Question: 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?

1. \(\frac{1}{3x}\)
2. \(\frac{3x}{(x-3)}\)
3. \(\frac{(x-3)}{3x}\)
4. \(\frac{x}{(x-3)}\)
5. \(\frac{(x-3)}{x}\)

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement states that:

Given:

• Lindsay can paint 1/x of a certain room in 20 minutes.
• Lindsay and Joseph can paint the room in an hour, working together at their respective rates.

Find Out:

• The fraction of the same room Joseph can paint in 20 minutes.

We know that Lindsay can paint 1/x of a certain room in 20 minutes i.e \(\frac{20}{60} = \frac{1}{3}\) hour

Therefore, in \(\frac{1}{3}\) hour, Lindsay can paint 1/x of a certain room
Then, in 1 hour, Lindsay can paint 3/x of a certain room.
Linsday and Joseph together paint the whole room in 1 hour

Then, in 1 hour, Joseph can paint = \(1-\frac{3}{x}\) of a room
=\(\frac{x-3}{x}\) of a room

Therefore, in 20 minutes or in ⅓ of an hour, Joseph can paint \(\frac{x-3}{3x}\) of a room

Approach Solution 2:

The problem statement informs that:

Given:

• Lindsay can paint 1/x of a certain room in 20 minutes.
• Lindsay and Joseph can paint the room in an hour, working together at their respective rates.

Asked:

• To find out the fraction of the same room Joseph can paint in 20 minutes.

Let's assume that the total unit of work is 1 unit.

Lindsay paints 1/x of certain paint in 20 minutes
Therefore, the efficiency of Lindsay is 1/20x

Together they complete the 1 unit of work in 60 minutes.
Hence, their combined efficiency is 1/60

Efficiency of Lindsay + efficiency of Joseph = 1/60
=> 1/20x + efficiency of Joseph = 1/60
=> efficiency of Joseph = 1/60 - 1/20x
=(x-3)/60x

Therefore, Joseph will complete the work in 60x/(x-3) minutes
Hence, in 20 minutes he will complete (x-3)/60x * 20 units of work = (x-3)/3x

Therefore, in 20 minutes Joseph can paint \(\frac{x-3}{3x}\) of a room.

Approach Solution 3:

The problem statement implies that:

Given:

• Lindsay can paint 1/x of a certain room in 20 minutes.
• Lindsay and Joseph can paint the room in an hour, working together at their respective rates.

Find out:

• The fraction of the same room Joseph can paint in 20 minutes.

Lindsay can paint 1/x of a room in 20 minutes
Therefore, she can paint 3/x of a room in 60 minutes (or in 1 hour).
Thus, her hourly rate is 3/x room/hr.

Together Linsday and Joseph can paint the entire room in 1 hour.

Let total work = 1
Let the number of hours Joseph takes to paint the room = j
Then Joseph’s rate = 1/j room/hr.

Therefore, we can say:
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 get:
=> 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.

We know that 20 minutes = 1/3 of an hour
We also know that work = rate x time
Then in 20 minutes, Joseph can complete = [(x - 3)/x](1/3) = (x - 3)/(3x) of the job.

Therefore, in 20 minutes Joseph can paint \(\frac{x-3}{3x}\) of a room.

Approach Solution 4:

The problem statement implies that:

Given:

• Lindsay can paint 1/x of a certain room in 20 minutes.
• Lindsay and Joseph can paint the room in an hour, working together at their respective rates.

Find out:

• The fraction of the same room Joseph can paint in 20 minutes.

Since Lindsay and Joseph, working together can paint the whole room in 1 hour.
Then in 20 minutes, they can paint 1/3 of the room.

Let the fraction of the room that Joseph can paint in 20 minutes be r.

Then we can say:
=> 1/x + r = 1/3
=> r = 1/3 - 1/x
Using a common denominator of (3x), we get:
=> r = (x - 3)/(3x)

Therefore, in 20 minutes Joseph can paint \(\frac{x-3}{3x}\) of a room.

“Lindsay can paint 1/x of a certain room in 20 minutes”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. To solve the GMAT Problem Solving questions, the candidates must have basic knowledge of calculations and mathematics. The candidates can follow GMAT Quant practice papers to go through several sorts of questions that will enable them to improve their mathematical knowledge.

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