Two pipes of different diameters may be used to fill a swimming pool. The price with the larger diameter working alone can fill the swimming pool 1.25 times faster than the other pipe when it works alone. One hour after the larger pipe is opened, the smaller pipe is opened, and the swimming pool is filled 5 hours later. Which equation could be used to find the number of hours, x, it would take for the larger pipe to fill the pool working alone?

Options

A. `((1)/(1.25x))5+((1)/(x))6=1`
B. `((1)/(x))5+((1)/(1.25x))6=1`
C. `((x)/(5))1.25+((x)/(6))=1`
D. `((x)/(5))+((x)/(6))1.25=1`

Answer

To solve the problem, we need to establish the relationship between the two pipes and how they fill the swimming pool. Let's break it down step by step.

### Step 1: Define Variables
Let \( x \) be the number of hours it takes for the larger pipe to fill the pool alone.

### Step 2: Determine the Time for the Smaller Pipe
Since the larger pipe is 1.25 times faster than the smaller pipe, the time it takes for the smaller pipe to fill the pool alone can be expressed as:
\[ 1.25x \]

### Step 3: Calculate Work Done by the Larger Pipe
The larger pipe works alone for 1 hour before the smaller pipe is opened. Therefore, in that hour, the larger pipe fills:
\[ \frac{1}{x} \text{ of the pool in 1 hour.} \]

### Step 4: Calculate Work Done by Both Pipes Together
After 1 hour, both pipes work together for the next 5 hours. During this time, the work done by the larger pipe in 5 hours is:
\[ \frac{5}{x} \]
And the work done by the smaller pipe in 5 hours is:
\[ \frac{5}{1.25x} = \frac{4}{x} \]

### Step 5: Set Up the Equation
The total work done by both pipes to fill the pool is the sum of the work done by the larger pipe and the smaller pipe. Therefore, we can set up the equation:
\[ \frac{1}{x} + \frac{5}{x} + \frac{4}{x} = 1 \]

### Step 6: Simplify the Equation
Combine the terms on the left side:
\[ \frac{1 + 5 + 4}{x} = 1 \]
This simplifies to:
\[ \frac{10}{x} = 1 \]

### Step 7: Rearranging the Equation
To isolate \( x \), multiply both sides by \( x \):
\[ 10 = x \]

### Final Equation
Thus, the equation that could be used to find the number of hours \( x \) it would take for the larger pipe to fill the pool working alone is:
\[ \frac{10}{x} = 1 \]