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Effective Rate Problems

Introduction

Effective rate वाले problems आपसे वह single annual rate निकालने के लिए कहते हैं जो पूरे period में अलग-अलग rates से मिले simple interest के बराबर interest दे। Exams में यह काफी आते हैं क्योंकि यह checks करते हैं कि आप different rates और time intervals को कितनी अच्छी तरह handle कर सकते हैं।

Pattern: Effective Rate Problems

Pattern

Key idea: पूरे period का total interest निकालें और फिर उसे 1 year के percentage के रूप में express करें।

Formula:
SI = (P × R × T) / 100
Effective Rate (Reff) = (Total SI ÷ P) × 100

Step-by-Step Example

Question

एक व्यक्ति पहले 6 months के लिए 8% p.a. simple interest पर पैसा lend करता है और अगले 6 months के लिए 10% p.a. पर। Effective annual rate निकालें।

Options:
A. 8%
B. 9%
C. 10%
D. 11%

Solution

  1. Step 1: आसान calculation के लिए principal assume करें

    P = ₹100 ले लेते हैं।

  2. Step 2: पहले 6 months का SI (8%)

    6 months = 0.5 year
    SI₁ = (100 × 8 × 0.5) / 100 = ₹4

  3. Step 3: अगले 6 months का SI (10%)

    6 months = 0.5 year
    SI₂ = (100 × 10 × 0.5) / 100 = ₹5

  4. Step 4: Total SI निकालें

    Total SI = 4 + 5 = ₹9

  5. Step 5: Effective annual rate निकालें

    Effective Rate = (9 ÷ 100) × 100 = 9%

  6. Final Answer:

    9% → Option B

  7. Quick Check:

    P = 100 रखने पर 9% का interest = ₹9 होता है → match करता है। ✅

Quick Variations

1. Time splits अलग हो सकते हैं जैसे 4 months + 8 months।

2. एक ही year में दो से ज्यादा rates भी हो सकते हैं।

3. कभी-कभी half-year या quarter-year का effective rate पूछा जाता है।

4. Time months या days में दिया हो तो उसे years में convert करना जरूरी है।

Trick to Always Use

  • Step 1 → P = 100 assume करें ताकि calculation आसान हो जाए।
  • Step 2 → Months को years में convert करें।
  • Step 3 → हर segment के लिए SI निकालें और add करें।
  • Step 4 → Effective Rate = (Total SI ÷ P) × 100।

Summary

Summary

  • Year को अलग-अलग भागों में बाँटकर हर rate के हिसाब से SI निकालें।
  • Calculation को आसान बनाने के लिए P = 100 use करें।
  • Time हमेशा years में convert करें।
  • Effective annual rate = (Total SI ÷ Principal) × 100।

Practice

(1/5)
1. A sum is lent at 6% p.a. for the first 6 months and at 8% p.a. for the next 6 months. Find the effective annual rate of interest.
easy
A. 7.00%
B. 6.50%
C. 7.50%
D. 8.00%

Solution

  1. Step 1: Assume principal for easy calculation

    Use P = ₹100 for easy percent interpretation.

  2. Step 2: Compute SI for the first 6 months

    First 6 months → T = 0.5 year. SI₁ = (100 × 6 × 0.5) / 100 = ₹3.00.

  3. Step 3: Compute SI for the next 6 months

    Next 6 months → T = 0.5 year. SI₂ = (100 × 8 × 0.5) / 100 = ₹4.00.

  4. Step 4: Add segment interests to get total annual SI

    Total interest for 1 year = 3.00 + 4.00 = ₹7.00.

  5. Final Answer:

    7.00% → Option A.

  6. Quick Check:

    For P = 100, 7% of 100 = 7, matches total SI (3 + 4). ✅
Hint: Take P = 100 and add the segment interests (SI = P×R×T/100).
Common Mistakes: Forgetting to convert months to years (6 months = 0.5 year).
2. A sum is lent at 7% p.a. for 4 months and at 9% p.a. for the remaining 8 months. Find the effective annual rate (rounded to two decimals).
easy
A. 8.33%
B. 8.25%
C. 8.50%
D. 9.00%

Solution

  1. Step 1: Assume principal for percent conversion

    Choose P = ₹100.

  2. Step 2: Compute SI for first 4 months

    4 months = 4/12 = 0.333333... year. SI₁ = (100 × 7 × 0.3333333) / 100 = ₹2.333333... (2.3333).

