Problem:
One hose can fill a swimming pool in 6 hours if it works by itself. A different hose needs 8 hours to fill the same pool. How long will it take to fill the pool if both hoses are used together?
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Solution:
#### Step 1: Define the rates of each hose
-
Hose 1 fills the pool in 6 hours. Therefore, its rate is:
$$
\text{Rate}_1 = \frac{1}{6} \quad \text{(pool per hour)}
$$
This means Hose 1 fills \( \frac{1}{6} \) of the pool in one hour.
-
Hose 2 fills the pool in 8 hours. Therefore, its rate is:
$$
\text{Rate}_2 = \frac{1}{8} \quad \text{(pool per hour)}
$$
This means Hose 2 fills \( \frac{1}{8} \) of the pool in one hour.
#### Step 2: Combined rate of both hoses
When both hoses work together, their combined rate is the sum of their individual rates:
$$
\text{Combined Rate} = \text{Rate}_1 + \text{Rate}_2 = \frac{1}{6} + \frac{1}{8}
$$
To add these fractions, we need a common denominator. The least common denominator (LCD) of 6 and 8 is 24. Rewrite the fractions with this common denominator:
$$
\frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}
$$
Now add the fractions:
$$
\frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24}
$$
So, the combined rate of both hoses is:
$$
\text{Combined Rate} = \frac{7}{24} \quad \text{(pool per hour)}
$$
#### Step 3: Determine the time to fill the pool
Let \( t \) be the time (in hours) it takes for both hoses to fill the pool together. Since the combined rate is \( \frac{7}{24} \) pools per hour, the relationship between the rate, time, and the total work (filling 1 pool) is given by:
$$
\text{Rate} \times \text{Time} = \text{Work}
$$
Here, the work is filling 1 pool, so:
$$
\frac{7}{24} \cdot t = 1
$$
#### Step 4: Solve for \( t \)
To solve for \( t \), multiply both sides of the equation by the reciprocal of \( \frac{7}{24} \):
$$
t = 1 \div \frac{7}{24} = 1 \cdot \frac{24}{7} = \frac{24}{7}
$$
Convert \( \frac{24}{7} \) to a decimal:
$$
\frac{24}{7} \approx 3.4286 \quad \text{(approximately 3.4 hours)}
$$
#### Final Answer:
The time it takes to fill the pool when both hoses are used together is:
$$
\boxed{3.4}
$$
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This concludes the solution. If you have any further questions, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of rational equation word problems worksheet.
