Solve the word problem. Lee can frame a cabin in 4 days less than Ron. When they work together, they will do the job in 4 4/5 days. How long would each of them take to frame the cabin alone?

Let L represent the time it takes Lee to frame the cabin. Then it takes L+4 days for Ron to frame the cabin. Working together, the framing time is ...

L(L+4)/(L +(L+4)) = 4 4/5

Multiplying by 5(2L+4), we get

5L(L +4) = 24(2L +4)

5L^2 +20L = 48L +96 . . . . . eliminate parentheses

5L^2 -28L -96 = 0 . . . . . . . . subtract 48L+96

(5L +12)(L -8) = 0 . . . . . . . . . factor

The solution of interest is one that makes a factor be zero:

L = 8

It takes Lee 8 days to frame the cabin and Ron 12 days when they each work alone.

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For "working together" problems where rates are given as "time per job", it is usually useful to invert the numbers to give "jobs per time". Here, Lee can complete 1/L framing jobs per day, and Ron can complete 1/(L+4) framing jobs per day. Working together, they complete a total of ...

1/L + 1/(L+4) = ((L+4) +(L))/(L(L+4)) . . . . framing jobs per day

Inverting that result gives the "days per job" when they work together:

days per job = L(L+4)/(2L+4) . . . . . . the formula we used above