  3. Step 3: Compute SI for next 8 months

    8 months = 8/12 = 0.666666... year. SI₂ = (100 × 9 × 0.6666667) / 100 = ₹6.00.

  4. Step 4: Sum segment interests and round

    Total SI = 2.333333... + 6.00 = 8.333333... → rounded to two decimals = 8.33%.

  5. Final Answer:

    8.33% → Option A.

  6. Quick Check:

    Exact fraction = 8 + 1/3% = 8.333...%, rounding gives 8.33% ✅
Hint: Compute each segment with months/12, sum interests, then divide by P (100) to get percent.
Common Mistakes: Rounding too early or using 0.4 instead of 4/12 for 4 months.
3. A sum is lent at 8% p.a. for 3 months and at 10% p.a. for the next 9 months. Find the effective annual rate.
easy
A. 9.00%
B. 9.50%
C. 9.75%
D. 10.00%

Solution

  1. Step 1: Assume principal for calculation

    Take P = ₹100.

  2. Step 2: Compute SI for first 3 months

    3 months = 0.25 year. SI₁ = (100 × 8 × 0.25) / 100 = ₹2.00.

  3. Step 3: Compute SI for next 9 months

    9 months = 0.75 year. SI₂ = (100 × 10 × 0.75) / 100 = ₹7.50.

  4. Step 4: Sum segment interests to get effective rate

    Total SI = 2.00 + 7.50 = ₹9.50 → Effective rate = 9.50%.

  5. Final Answer:

    9.50% → Option B.

  6. Quick Check:

    For P = 100, 9.5% of 100 = 9.5, equals total SI (2 + 7.5). ✅
Hint: Convert months to fraction of year (3 months = 0.25) before using SI formula.
Common Mistakes: Using whole-year rates with monthly durations without conversion.
4. Money is lent at 7% p.a. for 4 months, at 10% p.a. for 5 months, and at 8% p.a. for 3 months. Find the effective annual rate (to two decimals).
medium
A. 8.00%
B. 8.25%
C. 8.50%
D. 8.75%

Solution

  1. Step 1: Assume principal for percent conversion

    Assume P = ₹100.

  2. Step 2: Compute SI for first 4 months

    4 months = 4/12 = 0.333333... year. SI₁ = (100 × 7 × 0.3333333) / 100 = ₹2.333333... (2.3333).

  3. Step 3: Compute SI for next 5 months

    5 months = 5/12 ≈ 0.4166667 year. SI₂ = (100 × 10 × 0.4166667) / 100 = ₹4.166667 (4.1667).

  4. Step 4: Compute SI for final 3 months

    3 months = 0.25 year. SI₃ = (100 × 8 × 0.25) / 100 = ₹2.00.

  5. Step 5: Sum segment interests and report

    Total SI = 2.333333... + 4.166667 + 2.00 = 8.50 → Effective rate = 8.50%.

  6. Final Answer:

    8.50% → Option C.

  7. Quick Check:

    Sum of segment interests = 8.5 → for P=100 this is 8.50% ✅
Hint: Use P = 100 and carry fractional months precisely (e.g., 5/12 = 0.4166667).
Common Mistakes: Rounding the fractional-year values too early.
5. Money is lent at 7% p.a. for 2 months, at 8% p.a. for 6 months and at 9% p.a. for 4 months. Find the effective annual rate (rounded to two decimals).
medium
A. 7.80%
B. 8.00%
C. 8.10%
D. 8.17%

Solution

  1. Step 1: Assume principal for percent conversion

    Take P = ₹100.

  2. Step 2: Compute SI for first 2 months

    2 months = 2/12 = 0.1666667 year. SI₁ = (100 × 7 × 0.1666667) / 100 ≈ ₹1.1666667.

  3. Step 3: Compute SI for next 6 months

    6 months = 0.5 year. SI₂ = (100 × 8 × 0.5) / 100 = ₹4.00.

  4. Step 4: Compute SI for final 4 months

    4 months = 4/12 = 0.3333333 year. SI₃ = (100 × 9 × 0.3333333) / 100 = ₹3.00.

  5. Step 5: Sum segment interests and round

    Total SI ≈ 1.1666667 + 4.00 + 3.00 = 8.1666667 → rounded to two decimals = 8.17%.

  6. Final Answer:

    8.17% → Option D.

  7. Quick Check:

    Total interest ≈ 8.1667 for P=100 → 8.17% after rounding ✅
Hint: Compute each segment with months/12, sum interests and round only at the end.
Common Mistakes: Using 0.33 for 4/12 instead of 1/3 (introduces extra rounding error).

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