EMI Made Easy — Formula, Intuition, and a CAT-style Question

You’ll learn: (1) where the EMI formula comes from, (2) the one-line “sanity bounds” to beat options, and (3) a 12% p.a., 4-month example solved cleanly.

1) What is EMI? (30-second intuition)

Loan of \(L\) is repaid in \(n\) equal monthly payments (EMIs). Each month, the bank charges interest on the remaining principal at the monthly rate \(r\) (if annual rate is \(R\%\), then \(r=\tfrac{R}{12\cdot 100}\)).

Key idea (present value). The present value of all EMIs equals the loan: \[ L=\frac{E}{(1+r)^1}+\frac{E}{(1+r)^2}+\cdots+\frac{E}{(1+r)^n}. \] This is a finite GP. Sum it to get EMI \(E\).

2) Deriving the formula (no magic)

Let \(E\) be the EMI. Using the GP sum \(S=a(1-q^n)/(1-q)\) with \(a=\tfrac{E}{1+r}\) and \(q=\tfrac{1}{1+r}\):

\[ L=E\cdot \frac{\dfrac{1}{1+r}\left(1-\left(\tfrac{1}{1+r}\right)^n\right)}{1-\tfrac{1}{1+r}} =E\cdot \frac{1-(1+r)^{-n}}{r}. \] Hence \[ \boxed{\;E = L\; \frac{r}{1-(1+r)^{-n}} = L\; \frac{r(1+r)^n}{(1+r)^n-1}\;} \] where \(r=\dfrac{R}{12\cdot100}\) and \(n=\) number of months.

You can memorize either form. The second is numerically stable.

3) Cheat codes (fast option killing)

Monthly rate: \(r=\dfrac{R}{1200}\). For \(R=12\%\), \(r=0.01\) (1%).
Interest bounds (reducing balance): \[ \underbrace{L\cdot r\cdot \frac{n}{2}}_{\text{lower (avg balance)}} \;\lesssim\; \text{Total interest } (nE-L)\; \lesssim \; \underbrace{L\cdot r\cdot n}_{\text{upper (simple interest)}}. \] This lets you sanity-check options in seconds.
Tiny \(r\), small \(n\): \((1+r)^n\approx 1+nr\) (binomial). Useful for quick estimates.
Interest per month ≈ \(r \times\) current balance. If one option implies crazy total interest, discard it.

Visual scratch (optional)

Scratch hint: total interest bounds — Total interest is \(nE-L\); it must sit between \(Lrn/2\) and \(Lrn\).

4) CAT-style question (full work)

Question. Mr. Sharma spends ₹24,000 on his credit card. The bank converts this into an EMI plan at 12% p.a., repayable in 4 months via equal monthly EMIs. What is the approximate EMI (₹)?

Click for solution

Step-1: Identify \(L, r, n\)

Loan \(L=24000\). Annual \(R=12\%\Rightarrow r=\frac{12}{1200}=0.01\) (1% per month). \(n=4\) months.

Step-2: Apply EMI formula

\[ E=L\;\frac{r(1+r)^n}{(1+r)^n-1} =24000\cdot \frac{0.01\,(1.01)^4}{(1.01)^4-1}. \]

Step-3: Quick compute

\((1.01)^4\approx 1.040604\). Numerator \(=24000\times 0.01\times 1.040604\approx 249.744\).
Denominator \(=1.040604-1=0.040604\). Therefore \(E\approx \dfrac{249.744}{0.040604}\approx 6154\Rightarrow \boxed{6150}\) (to nearest 50).

Step-4: Sanity by bounds

Total interest \(=4E-24000\approx 24616-24000\approx 616\). Bounds: \(Lrn/2=24000\times 0.01\times 2=480\) and \(Lrn=960\). \(616\) lies comfortably between \(480\) and \(960\) ⇒ answer consistent.

Worked board for the EMI question — Solution board (revealed on click only).

5) One-screen recap

EMI: \( \displaystyle E=L\,\frac{r(1+r)^n}{(1+r)^n-1}\), \(r=\frac{R}{1200}\), \(n\) months.
Bounds: \(Lrn/2 \lesssim nE-L \lesssim Lrn\).
Micro-tip: For tiny \(r\) and small \(n\), \((1+r)^n\approx 1+nr\) speeds mental math.

EMI Made Easy — Formula, Intuition & Quick Checks