Compound Interest and Depreciation Explained
Compound interest questions become easier when you understand what changes each period and what the question asks you to find. For Winnipeg students reviewing financial math, this guide explains how to calculate investment growth, solve for time or an interest rate, and work with depreciation.
We will connect the formulas to geometric sequences and use four worked examples to show how to set up a TVM calculator. Each investment example assumes one initial deposit, a constant interest rate, and no further deposits or withdrawals.
How Compound Interest Works
Compound interest is interest calculated on the original investment and on interest already added to the account. The original investment is called the principal.
Suppose you invest $1,000 at 5% interest compounded annually.
After one year:
$1,000 × 1.05 = $1,050
After two years:
$1,050 × 1.05 = $1,102.50
The second year earns $52.50 because the 5% applies to the updated balance. Multiplying by 1.05 means keeping 100% of the balance and adding another 5%.
The Connection to Geometric Sequences
The balances $1,000, $1,050, $1,102.50, … form a geometric sequence with a common ratio of 1.05. Each new balance is found by multiplying the previous balance by the same factor.
After n years, the balance is:
1,000(1.05)ⁿ
The exponent is n because n years means n interest calculations. The initial deposit occurs at time zero.
In a sequence formula such as aₙ = a₁qⁿ⁻¹, the exponent instead counts the steps from term 1 to term n.
The Compound Interest Formula
A = P(1 + r/m)ᵐᵗ
Here:
A is the final balance, including the original investment.
P is the principal.
r is the annual interest rate as a decimal.
m is the number of compounding periods per year.
t is the time in years.
For annual compounding, m = 1. For monthly compounding, m = 12.
The expression r/m means r divided by m and gives the rate per compounding period. The exponent mt means m multiplied by t and gives the total number of periods.
For example, 6% per year compounded monthly gives:
0.06 ÷ 12 = 0.005
That is 0.5% per month. Over four years, there are:
12 × 4 = 48 compounding periods
If asked for the interest earned, subtract the principal from the final balance:
Interest earned = A − P
How to Set Up a TVM Calculator
A time value of money calculator solves for a missing financial variable. The examples below use common TI-style field labels; menus vary by calculator.
These are the fields you need to recognize:
N: Total number of payment periods.
I%: Annual interest rate entered as a percentage.
PV: Present value or starting amount.
PMT: Regular payment or deposit each period.
FV: Future value or final amount.
P/Y: Number of payment periods per year.
C/Y: Number of compounding periods per year.
For these single-deposit examples, enter PMT = 0.
Set P/Y and C/Y both to 1 for annual compounding, or both to 12 for monthly compounding. This makes N count years or months, respectively, even though there are no regular payments.
Use a negative PV for money invested and a positive FV for money received.
For a 5% annual rate, enter 5 in I%; use 0.05 in the formula. Check every field before solving so old settings do not carry into a new question.
Example 1: Finding Time with Annual Compounding
You invest $800 at 5% interest compounded annually. How long will it take to grow to $1,200?
Set Up the Equation
Here, P = 800, A = 1,200, r = 0.05, and m = 1. Substitute into the formula:
1,200 = 800(1.05)ᵗ
Divide both sides by 800:
1.5 = (1.05)ᵗ
The unknown is an exponent, so ordinary division will not isolate it. Use the TVM solver to find N, which equals t with these annual settings.
Enter the TVM Values
N: Solve
I%: 5
PV: −800
PMT: 0
FV: 1,200
P/Y: 1
C/Y: 1
The calculator gives:
N ≈ 8.31 years
If you have learned logarithms, the same result comes from:
t = log(1.5) ÷ log(1.05) ≈ 8.31 years
Interpret the Time Correctly
The equation gives approximately 8.31 years. If the question asks for the first completed annual compounding period when the balance reaches at least $1,200, check the adjacent whole years.
After 8 years:
800(1.05)⁸ ≈ $1,181.96
After 9 years:
800(1.05)⁹ ≈ $1,241.06
Therefore, the first qualifying year-end balance occurs after 9 years.
Follow the question’s wording when deciding whether to report the calculated time or a completed compounding period.
Example 2: Finding Time with Monthly Compounding
You invest $1,500 at 6% interest per year compounded monthly. How long will it take to grow to $2,000?
