If $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009}$ =1, then the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ is:

Q: If $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009}$ =1, then the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ is:

To solve the problem, we need to find the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ given that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1$. Let's start by analyzing the given equation: \[ \frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1 \] We can express each term in terms of its complement to 1: 1. $\frac{x}{x+1} = 1 - \frac{1}{x+1}$ 2. $\frac{y}{y+2} = 1 - \frac{2}{y+2}$ 3. $\frac{z}{z+1009} = 1 - \frac{1009}{z+1009}$ Substituting these into the expression we want to evaluate: \[ \frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009} = \left(1 - \frac{1}{x+1}\right) + \left(1 - \frac{2}{y+2}\right) + \left(1 - \frac{1009}{z+1009}\right) \] Simplifying this expression: \[ = 3 - \left(\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009}\right) \] Given that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1$, we substitute this into the equation: \[ = 3 - 1 = 2 \] Thus, the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ is indeed 2, which corresponds to Option 2. **Explanation for Other Options:** - **Option 1:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 2$, which contradicts the given condition that it equals 1. - **Option 3:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 0$, which is also a contradiction. - **Option 4:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = -3$, which is not possible given the positive nature of the fractions involved. Therefore, Option 2 is the only correct answer, as it aligns perfectly with the given condition and logical deductions.

Question

Accepted Answer

To solve the problem, we need to find the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ given that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1$.

Let's start by analyzing the given equation:

$$
\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1
$$

We can express each term in terms of its complement to 1:

1. $\frac{x}{x+1} = 1 - \frac{1}{x+1}$
2. $\frac{y}{y+2} = 1 - \frac{2}{y+2}$
3. $\frac{z}{z+1009} = 1 - \frac{1009}{z+1009}$

Substituting these into the expression we want to evaluate:

$$
\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009} = \left(1 - \frac{1}{x+1}\right) + \left(1 - \frac{2}{y+2}\right) + \left(1 - \frac{1009}{z+1009}\right)
$$

Simplifying this expression:

$$
= 3 - \left(\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009}\right)
$$

Given that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 1$, we substitute this into the equation:

$$
= 3 - 1 = 2
$$

Thus, the value of $\frac{x}{x+1} + \frac{y}{y+2} + \frac{z}{z+1009}$ is indeed 2, which corresponds to Option 2.

**Explanation for Other Options:**

- **Option 1:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 2$, which contradicts the given condition that it equals 1.
- **Option 3:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = 0$, which is also a contradiction.
- **Option 4:** This would imply that $\frac{1}{x+1} + \frac{2}{y+2} + \frac{1009}{z+1009} = -3$, which is not possible given the positive nature of the fractions involved.

Therefore, Option 2 is the only correct answer, as it aligns perfectly with the given condition and logical deductions.