If at same rate of interest, in 2 years, the simple interest is ₹56 and compound interest is ₹58, then what is

Question

If at same rate of interest, in 2 years, the simple interest is ₹56 and compound interest is ₹58, then what is the principal (in ₹)?

Examoogle · Accepted Answer

The correct answer is Option A: 392. In a 2-year period, simple interest (SI) is earned only on the principal, while compound interest (CI) is earned on both the principal and the interest from the first year. The difference between CI and SI for 2 years represents the interest earned on the first year's interest, allowing us to find the rate and subsequently the principal.

Step-by-Step Solution & Concepts:
In a 2-year period, simple interest (SI) is earned only on the principal, while compound interest (CI) is earned on both the principal and the interest from the first year. The difference between CI and SI for 2 years represents the interest earned on the first year's interest, allowing us to find the rate and subsequently the principal.

Working Principle: For a principal $P$ and rate $R$, the SI for 2 years is $SI = \frac{P 	imes R 	imes 2}{100} = 56$. The CI for 2 years is $CI = P(1 + \frac{R}{100})^2 - P = 58$. The difference $CI - SI$ is equivalent to the interest on the first year's interest ($SI_1 = \frac{56}{2} = 28$). Thus, $58 - 56 = 2$ is the interest on $28$, implying a rate of $R = \frac{2 	imes 100}{28} = \frac{100}{14} = \frac{50}{7} %$. Substituting this into the SI formula gives $56 = \frac{P 	imes (50/7) 	imes 2}{100}$, leading to $P = 392$.

Key Formulas: $SI = \frac{P 	imes R 	imes T}{100}$ — Simple Interest formula; $CI = P(1 + \frac{R}{100})^T - P$ — Compound Interest formula; $Difference = P 	imes (\frac{R}{100})^2$ — Difference for 2 years

Key Points: Simple interest remains constant each year.; Compound interest grows because interest is added to the principal.; The difference for 2 years is always the interest on the first year's interest.

Advantages: Mathematically precise calculation.; Direct derivation from interest definitions.

Disadvantages: Requires careful algebraic manipulation.; Prone to calculation errors if the rate is a fraction.

Applications: Banking and financial planning.; Investment analysis for debt instruments.

Additional Info: The rate of interest $R$ is approximately $7.14\%$.. Option B (385), C (396), and D (387) are mathematically incorrect based on the given interest values.

Answer: A is correct — The principal is determined by calculating the interest on the first year's simple interest, which is ₹2, leading to the principal amount of ₹392.

Exam Tip: For 2-year problems, always remember that $Difference = \frac{SI 	imes R}{200}$ is a very quick shortcut to find the rate!

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