If ` (1)/( x + 2) = (3)/( y + 3) = (1331)/( z + 1331) = (1)/(3)` then what is the value of ` ( x)/( x + 1) + (4)/( y + 2) + (z )/( z + 2662)` ?

Options

A. 0
B. 1
C. `(3)/(2)`
D. 3

Answer

To solve the equation given in the problem, we need to find the values of \( x \), \( y \), and \( z \) based on the relationships provided.

We start with the equation:

\[ \frac{1}{x + 2} = \frac{3}{y + 3} = \frac{1331}{z + 1331} = \frac{1}{3} \]

### Step 1: Solve for \( x \)

From the first part of the equation, we have:

\[ \frac{1}{x + 2} = \frac{1}{3} \]

Cross-multiplying gives:

\[ 1 \cdot 3 = 1 \cdot (x + 2) \]

This simplifies to:

\[ 3 = x + 2 \]

Subtracting 2 from both sides:

\[ x = 3 - 2 = 1 \]

**Hint:** To isolate \( x \), move the constant to the other side of the equation.

### Step 2: Solve for \( y \)

Next, we use the second part of the equation:

\[ \frac{3}{y + 3} = \frac{1}{3} \]

Cross-multiplying gives:

\[ 3 \cdot 3 = 1 \cdot (y + 3) \]

This simplifies to:

\[ 9 = y + 3 \]

Subtracting 3 from both sides:

\[ y = 9 - 3 = 6 \]

**Hint:** Remember to cross-multiply and then isolate \( y \) by moving constants to the other side.

### Step 3: Solve for \( z \)

Now, we look at the third part of the equation:

\[ \frac{1331}{z + 1331} = \frac{1}{3} \]

Cross-multiplying gives:

\[ 1331 \cdot 3 = 1 \cdot (z + 1331) \]

This simplifies to:

\[ 3993 = z + 1331 \]

Subtracting 1331 from both sides:

\[ z = 3993 - 1331 = 2662 \]

**Hint:** Again, use cross-multiplication and isolate \( z \) by moving the constant to the other side.

### Step 4: Calculate the final expression

Now we need to find the value of:

\[ \frac{x}{x + 1} + \frac{4}{y + 2} + \frac{z}{z + 2662} \]

Substituting the values of \( x \), \( y \), and \( z \):

1. For \( \frac{x}{x + 1} \):

\[ \frac{1}{1 + 1} = \frac{1}{2} \]

2. For \( \frac{4}{y + 2} \):

\[ \frac{4}{6 + 2} = \frac{4}{8} = \frac{1}{2} \]

3. For \( \frac{z}{z + 2662} \):

\[ \frac{2662}{2662 + 2662} = \frac{2662}{5324} = \frac{1}{2} \]

Now, adding these values together:

\[ \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2} \]

### Final Answer

The value of \( \frac{x}{x + 1} + \frac{4}{y + 2} + \frac{z}{z + 2662} \) is:

\[ \frac{3}{2} \]

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