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Quantitative Aptitude

Simple & Compound Interest

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Master Compound Interest (CI) and Instalments for SSC CGL. Focus on core formulas, effective rates, and quick calculation methods to ace these critical quantitative aptitude questions.

Core Formula

Compound Interest (CI) is interest calculated on the initial principal and also on the accumulated interest of previous periods. The core formula for Compound Amount (A) is: A = P(1 + R/100)^T Where:

  • A = Amount after T years
  • P = Principal (initial investment/loan)
  • R = Annual Rate of Interest (in %)
  • T = Time period (in years)

Compound Interest (CI) = A - P

For Compounding Half-Yearly: Rate becomes R/2 and Time becomes 2T. For Compounding Quarterly: Rate becomes R/4 and Time becomes 4T.

Instalments (CI): For a loan of Present Worth (P) to be repaid in 'n' equal annual instalments of 'x' at R% CI, the formula is: P = x / (1 + R/100) + x / (1 + R/100)^2 + ... + x / (1 + R/100)^n

Worked Example 1

Q: Find the compound interest on Rs. 8000 for 2 years at 10% per annum, compounded annually. A:

  1. Identify values: P = 8000, R = 10%, T = 2 years.
  2. Calculate Amount (A): A = P(1 + R/100)^T A = 8000(1 + 10/100)^2 A = 8000(1 + 1/10)^2 A = 8000(11/10)^2 A = 8000 * (121/100) A = 80 * 121 = 9680
  3. Calculate CI: CI = A - P = 9680 - 8000 = Rs. 1680

Worked Example 2

Q: A loan of Rs. 2100 is to be paid back in two equal annual instalments. If the rate of interest is 10% per annum compounded annually, find the amount of each instalment. A:

  1. Identify values: P = 2100, R = 10%, n = 2 instalments.
  2. Use Instalment Formula: P = x / (1 + R/100) + x / (1 + R/100)^2 2100 = x / (1 + 10/100) + x / (1 + 10/100)^2 2100 = x / (11/10) + x / (121/100) 2100 = (10x/11) + (100x/121)
  3. Solve for x: 2100 = (110x + 100x) / 121 2100 = 210x / 121 x = (2100 * 121) / 210 x = 10 * 121 = Rs. 1210 Each instalment is Rs. 1210.

Shortcuts & Tricks

  1. Effective Rate for 2 years: For R% CI annually, the effective rate for 2 years is (2R + R^2/100)%. For Ex1 (10% for 2 years): (2*10 + 10^2/100)% = (20 + 1)% = 21%. CI = 21% of 8000 = Rs. 1680. (Much faster!)
  2. Ratio Method (for 2 years): If R = 10% = 1/10. Principal:Amount ratio for 1 year is 10:11. For 2 years, it's (10)^2 : (11)^2 = 100:121. If 100 units = 8000, then 1 unit = 80. CI = (121-100) units = 21 units = 21 * 80 = Rs. 1680.
  3. Tree Method: For 2 years at 10% on 8000:
    • Year 1 Interest: 10% of 8000 = 800
    • Year 2 Interest: 10% of 8000 (on principal) + 10% of 800 (on Year 1 interest) = 800 + 80 = 880
    • Total CI = 800 + 880 = 1680.
  4. Instalment Shortcut (for 2 years): If R = 10% = 1/10. The ratio of Principal to Instalment is (10/11) + (10/11)^2 = 10/11 + 100/121 = (110+100)/121 = 210/121. If Principal = 2100, then 2100 = (210/121) * x => x = 1210.

Common Mistakes

  1. Confusing SI and CI: Always read carefully whether it's Simple Interest or Compound Interest. The calculation methods are distinct.
  2. Incorrect Compounding Period: For half-yearly, remember to halve the rate and double the time. For quarterly, quarter the rate and quadruple the time. Not adjusting these is a frequent error.
  3. Calculation Errors: Especially with fractions or decimals, ensure precision. Using shortcuts correctly can minimize these.
  4. Instalment Misunderstanding: Many students confuse CI instalment with SI instalment or simply divide the total amount by the number of instalments. Remember CI instalments account for interest on the outstanding balance.

