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Comparison of SI & CI

Introduction

In bank or exam problems, you may be asked to compare Simple Interest (SI) and Compound Interest (CI). SI is always calculated on the original principal, but CI is calculated on the new balance every year. That’s why CI is always greater than SI for the same time and rate (except for 1 year).

Pattern: Comparison of SI & CI

Pattern

Key concept: SI grows in a straight line, CI grows with compounding.

- SI Formula: SI = (P × R × T) / 100
- CI Formula: CI = P(1 + R/100)^T - P
- Shortcut: • For 2 years → CI - SI = P × (R/100)² • For 3 years → CI - SI = P × (R/100)² × (3 + R/100)

Step-by-Step Example

Question

Find the difference between CI and SI on ₹5,000 at 10% per annum for 2 years.

Options:
A. ₹0
B. ₹50
C. ₹100
D. ₹150

Solution

  1. Step 1: Write down the known values

    Principal (P) = 5,000, Rate (R) = 10%, Time (T) = 2 years.

  2. Step 2: Calculate SI using the SI formula

    SI = (P × R × T) / 100 = (5,000 × 10 × 2) / 100 = 1,000.

  3. Step 3: Calculate CI year-by-year (beginner method)

    End of 1st year: Interest = (5,000 × 10) / 100 = 500 → Amount = 5,500.
    End of 2nd year: Interest = (5,500 × 10) / 100 = 550 → Amount = 6,050.
    CI = 6,050 - 5,000 = 1,050.

  4. Step 4: Subtract SI from CI to get the difference

    Difference = CI - SI = 1,050 - 1,000 = 50.

  5. Final Answer:

    ₹50 → Option B

  6. Quick Check:

    Shortcut formula for 2 years: CI - SI = P × (R/100)² = 5,000 × (10/100)² = 5,000 × 0.01 = 50 ✅

Quick Variations

1. For 1 year → SI = CI (difference = 0).

2. For 2 years → Difference = P × (R/100)².

3. For 3 years → Difference = P × (R/100)² × (3 + R/100).

4. For longer years → Use formula CI = P(1+R/100)^T - P and subtract SI.

Trick to Always Use

  • Step 1 → Calculate SI directly with (P×R×T)/100.
  • Step 2 → For CI, use year-by-year method if a beginner, or direct formula if comfortable.
  • Step 3 → Subtract SI from CI to get the difference.

Summary

Summary

  • Recognise that SI is always on the original principal; CI compounds on the growing amount each period.
  • Use the shortcut CI - SI = P × (R/100)² for 2 years to save time in exams.
  • For multi-year problems, prefer CI = P(1+R/100)^T - P then subtract SI for accuracy.
  • Always quick-check with the shortcut (where applicable) or recalculate yearly interest to verify.

Example to remember:
For P = ₹5,000, R = 10%, T = 2 → CI - SI = ₹50 (Option B).

Practice

(1/5)
1. Find the difference between SI and CI on ₹2000 at 10% per annum for 2 years.
easy
A. ₹20
B. ₹25
C. ₹30
D. ₹40

Solution

  1. Step 1: Write given values

    P = 2000, R = 10%, T = 2 years.

  2. Step 2: Compute simple interest

    SI = (P × R × T)/100 = (2000 × 10 × 2)/100 = 400.

  3. Step 3: Apply 2-year shortcut

    Difference = P × (R/100)^2 = 2000 × (0.1 × 0.1) = 2000 × 0.01 = 20.

  4. Step 4: Optional CI computation

    CI = SI + Difference = 400 + 20 = 420.

  5. Final Answer:

    ₹20 → Option A.

  6. Quick Check:

    Year 1 interest = 200 → amount 2200. Year 2 interest = 220 → CI = 420. CI - SI = 20 ✅
Hint: For 2 years, use Difference = P × (R/100)^2.
Common Mistakes: Forgetting to square (R/100) or using SI instead of difference.
2. On ₹5000 at 8% per annum for 2 years, find the difference between CI and SI.
easy
A. ₹28
B. ₹32
C. ₹30
D. ₹34

Solution

  1. Step 1: Note principal, rate, and time

    P = 5000, R = 8%, T = 2 years.

