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CI vs SI Difference (2–3 Years)

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Introduction

कई exam problems में Compound Interest (CI) और Simple Interest (SI) के बीच का फर्क 2 या 3 साल की छोटी अवधि के लिए पूछा जाता है। 2-year और 3-year वाले direct shortcuts जानने से समय बचता है और लंबी calculations से बचा जा सकता है।

Pattern: CI vs SI Difference (2–3 Years)

Pattern: CI vs SI Difference (2–3 Years)

Key concept: छोटी duration में कुछ आसान shortcuts मिल जाते हैं - फर्क सिर्फ P और rate की powers पर निर्भर करता है।

मान लें r = R/100 (R percent में)।
General: Difference = CI - SI = P[(1 + r)T - 1 - Tr].
2 साल के लिए (T = 2): Difference = P × r2 = P × R2 / 10,000।
3 साल के लिए (T = 3): Difference = P × (3r2 + r3) = P × (3R2/10,000 + R3/1,000,000)।

Step-by-Step Example

Question

₹5,000 पर 10% वार्षिक दर से 2 साल के लिए CI और SI का फर्क निकालें।

Solution

  1. Step 1: Values पहचानें

    P = ₹5,000; R = 10%; r = 0.10; T = 2 साल।

  2. Step 2: 2-year shortcut

    Difference = P × r2 = P × (R2 / 10,000)

  3. Step 3: Calculation

    R2 = 102 = 100 → P × R2 = 5,000 × 100 = 500,000।
    10,000 से divide करें → Difference = 500,000 / 10,000 = ₹50।

  4. Final Answer:

    Difference (CI - SI) = ₹50।

  5. Quick Check:

    SI = PRT/100 = 5,000 × 10 × 2 / 100 = ₹1,000।
    CI = P[(1.1)2 - 1] = 5,000 × (1.21 - 1) = 5,000 × 0.21 = ₹1,050।
    CI - SI = 1,050 - 1,000 = ₹50 ✅

Question

₹2,000 पर 10% वार्षिक दर से 3 साल के लिए CI और SI का फर्क निकालें।

Solution

  1. Step 1: Values पहचानें

    P = ₹2,000; R = 10%; r = 0.10; T = 3 साल।

  2. Step 2: 3-year shortcut

    Difference = P × (3r2 + r3)

  3. Step 3: Calculation

    r2 = 0.01; r3 = 0.001 → 3r2 + r3 = 0.031।
    Difference = 2,000 × 0.031 = ₹62।

  4. Final Answer:

    Difference (CI - SI) = ₹62।

  5. Quick Check:

    SI = PRT/100 = 2,000 × 10 × 3 / 100 = ₹600।
    CI = P[(1.1)3 - 1] = 2,000 × (1.331 - 1) = 2,000 × 0.331 = ₹662।
    CI - SI = 662 - 600 = ₹62 ✅

Quick Variations

1. Rate छोटा हो तो 2-year shortcut जल्दी estimate देता है।

2. 3 साल में r³ term जोड़ना जरूरी है ताकि exact answer मिले।

3. 3 साल से ज्यादा के लिए shortcuts की बजाय general CI formula इस्तेमाल करें।

4. इन shortcuts का उपयोग करते समय हमेशा annual compounding मानें।

Trick to Always Use

  • Step 1: 2 साल के लिए Difference = P × R2 / 10,000।
  • Step 2: 3 साल के लिए Difference = P × (3r2 + r3) जहाँ r = R/100।

Summary

  • CI और SI का फर्क interest-on-interest effect की वजह से आता है।
  • General formula: Difference = P[(1 + r)T - 1 - Tr]।
  • 2 साल के लिए: P × R² / 10,000।
  • 3 साल के लिए: P × (3r² + r³)।
  • Quick confirmation के लिए CI और SI अलग-अलग निकालकर verify कर सकते हैं।

Practice

(1/5)
1. Find the difference between Compound Interest and Simple Interest on ₹4,000 at 10% per annum for 2 years.
easy
A. ₹40
B. ₹50
C. ₹60
D. ₹70

Solution

  1. Step 1: Identify given values

    P = ₹4,000; R = 10%; T = 2 years.

  2. Step 2: Apply 2-year shortcut formula

    Difference = P × R² / 10,000.

  3. Step 3: Compute

    R² = 100 → Difference = 4,000 × 100 / 10,000 = ₹40.

  4. Final Answer:

    Difference = ₹40 → Option A.

