Pipe A can fill a water tank in x hours, where x > 0. Pipe B can fill the same water tank … — ACT

Pipe A can fill a water tank in $x$ hours, where $x > 0$ . Pipe B can fill the same water tank in $x + 3$ hours. If both pipes are opened at the same time, they can fill the tank together in exactly 2 hours. What is the value of $x$ ?

A
0.5
B
1
C
2
3Answer
E
6

Answer

The value of

x

is 3.

The correct value is 3. The work rate equation for the two pipes is

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

. Combining the fractions on the left gives

\frac{2x+3}{x^2+3x} = \frac{1}{2}

. Cross-multiplying yields

4x + 6 = x^2 + 3x

, which simplifies to the quadratic equation

x^2 - x - 6 = 0

. Factoring this quadratic equation gives

(x-3)(x+2) = 0

, which has the solutions

x = 3

and

x = -2

. Since

x

represents a positive number of hours, the only valid solution is 3.

Step-by-Step Solution

Set up the work rate equation based on the individual and combined rates.

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

The sum of the individual rates (in tanks per hour) equals the combined rate of the two pipes.

Find a common denominator to combine the fractions on the left side of the equation.

\frac{(x+3) + x}{x(x+3)} = \frac{1}{2} \implies \frac{2x+3}{x^2+3x} = \frac{1}{2}

To add fractions, they must share a common denominator, which is

x(x+3)

Cross-multiply to eliminate the denominators and form a quadratic equation.

2(2x+3) = x^2+3x \implies 4x + 6 = x^2 + 3x \implies x^2 - x - 6 = 0

Cross-multiplication converts the rational equation into a polynomial equation.

Factor the quadratic equation to find the solutions for

x

(x-3)(x+2) = 0 \implies x = 3

x = -2

Since

x

represents time, it must be positive (

x > 0

), so the only valid solution is

x = 3

Key Concept

Solving rational equations in the context of work-rate problems

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