Calculating Simple Interest Ratio: Rahul vs Shyam

This question asks us to find the ratio of simple interest earned by two individuals, Rahul and Shyam, who invested the same principal amount at the same interest rate but for different time periods. We will use the formula for simple interest to calculate the interest earned by each person and then find the ratio.

Understanding Simple Interest

Simple interest (SI) is calculated only on the principal amount. The formula for simple interest is:

$\text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$

Where:

• Principal (P) is the initial amount invested.
• Rate (R) is the annual interest rate (in percent).
• Time (T) is the period for which the amount is invested (in years).

Simple Interest Calculation for Rahul

Rahul invested X rupees for 3 years at a rate of y% per annum.

Principal for Rahul = X rupees

Rate of Interest for Rahul = y%

Time Period for Rahul = 3 years

Simple Interest earned by Rahul ($SI_{\text{Rahul}}$) is:

$SI_{\text{Rahul}} = \frac{X \times y \times 3}{100}$

Simple Interest Calculation for Shyam

Shyam invested the same amount (X rupees) at the same rate (y%) for 12 years.

Principal for Shyam = X rupees

Rate of Interest for Shyam = y%

Time Period for Shyam = 12 years

Simple Interest earned by Shyam ($SI_{\text{Shyam}}$) is:

$SI_{\text{Shyam}} = \frac{X \times y \times 12}{100}$

Finding the Ratio of Simple Interests

We need to find the ratio of simple interest earned by Rahul to simple interest earned by Shyam. This can be written as $SI_{\text{Rahul}} : SI_{\text{Shyam}}$ or $\frac{SI_{\text{Rahul}}}{SI_{\text{Shyam}}}$.

$\frac{SI_{\text{Rahul}}}{SI_{\text{Shyam}}} = \frac{\frac{X \times y \times 3}{100}}{\frac{X \times y \times 12}{100}}$

To simplify this expression, we can cancel out the common terms in the numerator and the denominator. The terms X, y, and 100 are present in both the numerator and the denominator.

$\frac{SI_{\text{Rahul}}}{SI_{\text{Shyam}}} = \frac{X \times y \times 3}{X \times y \times 12}$

Cancel out X and y:

$\frac{SI_{\text{Rahul}}}{SI_{\text{Shyam}}} = \frac{3}{12}$

Now, simplify the fraction $\frac{3}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$\frac{3 \div 3}{12 \div 3} = \frac{1}{4}$

So, the ratio of simple interest earned by Rahul to simple interest earned by Shyam is 1:4.

Conclusion

The ratio of simple interest earned by Rahul to simple interest earned by Shyam is 1 : 4.

Ratio = $\frac{SI_{\text{Rahul}}}{SI_{\text{Shyam}}} = \frac{3xy/100}{12xy/100} = \frac{3}{12} = \frac{1}{4}$

Revision Table: Simple Interest Key Points

Additional Information: Simple Interest vs Compound Interest

It's important to distinguish between simple interest and compound interest.

• Simple Interest: Calculated only on the initial principal amount. The interest earned each period remains constant if P and R are fixed.
• Compound Interest: Calculated on the initial principal and also on the accumulated interest from previous periods. Interest earns interest, leading to faster growth over time compared to simple interest for the same principal, rate, and time (greater than 1 year).

This question specifically deals with simple interest, making the calculation straightforward based on the formula SI = PRT/100.