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Yearly/Quarterly/Half-Yearly Compounding

Introduction

Interest अलग-अलग intervals पर compound हो सकता है - सिर्फ साल में एक बार नहीं। Banks और finance से जुड़े questions में अक्सर half-yearly या quarterly compounding मिलता है। हर case में rate और time कैसे adjust करना है, यह समझने से compound interest के सभी प्रकार के questions आसानी से solve हो जाते हैं।

Pattern: Yearly/Quarterly/Half-Yearly Compounding

Pattern

Key concept: Compounding frequency के हिसाब से rate और time को adjust करें।

Compound Interest का general formula है:
A = P × (1 + R / (100·n))nT
CI = A - P

जहाँ:
P = Principal amount
R = Annual rate of interest (%)
T = समय (years में)
n = एक साल में कितनी बार compounding होता है

  • Yearly compounding → n = 1
  • Half-Yearly compounding → n = 2
  • Quarterly compounding → n = 4

Step-by-Step Example

Question

₹8,000 पर 10% वार्षिक दर से 1 साल के लिए, half-yearly compounding होने पर compound interest निकालें।

Solution

  1. Step 1: Values पहचानें

    P = ₹8,000; R = 10%; T = 1 साल; n = 2 (half-yearly)。

  2. Step 2: Rate और time adjust करें

    Effective rate प्रति half-year = R/n = 10/2 = 5%।
    Total half-years = n × T = 2 × 1 = 2।

  3. Step 3: Formula लगाएँ

    A = 8,000 × (1 + R / (100·n))^{nT} = 8,000 × (1 + 10 / (100·2))^{2} = 8,000 × (1.05)^{2} = 8,000 × 1.1025 = ₹8,820.00।

  4. Step 4: CI निकालें

    CI = A - P = 8,820 - 8,000 = ₹820.00।

  5. Final Answer:

    Compound Interest = ₹820.00

  6. Quick Check:

    दो बार 5% → 8,000 → 8,400 → 8,820 → gain = 820 ✅

Question

₹16,000 पर 12% वार्षिक दर से 1 साल के लिए, quarterly compounding होने पर amount निकालें।

Solution

  1. Step 1: Values पहचानें

    P = ₹16,000; R = 12%; T = 1 साल; n = 4 (quarterly)।

  2. Step 2: Rate और time adjust करें

    Effective rate प्रति quarter = R/n = 12/4 = 3%।
    Total quarters = n × T = 4 × 1 = 4।

  3. Step 3: Formula लगाएँ

    A = 16,000 × (1 + R / (100·n))^{nT} = 16,000 × (1 + 12 / (100·4))^{4} = 16,000 × (1.03)^{4} ≈ 16,000 × 1.12550881 = ₹18,008.00 (approx)।

  4. Step 4: CI निकालें

    CI = A - P = 18,008 - 16,000 = ₹2,008.00।

  5. Final Answer:

    Compound Interest = ₹2,008.00

  6. Quick Check:

    Quarterly compounding से effective yearly return ~12.55% होता है → 16,000 × 0.1255 ≈ 2,008 ✅

Quick Variations

1. कभी-कभी question में “compounded half-yearly” होता है, पर answer पूरी years के हिसाब से चाहिए - हमेशा time को total half-years में बदलें।

2. Quarterly या monthly compounding से amount थोड़ा ज्यादा मिलता है क्योंकि interest बार-बार add होता है।

3. Exams में money-based answers को दो decimal places तक round करना पड़ सकता है।

Trick to Always Use

  • Step 1: Compounding type पहचानें → yearly (n=1), half-yearly (n=2), quarterly (n=4)।
  • Step 2: R को adjust करें → R / (100·n) और T → nT।
  • Step 3: Formula इस्तेमाल करें A = P × (1 + R / (100·n))nT और CI = A - P निकालें।
  • Step 4: Accuracy check करने के लिए effective rate जरूर देखें।

Summary

Summary

  • जब compounding साल में एक से ज्यादा बार होता है, तो R को (100·n) से divide करें और T को n से multiply करें।
  • Formula: A = P × (1 + R / (100·n))nT
  • n = 1 (yearly), 2 (half-yearly), 4 (quarterly)।
  • जितनी ज्यादा बार compounding होगा → उतना ज्यादा CI मिलेगा क्योंकि “interest on interest” बढ़ता है।

Practice

(1/5)
1. Find the compound interest on ₹5,000 at 10% per annum for 1 year, compounded half-yearly.
easy
A. ₹512.50
B. ₹505.50
C. ₹500.50
D. ₹515.50

Solution

  1. Step 1: Identify values

    P = ₹5,000; R = 10% p.a.; T = 1 year; n = 2 (half-yearly).

  2. Step 2: Adjust rate and time

    Rate per half-year = R/n = 10/2 = 5%; number of periods = n × T = 2.

  3. Step 3: Apply formula

    A = 5,000 × (1 + 5/100)^2 = 5,000 × (1.05)^2 = 5,000 × 1.1025 = ₹5,512.50.

