Pipes X, Y, and Z together can fill a tank in 8 hours. If all three work together for 3 hours, and then pipe Z is closed, the rem…

Question

Pipes X, Y, and Z together can fill a tank in 8 hours. If all three work together for 3 hours, and then pipe Z is closed, the remaining tank is filled by X and Y in 10 hours. In how many hours can pipe Z alone fill the empty tank?

1. A. 18
2. B. 16 (Correct answer)
3. C. 32
4. D. 24

Correct Answer

Option B — 16

Detailed Solution & Explanation

The correct answer is 16.

Shortcut Trick

• Work in rates per hour rather than times.
• All three together: $\dfrac{1}{8}$ of the tank per hour, so in 3 hours they fill $\dfrac{3}{8}$.
• Remaining $\dfrac{5}{8}$ takes X and Y ten hours $\Rightarrow$ their combined rate is $\dfrac{5/8}{10} = \dfrac{1}{16}$.
• $Z = \left(X+Y+Z\right) - \left(X+Y\right) = \dfrac{1}{8} - \dfrac{1}{16} = \dfrac{1}{16}$, so Z alone takes 16 hours.

Alternate Method

• Take the tank as 16 units (LCM of the times involved).
  • X + Y + Z together: $\dfrac{16}{8} = 2$ units/hour
  • In 3 hours they fill $6$ units, leaving $10$ units
  • X + Y fill those 10 units in 10 hours $\Rightarrow$ 1 unit/hour
  • Z alone $= 2 - 1 = 1$ unit/hour $\Rightarrow \dfrac{16}{1} = \textbf{16}$ hours

Additional Information

• Rates add, times do not. If A takes $a$ hours and B takes $b$ hours, together they take $\dfrac{ab}{a+b}$ hours — never $\dfrac{a+b}{2}$.
• The LCM method (taking the tank as a convenient number of units) removes fractions entirely and is usually the fastest route in this topic.
• An emptying pipe simply takes a negative rate, so a tank with inlets and an outlet is filled at the net rate.