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Equal Installments / EMI

Introduction

Loan repayment problems में borrower समान समयांतराल पर दिए जाने वाले equal periodic instalments (EMIs) भरता है। हर EMI में interest और principal repayment दोनों शामिल होते हैं। यह pattern आपको EMI निकालने, दिए गए EMI से principal खोजने, या समय के साथ total interest ज्ञात करने में मदद करता है।

ये problems bank loans, mortgages, और annuity concepts समझने के लिए बहुत जरूरी हैं, और aptitude तथा finance-related exams में अक्सर पूछे जाते हैं।

Pattern: Equal Installments / EMI

Pattern

EMI एक fixed payment होती है जिसे loan को interest सहित चुकाने के लिए periodically भरा जाता है। यह present value of an annuity formula से प्राप्त होती है।

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

Principal का Formula (जब EMI दिया हो):
P = EMI × [1 - (1 + r)^(-n)] / r

Step-by-Step Example

Question

₹5,00,000 के loan पर 10% प्रति वर्ष की दर से 5 साल के लिए (monthly payments) monthly EMI निकालें।

Solution

  1. Step 1: Identify Values

    Principal (P) = ₹5,00,000; Annual Rate (R) = 10%; Time = 5 साल; Monthly payments → 12 instalments per year.

  2. Step 2: Compute Periodic Rate and Total Periods

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

  3. Step 3: Apply EMI Formula

    (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 decimal तक rounded)。

  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 (10% for 5 years के अनुसार सही)।

Question

एक व्यक्ति 3 साल तक हर महीने ₹8,000 भरता है, 12% प्रति वर्ष (compounded monthly) की दर पर। Loan का principal amount निकालें।

Solution

  1. Step 1: Identify Values

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

  2. Step 2: Use Principal Formula

    P = EMI × [1 - (1 + r)^(-n)] / r

  3. Step 3: Substitute and Compute

    (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 तक rounded)。

  4. Final Answer:

    Principal ≈ ₹2,40,956.00

  5. Quick Check:

    इस P से EMI formula apply करने पर लगभग ₹8,000 आता है → सही ✅

Quick Variations

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

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

3. जब EMI, P, और R दिए हों तो time (n) logarithms से निकालें: n = ln(EMI / (EMI - P×r)) / ln(1 + r).

Trick to Always Use

  • Step 1: Annual rate को पहले periodic rate में convert करें।
  • Step 2: Formula में substitute करने से पहले हमेशा (1 + r)^n compute करें ताकि rounding errors न आएं।
  • Step 3: Total paid और interest amount देखकर answer verify करें।

Summary

Summary

  • EMI formula: EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ], जहाँ r periodic rate है और n total instalments।
  • Principal from EMI: P = EMI × [1 - (1 + r)^(-n)] / r.
  • Annual rate को हमेशा payment period के हिसाब से convert करें (monthly → divide by 12, quarterly → divide by 4).
  • जब EMI, P और r दिए हों तो n logs से निकालें: n = ln(EMI/(EMI - P·r)) / ln(1 + r).
  • Quick check: total paid = EMI × n; interest paid = total paid - principal - क्या rate और period के अनुसार reasonable है?

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.

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