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question 20
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katelyn can paint a small room in 18 hours. manuel can paint the same room in 9 hours. how long would it take for the two of them together to paint the room?
express your answer as an integer or a reduced fraction.
answer: enter your answer hours
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Explanation:

Step1: Calculate the work rate of each person

Work rate is defined as the amount of work done per unit time. If Katelyn can paint a room in \(18\) hours, her work - rate \(r_{K}\) is \(\frac{1}{18}\) of the room per hour. If Manuel can paint the same room in \(9\) hours, his work - rate \(r_{M}\) is \(\frac{1}{9}\) of the room per hour.

Step2: Calculate the combined work rate

When two people work together, their combined work rate \(r\) is the sum of their individual work rates. So \(r=r_{K}+r_{M}\). Substitute \(r_{K}=\frac{1}{18}\) and \(r_{M}=\frac{1}{9}\) into the formula:

$$
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Step3: Calculate the time taken when working together

We know that the formula for work is \(W = r\times t\), where \(W\) is the amount of work (in this case, \(W = 1\) room). Since \(r=\frac{1}{6}\) and \(W = 1\), we can solve for \(t\) using the formula \(t=\frac{W}{r}\). Substitute \(W = 1\) and \(r=\frac{1}{6}\) into the formula: \(t=\frac{1}{\frac{1}{6}}=6\)

Answer:

\(6\)