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Equal Installments / EMI

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Introduction

Loan repayment problems இல், borrower equal periodic instalments (EMIs) செலுத்துவார். ஒவ்வொரு EMI யிலும் interest மற்றும் principal repayment இரண்டும் அடங்கும். இந்த pattern மூலம் EMI கணக்கிடுவது, கொடுக்கப்பட்ட EMI யிலிருந்து principal ஐ கண்டறிதல், அல்லது காலப்பகுதியில் செலுத்தப்படும் மொத்த interest ஐ கணக்கிடுவது ஆகியவற்றை கற்றுக்கொள்ளலாம்.

இவை bank loans, mortgages, மற்றும் annuity concepts புரிந்து கொள்ள மிகவும் முக்கியமானவை, மேலும் aptitude மற்றும் finance-related exams இல் அடிக்கடி கேட்கப்படுகின்றன.

Pattern: Equal Installments / EMI

Pattern: Equal Installments / EMI

EMI என்பது interest உடன் loan ஐ திருப்பிச் செலுத்த, ஒவ்வொரு period இலும் செலுத்தப்படும் fixed payment ஆகும். இது annuity யின் present value formula இலிருந்து பெறப்படுகிறது.

Let:
P = Principal (loan amount)
R = Annual rate of interest (%)
n = Total number of instalments
r = Rate per period = R / (100 × periods per year)

EMI Formula:
EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ]

Principal Formula (EMI known):
P = EMI × [1 - (1 + r)^(-n)] / r

Step-by-Step Example

Question

₹5,00,000 loan க்கு, வருடத்திற்கு 10% வட்டியில், 5 years (monthly payments) க்கு monthly EMI ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

    Principal (P) = ₹5,00,000; Annual Rate (R) = 10%; Time = 5 years; Monthly payments → ஒரு வருடத்திற்கு 12 instalments.

  2. Step 2: Periodic Rate மற்றும் Total Periods கணக்கிடல்

    Periodic rate (r) = R / (100 × 12) = 0.10 / 12 = 0.008333333333333333.
    Number of instalments (n) = 5 × 12 = 60.

  3. Step 3: EMI Formula பயன்படுத்துதல்

    (1 + r)^n = (1.0083333333)^60 ≈ 1.6453089348.
    Numerator = P × r × (1 + r)^n ≈ 500,000 × 0.008333333333333333 × 1.6453089348 ≈ 6,855.4538949.
    Denominator = (1 + r)^n - 1 ≈ 1.6453089348 - 1 = 0.6453089348.
    EMI = Numerator / Denominator ≈ 6,855.4538949 / 0.6453089348 = ₹10,623.52 (2 dp வரை round செய்தது).

  4. Final Answer:

    Monthly EMI ≈ ₹10,623.52

  5. Quick Check:

    Total paid = EMI × n ≈ 10,623.52 × 60 = ₹6,37,411.20 → Interest ≈ 6,37,411.20 - 5,00,000 = ₹1,37,411.20 (5 years க்கு 10% வட்டியில் reasonable).

Question

ஒருவர் 3 years க்கு, வருடத்திற்கு 12% வட்டியில் (monthly compounding), மாதம் ₹8,000 செலுத்துகிறார். Loan இன் principal ஐ காண்க.

Solution

  1. Step 1: Values ஐ கண்டறிதல்

    EMI = ₹8,000; R = 12% p.a.; r = 0.12 / 12 = 0.01; n = 3 × 12 = 36.

  2. Step 2: Principal Formula பயன்படுத்துதல்

    P = EMI × [1 - (1 + r)^(-n)] / r

  3. Step 3: Substitute செய்து Compute செய்தல்

    (1 + r)^(-n) = (1.01)^(-36) ≈ 0.698805 → 1 - 0.698805 = 0.301195.
    0.301195 / 0.01 = 30.1195 → P = 8,000 × 30.1195 ≈ ₹2,40,956.00 (nearest rupee வரை round செய்தது).

