Math Problem Statement

a large pipe can empty a pool in 6 hours, and a small pipe can empty the same pool in 9 hours. how long will it take for both pipes to empty the pool?

Solution

To solve this problem, we need to determine the rate at which both pipes empty the pool together.

Step 1: Find the individual rates of the pipes.

  • The large pipe can empty the pool in 6 hours. Therefore, the rate of the large pipe is 16\frac{1}{6} of the pool per hour.
  • The small pipe can empty the pool in 9 hours. Therefore, the rate of the small pipe is 19\frac{1}{9} of the pool per hour.

Step 2: Add the rates of the two pipes.

The combined rate when both pipes are working together is the sum of their individual rates: Combined Rate=16+19\text{Combined Rate} = \frac{1}{6} + \frac{1}{9} To add these fractions, find the least common denominator (LCD), which is 18: 16=318,19=218\frac{1}{6} = \frac{3}{18}, \quad \frac{1}{9} = \frac{2}{18} So, the combined rate is: Combined Rate=318+218=518\text{Combined Rate} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18} This means that together, the pipes empty 518\frac{5}{18} of the pool per hour.

Step 3: Find the time taken to empty the pool.

The time taken to empty the entire pool when both pipes are working together is the reciprocal of their combined rate: Time=1Combined Rate=1518=185 hours\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{18}} = \frac{18}{5} \text{ hours} Therefore, it will take 185\frac{18}{5} hours, or 3.6 hours, for both pipes to empty the pool together.

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:

  1. If the large pipe could empty the pool in 4 hours, how long would it take for both pipes to empty the pool?
  2. How long would it take for only half of the pool to be emptied if both pipes are working together?
  3. If a third pipe that empties the pool in 12 hours is added, how long will it take to empty the pool with all three pipes working together?
  4. What fraction of the pool would be emptied in 1 hour by both pipes together?
  5. If the small pipe gets partially blocked and its time to empty the pool doubles, how long would it take for both pipes to empty the pool?

Tip: When solving rate problems, always express rates in terms of a single unit (e.g., per hour) and add them directly to find the combined rate.

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Math Problem Analysis

Mathematical Concepts

Rate problems
Fractions
Addition of fractions

Formulas

Rate of a pipe = 1 / (time to empty pool)
Combined rate = rate of large pipe + rate of small pipe
Time to empty pool = 1 / (combined rate)

Theorems

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Suitable Grade Level

Grades 6-8