Word Problems

[Awesome1]

I have been trying to solve this problem but am not sure of my result.

Here is the problem:
Twin Brothers, Kenny and Isaiah, live in Small Town, USA where it snows an average of 55 days each winter. They take turns shoveling the walkway after each snowfall. Kenny can shovel the walkway in 22 minutes, while it takes Isaiah only 14 minutes to shovel the same walkway. How many minutes will it take to shovel the walkway if they work together? Express your answer as a decimal to the nearest tenth.

Note:
I came up with an average time of 18 minutes but this doesn't seem correct.

 

[tkhunny]

Who said it was an average? It MUST be faster than the faster time. If your solution doesn't do that, something is wrong.

Think more about the task. How much of the task can be accomplished in a given time!

"Kenny can shovel the walkway in 22 minutes"

Kenny can shovel only 1/22 of the walk in one (1) minute.

"It takes Isaiah only 14 minutes to shovel the same walkway"

Isaiah can shovel a whopping 1/14 of the walk in one (1) minute.

Thus, working together, they can shovel [(1/22)+(1/14)] of the walk in one (1) minute.

Now what?

 

[soroban]

Hello, Awesome1!

This is a "Work" problem which requires special handling.

Actually, there are a number of approaches.
I'll show you one of them.

Twin brothers, Kenny and Isaiah, take turns shoveling the walkway after each snowfall.
Kenny can shovel the walkway in 22 minutes, while it takes Isaiah only 14 minutes to shovel the same walkway.
How many minutes will it take to shovel the walkway if they work together?
Express your answer as a decimal to the nearest tenth.

\(\displaystyle \text{Kenny can do the job in 22 minutes.}\)
. . \(\displaystyle \text{In one minute, he can do }\tfrac{1}{22}\text{ of the job.}\)
\(\displaystyle \text{In }x\text{ minutes, he can do }\frac{x}{22}\text{ of the job.}\)

\(\displaystyle \text{Isaiah can do the job in 15 minutes.}\)
. . \(\displaystyle \text{In one minute, he can do }\tfrac{1}{15}\text{ of the job.}\)
\(\displaystyle \text{In }x\text{ minutes, he can do }\frac{x}{15}\text{ of the job.}\)

\(\displaystyle \text{Working together for }x\text{ minutes, they can do: }\:\frac{x}{22} + \frac{x}{15}\text{ of the job.}\)

\(\displaystyle \text{But in }x\text{ minutes, we expect to complete the whole job (1 job).}\)

. . \(\displaystyle \text{There is our equation! }\quad \frac{x}{22} + \frac{x}{15} \:=\:1\)


\(\displaystyle \text{Multiply through by }330\!:\;\;330\left(\frac{x}{22}\right) \,+\, 330\left(\frac{x}{15}\right) \;=\;330(1) \quad\Rightarrow\quad 15x\,+\,22x \:=\:330\)

. . . . . . . \(\displaystyle 37x \:=\:330 \quad\Rightarrow\quad x \:=\:\frac{330}{37} \;=\;8.9289189...\)


\(\displaystyle \text{Therefore, working together it will take them about }8.9\text{ minutes.}\)

 

[Awesome1]

Thanks for the help. I see my mistake. The average didn't make sense when I did it but couldn't see the common denominator. When I look at how it was worked out it makes sense and seems so simple.

Thanks Again...