Work problem: Representations in the problem

Alex can pour on a walkway in x hours alone while Andy can pour concrete on the same way in two more hours than Alex.

a. How fast can they pour concrete on the walkway if they work together?

b. If Emman can pour concrete on the same walkway in one more hour than Alex, and Roger can pour the same walkway in one hour less than Andy, who must work together to finish the job with the least time?

My answer:

the work is the time times rate... therefore

a. Alex + Andy = 1/x + 1/(x+2) ...

b. Emman + Roger = 1/ [(x+2)+1] + 1/[(x+2) - 1] or 1/(x+3) + 1/(x+1)

=== I think the pair who can work with the lesser time is Alex and Andy.

Hope you guys will check my work.. thanks a lot ...

God Bless. Thank you.

Re: Work problem: Representations in the problem

I would attempt to use ratios to solve this problem. Set up a ratio " Time : Proportion of walkway completed " for each person.

So for Alex, the ratio is $\displaystyle \displaystyle \begin{align*} x : 1 \end{align*}$ or $\displaystyle \displaystyle \begin{align*} 1 : \frac{1}{x} \end{align*}$, and for Andy the ratio is $\displaystyle \displaystyle \begin{align*} x + 2 : 1 \end{align*}$, or $\displaystyle \displaystyle \begin{align*} 1 : \frac{1}{x + 2} \end{align*}$.

That means if they were working together, then in an hour the ratio for them both would be

$\displaystyle \displaystyle \begin{align*} 1 &: \frac{1}{x} + \frac{1}{x + 2} \end{align*}$

So I agree with your answer that together, Alex and Andy would be working at the rate of $\displaystyle \displaystyle \begin{align*} \frac{1}{x} + \frac{1}{x + 2} \end{align*}$ per hour.

Re: Work problem: Representations in the problem

It takes Emman one hour more than Alex (x+1), not (x+2)+1. Roger works one hour faster than Andy so I agree he would take (x+2)-1 hours = x+1 hours. I think if all four work together, they will finish in the least amount of time. If you may only choose two of them, then I would think Alex and either Emman or Roger could finish the job equally quickly.

Oh, and if you must choose between the pair Alex and Andy or Emman and Roger, then Alex and Andy can finish the job in $\displaystyle \dfrac{2(x+1)}{x^2+2x}$ hours while Emman and Roger can finish in $\displaystyle \dfrac{2(x+1)}{(x+1)^2} = \dfrac{2}{x+1} < \dfrac{2(x+1)}{x^2+2x}$. So, Emman and Roger would work faster.