https://file.aiquickdraw.com/mathbot/file/1726131361512ozusfeb8.jpg

Let's break down the problem presented in the image step by step and solve it:

### Problem Breakdown:
- There are three pipes that can fill a tank:
  1. The first two pipes, when working together, can fill the tank in the same time that the third pipe can fill it alone.
  2. The second pipe fills the tank 5 hours faster than the first pipe.
  3. The third pipe fills the tank 4 hours slower than the second pipe.

### Let:
- \( t_1 \): Time taken by the first pipe to fill the tank.
- \( t_2 = t_1 - 5 \): Time taken by the second pipe (5 hours faster than the first pipe).
- \( t_3 = t_2 + 4 = t_1 - 1 \): Time taken by the third pipe (4 hours slower than the second pipe).

### Key Equation:
According to the problem, the first two pipes working together fill the tank in the same time as the third pipe alone. This gives the equation:
\[
\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t_3}
\]
Substitute \( t_2 = t_1 - 5 \) and \( t_3 = t_1 - 1 \) into this equation:
\[
\frac{1}{t_1} + \frac{1}{t_1 - 5} = \frac{1}{t_1 - 1}
\]

Now let's solve this equation to find \( t_1 \).It seems the equation has resulted in complex numbers, which indicates that there might be an error in the formulation of the variables. Let's reconsider the equation setup and attempt a different approach to solve this algebraic problem.

I'll first simplify the equation more carefully:

\[
\frac{1}{t_1} + \frac{1}{t_1 - 5} = \frac{1}{t_1 - 1}
\]

Multiplying both sides by \( t_1(t_1 - 5)(t_1 - 1) \) will clear the denominators, and we can solve it step by step. Let me attempt this next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, based on the equation setup we derived earlier, I can guide you through solving it manually if you'd like! Would you prefer I explain how to manually simplify and solve this step by step, or would you like to upload another query?

Here are some related questions for further practice:
1. If the second pipe fills the tank 6 hours faster than the first, how would that change the problem?
2. What if the third pipe is twice as slow as the second? How would you approach this?
3. How can you solve pipe and tank problems using work rate formulas?
4. How would you solve the problem if the pipes had variable flow rates instead of uniform flow?
5. How does the relationship between the pipes' speeds affect the total time taken?

**Tip:** When dealing with multiple pipes or workers filling a tank, always start by converting the time taken into rates (work done per hour). This makes the equations easier to solve.

A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is:

This problem focuses on solving a work and time equation involving three pipes filling a tank. Two pipes work together while the third works alone, and the relationship between their speeds is provided. The problem involves setting up algebraic equations based on work rates and solving for the time required for the first pipe to fill the tank.

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