Finding the Ratio of Numbers After Operations

This problem involves finding an unknown number 'y' based on changes in the ratio of two initial numbers and then calculating a new ratio using that value of 'y'. Let's break it down step by step.

Step 1: Represent the Initial Numbers

The initial ratio of the two numbers is given as 5:4. We can represent these numbers algebraically. Let the two numbers be $5x$ and $4x$, where $x$ is a common multiplier.

Step 2: Apply the Subtraction and Formulate the Equation

A number $y$ is subtracted from each of these initial numbers. The new numbers become $(5x - y)$ and $(4x - y)$.

The ratio of these resultant numbers is given as 2:1. We can write this as an equation:

$ \frac{5x - y}{4x - y} = \frac{2}{1} $

Step 3: Solve for the Relationship Between x and y

To find the value of $y$ in terms of $x$, we cross-multiply the equation from Step 2:

$ 1 \times (5x - y) = 2 \times (4x - y) $ $ 5x - y = 8x - 2y $

Now, rearrange the terms to solve for $y$:

$ 2y - y = 8x - 5x $ $ y = 3x $

This tells us that the number subtracted, $y$, is equal to 3 times the common multiplier $x$.

Step 4: Calculate the New Ratio After Addition

The question asks for the ratio of the resultant numbers when the same number $y$ is *added* to each of the two initial numbers ($5x$ and $4x$).

The new numbers will be $(5x + y)$ and $(4x + y)$.

We need to find the ratio:

$ \frac{5x + y}{4x + y} $

Substitute the value of $y$ we found ($y = 3x$) into this expression:

$ \frac{5x + (3x)}{4x + (3x)} $ $ \frac{8x}{7x} $

Step 5: Simplify the Final Ratio

The $x$ terms cancel out, leaving the final ratio:

$ \frac{8}{7} $

Therefore, the ratio of the resultant numbers when $y$ is added to each of the initial numbers is 8:7.

Summary Table

Step	Description	Result
1	Initial Numbers Representation	$5x, 4x$
2	Subtraction and Ratio Given	$ \frac{5x - y}{4x - y} = \frac{2}{1} $
3	Solving for y	$y = 3x$
4	Addition and New Ratio Expression	$ \frac{5x + y}{4x + y} $
5	Final Ratio Calculation	8:7