SOLUTION: With this question: "If 6 men can do a piece of work in 14 days, how many men are needed to do the work in 21 days?" How do I know when to solve it with proportions (x=9 version 1)

Question 934502: With this question: "If 6 men can do a piece of work in 14 days, how many men are needed to do the work in 21 days?" How do I know when to solve it with proportions (x=9 version 1) versus with man-days (x=4 version 2) as shown below?
Version 1:
if 6 men/14 days than, than x men/21 days [set-up as a proportion problem]
6 men/14 days = x men/21 days [set the ratios equal to each other and cross-multiply]
(6)(21)=14x
126=14x [solve for x]
x=9
]
Version 2:
If 6 men can do the piece of work in 14 days, then one man requires (6)(14) or 84 days to do it. In
other words, the job takes 84 man-days
1 man takes 84 days
2 men takes 42 days
4 men takes 21 days
Let x= number of men required to do the job in 21 days
21 times x = number of man-days required to do the job and that equals 84
21x=84
x=4
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
Part of solving this one is to be able to recognize that it is a UNIFORM RATES problem. It takes the form of R*T=J for rate, time, job. The rate also depends on how many agents are participating to do any part of the job.

Assume all men work individually at an equal rate, calling this rate r, in units of JOBS PER DAYS. The 1 job is "a piece of work". The men being described will in the way they are distributed, complete ONE job.

"If 6 men can do a piece of work in 14 days, how many men are needed to do the work in 21 days?"

Let r be the rate for one man.
Let n be the unknown number of men to find.
Let j be 1 for ONE WHOLE JOB.
Uniform rates rule here would be of the form, R*t=j, where R depends on the number of men, workers, or agents doing any quantity of job.

That should now make sense.
Solve for n.

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