Problems solving using rational equations

Question

An elementary school teacher can assemble and staple the weekly newsletter two times faster than her student aide. Working together, they can assemble and staple the letters in 18 minutes. How long would it take each, working alone, to complete the job

Answer 1

If we let "x" be the time it takes the teacher to assemble and staple the letters, then "2x" would be the time it takes her aide. So the teacher completes (1/x) of the newsletter per minute, and the aide completes (1/2x) of the newsletter per minute.

 

Together, the student and aide complete (1/18) of the newsletter per minute.

 

So if we let (1/x) + (1/2x) = (1/18)

Which represents the rate of the teacher's work per minute plus the rate of the aide's work per minute equaling the total rate of work per minute.

 

If we multiply every term by (18x) we get

(1/x)(18x) + (1/2x)(18x) = (1/18)(18x)

 

Cancelling like terms gives us:

18 + (18/2) = x

 

So:

 

18 + 9 = x

27 = x

 

Which means the teacher completes (1/27) newsletters per minute, and the aide completes (1/(27*2)) or (1/54) newsletters per minute. 

 

Since we want to know how long it would take each to complete the job, we simply use the "x" and "2x" values, meaning that the teacher takes 27 minutes to complete the task, and the aide takes 54 minutes.

Answer 2

What is the time for the student?  Let's say it takes the student s minutes to do the job alone.

 

Can the student do the whole job in one minute?

 

What fraction of the job can the student do in one minute?

 

What fraction of the job can the teacher do in one minute?  Answer is 2/s.

 

What fraction of the job can the student do in 18 minutes?  Answer is 18/s.

 

What fraction of the job can the teacher do in 18 minutes?  Answer is twice as much as the student or, ???

 

Fraction the student does in 18 min.  +  fraction the teacher does in 18 min.  =  what total fraction of the whole job?

 

solve for s.