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Fractional Time (Compound Interest)

Introduction

Many compound-interest problems involve fractional years (for example 1.5 years or 2.25 years). The primary (mathematical) approach is to convert the fractional time into the appropriate exponent or into total compounding periods and use the compound-interest formula directly. Exams/textbooks sometimes use a convenient convention for fractional leftover periods - that note is provided separately.

Pattern: Fractional Time (Compound Interest)

Pattern

Key concept: Convert the fractional years into total compounding periods and use

General formula:
A = P × (1 + R/(100 × n))^(nT) → CI = A - P.
Where:

  • P = Principal
  • R = Annual nominal rate (in %)
  • T = Time in years (may be fractional)
  • n = Number of compounding periods per year (1 for annual, 2 for half-yearly, 4 for quarterly, ...)

Note: nT is the total number of compounding periods. It may be integer (e.g., 1.5 years with half-yearly → 3 periods) or non-integer (e.g., 1.5 years with annual → exponent 1.5). The mathematical formula with fractional exponents is valid; when the compounding frequency is inherently discrete and you must follow textbook/exam convention, treat leftover partial periods as described in the note section below.

Step-by-Step Example

Question

Find the compound interest on ₹12,000 at 8% per annum for 1.5 years (compounded annually).

Solution

  1. Step 1: Identify values

    P = ₹12,000; R = 8% p.a.; T = 1.5 years; n = 1 (annual).

  2. Step 2: Apply formula with fractional exponent

    A = P × (1 + R/(100 × n))^(nT) = 12,000 × (1 + 0.08)^{1.5} = 12,000 × (1.08)^{1.5}.

  3. Step 3: Compute precisely

    (1.08)^{1.5} = 1.08 × √1.08 ≈ 1.08 × 1.03923048454 ≈ 1.12231892331.
    A ≈ 12,000 × 1.12231892331 = ₹13,467.82.

  4. Step 4: Find CI

    CI = A - P ≈ 13,467.82 - 12,000 = ₹1,467.82.

  5. Final Answer (Mathematical):

    CI ≈ ₹1,467.82.

  6. Quick Check:

    1.5 years growth factor ≈ 12.2319% total → 12,000 × 1.122319 ≈ 13,467.82 ✅

Solution

  1. Step A:

    Compute amount for whole years (1 year): A₁ = 12,000 × 1.08 = 12,960.

  2. Step B:

    For the remaining half-year (f = 0.5), compute simple interest on A₁: SI = A₁ × R × f / 100 = 12,960 × 8% × 0.5 = 12,960 × 0.04 = ₹518.40.

  3. Step C:

    A_total = 12,960 + 518.40 = ₹13,478.40 → CI = 13,478.40 - 12,000 = ₹1,478.40.

  4. Note:

    This textbook/exam convention (compound for whole periods + simple interest on leftover fraction) is commonly used in many curricula. It gives a slightly different numeric result from the pure fractional-exponent method.

Quick Variations

1. Annual compounding with fractional years → use exponent T (mathematical) or textbook convention (compound whole years + SI for remainder).

2. Half-yearly / quarterly → if nT is integer, use direct formula; if nT is non-integer, follow curriculum convention (compound whole periods + SI on leftover) unless the question explicitly asks for fractional exponent.

3. When compounding frequency is high and time is fractional, prefer the mathematical fractional-exponent method for precise results unless exam instructions state otherwise.

Trick to Always Use

  • Step 1: Compute r_per = R / (100 × n) and total periods = nT.
  • Step 2: If total periods is integer → A = P × (1 + r_per)^{nT} (use this directly).
  • Step 3: If total periods is fractional and the problem is mathematical → you may use the fractional exponent directly: A = P × (1 + r_per)^{nT}.
  • Step 4: If total periods is fractional and the exam/textbook expects discrete compounding → compound for the integer number of periods, then apply simple interest for the leftover fractional period on the amount after integer periods.

Summary

Summary

  • General formula: A = P × (1 + R/(100×n))^(nT) → CI = A - P.
  • Annual compounding with fractional years: use exponent T (mathematical). Textbook convention: compound whole years then apply SI on leftover fraction.
  • Half-yearly/quarterly: compute r_per = R/n and total periods = nT - if nT is integer, use direct compounding; if not integer, follow instruction/convention for leftover fraction.
  • When in doubt, state which method you use (mathematical fractional exponent vs. textbook discrete-period + SI) and show computation accordingly.

Practice

(1/5)
1. Find the compound interest on ₹14,000 at 7% per annum for 1.5 years (compounded annually).
easy
A. ₹1,495.43
B. ₹1,500.00
C. ₹1,512.68
D. ₹1,506.00

Solution

  1. Step 1: Identify values

    Identify values: P = ₹14,000; R = 7% p.a.; T = 1.5 years; n = 1 (annual).

  2. Step 2: Set up fractional-exponent formula

    Apply fractional-exponent formula: A = P × (1 + R/100)^T = 14,000 × (1.07)^{1.5}.

  3. Step 3: Compute A using the fractional exponent

    Compute: (1.07)^{1.5} = 1.07 × √1.07 ≈ 1.1068179 → A ≈ 14,000 × 1.1068179 = ₹15,495.43.