  5. Quick Check:

    SI = 4,000 × 10 × 2 /100 = ₹800; CI = 4,000 × (1.1² - 1) = 4,000 × 0.21 = ₹840; CI - SI = 840 - 800 = ₹40 ✅
Hint: For 2 years, difference = P × R² / 10,000.
Common Mistakes: Using full CI formula unnecessarily or missing the square of R.
2. Find the difference between CI and SI on ₹5,000 at 8% per annum for 2 years.
easy
A. ₹30
B. ₹32
C. ₹35
D. ₹40

Solution

  1. Step 1: Identify given values

    P = ₹5,000; R = 8%; T = 2 years.

  2. Step 2: Apply 2-year difference formula

    Difference = P × R² / 10,000.

  3. Step 3: Compute

    R² = 64 → Difference = 5,000 × 64 / 10,000 = ₹32.

  4. Final Answer:

    Difference = ₹32 → Option B.

  5. Quick Check:

    SI = 5,000 × 8 × 2 /100 = ₹800; CI = 5,000 × (1.08² - 1) = 5,000 × 0.1664 = ₹832; CI - SI = 832 - 800 = ₹32 ✅
Hint: Remember: CI - SI for 2 years = P × R² / 10,000.
Common Mistakes: Calculating SI for wrong number of years or forgetting to square R.
3. Find the difference between CI and SI on ₹10,000 at 10% per annum for 3 years.
easy
A. ₹310
B. ₹320
C. ₹330
D. ₹331

Solution

  1. Step 1: Identify given values

    P = ₹10,000; R = 10%; T = 3 years; r = R/100 = 0.10.

  2. Step 2: Apply 3-year shortcut formula

    Difference = P × (3r² + r³).

  3. Step 3: Compute

    r² = 0.01; r³ = 0.001 → 3r² + r³ = 0.031 → Difference = 10,000 × 0.031 = ₹310.

  4. Final Answer:

    Difference = ₹310 → Option A.

  5. Quick Check:

    SI = 10,000 × 10 × 3 /100 = ₹3,000; CI = 10,000 × (1.1³ - 1) = 10,000 × 0.331 = ₹3,310; CI - SI = 3,310 - 3,000 = ₹310 ✅
Hint: For 3 years use r = R/100 and compute P × (3r² + r³).
Common Mistakes: Applying the 2-year shortcut to a 3-year problem.
4. Find the difference between CI and SI on ₹8,000 at 12% per annum for 2 years.
medium
A. ₹110.40
B. ₹116.20
C. ₹115.20
D. ₹130.00

Solution

  1. Step 1: Identify given values

    P = ₹8,000; R = 12%; T = 2 years.

  2. Step 2: Apply 2-year difference formula

    Difference = P × R² / 10,000.

  3. Step 3: Compute

    R² = 144 → Difference = 8,000 × 144 / 10,000 = 1,152,000 / 10,000 = ₹115.20.

  4. Final Answer:

    Difference = ₹115.20 → Option C.

  5. Quick Check:

    SI = 8,000 × 12 × 2 /100 = ₹1,920; CI = 8,000 × (1.12² - 1) = 8,000 × 0.2544 = ₹2,035.20; CI - SI = 2,035.20 - 1,920 = ₹115.20 ✅
Hint: Compute R² first; then multiply by P/10,000.
Common Mistakes: Using R instead of R² while computing difference.
5. Find the difference between CI and SI on ₹6,000 at 10% per annum for 3 years.
medium
A. ₹180
B. ₹186.60
C. ₹200
D. ₹186.00

Solution

  1. Step 1: Identify given values

    P = ₹6,000; R = 10%; T = 3 years; r = 0.10.

  2. Step 2: Apply 3-year shortcut formula

    Difference = P × (3r² + r³).

  3. Step 3: Compute

    r² = 0.01; r³ = 0.001 → 3r² + r³ = 0.031 → Difference = 6,000 × 0.031 = ₹186.00.

  4. Final Answer:

    Difference = ₹186.00 → Option D.

  5. Quick Check:

    SI = 6,000 × 10 × 3 /100 = ₹1,800; CI = 6,000 × (1.1³ - 1) = 6,000 × 0.331 = ₹1,986; CI - SI = 1,986 - 1,800 = ₹186 ✅
Hint: 3-year difference = 3.1% of P when R = 10%.
Common Mistakes: Using 2-year formula or forgetting the r³ term.