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Fractional Time (Compound Interest)

Introduction

कई compound-interest वाले सवालों में fractional साल शामिल होते हैं (जैसे 1.5 साल या 2.25 साल)। मुख्य (mathematical) तरीका यह है कि fractional time को सही exponent या total compounding periods में बदलकर compound-interest का formula सीधे इस्तेमाल किया जाए। Exams/textbooks कभी-कभी fractional leftover periods के लिए आसान convention इस्तेमाल करते हैं - वह note अलग से दिया गया है।

Pattern: Fractional Time (Compound Interest)

Pattern

मुख्य concept: fractional सालों को total compounding periods में बदलें और उपयोग करें

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

  • P = Principal
  • R = Annual nominal rate (in %)
  • T = सालों में समय (fractional भी हो सकता है)
  • n = प्रति वर्ष compounding periods की संख्या (1 annual के लिए, 2 half-yearly के लिए, 4 quarterly के लिए, ...)

Note: nT कुल compounding periods हैं। यह integer हो सकता है (जैसे 1.5 साल और half-yearly → 3 periods) या non-integer (जैसे 1.5 साल और annual → exponent 1.5)। Fractional exponents वाला mathematical formula valid है; जब compounding frequency पूरी तरह discrete हो और आपको textbook/exam convention follow करना हो, तो leftover partial periods को नीचे दिए गए note अनुसार treat करें।

Step-by-Step Example

Question

₹12,000 पर 8% प्रति वर्ष की दर से 1.5 साल (annual compounding) के लिए compound interest निकालें।

Solution

  1. Step 1: Values पहचानें

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

  2. Step 2: Fractional exponent के साथ formula लगाएँ

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

  3. Step 3: Accurate calculation

    (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: CI निकालें

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

  5. Final Answer (Mathematical):

    CI ≈ ₹1,467.82.

  6. Quick Check:

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

Solution

  1. Step A:

    पूरे 1 साल के लिए amount: A₁ = 12,000 × 1.08 = 12,960.

  2. Step B:

    बचे हुए आधे साल (f = 0.5) के लिए A₁ पर simple interest: 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:

    यह textbook/exam convention (पूरे periods पर compounding + leftover fraction पर SI) कई syllabuses में इस्तेमाल होता है। इससे mathematical fractional-exponent method से थोड़ा अलग numerical result आता है।

Quick Variations

1. Annual compounding और fractional साल → exponent T (mathematical) या textbook convention (पूरे साल compound + बाकी fraction पर SI)।

2. Half-yearly / quarterly → अगर nT integer है तो सीधा formula; अगर nT non-integer है, तो curriculum convention follow करें (whole periods compound + leftover fraction पर SI), जब तक सवाल explicit तौर पर fractional exponent नहीं मांगता।

3. जब compounding frequency ज्यादा हो और time fractional हो, तो precise result के लिए mathematical fractional-exponent method बेहतर है, जब तक exam instructions कुछ और न कहें।

Trick to Always Use

  • Step 1: r_per = R / (100 × n) और total periods = nT निकालें।
  • Step 2: अगर total periods integer है → A = P × (1 + r_per)^{nT} (सीधा इस्तेमाल करें)।
  • Step 3: अगर total periods fractional है और problem mathematical है → fractional exponent सीधे इस्तेमाल करें: A = P × (1 + r_per)^{nT}।
  • Step 4: अगर total periods fractional है और exam/textbook discrete compounding की उम्मीद करता है → integer periods तक compound करें, फिर leftover fractional period पर simple interest लगाएँ।

Summary

Summary

  • General formula: A = P × (1 + R/(100×n))^(nT) → CI = A - P.
  • Annual compounding + fractional years: exponent T (mathematical) का उपयोग करें। Textbook convention: पूरे साल compound करें और leftover fraction पर SI।
  • Half-yearly/quarterly: r_per = R/n और total periods = nT निकालें - nT integer हो तो direct compounding; नहीं तो leftover fraction के लिए instruction/convention follow करें।
  • Confusion हो तो यह बताएं कि कौन-सा method उपयोग किया है (mathematical fractional exponent vs. textbook discrete-period + SI) और उसी अनुसार calculation दिखाएँ।

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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