  4. Step 4: Subtract principal to get CI

    CI = A - P = 15,495.43 - 14,000 = ₹1,495.43.

  5. Final Answer:

    ₹1,495.43 → Option A.

  6. Quick Check:

    1.5 years growth ≈ 10.68% → 14,000 × 0.1068 ≈ 1,495 ✅
Hint: Use A = P(1 + R/100)^T for fractional annual years.
Common Mistakes: Using simple interest for the fractional part instead of fractional-exponent compounding.
2. Find the amount on ₹10,000 at 10% per annum for 1.5 years, compounded half-yearly.
easy
A. ₹11,550.00
B. ₹11,576.25
C. ₹11,600.00
D. ₹11,500.10

Solution

  1. Step 1: Identify values

    Identify values: P = ₹10,000; R = 10% p.a.; T = 1.5 years; n = 2 (half-yearly).

  2. Step 2: Convert to per-period rate and total periods

    Rate per half-year r = R/(100×n) = 0.10/2 = 0.05; total periods = nT = 2 × 1.5 = 3.

  3. Step 3: Apply per-period compounding formula

    Apply formula: A = 10,000 × (1 + 0.05)^3 = 10,000 × 1.157625 = ₹11,576.25.

  4. Step 4: (Optional) Compute CI

    CI = A - P = 11,576.25 - 10,000 = ₹1,576.25.

  5. Final Answer:

    ₹11,576.25 → Option B.

  6. Quick Check:

    Half-year steps: 10,000 → 10,500 → 11,025 → 11,576.25 ✅
Hint: Convert annual rate to per-period (R/2) and use (1 + r)^{periods}.
Common Mistakes: Using annual 10% for each half-year period instead of 5% per half-year.
3. Find the compound interest on ₹8,000 at 8% p.a. for 0.75 years, compounded quarterly.
easy
A. ₹489.66
B. ₹480.33
C. ₹500.66
D. ₹495.33

Solution

  1. Step 1: Identify values

    Identify values: P = ₹8,000; R = 8% p.a.; T = 0.75 years; n = 4 (quarterly).

  2. Step 2: Compute per-period rate and number of periods

    Rate per quarter r = R/(100×n) = 0.08/4 = 0.02; total quarters = nT = 4 × 0.75 = 3.

  3. Step 3: Apply quarterly compounding

    A = 8,000 × (1.02)^3 = 8,000 × 1.061208 = ₹8,489.66.

  4. Step 4: Subtract principal to get CI

    CI = A - P = 8,489.66 - 8,000 = ₹489.66.

  5. Final Answer:

    ₹489.66 → Option A.

  6. Quick Check:

    Three quarters at 2% each ≈ 6.1208% total → 8,000 × 0.061208 ≈ 489.66 ✅
Hint: Compute periods = 4T and use r = R/400 for quarterly compounding.
Common Mistakes: Mistaking 0.75 years for a full year or using annual rate directly.
4. Find the compound interest on ₹50,000 at 6% p.a. for 2.25 years (compounded annually).
medium
A. ₹6,000.00
B. ₹7,100.00
C. ₹7,004.38
D. ₹7,013.25

Solution

  1. Step 1: Identify values

    Identify values: P = ₹50,000; R = 6% p.a.; T = 2.25 years; n = 1 (annual).

  2. Step 2: Set up fractional-exponent compounding

    Apply fractional-exponent formula: A = 50,000 × (1.06)^{2.25}.

  3. Step 3: Compute A and CI

    A ≈ 50,000 × 1.1400875336 = ₹57,004.38 → CI = A - P = 57,004.38 - 50,000 = ₹7,004.38.

  4. Final Answer:

    ₹7,004.38 → Option C.

  5. Quick Check:

    Approximately 14.01% growth over 2.25 years → 50,000 × 0.14009 ≈ 7,004 ✅
Hint: Use A = P(1 + R/100)^T for fractional-year exponents.
Common Mistakes: Switching to SI for fractional part when fractional-exponent compounding is required.
5. Find the compound interest on ₹15,000 at 9% p.a. for 1.25 years, compounded half-yearly.
medium
A. ₹1,720.28
B. ₹1,750.44
C. ₹1,740.67
D. ₹1,744.88

Solution

  1. Step 1: Identify values

    Identify values: P = ₹15,000; R = 9% p.a.; T = 1.25 years; n = 2 (half-yearly).

  2. Step 2: Compute half-year rate and total periods

    Rate per half-year r = R/(100×n) = 0.09/2 = 0.045; total periods = nT = 2 × 1.25 = 2.5.

  3. Step 3: Apply per-period compounding with fractional periods

    Apply fractional-exponent formula: A = 15,000 × (1.045)^{2.5} ≈ 15,000 × 1.1163251935 = ₹16,744.88.

  4. Step 4: Subtract principal to get CI

    CI = A - P = 16,744.88 - 15,000 = ₹1,744.88.

  5. Final Answer:

    ₹1,744.88 → Option D.

  6. Quick Check:

    Total growth ≈ 11.63% → 15,000 × 0.11643 ≈ 1,744.88 ✅
Hint: Use r = R/(100n) and A = P(1 + r)^{nT} even when nT is fractional.
Common Mistakes: Converting only whole periods and applying SI to remaining fraction when fractional-exponent is intended.

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