t & w 14

Saumyojit

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Two men and a woman are entrusted with a task. The second man needs three hours more to cope with the job than the first man and the woman would need working together. The first man, working alone, would need as much time as the second man and the woman working together. The first man, working alone, would spend eight hours less than the double period of time the second man would spend working alone. How much time would the two men and the woman need to complete the task if they all worked together?
  1. 2 hours
  2. 3 hours
  3. 4 hours
  4. 5 hours




W m1 , m2, w are the work rate
per hr of man1 , man2 , women

x hrs is the time taken both by man 2 and women to complete the work .

The second man needs three hours more to cope with the job than the first man and the woman would need working together

Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)

Wm2 x + 3Wm2 = Wm2 x + Ww x

Wm2= ( Ww x ) / 3

k hrs is the time taken both by man 2 and women & man 1 Alone to complete the work .

The first man, working alone, would need as much time as the second man and the woman working together.

Wm1 * k hrs = ( Wm2 )* k + Ww k ---(ii)



Substitution of Wm2= ( Ww x ) / 3 in (ii)

Wm1 k = (Ww * x * k) / 3 + Ww k



y hrs is the time taken by man 2 to complete the task alone .

The first man, working alone, would spend eight hours less than the double period of time the second man would spend working alone.

Wm1 ( 2y - 8 hrs ) = Wm2 * y hrs ---(iii)


First we need to find the work rate of all persons ...


Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)
Wm1 k = ( Wm2 ) k + Ww k ---(ii)
Wm1 ( 2y - 8) = Wm2 * y ---(iii)





















 

lev888

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Two men and a woman are entrusted with a task. The second man needs three hours more to cope with the job than the first man and the woman would need working together. The first man, working alone, would need as much time as the second man and the woman working together. The first man, working alone, would spend eight hours less than the double period of time the second man would spend working alone. How much time would the two men and the woman need to complete the task if they all worked together?

  1. 2 hours
  2. 3 hours
  3. 4 hours
  4. 5 hours

W m1 , m2, w are the work rate per hr of man1 , man2 , women

x hrs is the time taken both by man 2 and women to complete the work .

The second man needs three hours more to cope with the job than the first man and the woman would need working together

Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)

Wm2 x + 3Wm2 = Wm2 x + Ww x

Wm2= ( Ww x ) / 3

k hrs is the time taken both by man 2 and women & man 1 Alone to complete the work .

The first man, working alone, would need as much time as the second man and the woman working together.

Wm1 * k hrs = ( Wm2 )* k + Ww k ---(ii)

Substitution of Wm2= ( Ww x ) / 3 in (ii)

Wm1 k = (Ww * x * k) / 3 + Ww k


y hrs is the time taken by man 2 to complete the task alone .

The first man, working alone, would spend eight hours less than the double period of time the second man would spend working alone.

Wm1 ( 2y - 8 hrs ) = Wm2 * y hrs ---(iii)


First we need to find the work rate of all persons ...

Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)
Wm1 k = ( Wm2 ) k + Ww k ---(ii)
Wm1 ( 2y - 8) = Wm2 * y ---(iii)
A couple of suggestions.
1. The font you use in parts of your post appears very small - hard to read.
2. Your variable names are maybe be convenient for you but again, hard to read.
Compare:
Wm1 * k hrs = ( Wm2 )* k + Ww k
vs
A * k hrs = ( B )* k + C k

Which one is easier to read and comprehend?

The problem is that Ww looks like W times w. Which is why single letter names are used in algebra.
Also note the inconsistent use of the multiplication sign: Wm1 * k vs Ww k
 

Dr.Peterson

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Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)
Wm1 k = ( Wm2 ) k + Ww k ---(ii)
Wm1 ( 2y - 8) = Wm2 * y ---(iii)
I see six variables here. You need six equations. Where are the other three?

Again, I would use only three variables, which would be the time each needs, maybe M, N, W, or a, b, c, or x, y, z. You might add some extras temporarily, but make sure they are accompanied by extra equations.

