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Chapter 7: Problem 10

A community water tank can be filled in 18 hr by the town office well alone and in 22 hr by the high school well alone. How long will it take to fill the tank if both wells are working?

Short Answer

Expert verified
9.9 hours

Step by step solution

01

Determine the Rate of Each Well

First, find the rate at which each well fills the water tank. The town office well fills the tank in 18 hours, so its rate is \(\frac{1}{18}\) of the tank per hour. Similarly, the high school well fills the tank in 22 hours, so its rate is \(\frac{1}{22}\) of the tank per hour.
02

Add the Rates Together

Next, add the rates of the two wells to find the combined rate. The combined rate is the sum of their individual rates: \[ \text{Combined rate} = \frac{1}{18} + \frac{1}{22} \]
03

Find the Common Denominator

To add the fractions, find a common denominator. The least common multiple (LCM) of 18 and 22 can be found. The LCM of 18 and 22 is 198. So, express each fraction with a common denominator: \[ \frac{1}{18} = \frac{11}{198} \] \[ \frac{1}{22} = \frac{9}{198} \]
04

Add the Fractions

Add the two fractions: \[ \frac{11}{198} + \frac{9}{198} = \frac{20}{198} = \frac{10}{99} \] So, the combined rate is \(\frac{10}{99}\) of the tank per hour.
05

Calculate the Time to Fill the Tank

The time to fill the tank when both wells are working together is the reciprocal of the combined rate: \[ \text{Time} = \frac{99}{10} = 9.9 \text{ hours} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

combined rates
When looking at work rate problems, understanding how to combine rates is crucial. If we have two machines, people, or in this case, wells, working together, their combined rate is the sum of their individual rates. For the town office well and the high school well, their rates were \(\frac{1}{18}\) and \(\frac{1}{22}\), respectively.

When combined, the total rate is simply these rates added together:

\(\frac{1}{18} + \frac{1}{22}\).

This step is straightforward but forms the basis of solving these types of problems.
least common denominator
To add the fractions representing the rates of the wells, identifying the least common denominator (LCD) is necessary. The LCD of the rates \(\frac{1}{18}\) and \(\frac{1}{22}\) is found by determining the least common multiple (LCM) of 18 and 22.

The LCM ensures that we express both fractions with a common bottom number, making addition possible.

The LCM of 18 and 22 is 198. Once we have this common denominator, we rewrite the fractions:
  • \(\frac{1}{18} = \frac{11}{198}\)
  • \(\frac{1}{22} = \frac{9}{198}\)
The common denominator allows us to move on to the next step of addition.
fraction addition
Adding fractions involves combining their numerators while keeping the denominator the same.

Once the rates from each well are expressed with a common denominator (\frac{11}{198} and \frac{9}{198}), we add these fractions:

This results in a simplified combined rate of \(\frac{10}{99}\) once properly reduced. Understanding fraction addition is essential to progressing through work rate problems.
reciprocals in algebra
The final concept in solving work-rate problems is using reciprocals. In this problem, to determine how long it takes for both wells to fill the tank, we need the reciprocal of their combined rate.

The combined rate was found as \(\frac{10}{99}\).

Taking the reciprocal flips this fraction:
  • The reciprocal of \(\frac{10}{99}\) is \(\frac{99}{10}\).
This helps us find the final time it takes to fill the tank when both wells work together:

\(\frac{99}{10} = 9.9 \text{ hours}.\) Knowing how to work with reciprocals is key in converting a rate into a useful measure of time.

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