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Effective Rate Problems

Introduction

Effective rate problems ask you to find the single annual rate (or equivalent rate for a given period) that produces the same simple interest as different rates applied over parts of the period. These problems are common in exams because they test how well you handle different rates and time intervals.

Pattern: Effective Rate Problems

Pattern

Key idea: Calculate total interest for the entire period, then express it as a percentage of the principal for 1 year.

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

Step-by-Step Example

Question

A person lends money at 8% p.a. simple interest for the first 6 months and at 10% p.a. for the next 6 months. Find the effective annual rate of interest.

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

Solution

  1. Step 1: Assume a convenient principal

    Take P = ₹100 for easy calculation.

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

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

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

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

  4. Step 4: Add both interest amounts

    Total SI = 4 + 5 = ₹9

  5. Step 5: Convert total interest into effective annual rate

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

  6. Final Answer:

    9% → Option B

  7. Quick Check:

    If P = ₹100, 9% = 9 which matches total interest. ✅

Quick Variations

1. Different time splits, e.g., 4 months + 8 months.

2. More than two different rates in one year.

3. Effective rate for half-year or quarter-year.

4. Durations may be given in days or months - always convert to years.

Trick to Always Use

  • Step 1 → Assume P = 100 for simplicity.
  • Step 2 → Convert months into years.
  • Step 3 → Compute SI for each segment and add them.
  • Step 4 → Effective rate = (Total SI ÷ P) × 100.

Summary

Summary

  • Break the year into segments and calculate SI for each.
  • Use P = 100 to make calculations simpler.
  • Convert time correctly into years.
  • 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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