Understanding the Simple Interest Problem

This problem asks us to find the ratio of two principal amounts, ₹x and ₹y, given that they earn the same simple interest under different conditions of interest rate and time period.

Simple interest is calculated using the formula:

\( \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} \)

Here, R is the annual interest rate in percentage and T is the time period in years.

Calculating Simple Interest for Investment x

For the first investment:

• Principal (P\(_1\)) = ₹x
• Rate (R\(_1\)) = 9% per annum
• Time (T\(_1\)) = 5 years

Using the simple interest formula, the interest earned on ₹x is:

\( \text{SI}_1 = \frac{x \times 9 \times 5}{100} \)

\( \text{SI}_1 = \frac{45x}{100} \)

Calculating Simple Interest for Investment y

For the second investment:

• Principal (P\(_2\)) = ₹y
• Rate (R\(_2\)) = 7.5% per annum
• Time (T\(_2\)) = 4 years

Using the simple interest formula, the interest earned on ₹y is:

\( \text{SI}_2 = \frac{y \times 7.5 \times 4}{100} \)

\( \text{SI}_2 = \frac{30y}{100} \)

Finding the Investment Ratio x:y

The problem states that the interest earned from both investments is the same. Therefore, we can set SI\(_1\) equal to SI\(_2\):

\( \text{SI}_1 = \text{SI}_2 \)

\( \frac{45x}{100} = \frac{30y}{100} \)

To solve for the ratio \(x : y\), we can first multiply both sides of the equation by 100 to eliminate the denominators:

\( 45x = 30y \)

Now, we need to isolate the ratio \( \frac{x}{y} \). We can do this by dividing both sides by \(y\) and then by 45:

\( \frac{x}{y} = \frac{30}{45} \)

To simplify the ratio \( \frac{30}{45} \), we find the greatest common divisor (GCD) of 30 and 45. The GCD is 15.

Divide the numerator and the denominator by 15:

\( \frac{x}{y} = \frac{30 \div 15}{45 \div 15} = \frac{2}{3} \)

So, the ratio \( x : y \) is \( 2 : 3 \).

Summary of Simple Interest Calculation Steps

Here is a summary of the steps taken to find the ratio x:y:

1. Write down the simple interest formula: \( SI = \frac{P \times R \times T}{100} \).
2. Calculate the simple interest for the first investment (SI\(_1\)) using P=x, R=9, T=5.
3. Calculate the simple interest for the second investment (SI\(_2\)) using P=y, R=7.5, T=4.
4. Equate the two simple interests: SI\(_1\) = SI\(_2\).
5. Solve the resulting equation for the ratio \( \frac{x}{y} \).
6. Simplify the fraction to its lowest terms.

Revision Table: Key Simple Interest Concepts

Additional Information on Simple Interest Calculations

Simple interest is a basic concept in finance. It is different from compound interest, where interest is calculated on the principal amount plus any accumulated interest from previous periods. Simple interest is easier to calculate but results in lower earnings over time compared to compound interest, assuming the same rate and principal.

When solving problems involving simple interest, it is crucial to ensure that the time period (T) and the rate (R) are consistent. If the rate is given per annum, the time should be in years. If the time is given in months or days, it needs to be converted to years before using the formula.

For example:

• If time is given in months, divide the number of months by 12 to convert it to years.
• If time is given in days, divide the number of days by 365 (or 366 in a leap year) to convert it to years.

In this specific problem, the time periods were already given in years (5 years and 4 years), so no conversion was needed.