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Yearly/Quarterly/Half-Yearly Compounding

Introduction

Interest என்பது வருடத்திற்கு ஒருமுறை மட்டுமல்ல; வெவ்வேறு இடைவெளிகளில் compounding செய்யப்படலாம். வங்கிகள் மற்றும் finance questions இல் half-yearly அல்லது quarterly compounding அடிக்கடி பயன்படுத்தப்படுகின்றன. ஒவ்வொரு case க்கும் rate மற்றும் time ஐ எவ்வாறு adjust செய்வது என்று புரிந்துகொண்டால், எல்லா வகையான Compound Interest problems ஐ எளிதாக கையாளலாம்.

Pattern: Yearly/Quarterly/Half-Yearly Compounding

Pattern

Key concept: Compounding frequency அடிப்படையில் rate மற்றும் time ஐ adjust செய்ய வேண்டும்.

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

இதில்:
P = Principal amount
R = Annual rate of interest (in %)
T = Time in years
n = ஒரு வருடத்தில் interest compound செய்யப்படும் முறை எண்ணிக்கை

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

Step-by-Step Example

Question

₹8,000 மீது, வருடத்திற்கு 10% வட்டி விகிதத்தில் 1 year க்கு, half-yearly compounding ஆக Compound Interest ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

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

  2. Step 2: Rate மற்றும் Time ஐ adjust செய்தல்

    ஒவ்வொரு half-year க்கான effective rate = R/n = 10/2 = 5%.
    மொத்த half-years = n × T = 2 × 1 = 2.

  3. Step 3: Formula பயன்படுத்துதல்

    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 ஐ காண்க

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

  5. Final Answer:

    Compound Interest = ₹820.00

  6. Quick Check:

    இரண்டு 5% periods → 8,000 → 8,400 → 8,820 → gain = 820 ✅

Question

₹16,000 மீது, வருடத்திற்கு 12% வட்டி விகிதத்தில் 1 year க்கு, quarterly compounding ஆக amount ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

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

  2. Step 2: Rate மற்றும் Time ஐ adjust செய்தல்

    ஒவ்வொரு quarter க்கான effective rate = R/n = 12/4 = 3%.
    மொத்த quarters = n × T = 4 × 1 = 4.

  3. Step 3: Formula பயன்படுத்துதல்

    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 ஐ காண்க

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

  5. Final Answer:

    Compound Interest = ₹2,008.00

  6. Quick Check:

    3% ஐ 4 முறை compounding → ~12.55% effective yearly return → 16,000 × 0.1255 ≈ 2,008 ✅

Quick Variations

1. சில questions இல் “compounded half-yearly” என்று கொடுத்து, முழு years க்கான amount கேட்கப்படும் - எப்போதும் time ஐ total half-years ஆக மாற்றுங்கள்.

2. Quarterly அல்லது monthly compounding இல், interest அடிக்கடி சேர்வதால் amount சற்றே அதிகமாகும்.

3. Exams இல், money-based answers க்கு இரண்டு decimal places வரை round செய்ய வேண்டியிருக்கும்.

Trick to Always Use

  • Step 1: Compounding type ஐ அடையாளம் காணுங்கள் → yearly (n=1), half-yearly (n=2), quarterly (n=4).
  • Step 2: R ஐ → R / (100·n) ஆகவும், T ஐ → nT ஆகவும் adjust செய்யுங்கள்.
  • Step 3: A = P × (1 + R / (100·n))nT பயன்படுத்தி, CI = A - P ஐ காண்க.
  • Step 4: Accuracy உறுதி செய்ய effective rate of interest ஐ எப்போதும் check செய்யுங்கள்.

Summary

Summary

  • ஒரு வருடத்தில் ஒன்றுக்கு மேற்பட்ட முறை compounding நடந்தால், R ஐ (100·n) ஆல் வகுத்து, T ஐ n ஆக பெருக்க வேண்டும்.
  • Formula: A = P × (1 + R / (100·n))nT
  • n = 1 (yearly), 2 (half-yearly), 4 (quarterly).
  • Compounding அதிகமாக இருந்தால் → “interest on interest” காரணமாக CI சற்றே அதிகமாகும்.

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.

Mock Test

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