Simple and Compound Interest
Overview
Simple and Compound Interest is a high-scoring topic in HSSC CET Quantitative Ability. Questions typically involve straightforward formula application, but examiners test your understanding through word problems involving loans, deposits, population growth, and depreciation.
The core difference is simple: in Simple Interest (SI), interest is calculated only on the principal throughout the period. In Compound Interest (CI), interest is calculated on the principal plus accumulated interest—interest earns interest. Most exam questions test your ability to apply formulas quickly, find the difference between SI and CI, or work backwards to find principal, rate, or time.
Focus areas for HSSC CET include: basic SI/CI calculations, finding the difference between CI and SI for 2–3 years, half-yearly/quarterly compounding, and real-world applications like population growth and depreciation.
Key Concepts
- Principal (P): The initial amount of money borrowed or invested.
- Rate (R): The percentage of interest charged per time period (usually per annum).
- Time (T/n): The duration for which money is borrowed or invested.
- Simple Interest: Interest remains constant each year because it's always calculated on the original principal only.
- Compound Interest: Interest is added to principal at the end of each compounding period, so next period's interest is higher.
- Compounding Frequency: Interest may be compounded annually, half-yearly (2 times/year), quarterly (4 times/year), or monthly (12 times/year).
- Amount (A): Total money at the end = Principal + Interest.
- CI and SI are equal for the first year when compounding is annual and rate remains constant.
Formulas / Key Facts
Simple Interest:
- SI = (P × R × T) / 100
- Amount (A) = P + SI = P(1 + RT/100)
Compound Interest:
- Amount = P × (1 + R/100)ⁿ, where n = number of years
- CI = Amount – P = P × [(1 + R/100)ⁿ – 1]
Half-yearly Compounding:
- Rate becomes R/2, Time becomes 2n
- Amount = P × (1 + R/200)^(2n)
Quarterly Compounding:
- Rate becomes R/4, Time becomes 4n
- Amount = P × (1 + R/400)^(4n)
Difference between CI and SI:
- For 2 years: CI – SI = P × (R/100)²
- For 3 years: CI – SI = P × (R/100)² × (3 + R/100)
Population/Growth Formula:
- After n years: P × (1 + R/100)ⁿ (for growth)
- After n years: P × (1 – R/100)ⁿ (for depreciation)
When Amount doubles:
- Under SI: T = 100/R years
- Under CI: Use (1 + R/100)ⁿ = 2
Key Fact: At the same rate, CI > SI for any period greater than 1 year.
Worked Examples
Example 1: Basic Simple Interest
Find the SI on ₹8,000 at 12% per annum for 3 years.
SI = (P × R × T) / 100 SI = (8000 × 12 × 3) / 100 SI = 288000 / 100 = ₹2,880
Amount = 8000 + 2880 = ₹10,880
Example 2: Compound Interest Calculation
Find CI on ₹5,000 at 10% per annum for 2 years, compounded annually.
Amount = P × (1 + R/100)ⁿ Amount = 5000 × (1 + 10/100)² Amount = 5000 × (1.1)² Amount = 5000 × 1.21 = ₹6,050
CI = 6050 – 5000 = ₹1,050
Example 3: Difference between CI and SI for 2 years
The difference between CI and SI on a sum at 5% for 2 years is ₹20. Find the sum.
Using formula: CI – SI = P × (R/100)² 20 = P × (5/100)² 20 = P × (1/400) P = 20 × 400 = ₹8,000
Example 4: Half-yearly Compounding
Find the amount on ₹10,000 at 8% per annum for 1 year, compounded half-yearly.
Rate per half-year = 8/2 = 4% Number of periods = 2
Amount = 10000 × (1 + 4/100)² Amount = 10000 × (1.04)² Amount = 10000 × 1.0816 = ₹10,816
Example 5: Population Growth
The population of a town is 50,000. It increases by 10% in the first year and 20% in the second year. Find the population after 2 years.
After Year 1: 50000 × (1 + 10/100) = 50000 × 1.1 = 55,000 After Year 2: 55000 × (1 + 20/100) = 55000 × 1.2 = 66,000
Final population = 66,000
Common Mistakes
- Forgetting to convert time to years: If time is given in months, divide by 12. Many students use months directly in the formula. → Correct fix: Always convert time to years before applying SI/CI formulas.
- Using annual rate for half-yearly problems: When compounding is half-yearly, you must halve the rate and double the time period. → Correct fix: For half-yearly, use R/2 and 2n; for quarterly, use R/4 and 4n.
- Confusing CI formula with Amount formula: Students sometimes calculate Amount but report it as CI. → Correct fix: CI = Amount – Principal. Always subtract principal to get interest.
- Applying CI–SI difference formula for 3 years incorrectly: The 2-year formula P(R/100)² doesn't work for 3 years. → Correct fix: For 3 years, use P × (R/100)² × (3 + R/100).
- Ignoring different rates for different years: When rates vary each year, you cannot use standard CI formula. → Correct fix: Calculate year by year: Amount after Year 1 becomes Principal for Year 2.
Quick Reference
- SI = (P × R × T) / 100 — interest on original principal only
- CI Amount = P × (1 + R/100)ⁿ — interest on interest
- CI – SI (2 years) = P × (R/100)² — quick trick for exam
- Half-yearly: rate ÷ 2, time × 2
- Doubling time under SI = 100/R years
- For growth use (1 + R/100)ⁿ; for depreciation use (1 – R/100)ⁿ