  4. Final Answer:

    Principal ≈ ₹2,40,956.00

  5. Quick Check:

    இதே P வைத்து EMI மீண்டும் கணக்கிடும் போது → ≈ ₹8,000 (close) ✅

Quick Variations

1. Quarterly instalments க்கு: r = R/400, n = years × 4.

2. Annual instalments க்கு: r = R/100, n = years.

3. EMI, P, R கொடுக்கப்பட்டால், time (n) ஐ logarithms மூலம் காணலாம்: n = ln(EMI / (EMI - P×r)) / ln(1 + r).

Trick to Always Use

  • Step 1: Annual rate ஐ substitution முன் periodic rate ஆக மாற்றுங்கள்.
  • Step 2: Rounding errors தவிர்க்க, formula இல் substitute செய்வதற்கு முன் (1 + r)^n ஐ முதலில் கணக்கிடுங்கள்.
  • Step 3: Total paid மற்றும் interest amount ஐ ஒப்பிட்டு answer ஐ verify செய்யுங்கள்.

Summary

  • EMI formula: EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ]; இங்கு r = periodic rate, n = total instalments.
  • EMI யிலிருந்து Principal: P = EMI × [1 - (1 + r)^(-n)] / r.
  • Payments period உடன் ஒத்ததாக annual rate ஐ மாற்றுங்கள் (monthly → 12 ஆல், quarterly → 4 ஆல் divide).
  • n ஐ கண்டறிய logs பயன்படுத்தலாம்: n = ln(EMI/(EMI - P·r)) / ln(1 + r).
  • Quick check: total paid = EMI × n; interest = total paid - principal - rate மற்றும் period க்கு பொருந்துகிறதா என்று பார்க்கவும்.

Practice

(1/5)
1. A loan of ₹100,000 is to be repaid in 2 annual instalments at 10% per annum. What is the annual instalment (EMI)?
easy
A. ₹57,619.05
B. ₹55,000.15
C. ₹60,000.55
D. ₹50,000.25

Solution

  1. Step 1: Identify given values

    P = ₹100,000; R = 10% p.a.; annual payments → r = 0.10; n = 2.

  2. Step 2: State the EMI formula

    EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ].

  3. Step 3: Compute numerator and denominator

    (1 + r)^n = (1.10)^2 = 1.21. Numerator = 100,000 × 0.10 × 1.21 = 12,100. Denominator = 1.21 - 1 = 0.21. EMI = 12,100 / 0.21 = ₹57,619.05.

  4. Final Answer:

    Annual instalment = ₹57,619.05 → Option A.

  5. Quick Check:

    Two instalments of ₹57,619.05 → total ≈ ₹1,15,238.10; interest ≈ ₹15,238.10 which is reasonable for 10% over 2 years ✅
Hint: Use the annuity formula with r = R/100 for annual instalments.
Common Mistakes: Using simple-interest split (P×R) instead of the annuity (EMI) formula.
2. Find the monthly EMI on a loan of ₹3,00,000 at 9% per annum for 3 years (monthly payments).
easy
A. ₹9,000.84
B. ₹9,539.92
C. ₹10,000.24
D. ₹8,750.18

Solution

  1. Step 1: Identify given values

    P = ₹3,00,000; R = 9% p.a.; monthly → r = 0.09/12 = 0.0075; n = 3 × 12 = 36.

  2. Step 2: State the EMI formula

    EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ].

  3. Step 3: Compute powers and EMI

    (1 + r)^n ≈ (1.0075)^36 ≈ 1.3086453709. Numerator ≈ 300,000 × 0.0075 × 1.3086453709 ≈ 2,944.45. Denominator ≈ 0.3086453709. EMI ≈ 2,944.45 / 0.3086453709 = ₹9,539.92.

  4. Final Answer:

    Monthly EMI ≈ ₹9,539.92 → Option B.

