Step-by-Step Solution

Topic: Financial Mathematics (EMI)

(I) EMI FORMULA

The formula for Equated Monthly Installment (EMI) using the reducing balance method is:

E = P × i × [ (1 + i)n / ((1 + i)n – 1) ]

Alternative Form (using negative power):

E = (P × i) / (1 – (1 + i)-n)

(This second form is easier to use with the values given in the question context).

(II) IDENTIFYING VALUES (P, i, n)

  • P (Principal): ₹ 7,00,000
  • i (Monthly Interest): Rate is 12% per annum.

    ⇒ i = 12 / (100 × 12) = 0.01
  • n (Total Months): 30 years.

    ⇒ n = 30 × 12 = 360 months

(III)(A) EMI CALCULATION FOR 30 YEARS

Using the formula: E = (P × i) / (1 – (1 + i)-n)

Substitute the values:

  • Numerator = 7,00,000 × 0.01 = 7,000
  • Denominator = 1 – (1.01)-360
  • Given value: (1.01)-360 = 0.02781668
  • Denominator = 1 – 0.02781668 = 0.97218332
EMI = 7000 / 0.97218332 ≈ ₹ 7,200.28

(III)(B) EMI CALCULATION FOR 20 YEARS

New n: 20 × 12 = 240 months.

Substitute values into the same formula:

  • Numerator = 7,00,000 × 0.01 = 7,000
  • Denominator = 1 – (1.01)-240
  • Given value: (1.01)-240 = 0.09180584
  • Denominator = 1 – 0.09180584 = 0.90819416
EMI = 7000 / 0.90819416 ≈ ₹ 7,707.59

⚠️ Silly Mistake Audit: Interest Rate Conversion

The most common error is using the annual rate (12%) directly in the monthly formula. You must divide the annual rate by 12 and divide by 100 to get the decimal monthly rate (0.01). Using 0.12 instead of 0.01 will make the answer completely wrong!

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