Question

XThree janitors work the night shift at the local hospital. Working alone, it takes Marla nine more hours1than it takes Tom, and it takes Bob twice as long as it takes Marla. So in one hour, Tom can clean - of11the building. Marla can clean of the building and Bob can clean of the building. If allX + 92x + 18three janitors work together, how much of the building can they clean in one hour?

Answer 1

When adding the cleaning rates of Tom, Marla, and Bob, we must correct the fractions to accurately represent the time it takes each to clean. Marla cleans 1/(x+9) in an hour and Bob cleans 1/(2*(x+9)) in one hour. Their combined cleaning rate in one hour is the sum of their individual rates.

Since Marla takes nine more hours to clean than Tom, if Tom can clean 1/x of the building in one hour, Marla can clean 1/(x+9) of the building in the same time, not 1/x+9 as the typo suggests.

For Bob, who takes twice as long as Marla, it would be correct to state Bob cleans 1/(2*(x+9)) of the building in one hour, not 1/2x+18.

When all three cleaners are working together, their individual cleaning fractions per hour should be added together to determine how much of the building they can clean in one hour:

Total cleaning rate = (1/x) + (1/(x+9)) + (1/(2*(x+9)))

This common denominator for this expression would be x(x+9)*2, and so the total sum needs to be calculated with respect to this common denominator in order to find their combined cleaning rate. Calculating this sum will provide the fraction of the building all three can clean in one hour together.

Answer 2

Let us begin by extracting the required information

In 1 hour,

`\begin{gathered} Tom\text{ can clean }(1)/(x)\text{ }of\text{ the building} \\ Maria\text{ can clean }\frac{1}{x\text{ + 9}}\text{ of the building} \\ Bob\text{ can clean }\frac{1}{2\text{x + 18}}\text{ of the building} \end{gathered}`

We are required the size of the building all three janitors can clean in 1 hour

To calculate this, we add the fractions for each janitor:

`=\text{ }(1)/(x)\text{ + }\frac{1}{x\text{ + 9}}+\text{ }\frac{1}{2x\text{ + 18}}`

Simplifying, we have:

`\begin{gathered} \text{= }\frac{(x+9)(2x+18)\text{ + x(2x+18) + x(x+9)}}{x(x\text{ + 9)(2x + 18)}} \\ =\text{ }\frac{2x^2+18x+18x+162+2x^2+18x+x^2\text{ + 9x}}{x(x\text{ + 9)(2x + 18)}} \\ =\frac{5x^2\text{ + 63x + 162}}{x(x\text{ +9)(2x + 18)}} \end{gathered}`

Hence, the fraction of the building they can clean in 1 hour is:

`=\frac{5x^2\text{ + 63x + 162}}{x(x\text{ +9)(2x + 18)}}`