If `x = ay, y= bx,` ten the value of `(1)/(1 + a) + (1)/(1 + b)` is :

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem, we start with the equations given:

1. \( x = ay \)
2. \( y = bx \)

We need to find the value of \( \frac{1}{1 + a} + \frac{1}{1 + b} \).

### Step 1: Express \( a \) and \( b \) in terms of \( x \) and \( y \)

From the first equation \( x = ay \), we can express \( a \) as:
\[ a = \frac{x}{y} \]

From the second equation \( y = bx \), we can express \( b \) as:
\[ b = \frac{y}{x} \]

### Step 2: Substitute \( a \) and \( b \) into the expression

Now we substitute \( a \) and \( b \) into the expression \( \frac{1}{1 + a} + \frac{1}{1 + b} \):
\[ \frac{1}{1 + a} = \frac{1}{1 + \frac{x}{y}} = \frac{1}{\frac{y + x}{y}} = \frac{y}{x + y} \]

\[ \frac{1}{1 + b} = \frac{1}{1 + \frac{y}{x}} = \frac{1}{\frac{x + y}{x}} = \frac{x}{x + y} \]

### Step 3: Combine the fractions

Now we can add the two fractions:
\[ \frac{1}{1 + a} + \frac{1}{1 + b} = \frac{y}{x + y} + \frac{x}{x + y} \]

Since both fractions have the same denominator, we can combine them:
\[ \frac{y + x}{x + y} = \frac{x + y}{x + y} = 1 \]

### Conclusion

Thus, the value of \( \frac{1}{1 + a} + \frac{1}{1 + b} \) is:
\[ \boxed{1} \]