Finding the Approximate Principal Sum using Simple and Compound Interest Difference

This problem asks us to find the approximate original sum of money (the principal) given the difference between the Simple Interest (SI) earned over a period and the Compound Interest (CI) earned over a different period on the same sum, but possibly at different rates.

Understanding Simple Interest (SI)

Simple Interest is calculated only on the principal amount. It does not compound, meaning interest earned in previous periods is not added to the principal for calculating interest in the next period.

The formula for Simple Interest is:

SI = $\frac{P \times R \times T}{100}$

• P is the Principal sum.
• R is the annual rate of interest.
• T is the time period in years.

Understanding Compound Interest (CI)

Compound Interest is calculated on the principal amount and also on the accumulated interest from previous periods. This means the interest "compounds" over time.

The formula for the total amount (Principal + CI) after T years compounded annually is:

Amount = $P \left(1 + \frac{R}{100}\right)^T$

The Compound Interest is then:

CI = Amount - P = $P \left(1 + \frac{R}{100}\right)^T - P$

Applying the Formulas to the Problem

Let the principal sum be P.

Calculate Simple Interest (SI)

The SI is for 4 years at 4 percent per annum.

• P = P
• R = 4%
• T = 4 years

SI = $\frac{P \times 4 \times 4}{100} = \frac{16P}{100} = 0.16P$

Calculate Compound Interest (CI)

The CI is for 3 years at 5 percent per annum.

• P = P
• R = 5%
• T = 3 years

CI = $P \left(1 + \frac{5}{100}\right)^3 - P$

CI = $P \left(1 + 0.05\right)^3 - P$

CI = $P (1.05)^3 - P$

Calculate $(1.05)^3$:

• $1.05 \times 1.05 = 1.1025$
• $1.1025 \times 1.05 = 1.157625$

So, CI = $P (1.157625) - P = P (1.157625 - 1) = 0.157625P$

Setting up the Difference Equation

The problem states that the SI exceeds the CI by Rs. 57.

SI - CI = 57

Substitute the calculated values of SI and CI:

$0.16P - 0.157625P = 57$

Solving for the Principal (P)

Combine the terms involving P:

$(0.16 - 0.157625)P = 57$

$0.002375P = 57$

Now, solve for P:

$P = \frac{57}{0.002375}$

Let's perform the division:

$P \approx 24000$

The approximate sum is Rs. 24000.

Verification with Options

Let's quickly verify the options:

The calculation with P = 24000 yields a difference of 57, matching the problem statement exactly. Therefore, the approximate sum is Rs. 24000.

Revision Table: Key Concepts

Additional Information: Approximations in Math Problems

Some problems in mathematics, especially those involving compound interest over several periods, can result in complex calculations. When asked for an "approximate sum," this indicates that the final answer might be rounded or that intermediate calculations could involve some level of rounding.

In this specific problem, calculating $(1.05)^3$ yields a precise value of 1.157625. The difference $0.16 - 0.157625 = 0.002375$ is also precise. Dividing 57 by 0.002375 gives exactly 24000. So, in this case, the "approximate" sum is actually the exact value among the given options. However, in other problems, you might need to round your final calculated value to match the closest option provided.

Always read the question carefully to understand if an exact value or an approximation is required. If options are given, they can help guide your level of precision in calculations.