And don't include units in the equations you plan to solve; they are used only in setting up the equations to make sure they are sensible.
 

Dr.Peterson

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my brain could come up with this three only

Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)
Wm1 k = ( Wm2 ) k + Ww k ---(ii)
Wm1 ( 2y - 8) = Wm2 * y ---(iii)
Since you chose the variables, it's your job to use them. Look at where you first introduced each of the extra variables:

x hrs is the time taken both by man 2 and women to complete the work .
k hrs is the time taken both by man 2 and women & man 1 Alone to complete the work .
y hrs is the time taken by man 2 to complete the task alone .
Each of those is an equation! (And in each case, I would have just used the resulting expression.)

But look closely at equation (ii) before you worry too much about k! And don't forget to check that each equation means what you want it to mean; I haven't done that, so I give no guarantees.
 

Saumyojit

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Each of those is an equation! (And in each case, I would have just used the resulting expression.)
( Wm2 + Ww ) * x hrs = 1 --(iv)

Wm1 k = 1 => Wm1 = 1/k ---(v)

Wm2 * y = 1 => y = 1/ Wm2 (vi)



Wm2 ( x + 3hr) = 1 , ( Wm2 ) k + Ww k = 1 , Wm1 ( 2y - 8) =1

then ?


look closely at equation (ii)

k hrs is the time taken both by man 2 and women & man 1 Alone to complete the work .

The first man, working alone, would need as much time as the second man and the woman working together.


Wm1 * k = ( Wm2 + Ww) * k ---(ii)
 

Dr.Peterson

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Wm1 * k = ( Wm2 + Ww) * k ---(ii)
Do you not see that you can divide by k and eliminate k entirely?

The first step of solving a system of equations is to look for "low-hanging fruit", bits that can be done easily, before getting into the hard parts.

Then ... start solving! You don't need others to do everything for you!

That means trying to eliminate one variable at a time.

Well, there's one thing to be done first: Remove all those units from the equation, which just get in the way. I've already told you, units can help in setting up equations, but are not part of the algebra.
 

Saumyojit

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Do you not see that you can divide by k and eliminate k entirely?
Wm1= ( Wm2 + Ww) ---(ii) after dividing by k .

Wm2 = Wm1 - Ww


Wm2 ( x + 3 )= ( Wm2 + Ww ) * x ---(i)

Substitution of Wm2 ,

( Wm1 - Ww ) * (x + 3) = ( Wm2 + Ww ) * x

( Wm1 * x - Ww x ) + ( Wm1 * 3 - Ww 3 ) = ( Wm2 + Ww ) * x

right?
 

Dr.Peterson

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Wm1= ( Wm2 + Ww) ---(ii) after dividing by k .

Wm2 = Wm1 - Ww


Wm2 ( x + 3 )= ( Wm2 + Ww ) * x ---(i)

Substitution of Wm2 ,

( Wm1 - Ww ) * (x + 3) = ( Wm2 + Ww ) * x

( Wm1 * x - Ww x ) + ( Wm1 * 3 - Ww 3 ) = ( Wm2 + Ww ) * x

right?
No. The reason to do a substitution is to eliminate a variable, and you still have Wm2 there! When you substitute, you need to substitute everywhere.

Always keep the goal in mind. Never do anything that accomplishes nothing toward that goal.

You have learned about solving systems of equations, right? Do what you learned. Don't just make things up as you go.
 

Saumyojit

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The reason to do a substitution is to eliminate a variable, and you still have Wm2 there! When you substitute, you need to substitute everywhere.
( Wm1 * x - Ww x ) + ( Wm1 * 3 - Ww 3 ) = ( Wm2 + Ww ) * x

Substituting Wm2 = Wm1 - Ww

( Wm1 * x - Ww x ) + ( Wm1 * 3 - Ww 3 ) = ( Wm1 - Ww + Ww ) * x


( Wm1 * x - Ww x ) + ( Wm1 * 3 - Ww 3 ) = Wm1 * x

right?