  5. Quick Check:

    Total paid ≈ 9,539.92 × 36 ≈ ₹3,43,437 → interest ≈ ₹43,437 (reasonable for 9% over 3 years) ✅
Hint: Convert annual rate to monthly (divide by 12) and use n = years×12.
Common Mistakes: Using annual r without dividing by 12 for monthly EMI.
3. A borrower pays a monthly EMI of ₹20,000 for 2 years at 12% per annum (monthly compounding). What principal is being repaid?
easy
A. ₹4,50,000.30
B. ₹4,20,000.55
C. ₹4,24,867.75
D. ₹4,30,000.10

Solution

  1. Step 1: Identify given values

    EMI = ₹20,000; R = 12% p.a.; monthly r = 0.12/12 = 0.01; n = 2 × 12 = 24.

  2. Step 2: State principal formula

    P = EMI × [1 - (1 + r)^(-n)] / r.

  3. Step 3: Compute discount factor and P

    (1 + r)^(-n) = (1.01)^(-24) ≈ 0.788726 → 1 - 0.788726 = 0.211274. Divide by r: 0.211274 / 0.01 = 21.1274. P = 20,000 × 21.1274 ≈ ₹4,24,867.75.

  4. Final Answer:

    Principal ≈ ₹4,24,867.75 → Option C.

  5. Quick Check:

    Recompute EMI from this P using EMI formula → ≈ ₹20,000 (matches) ✅
Hint: Compute the discount factor [1 - (1+r)^(-n)]/r first, then multiply by EMI to get P.
Common Mistakes: Forgetting the negative exponent in (1 + r)^(-n) when computing the bracket.
4. A loan of ₹2,50,000 is repaid by monthly instalments of ₹7,000 at 10% per annum. Approximately how many years will it take to clear the loan?
medium
A. 3.40 years
B. 4.00 years
C. 3.00 years
D. 3.55 years

Solution

  1. Step 1: Identify given values

    P = ₹2,50,000; EMI = ₹7,000; R = 10% p.a.; monthly r = 0.10/12 ≈ 0.0083333333.

  2. Step 2: State formula for number of periods

    n = ln(EMI/(EMI - P·r)) / ln(1 + r).

  3. Step 3: Compute n and convert to years

    P·r = 250,000 × 0.0083333333 = 2,083.3333 → EMI/(EMI - P·r) = 7,000 / (7,000 - 2,083.3333) ≈ 1.423529. n = ln(1.423529) / ln(1.0083333333) ≈ 0.35345 / 0.008291 ≈ 42.57 months → years = 42.57 / 12 ≈ 3.55 years.

  4. Final Answer:

    Time ≈ 3.55 years → Option D.

  5. Quick Check:

    42-43 monthly payments of ₹7,000 → total paid ≈ ₹2,94,000-3,01,000; interest ≈ ₹44,000-51,000 which is reasonable at 10% for ~3.5 years ✅
Hint: Use the log formula for n once you compute EMI/(EMI - P·r).
Common Mistakes: Using simple-interest time formula or forgetting to convert annual rate to monthly.
5. A borrower pays ₹15,000 every quarter for 3 years at 8% per annum (quarterly compounding). What principal is being repaid?
medium
A. ₹1,58,630.12
B. ₹1,60,000.12
C. ₹1,50,000.12
D. ₹1,55,000.12

Solution

  1. Step 1: Identify given values

    EMI (quarterly) = ₹15,000; R = 8% p.a.; quarterly r = 0.08/4 = 0.02; n = 3 × 4 = 12.

  2. Step 2: State principal formula

    P = EMI × [1 - (1 + r)^(-n)] / r.

  3. Step 3: Compute discount factor and P

    (1 + r)^(-n) = (1.02)^(-12) ≈ 0.787053 → 1 - 0.787053 = 0.212947. Divide by r: 0.212947 / 0.02 = 10.64735. P = 15,000 × 10.64735 ≈ ₹1,58,630.12.

  4. Final Answer:

    Principal ≈ ₹1,58,630.12 → Option A.

  5. Quick Check:

    Recompute quarterly EMI from this P → ~₹15,000 (matches) ✅
Hint: For quarterly payments divide R by 4 and use n = years×4 in the principal formula.
Common Mistakes: Using annual r directly instead of r per quarter, or using n = years instead of years×4.