Derivation (brief)

Compound Interest builds upon the concept of Simple Interest. For the first period, CI is the same as SI. However, for subsequent periods, the interest earned in the previous period is added to the principal, and interest is then calculated on this new, larger principal. This 'interest on interest' is what makes CI grow exponentially.

  • Year 1 Amount: P(1 + R/100)
  • Year 2 Amount: [P(1 + R/100)] * (1 + R/100) = P(1 + R/100)^2
  • Year T Amount: P(1 + R/100)^T

Advanced Examples

Q: A sum of money doubles itself at CI in 5 years. In how many years will it become 8 times itself at the same rate? A:

  1. Understanding the pattern: If it doubles in 5 years, it means (1 + R/100)^5 = 2.
  2. Target: We want it to become 8 times, which is 2^3.
  3. Apply the pattern: If (1 + R/100)^5 = 2, then to get 2^3, we need to raise both sides to the power of 3. [(1 + R/100)^5]^3 = 2^3 (1 + R/100)^(5*3) = 8 So, it will become 8 times in 5 * 3 = 15 years.

Variation Types

  1. Population Growth/Depreciation: These are direct applications of the CI formula. Growth uses +R, Depreciation uses -R. P_final = P_initial (1 ± R/100)^T.
  2. Varying Rates: If rates are R1, R2, R3 for different years, A = P(1 + R1/100)(1 + R2/100)(1 + R3/100).
  3. 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).
  4. Instalments with different periods: The fundamental present worth concept remains, just adjust the number of terms and the power for each term.

Time-Saving Methods

  1. Approximation: For complex rates or large numbers, sometimes approximating (1+R/100)^T can save time, especially if options are far apart. For example, (1.05)^2 is roughly 1.1025.
  2. Memorize common fractions: Ratios like 10% = 1/10, 20% = 1/5, 25% = 1/4, 5% = 1/20 are very useful for the ratio method.
  3. Focus on the last digit: In multiple-choice questions, sometimes calculating just the last digit of the amount can eliminate options.
  4. Work backwards for instalments: If you know the instalment amount, you can work backwards using the present worth formula to find the principal. This is often easier than solving for 'x' directly.
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Simple Interest (SI) is calculated only on the principal amount. Master its formula and percentage-based shortcuts for quick, accurate calculations in SSC CGL exams.

Core Formula

Simple Interest (SI) is the interest calculated only on the principal amount. It's a fundamental concept for SSC CGL. The core formula for Simple Interest is: SI = (P * R * T) / 100 Where:

  • P = Principal amount (the initial sum of money)
  • R = Rate of interest per annum (in percentage)
  • T = Time period (in years)

The total amount (A) to be repaid or received is the sum of the Principal and the Simple Interest: Amount (A) = P + SI

Worked Example 1

Question: Find the Simple Interest on INR 8,000 at 12% per annum for 3 years. Also, find the total amount. Solution: Given: P = INR 8,000, R = 12%, T = 3 years Using the formula: SI = (P * R * T) / 100 SI = (8000 * 12 * 3) / 100 SI = (80 * 12 * 3) SI = 80 * 36 SI = INR 2,880

Amount (A) = P + SI A = 8000 + 2880 A = INR 10,880

Worked Example 2

Question: A sum of money fetches INR 1,500 as Simple Interest at 10% per annum in 5 years. Find the principal amount. Solution: Given: SI = INR 1,500, R = 10%, T = 5 years We need to find P. Using the formula: SI = (P * R * T) / 100 1500 = (P * 10 * 5) / 100 1500 = (P * 50) / 100 1500 = P / 2 P = 1500 * 2 P = INR 3,000