Always keep the goal in mind. Never do anything that accomplishes nothing toward that goal.

we need to find the work rate of all persons
Wm2 , Wm1 , Ww


Wm2 ( x + 3hr) = ( Wm2 + Ww ) * x hrs ---(i)
Wm1 k = ( Wm2 ) k + Ww k ---(ii)
Wm1 ( 2y - 8) = Wm2 * y ---(iii)

These were the first and foremost 3 imp equations .
 

Dr.Peterson

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I have no idea whether this is right, because I am not inclined to read through your ugly equations in detail. (They are entirely different from mine, so I have nothing to compare to, since as I've said before, I usually prefer using times rather than rates as variables, and I don't insert extra variables unless absolutely necessary.)

Please follow the suggestions in #2 and #3 to make it readable, and attract more help.

In any case, your current goal is to eliminate one variable from your five equations, leaving four equations in four variables. What are they?
 

Dr.Peterson

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I got it that single letter names are used in algebra .
Subscripts are allowed, but you must actually write them as subscripts, and even then, they make things both harder to write and harder to read. Please be respectful of those who offer to help you, by making their lives easier rather than harder.
i tried but did not figure out
Then try again. Telling us you can't do something gives us nothing new to work with. You are supposed to be learning something here; so learn!

The least you should be learning is to be orderly in your work so that you can find things. Start fresh and do so.
 

Saumyojit

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ok i saw the solution which is

The correct option is 2 hours

In order to solve this question, if we look at the first statement, we could think of the following scenarios: If the time taken by the first man and the woman is 1 hour (100% work per hour), the time taken by the second man would be 4 hours (25% work per hour).

In such a case, the total time taken by all three to complete the task would be 100/125=0.8 hours. But this value is not there in the options. Hence, we reject this set of values.


I did not understand this paragraph . How and why 125 . What is 100 / 125 ? I don't understand the logic

lets go step by step
 

Dr.Peterson

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Do you notice that they are not using algebra at all, but trial and error? This is a test-taking strategy that multiple-choice problems allow; I would never approach this problem that way, because the method will only work if the problem is designed with exceptionally simple numbers. I will not guide you through this entire answer, because it will not teach you what you need to know.

On the other hand, it is not a bad idea to practice with specific numbers in order to get a sense of how a problem works before using variables. And the fact that you are confused already at this point suggests it is a good idea!

In this scenario, the three people working together will have a total rate of 125% per hour, which I would write as 1.25 jobs per hour; the time taken is the reciprocal of the rate, namely 1/1.25 hours per job. That is what they are doing, but in terms of percent: taking the rate as 125/100 per hour, so the time is 100/125 hours per job. Do you not know how these rates work, or is it just their use of percentages that confuses you?

Following this example, if we use variables representing times, if the three people take A, B, and C hours, then their rates are 1/A, 1/B, and 1/C; the rate for A and C together is 1/A + 1/C, and the time they take together is 1/(1/A + 1/C) = AC/(A + C). This is what I did in my solution.
 

Saumyojit

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Do you notice that they are not using algebra at all, but trial and error? This is a test-taking strategy that multiple-choice problems allow; I would never approach this problem that way, because the method will only work if the problem is designed with exceptionally simple numbers. I will not guide you through this entire answer, because it will not teach you what you need to know.
I got it that How they deduced it using hit and trial to find the total time taken by all three to complete the task would be 100 / 50=2 hours. ok .

but , I could not understand that
" we should try to see whether this set of values meets the other conditions in the question. Since the work of all three is 50%, this means that the work of the first man is 25%. Consequently, the work of the woman is 8.33%. "


work of all three is 50% . HOW?

 

Dr.Peterson

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Try explaining it yourself. You need this practice thinking, rather than depending on others. Don't just say you CAN'T understand; TRY until you do.

If the first along takes as long as the other two together, how do their rates compare?
 

Saumyojit

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Try explaining it yourself. You need this practice thinking, rather than depending on others. Don't just say you CAN'T understand; TRY until you do.

If the first along takes as long as the other two together, how do their rates compare?
done.
 
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