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Equal Installments / EMI

Introduction

In loan repayment problems, the borrower pays equal periodic instalments (EMIs). Each EMI includes both interest and principal repayment. This pattern helps you calculate EMIs, find the principal from a given EMI, or determine the total interest paid over time.

These problems are essential in understanding bank loans, mortgages, and annuity concepts, which are frequently asked in aptitude and finance-related exams.

Pattern: Equal Installments / EMI

Pattern

The EMI is a fixed payment made periodically to repay a loan with interest. It is derived from the present value of an annuity formula.

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

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

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

Step-by-Step Example

Question

Find the monthly EMI on a loan of ₹5,00,000 at 10% per annum for 5 years (monthly payments).

Solution

  1. Step 1: Identify Values

    Principal (P) = ₹5,00,000; Annual Rate (R) = 10%; Time = 5 years; 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

    Compute (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 (rounded to 2 dp).

  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 (reasonable for 10% over 5 years).

Question

A person pays ₹8,000 every month for 3 years at 12% per annum compounded monthly. Find the principal amount of the loan.

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 (rounded to nearest rupee).

  4. Final Answer:

    Principal ≈ ₹2,40,956.00

  5. Quick Check:

    Recalculate EMI from this P using EMI formula → ≈ ₹8,000 (close) ✅

Quick Variations

1. For quarterly instalments: r = R/400, n = years × 4.

2. For annual instalments: r = R/100, n = years.

3. If EMI, P, and R are known, time (n) can be found using logarithms: n = ln(EMI / (EMI - P×r)) / ln(1 + r).

Trick to Always Use

  • Step 1: Convert annual rate into periodic rate before substitution.
  • Step 2: Always compute (1 + r)^n before substituting into formula to avoid rounding errors.
  • Step 3: Verify results by comparing total paid and interest amount.

Summary

Summary

  • EMI formula: EMI = P × r × (1 + r)^n / [ (1 + r)^n - 1 ], where r is periodic rate and n is total instalments.
  • Principal from EMI: P = EMI × [1 - (1 + r)^(-n)] / r.
  • Always convert the annual rate to the same period as payments (monthly → divide by 12, quarterly → divide by 4).
  • Use logs to solve for n when EMI, P and r are known: n = ln(EMI/(EMI - P·r)) / ln(1 + r).
  • Quick check: total paid = EMI × n; interest paid = total paid - principal - does it match expectations for the rate and period?

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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