Work Problem

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Lindsay can paint 1/x of a certain room in 20 minutes. What fraction of the same room can Joseph paint in 20 minutes if the two of them can paint the room in an hour, working together at their respective rates?

a)1/3x

b)3x/x-3

c)x-3/3x

d)x/x-3

e)x-3/x

[spoiler]
OA - c [/spoiler]

[pepeprepa]

Need to use the method for work problem.

Task/Time

Lindsay can paint 1/x of a certain room in 20 minutes, which is written like that: (1/x)/20
We look for the fraction of the room (1/t) John can make in 20 minutes: (1/t)/20
The two of them can paint the room in an hour, the result is 1/60, 1 means the work is completed, 60 is the number of minutes in one hour.


(1/x)/20 + (1/t)/20 = 1/60

3/x + 3/t = 1
3/t = 1-(3/x)
1/t = (1-(3/x)) /3
1/t = (x-3) / 3x

[santhosh_katkurwar]

Could someone explain how to solve this using values?

[Brent@GMATPrepNow]

Lindsay can paint 1/x of a certain room in one hour. If Lindsay and Joseph, working together at their respective rates, can paint the room in one hour, what fraction of the room can Joseph paint in 20 minutes?

1/(3x)

x/(x - 3)

(x - 1)/3x

x/(x - 1)

(x - 1)/x

NOTE: I added some brackets to your answer choices to avoid confusion.

This approach is similar to raul_200435's great solution. I've just filled in a few extra details.

Given: Lindsay and Joseph can paint the room in one hour.
During that one hour, Lindsay can paint 1/x of the room.
So, during that one hour, Joseph must paint the rest (whatever Lindsay did not paint)
So, during the one hour, the fraction of the room that Joseph paints = 1 - 1/x
= x/x - 1/x
= (x-1)/x

So, (x-1)/x = the fraction of the room that Joseph paints in one hour.
Since 20 minutes = 1/3 of an hour, Joseph can paint (1/3)[(x-1)/x] of the room in 20 minutes.
(1/3)[(x-1)/x] = [spoiler](x-1)/3x = C[/spoiler]

Cheers,
Brent

[GMATGuruNY]

Lindsay can paint 1/x of a certain room in one hour. If Lindsay and Joseph, working together at their respective rates, can paint the room in one hour, what fraction of the room can Joseph paint in 20 minutes?

1/3x

x/x - 3

(x - 1)/3x

x/x - 1

(x - 1)/x

Let the room = 20 units.
Let x = 4.

Since Lindsay paints 1/x of the room in 1 hour, Lindsay's rate = (1/4) * 20 = 5 units per hour.
Since Lindsay and Joseph paint the entire room in 1 hour, their combined rate = 20 units per hour.
Thus, Joseph's rate = (combined rate) - (Lindsay's rate) = 20-5 = 15 units per hour.

In 20 minutes -- the equivalent of 1/3 of an hour -- the amount of work produced by Joseph = r*t = 15 * (1/3) = 5 units.
Thus, the fraction painted by Joseph = 5/20 = 1/4. This is our target.

Now we plug x=4 into the answers to see which yields our target of 1/4.
Only C works:
(x-1)/3x = (4-1)/(3*4) = 3/12 = 1/4.

The correct answer is C.

[Scott@TargetTestPrep]

Lindsay can paint 1/x of a certain room in 20 minutes. What fraction of the same room can Joseph paint in 20 minutes if the two of them can paint the room in an hour, working together at their respective rates?

a)1/3x

b)3x/x-3

c)x-3/3x

d)x/x-3

e)x-3/x

[spoiler]
OA - c [/spoiler]

We are given that Lindsay can paint 1/x of a room in 20 minutes; thus, she can paint 3/x of a room in 60 minutes (or in 1 hour). Thus, her hourly rate is 3/x room/hr. We are also given that when she works with Joseph, they can paint the entire room in 1 hour. If we let total work = 1 and j = the number of hours it takes Joseph to paint the room by himself, then Joseph's rate = 1/j room/hr. We can create the following equation and isolate j:

work of Lindsay + work of Joseph = 1

(3/x)(1) + (1/j)(1) = 1

3/x + 1/j = 1

Multiplying the entire equation by xj, we obtain:

3j + x = xj

x = xj - 3j

x = j(x - 3)

x/(x - 3) = j

Since j = x/(x - 3) and 1/j = Joseph's rate, then Joseph's rate, in terms of x, is (x - 3)/x.

Since 20 minutes = 1/3 of an hour, and since work = rate x time, Joseph can complete:

[(x - 3)/x](1/3) = (x - 3)/(3x) of the job in 20 minutes.

Alternate Solution:

Since Lindsay and Joseph, working together, can paint the entire room in 1 hour, then in 20 minutes, they can paint 1/3 of the room. If we let r be the fraction of the room that Joseph can paint in 20 minutes, then it must be true that:

1/x + r = 1/3

r = 1/3 - 1/x

r = x/(3x) - 3/(3x)

r = (x - 3)/(3x)

Answer: C