Convert the Rate and Count the Periods
The monthly rate is:
0.06 ÷ 12 = 0.005
If t is in years, there are 12t monthly periods:
2,000 = 1,500(1.005)¹²ᵗ
Dividing both sides by 1,500 gives:
4/3 = (1.005)¹²ᵗ
Here, 4/3 means four divided by three. Keep this fraction exact when entering it into a calculator.
Enter the TVM Values
Keep I% at 6 because this field takes the annual rate; the calculator handles the monthly conversion.
N: Solve
I%: 6
PV: −1,500
PMT: 0
FV: 2,000
P/Y: 12
C/Y: 12
Convert the Answer to Years
The calculator gives N ≈ 57.68. With P/Y = 12, this means 57.68 months.
Convert to years:
t = N ÷ 12
t ≈ 57.68 ÷ 12 ≈ 4.81 years
If using logarithms, the number of months is:
N = log(4/3) ÷ log(1.005) ≈ 57.68 months
For the first completed monthly compounding period that reaches the target, round up to 58 months, or 4 years and 10 months.
After 57 months:
1,500(1.005)⁵⁷ ≈ $1,993.23
After 58 months:
1,500(1.005)⁵⁸ ≈ $2,003.19
These values confirm that the investment first reaches the target at the end of month 58.
Example 3: Finding the Required Interest Rate
An investment of $600 must grow to $900 in 8 years. What annual interest rate is required if interest is compounded annually?
Isolate the Growth Factor
Substitute the known values:
900 = 600(1 + r)⁸
Divide both sides by 600:
1.5 = (1 + r)⁸
Take the eighth root of both sides.
1 + r = ⁸√(1.5)
Subtract 1:
r = ⁸√(1.5) − 1
r ≈ 0.0519895
To express this decimal as a percentage, multiply by 100:
0.0519895 × 100 = 5.19895
The required annual interest rate is approximately 5.20%.
Check Using the TVM Solver
N: 8
I%: Solve
PV: −600
PMT: 0
FV: 900
P/Y: 1
C/Y: 1
The solver gives I% ≈ 5.19895, confirming an annual rate of approximately 5.20%.
Do not multiply the TVM result by 100 again: I% is already a percentage.
Example 4: Calculating Depreciation
Depreciation is a decrease in an asset’s value. In a constant-percentage depreciation model, the same percentage is lost from the remaining value each period.
For annual depreciation, use:
V = P(1 − d)ᵗ
Here:
V is the remaining value.
P is the initial value.
d is the annual depreciation rate as a decimal.
t is the number of years.
A car is worth $24,000 and depreciates by 18% each year. What will it be worth after 5 years?
Find the Percentage Retained
The car retains:
100% − 18% = 82%
Its multiplier is therefore 0.82.
Substitute into the formula:
V = 24,000(1 − 0.18)⁵
V = 24,000(0.82)⁵
V ≈ $8,897.76
The car’s estimated value after 5 years is $8,897.76.
Its total loss in value is:
$24,000 − $8,897.76 = $15,102.24
Check Using the TVM Solver
Use a negative annual rate to model the decrease:
N: 5
I%: −18
PV: −24,000
PMT: 0
FV: Solve
P/Y: 1
C/Y: 1
The solver gives FV ≈ $8,897.76.
Multiplying the current value by 0.18 would calculate the amount lost in one year. Multiplying it by 0.82 calculates the amount remaining.
Common Mistakes in Compound Interest Questions
Mixing percentages and decimals. Enter 6 for a 6% rate in I%, but 0.06 in the formula.
Using an annual rate as a monthly rate. In the formula, divide the annual rate by 12 for monthly compounding. In the TVM solver, enter the annual rate and set the frequency fields correctly.
Confusing months and years. With the monthly settings used here, N counts months. Divide by 12 to express that time in years.
Leaving a previous payment in PMT. For a single initial deposit with no regular contributions, PMT must be zero.
Rounding too early. Keep full calculator precision during the solution. Round money to the nearest cent and interpret time according to the question.
Math Tutoring for Winnipeg Students
If compound interest or depreciation questions still feel confusing, individual tutoring can help you identify whether the difficulty is choosing a formula, setting up the calculator, or interpreting the answer.
At Tutor Advance, I offer online math tutoring for Winnipeg students, with worked examples and practice tailored to each student’s course. As your math tutor, I can help you understand the reasoning behind financial math and apply it independently.
Contact Tutor Advance to discuss your math course, learning goals, and current tutoring availability.