Shortcuts & Tricks

  1. Percentage Method (R% per year): Simple Interest is always (R * T)% of the Principal.
    • If R = 10%, T = 3 years, then SI = (10 * 3)% = 30% of P.
    • Example: P = 5000, R = 8%, T = 2 years. SI = (8*2)% of 5000 = 16% of 5000 = (16/100) * 5000 = 16 * 50 = INR 800.
  2. Ratio Method for 'n' times: If a sum becomes 'n' times itself in 'T' years, then SI = (n-1)P. Use (n-1) = (R * T) / 100 to find R or T quickly.
    • Example: A sum doubles in 5 years. This means SI = P (n=2). (2-1) = (R * 5) / 100 => 1 = 5R / 100 => R = 20%.

Common Mistakes

  1. Time Unit Conversion: Always convert the time period into years. If given in months, divide by 12. If given in days, divide by 365.
  2. Confusing SI with Amount: Remember that Amount is Principal + SI. Many students mistakenly report SI as the final amount or vice-versa.
  3. Calculation Errors: Simple arithmetic mistakes, especially with multiplication and division, can lead to incorrect answers. Practice mental math.

Derivation (brief)

Simple Interest is the most basic form of interest calculation, characterized by its linear growth. The interest is calculated solely on the initial principal amount. For every year the principal P is invested or borrowed, a fixed percentage R of P is earned as interest. This annual interest is (P * R) / 100. Since this amount remains constant each year, the total interest accumulated over T years is simply T times the annual interest. Thus, SI = T * [(P * R) / 100], which simplifies to the core formula SI = (P * R * T) / 100. The principal itself does not change for interest calculation purposes during the term.

Advanced Examples

Question: A sum of money is invested at a certain rate of simple interest for 2 years. If it had been invested at 3% higher rate, it would have fetched INR 300 more. Find the principal sum. Solution: Let the principal be P and the original rate be R. Original SI = (P * R * 2) / 100 New rate = (R + 3)% New SI = (P * (R + 3) * 2) / 100 According to the problem, New SI - Original SI = 300 [(P * (R + 3) * 2) / 100] - [(P * R * 2) / 100] = 300 [2P(R + 3) - 2PR] / 100 = 300 [2PR + 6P - 2PR] / 100 = 300 6P / 100 = 300 6P = 300 * 100 6P = 30000 P = 30000 / 6 P = INR 5,000

Variation Types

  1. "Sum becomes 'n' times itself" problems: If a sum P becomes nA (n times the principal) in T years at R% SI, the Simple Interest earned is nA - P = (n-1)P.
    • Using SI = (P * R * T) / 100, we get (n-1)P = (P * R * T) / 100.
    • This simplifies to (n-1) = (R * T) / 100. This is a powerful shortcut.
    • Example: A sum triples itself in 8 years. Find the rate of interest. Here, n=3. (3-1) = (R * 8) / 100 2 = (R * 8) / 100 200 = 8R R = 200 / 8 = 25%
  2. Interest as a fraction of Principal: If SI is x/y of the principal, then (x/y)P = (P * R * T) / 100.
    • This simplifies to x/y = (R * T) / 100.
    • Example: Simple interest on a sum is 4/9 of the principal. If the number of years is numerically equal to the rate percent per annum, find the rate. Given: SI = (4/9)P, T = R (4/9)P = (P * R * R) / 100 4/9 = R^2 / 100 R^2 = 400 / 9 R = sqrt(400/9) = 20/3 = 6 2/3 %

Time-Saving Methods

  1. Focus on the difference: In problems involving changes in rate or time, calculate the change in SI directly. For instance, if the rate increases by x%, the extra SI earned is simply (P * x * T) / 100. This avoids calculating two full SI values and subtracting them.
  2. Mental Calculation for Common Values: Develop the ability to quickly calculate percentages for common rates (e.g., 10% is 1/10th, 12.5% is 1/8th, 20% is 1/5th). This allows for faster mental computation of (R*T)% of the principal, significantly reducing reliance on pen and paper for simpler problems.
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