  2. Step 2: Calculate SI

    SI = (5000 × 8 × 2)/100 = 800.

  3. Step 3: Use 2-year shortcut

    Difference = P × (R/100)^2 = 5000 × (0.08 × 0.08) = 5000 × 0.0064 = 32.

  4. Step 4: Compute CI

    CI = 800 + 32 = 832.

  5. Final Answer:

    ₹32 → Option B.

  6. Quick Check:

    Year 1 interest = 400 → amount 5400. Year 2 interest = 5400 × 8% = 432 → CI = 832. CI - SI = 32 ✅
Hint: Compute P × (R/100)^2 directly for 2-year difference.
Common Mistakes: Using R instead of R/100 or forgetting to square it.
3. Find the difference between SI and CI on ₹4000 at 12% per annum for 2 years.
easy
A. ₹54.50
B. ₹56
C. ₹57.60
D. ₹60

Solution

  1. Step 1: List known values

    P = 4000, R = 12%, T = 2 years.

  2. Step 2: Calculate SI

    SI = (4000 × 12 × 2)/100 = 960.

  3. Step 3: Use 2-year shortcut for difference

    Difference = 4000 × (0.12 × 0.12) = 4000 × 0.0144 = 57.6 → ₹57.60.

  4. Step 4: Compute CI

    CI = 960 + 57.60 = 1,017.60.

  5. Final Answer:

    ₹57.60 → Option C.

  6. Quick Check:

    Year 1 interest = 480 → amount = 4,480. Year 2 interest = 4,480 × 12% = 537.6 → CI = 1,017.6. Difference = 1,017.6 - 960 = 57.6 ✅
Hint: Keep two-decimal accuracy for non-integer differences.
Common Mistakes: Rounding too early or dropping decimals.
4. On ₹6000 at 5% per annum for 3 years, what is the difference between CI and SI?
medium
A. ₹42.50
B. ₹45.00
C. ₹46.25
D. ₹45.75

Solution

  1. Step 1: Write principal, rate, and time

    P = 6000, R = 5% (0.05), T = 3 years.

  2. Step 2: Compute SI

    SI = (6000 × 5 × 3)/100 = 900.

  3. Step 3: Apply 3-year CI-SI formula

    Difference = P × (R/100)^2 × (3 + R/100).

  4. Step 4: Evaluate stepwise

    (R/100)^2 = 0.05 × 0.05 = 0.0025. P × 0.0025 = 6000 × 0.0025 = 15. Multiply by (3 + 0.05) = 3.05 → 15 × 3.05 = 45.75.

  5. Final Answer:

    ₹45.75 → Option D.

  6. Quick Check:

    CI = 6000(1.05)^3 - 6000 = 6945.75 - 6000 = 945.75; CI - SI = 45.75 ✅
Hint: For 3 years, include extra multiplier (3 + R/100).
Common Mistakes: Dropping the +R/100 term or rounding mid-steps.
5. Find the difference between CI and SI on ₹10000 at 15% per annum for 2 years.
medium
A. ₹225
B. ₹220
C. ₹230
D. ₹240

Solution

  1. Step 1: Identify P, R, and T

    P = 10000, R = 15%, T = 2 years.

  2. Step 2: Calculate SI

    SI = (10000 × 15 × 2)/100 = 3000.

  3. Step 3: Apply shortcut for 2 years

    Difference = P × (R/100)^2 = 10000 × (0.15 × 0.15) = 10000 × 0.0225 = 225.

  4. Step 4: Compute CI

    CI = 3000 + 225 = 3225.

  5. Final Answer:

    ₹225 → Option A.

  6. Quick Check:

    CI = 10000(1.15)^2 - 10000 = 13225 - 10000 = 3225; CI - SI = 225 ✅
Hint: For 2-year CI-SI difference, simply compute P × (R/100)^2.
Common Mistakes: Not squaring the rate fraction or confusing SI with CI.

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