Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5, then find the numbers.

Options

A. `x= 42, y= 44`
B. `x= 48, y=40`
C. `x=40, y=48`
D. `x= 50 , y=40`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the first number be \( x \) and the second number be \( y \).

### Step 2: Set Up the First Equation
According to the problem, the ratio of the two numbers is given as \( 5:6 \). This can be expressed mathematically as:
\[ \frac{x}{y} = \frac{5}{6} \]
Cross-multiplying gives us:
\[ 6x = 5y \]
This is our first equation:
\[ 6x - 5y = 0 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that if 8 is subtracted from each number, the new ratio becomes \( 4:5 \). This can be expressed as:
\[ \frac{x - 8}{y - 8} = \frac{4}{5} \]
Cross-multiplying gives us:
\[ 5(x - 8) = 4(y - 8) \]
Expanding this, we have:
\[ 5x - 40 = 4y - 32 \]
Rearranging gives us:
\[ 5x - 4y = 8 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We now have the following system of equations:
1. \( 6x - 5y = 0 \) (Equation 1)
2. \( 5x - 4y = 8 \) (Equation 2)

We can solve these equations using substitution or elimination. Here, we will use substitution.

From Equation 1, we can express \( y \) in terms of \( x \):
\[ 5y = 6x \implies y = \frac{6}{5}x \]

### Step 5: Substitute \( y \) in Equation 2
Now, substitute \( y = \frac{6}{5}x \) into Equation 2:
\[ 5x - 4\left(\frac{6}{5}x\right) = 8 \]
This simplifies to:
\[ 5x - \frac{24}{5}x = 8 \]
To eliminate the fraction, multiply the entire equation by 5:
\[ 25x - 24x = 40 \]
This simplifies to:
\[ x = 40 \]

### Step 6: Find \( y \)
Now that we have \( x \), we can find \( y \) using the equation \( y = \frac{6}{5}x \):
\[ y = \frac{6}{5} \times 40 = 48 \]

### Conclusion
The two numbers are:
- First number (\( x \)) = 40
- Second number (\( y \)) = 48