Question

select the correct answer:
a water tank can be filled using pipe - a, pipe - b, or both pipes. it takes pipe - a twice as long to fill the tank when working alone rather than both pipes are working. it takes only 4 hours. the expression (\frac{1}{2x}+\frac{1}{x}) represents the situation.
which expression also represents this situation?
options:
(\frac{4(x + 2)}{x})
(\frac{12}{x})
(\frac{x + 2}{2x})
(\frac{3}{x})

Explanation:

Step1: Define variables for rates

Let the volume of the tank be $V$. Rate of Pipe A: $r_A = \frac{V}{10}$. Rate of Pipe B: $r_B = \frac{V}{x}$. Combined rate: $r_{A+B} = \frac{V}{8}$.

Step2: Set up combined rate equation

Combined rate equals sum of individual rates:
$$\frac{V}{8} = \frac{V}{10} + \frac{V}{x}$$

Step3: Simplify equation (divide by $V$)

Cancel $V$ from all terms:
$$\frac{1}{8} = \frac{1}{10} + \frac{1}{x}$$

Step4: Rearrange to solve for $\frac{1}{x}$

Isolate $\frac{1}{x}$:
$$\frac{1}{x} = \frac{1}{8} - \frac{1}{10}$$

Step5: Find common denominator (40)

Compute the right-hand side:
$$\frac{1}{x} = \frac{5 - 4}{40} = \frac{1}{40}$$

Step6: Cross-multiply to solve for $x$

Rearrange to solve for $x$:
$$x = 40$$

Step7: Test given options

Substitute $x=40$ into the original expression $\frac{1}{10}+\frac{1}{x}$: $\frac{1}{10}+\frac{1}{40}=\frac{5}{40}=\frac{1}{8}$. Now test options:

• $\frac{40 - 8}{40} = \frac{32}{40} = \frac{4}{5}$ (no)
• $\frac{40}{8} = 5$ (no)
• $\frac{40 - 8}{80} = \frac{32}{80} = \frac{2}{5}$ (no)
• $\frac{40}{8x}$: Substitute $x=40$, $\frac{40}{8*40}=\frac{1}{8}$ (matches)

Answer:

D. $\boldsymbol{\frac{40}{8x}}$