  4. Final Answer:

    CI = A - P = 5,512.50 - 5,000 = ₹512.50 → Option A.

  5. Quick Check:

    Two 5% periods → 5,000 → 5,250 → 5,512.50; CI = 512.50 ✅
Hint: For half-yearly compounding over 1 year use (1 + R/200)^2 and subtract P.
Common Mistakes: Using annual rate directly instead of dividing by 2 for half-yearly compounding.
2. Find the compound interest on ₹8,000 at 12% per annum for 1 year, compounded quarterly.
easy
A. ₹980.07
B. ₹1,004.07
C. ₹990.07
D. ₹1,008.07

Solution

  1. Step 1: Identify values

    P = ₹8,000; R = 12% p.a.; T = 1 year; n = 4 (quarterly).

  2. Step 2: Adjust rate and time

    Rate per quarter = R/n = 12/4 = 3%; number of periods = 4.

  3. Step 3: Apply formula

    A = 8,000 × (1 + 3/100)^4 = 8,000 × (1.03)^4 = 8,000 × 1.12550881 = ₹9,004.07 (approx).

  4. Final Answer:

    CI = A - P = 9,004.07 - 8,000 = ₹1,004.07 → Option B.

  5. Quick Check:

    Quarterly 3% periods give effective yearly multiplier ≈ 1.1255 → 8,000 × 0.1255 ≈ ₹1,004.07 ✅
Hint: For quarterly compounding divide R by 4 and multiply T by 4, then use A = P(1 + R/4×100)^(4T).
Common Mistakes: Using the annual rate directly instead of dividing by 4 for quarterly compounding.
3. Find the amount on ₹10,000 at 8% per annum for 1 year, compounded half-yearly.
easy
A. ₹10,816.00
B. ₹10,804.00
C. ₹10,820.00
D. ₹10,830.00

Solution

  1. Step 1: Identify values

    P = ₹10,000; R = 8% p.a.; T = 1 year; n = 2 (half-yearly).

  2. Step 2: Adjust rate and time

    Rate per half-year = 8/2 = 4%; number of periods = 2.

  3. Step 3: Apply formula

    A = 10,000 × (1 + 4/100)^2 = 10,000 × (1.04)^2 = 10,000 × 1.0816 = ₹10,816.00.

  4. Final Answer:

    Amount = ₹10,816.00 → Option A.

  5. Quick Check:

    Two 4% periods → 10,000 → 10,400 → 10,816; matches computed amount ✅
Hint: For 1 year half-yearly compounding, use (1 + R/200)^2 and multiply by P.
Common Mistakes: Using (1 + R/100) for yearly compounding instead of adjusting to half-yearly periods.
4. Find the compound interest on ₹12,000 at 8% per annum for 1.5 years, compounded half-yearly.
medium
A. ₹1,480.00
B. ₹1,494.10
C. ₹1,498.37
D. ₹1,520.00

Solution

  1. Step 1: Identify values

    P = ₹12,000; R = 8% p.a.; T = 1.5 years; n = 2 (half-yearly).

  2. Step 2: Adjust rate and time

    Rate per half-year = 8/2 = 4%; number of half-year periods = 1.5 × 2 = 3.

  3. Step 3: Apply formula

    A = 12,000 × (1 + 4/100)^3 = 12,000 × (1.04)^3 = 12,000 × 1.124864 = ₹13,498.37 (approx).

  4. Final Answer:

    CI = A - P = 13,498.37 - 12,000 = ₹1,498.37 → Option C.

  5. Quick Check:

    Three 4% periods → effective gain ≈ 12.4864% → 12,000 × 0.124864 ≈ ₹1,498.37 ✅
Hint: Convert 1.5 years to 3 half-years and use 4% per period.
Common Mistakes: Not converting fractional years to the correct number of compounding periods.
5. Find the amount on ₹15,000 at 10% per annum for 1.5 years, compounded quarterly.
medium
A. ₹17,345.00
B. ₹17,348.50
C. ₹17,360.00
D. ₹17,395.40

Solution

  1. Step 1: Identify values

    P = ₹15,000; R = 10% p.a.; T = 1.5 years; n = 4 (quarterly).

  2. Step 2: Adjust rate and time

    Rate per quarter = 10/4 = 2.5%; number of quarters = 1.5 × 4 = 6.

  3. Step 3: Apply formula

    A = 15,000 × (1 + 2.5/100)^6 = 15,000 × (1.025)^6 = 15,000 × 1.159693383 = ₹17,395.40 (approx).

  4. Final Answer:

    Amount = ₹17,395.40 → Option D.

  5. Quick Check:

    Six 2.5% periods compound to ≈15.969% total gain → 15,000 × 0.159693 ≈ ₹2,395.40 → amount ≈ 17,395.40 ✅
Hint: For fractional years with quarterly compounding, convert time to quarters (T×4) and use R/4 per period.
Common Mistakes: Failing to multiply the time by 4 for quarterly compounding or dividing rate by 4.

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