → Lesson 10 (Finding the Regular Payment).
  • Changing several conditions at once, or confusing a lower payment with a cheaper loan → Lesson 11 (Changing the Conditions).
  • Decisions without dates, sources, or non-math factors → Lesson 12 (Making Saving, Investing and Borrowing Decisions in Canada).

Looking ahead: Grade 12 Advanced Functions (MHF4U) teaches logarithms, which solve for n directly, and Foundations for College Mathematics (MAP4C) revisits annuities and personal finance.