Understanding Interest
Interest is the cost of borrowing money or the return earned on money saved or invested. It's usually expressed as a percentage rate per period (often per year, "per annum" or "p.a.").
Key terms:
- Principal (P): The initial amount of money borrowed or invested.
- Interest Rate (R): The percentage at which interest is calculated, usually per year. Remember to convert this to a decimal for calculations (e.g., 5% = 0.05).
- Time (T): The duration for which the money is borrowed or invested, usually in years.
There are two main types of interest: Simple Interest and Compound Interest.
Simple Interest
Simple interest is calculated only on the original principal amount. The amount of interest earned or paid is the same for each period.
Simple Interest (I) = Principal (P) × Rate (R) × Time (T)
I = P × R × T
Total Amount (A) = Principal (P) + Simple Interest (I)
A = P + I
Worked Example: Simple Interest
Problem: Sarah invests £2000 in a savings account that pays 4% simple interest per year for 3 years.
a) How much simple interest will she earn?
b) What will be the total amount in her account after 3 years?
Given: P = £2000, R = 4% = 0.04, T = 3 years
a) Calculate Simple Interest (I):
I = P × R × T
I = £2000 × 0.04 × 3
I = £80 × 3 (Interest per year is £80)
I = £240
Answer (a): Sarah will earn £240 in simple interest.
b) Calculate Total Amount (A):
A = P + I
A = £2000 + £240
A = £2240
Answer (b): The total amount in her account will be £2240.
Compound Interest
Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means "interest on interest," so it grows faster than simple interest.
Total Amount (A) = Principal (P) × (1 + Rate (R))Time (T)
A = P(1 + R)T
Compound Interest (CI) = Total Amount (A) - Principal (P)
CI = A - P
Note: R is the interest rate per compounding period, and T is the total number of compounding periods. If interest is compounded annually, R is the annual rate and T is the number of years.
Worked Example: Compound Interest
Problem: John invests £2000 in an account paying 4% compound interest per year for 3 years.
a) What will be the total amount in his account after 3 years?
b) How much compound interest will he earn?
Given: P = £2000, R = 4% = 0.04, T = 3 years
a) Calculate Total Amount (A):
A = P(1 + R)T
A = £2000 × (1 + 0.04)³
A = £2000 × (1.04)³
A = £2000 × (1.04 × 1.04 × 1.04)
A = £2000 × 1.124864
A = £2249.728
Answer (a): The total amount will be £2249.73 (rounded to 2 decimal places).
b) Calculate Compound Interest (CI):
CI = A - P
CI = £2249.73 - £2000
CI = £249.73
Answer (b): John will earn £249.73 in compound interest.
Compare this to the simple interest of £240 earned over the same period. Compound interest earns more!
Exam-Style Interest Problems
1. Simple Interest Loan
David borrows £5000 for 4 years at a simple interest rate of 3.5% per annum. How much will he repay in total?
Answer: £5700
Explanation:
- Principal (P) = £5000, Rate (R) = 3.5% = 0.035, Time (T) = 4 years.
- Calculate Simple Interest (I):
I = P × R × T = £5000 × 0.035 × 4. I = £175 × 4 = £700.- Total Repayment (A) = Principal + Interest =
£5000 + £700 = £5700.
2. Compound Interest Investment Growth
Maria invests £1200 at an annual compound interest rate of 5%. How much will her investment be worth after 2 years?
Answer: £1323.00
Explanation:
- Principal (P) = £1200, Rate (R) = 5% = 0.05, Time (T) = 2 years.
- Formula for Total Amount (A):
A = P(1 + R)T. A = £1200 × (1 + 0.05)²A = £1200 × (1.05)²A = £1200 × 1.1025A = £1323.00.
Interactive Interest Calculators
Simple Interest Calculator
Simple Interest Earned: ?
Total Amount (P+I): ?
Compound Interest Calculator
E.g., 1 for annually, 2 for semi-annually, 4 for quarterly, 12 for monthly.
Total Amount (A): ?
Compound Interest Earned (CI): ?
Key Points for Interest
- Simple Interest is calculated only on the original principal amount.
- Compound Interest is calculated on the principal AND any accumulated interest. It grows faster!
- Always convert percentage rates to decimals for calculations (e.g., 5% = 0.05).
- Ensure the time period matches the interest rate period (e.g., if rate is per year, time should be in years).