1st tractor need 2 fewer hrs to plow field than 3rd, 2 more than 2nd. Together,...

[7Hero]

Hello Guys , could you help me resovle this simple(hard) problem ? : )

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The first tractor driver to plow a field needs 2 hours less than the third tractor driver and with one hour more than the second tractor driver. Working together, the first one and second one are plowing the field in 1 hour and 12 minutes. How long it takes for all of them to plow the field together?

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/english not my native language/

 

[Deleted member 4993]

The first tractor driver to plow a field needs 2 hours less than the third tractor driver and with one hour more than the second tractor driver.Working together , the first one and second one are plowing the field in 1 hour and 12 minutes.How long it takes for all of them to plow the field together?

/english not my native language/

Hint 1 hr 12 min = 1.2 hrs.

What are your thoughts?

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[HallsofIvy]

The first tractor driver to plow a field needs 2 hours less than the third tractor driver and with one hour more than the second tractor driver.Working together , the first one and second one are plowing the field in 1 hour and 12 minutes.How long it takes for all of them to plow the field together?

/english not my native language/

Let H be the number of hours required by the first driver to plow the field. The one hour more than required by the second driver so the second driver requires H- 1 hours. It is also two hours less than required by the third driver so the third driver requires H+ 2 hour. When a group of people work together their rates of work add. The first driver works at rate 1/H "fields per hour", the second driver works at rate 1/(H- 1) "fields per hour". And the third driver works at rate 1/(H+ 2) "fields per hour." The first and second together work at rate 1/(H- 1)+ 1/(H+ 2)= (H+2)/((H-1)(H+2)+ (H-1)/((H-1)(H+1)) = (2H+1)/((H-1)(H+2))= 1 field per 6/5 hours= 5/6 "field per hour". Solve (2H+1)/((H-1)(H+2))= 5/6 for H and use that to find their speed when all three work together.

 

[7Hero]

I made it my self and x is 2 hours( and it the right answer)
How I made it :
I made the speed equation for every tractor driver so => v(1td) = m/x v(2td) = m/x+1 v(3td)= m/x+3 m= the area they have to plow
so we know that the first and second plow m in 1.2 hours
v(1td)+v(2td) = m/x + m/x+1 = 2mx+m/x^2+x
And that's where I made the proportion we know that both together m in 1.2 and 2mx+m in x^2+x => m(x^2+x)=2.4mx+1.2m =>mx^2+mx-2.4m-1.2m=0 (m is a constant so i can replace it with any number ) so I replace m with 5 =>5x^2+5x - 12x-6=0 => 5x^2-7x-6=0 => x1= 2 x2=-1/2 => x=2 because the time can't be negative
So that we know x = 2 we can calculate how long it takes for all of them to plow the field together.
v(1td) = m/2 v(2td)=m/3 and v(3td)=m/5
v(1td) + v(2td) + v(3td) = m/2+m/3+m/5 = 31m/30 so we make the proportion again 31xm=30m (x is the time they plow the field together) =>30/31
in hours that is ~ 58 minutes.