Interest Applications
Calculate simple interest using I = PRT and compound interest using A = P(1 + r)n, and compare the two in real-world financial contexts.
Simple Interest
Simple interest is earned (or charged) only on the original amount of money, called the principal. The interest is the same every period, so growth is linear.
Simple Interest Formula
I = P × R × T
I — Interest
The amount earned or charged (in dollars)
P — Principal
The original amount invested or borrowed
R — Rate
Annual interest rate as a decimal (e.g., 5% = 0.05)
T — Time
The time period in years
To find the total amount (balance) after simple interest, add the interest to the principal:
A = P + I or A = P(1 + RT)
Simple Interest Growth: $1 000 at 10% p.a.
Linear growth: the same $100 interest is added each year.
Compound Interest
Compound interest is earned on both the original principal and the interest already accumulated. Each period, the balance grows by a percentage of the new total, producing exponential growth.
Compound Interest Formula
A = P(1 + r)n
A — Final Amount
Total value after interest (principal + interest)
P — Principal
Initial amount invested or borrowed
r — Interest rate per period
As a decimal (e.g., 6% p.a. = 0.06)
n — Number of periods
Total compounding periods (e.g., years)
To find the compound interest earned: Subtract the principal from the final amount.
Compound Interest = A − P = P(1 + r)n − P
Compound vs Simple Interest: $1 000 at 10% p.a.
Compound interest grows faster due to “interest on interest”.
Key Vocabulary
| Term | Definition |
|---|---|
| Principal (P) | The initial amount of money invested or borrowed, before any interest is added. |
| Interest Rate (r or R) | The percentage charged or earned per period, expressed as a decimal in calculations (e.g., 5% = 0.05). |
| Simple Interest | Interest calculated only on the original principal; grows linearly. Formula: I = PRT. |
| Compound Interest | Interest calculated on the principal plus previously earned interest; grows exponentially. Formula: A = P(1 + r)n. |
Worked Examples
Mia invests $5 000 at a simple interest rate of 4% per annum for 3 years. Calculate the interest earned and the total amount.
Identify: P = $5 000, R = 4% = 0.04, T = 3 years
Apply the formula:
I = P × R × T
I = 5 000 × 0.04 × 3
I = $600
Total amount:
A = P + I = 5 000 + 600 = $5 600
Mia earns $600 in interest, giving a total of $5 600.
Ben invests $2 000 at a compound interest rate of 6% per annum for 4 years. Calculate the final amount and the compound interest earned.
Identify: P = $2 000, r = 6% = 0.06, n = 4
Apply the formula:
A = P(1 + r)n
A = 2 000 × (1 + 0.06)4
A = 2 000 × (1.06)4
A = 2 000 × 1.26247696…
A ≈ $2 524.95
Compound interest earned:
Interest = A − P = 2 524.95 − 2 000 = $524.95
Ben earns approximately $524.95 in compound interest.
A bank offers two options for a $10 000 investment over 5 years: Option A pays 5% simple interest p.a.; Option B pays 5% compound interest p.a. Which option gives a greater return?
Option A — Simple Interest:
I = 10 000 × 0.05 × 5 = $2 500
A = 10 000 + 2 500 = $12 500
Option B — Compound Interest:
A = 10 000 × (1.05)5
A = 10 000 × 1.27628…
A ≈ $12 762.82
Option B gives $262.82 more. Compound interest always yields a greater return than simple interest for the same rate and time period greater than one year.
Knowledge Check
Select the best answer for each question.
Question 1
Ava borrows $3 000 at a simple interest rate of 8% per annum for 2 years. How much interest does she pay?
Question 2
What is the total amount after investing $4 000 at 5% simple interest per annum for 3 years?
Question 3
Liam invests $1 000 at a compound interest rate of 10% per annum for 2 years. What is the final amount?
Question 4
In the compound interest formula A = P(1 + r)n, what does n represent?
Question 5
$6 000 is invested for 5 years at 4% p.a. For a time period greater than 1 year, which method always produces a higher final value?
Key Concepts Summary
- ●Simple interest: I = P × R × T. Interest is calculated only on the principal; growth is linear.
- ●Total amount (simple): A = P + I = P(1 + RT).
- ●Compound interest: A = P(1 + r)n. Interest is earned on the growing balance; growth is exponential.
- ●Compound interest earned: Interest = A − P.
- ●For any rate and time period greater than 1 year, compound interest always gives a larger final amount than simple interest.