Simple Interest Ratio Calculation

The problem requires finding the ratio $x : y$ based on equal simple interest yields from two different investments.

Understanding Simple Interest

The formula for Simple Interest (SI) is:

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

Where:

• $P$ is the Principal amount
• $R$ is the Rate of interest per annum
• $T$ is the Time period in years

Calculating Interest Yields

Interest from investment ₹$x$:

• Principal ($P_x$) = $x$
• Rate ($R_x$) = 9%
• Time ($T_x$) = 5 years
• Interest ($SI_x$) = $ \frac{x \times 9 \times 5}{100} = \frac{45x}{100} $

Interest from investment ₹$y$:

• Principal ($P_y$) = $y$
• Rate ($R_y$) = 6.25%
• Time ($T_y$) = 8 years
• Interest ($SI_y$) = $ \frac{y \times 6.25 \times 8}{100} = \frac{50y}{100} $

Equating the Interests

The problem states that the interests are the same:

$ SI_x = SI_y $

$ \frac{45x}{100} = \frac{50y}{100} $

Finding the Ratio $x : y$

We can simplify the equation by cancelling out the denominator $100$:

$ 45x = 50y $

To find the ratio $x : y$, we rearrange the equation:

$ \frac{x}{y} = \frac{50}{45} $

Now, simplify the fraction:

$ \frac{x}{y} = \frac{50 \div 5}{45 \div 5} = \frac{10}{9} $

Therefore, the ratio $x : y$ is $10 : 9$.