`(1)/(x)+(3)/(x)=(1)/(2)`
Dennis is helping Billy assemble his new computer desk. Billy can put the desk together three times as fast as Dennis, and together Billy and Dennis can finish assembling the desk in 2 hours. The equation above represents this situation. Which of the following describes what the expression `(3)/(x)` represents in this equation?

Options

A. The fraction of the job that would be completed by Billy in 1 hour.
B. The fraction of the job that would be completed by Dennis in 1 hour.
C. The time, in hours, that it takes Billy to complete one third of the job.
D. The time, in hours that it takes Billy to assemble the desk by himself.

Answer

To solve the equation \(\frac{1}{x} + \frac{3}{x} = \frac{1}{2}\) and understand what the expression \(\frac{3}{x}\) represents, we can follow these steps:

### Step 1: Combine the Left Side of the Equation
The left side of the equation consists of two fractions with the same denominator. We can combine them:

\[ \frac{1}{x} + \frac{3}{x} = \frac{1 + 3}{x} = \frac{4}{x} \]

### Step 2: Rewrite the Equation
Now, we can rewrite the equation with the combined left side:

\[ \frac{4}{x} = \frac{1}{2} \]

### Step 3: Cross-Multiply to Solve for \(x\)
To eliminate the fractions, we can cross-multiply:

\[ 4 \cdot 2 = 1 \cdot x \]

This simplifies to:

\[ 8 = x \]

### Step 4: Determine the Rate of Work
Now that we have \(x = 8\), we can find the rates of work for Dennis and Billy. Since \(x\) represents the time it takes Dennis to complete the desk alone, Billy, who works three times as fast, will take \(\frac{x}{3} = \frac{8}{3}\) hours to complete the desk alone.

### Step 5: Interpret \(\frac{3}{x}\)
The expression \(\frac{3}{x}\) represents the fraction of the job that Billy can complete in one hour. Since we found \(x = 8\):

\[ \frac{3}{x} = \frac{3}{8} \]

This means that in one hour, Billy can complete \(\frac{3}{8}\) of the desk.

### Conclusion
Thus, the expression \(\frac{3}{x}\) represents the fraction of the job that Billy contributes in one hour.

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