Amrita buys a car for which she makes a down payment of ₹ 2,50,000 and the balance is to be paid in 2 years by monthly instalments of ₹ 25,448 each. If the financer charges interest at the rate of 20% p.a, find the actual price of the car. [Given $\left(\dfrac{61}{60}\right)^{-24} = 0.67253$]

> [!TLDR] > With $i=\tfrac1{60}$, $n=24$, the loan $P=\text{EMI}\times\dfrac{1-(1+i)^{-n}}{i}=25448\times60\times(1-0.67253)\approx₹5{,}00{,}000$; add the down payment to get $₹7{,}50{,}000$. > [!FORMULA] > $\text{EMI}=\dfrac{P\,i}{1-(1+i)^{-n}}$, so loan $P=\text{EMI}\times\dfrac{1-(1+i)^{-n}}{i}$, where $i$ is the monthly rate and $n$ the number of monthly instalments. Actual price $=$ down payment $+$ loan $P$. Given: down payment $=₹2{,}50{,}000$, EMI $=₹25{,}448$, term $=2$ years, rate $=20\%$ p.a. 1. Number of instalments: $n=2\times12=24$. 2. Monthly rate: $i=\dfrac{20}{12\times100}=\dfrac{20}{1200}=\dfrac{1}{60}$, so $1+i=\dfrac{61}{60}$. 3. Loan amount: $P=\text{EMI}\times\dfrac{1-(1+i)^{-n}}{i}=25448\times\dfrac{1-\left(\frac{61}{60}\right)^{-24}}{1/60}=25448\times60\times\left[1-\left(\dfrac{61}{60}\right)^{-24}\right]$. …

Amrita buys a car for which she makes a down payment of ₹ 2,50,000 and the balance is to be paid in 2 years by

Question

\left(\dfrac{61}{60}\right)^{-24} = 0.67253

CBSECBSE Class XII Board 2024Subjective· 5mImportance★★★★★

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Flat Interest Rate EMI

Flat Interest Rate EMI — From Intuition to Precision

Imagine you borrow ₹1,00,000 from a bank for 1 year. The bank says: "We charge 12% interest per year, flat." What does "flat" mean here?

The intuition: The bank calculates the total interest on the entire original loan amount for the full loan tenure, regardless of how much you've already repaid. You pay interest on the full ₹1,00,000 for the whole year — even though you're paying back the principal bit by bit each month.

Step-by-step breakdown

Let’s take a concrete example:

Loan amount (P): ₹1,00,000
Annual flat interest rate (R): 12%
Tenure (N): 1 year (12 months)

Step 1: Total interest for the year

Interest = Principal × Rate × Time

= ₹1,00,000 × (12/100) × 1 = ₹12,000

Step 2: Total amount to repay

Principal + Interest = ₹1,00,000 + ₹12,000 = ₹1,12,000

Step 3: Monthly EMI

EMI = Total amount ÷ Number of months

= ₹1,12,000 ÷ 12 = ₹9,333.33 per month

Note

Every month you pay ₹9,333.33. The interest portion in each EMI is not decreasing — it's a fixed ₹1,000 per month (₹12,000 ÷ 12). The principal repaid each month is also fixed: ₹8,333.33.

The precise formula

\text{EMI}_{\text{flat}} = \frac{P + (P \times R \times N)}{N}

Where:

$P$ = original loan principal
$R$ = annual flat interest rate (as a decimal, e.g., 0.12 for 12%)
$N$ = loan tenure in months (or years, as long as consistent)

Alternatively, in a cleaner form:

\text{EMI}_{\text{flat}} = \frac{P}{N} + \frac{P \times R}{12}

The first term is the fixed principal repayment per month. The second term is the fixed monthly interest.

Why "flat" is important — and a warning

Watch out

A flat interest rate looks lower than the equivalent reducing-balance rate, but it's actually more expensive.

In our example, 12% flat is equivalent to about 21.5% reducing-balance interest — nearly double. …

Start your 14-day free trial to unlock the full solution →

With $i=\tfrac1{60}$ , $n=24$ , the loan $P=\text{EMI}\times\dfrac{1-(1+i)^{-n}}{i}=25448\times60\times(1-0.67253)\approx₹5{,}00{,}000$ ; add the down payment to get $₹7{,}50{,}000$ .

\text{EMI}=\dfrac{P\,i}{1-(1+i)^{-n}}

Given: down payment $=₹2{,}50{,}000$ , EMI $=₹25{,}448$ , term $=2$ years, rate $=20\%$ p.a.

Number of instalments: $n=2\times12=24$ .
Monthly rate: $i=\dfrac{20}{12\times100}=\dfrac{20}{1200}=\dfrac{1}{60}$ , so $1+i=\dfrac{61}{60}$ .
Loan amount: $P=\text{EMI}\times\dfrac{1-(1+i)^{-n}}{i}=25448\times\dfrac{1-\left(\frac{61}{60}\right)^{-24}}{1/60}=25448\times60\times\left[1-\left(\dfrac{61}{60}\right)^{-24}\right]$ . …

Start your 14-day free trial to unlock the full solution →

🎓Unlock everything free for 14 days

Unlock everything free for 14 days