Question: Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

1. 1
2. 1.25
3. 2
4. 2.5
5. 4

Correct Answer: C
Solution and Explanation:

Approach Solution 1:

The problem statement states that:
Given:

• Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively.
• A starts working alone and 2 hours later B joins.
• After another 2 hours joins C.
• After that A, B, and C together complete the task in 15 minutes.

Find out:

• The value of x.

Since A starts working alone, so after 2 hours, work done by A =\( 2* \frac{1}{10}\) = \(\frac{1}{5}\)

Since after 2 hours B joins and after another 2 hours C joins, then we can say:
After 4 hours, the task will be done = \(\frac{1}{5}+ 2*(\frac{1}{10}+\frac{1}{5}) = \frac{4}{5}\)
The task left to be done = \(1-\frac{4}{5}= \frac{1}{5}\)

We are told that \(\frac{1}{5}\)th of the task is done in 15 minutes ( i.e 1/4th of an hour) by all three workers.
Therefore, we can say:

\(\frac{1}{4}*(\frac{1}{10}+\frac{1}{5}+\frac{1}{x}) = \frac{1}{5}\)

=> \(\frac{1}{10}+\frac{1}{5}+\frac{1}{x} = \frac{1}{5}*4\)

=> \(\frac{1}{x} = \frac{4}{5}-\frac{1}{10}-\frac{1}{5}\)

=> \(\frac{1}{x} = \frac{8 - 1 - 2}{10}\)

=> \(\frac{1}{x} = \frac{5}{10}\)

=> \(\frac{1}{x} = \frac{1}{2}\)
=> x = 2

Therefore, the value of x = 2 hours.

Approach Solution 2:

The problem statement informs that:
Given:

• Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively.
• A starts working alone and 2 hours later B joins.
• After another 2 hours joins C.
• After that A, B, and C together complete the task in 15 minutes.

Find out:

• The value of x.

We can solve the problem by using the LCM method.
Let's assume LCM for 10, 5 and x to be 10x. Our whole job will consist of 10x units

So:

A in 1 hour will make \(\frac{10x}{10}\) = x units

B in 1 hour will make \(\frac{10x}{5}\) = 2x units

C in 1 hour will make \(\frac{10x}{x}\) = 10 units

First 2 hours: A is working alone, so work done = 2x units
Following two hours: A and B are working, so work done = 2 * 3x = 6x units

Therefore, during first 4 hours, work done = 2x + 6x = 8x units
and work left to be done = 10x - 8x = 2x units.

A, B and C working together did the remaining 2x units in \(\frac{1}{4}\) hour

Therefore, we can say:
\(\frac{10 + 3x}{4} = 2x\)

=> 10 + 3x = 2x * 4
=> 10 + 3x = 8x
=> 8x - 3x = 10
=> 5x = 10
=> x = 2

Therefore, the value of x = 2 hours.

Approach Solution 3:

The problem statement informs that:
Given:

• Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively.
• A starts working alone and 2 hours later B joins.
• After another 2 hours joins C.
• After that A, B, and C together complete the task in 15 minutes.

Find out:

• The value of x.

Since 15 mins = 0.25 hours

According to the condition of the question we can say:
A works 2 + 2 + 0.25 hours = 4.25 hours
B works 2 + 0.25 hours = 2.25 hours
C works 0.25 hrs

A starts for 2 hours, he works at rate 1/10 = \(2*\frac{1}{10}\) = 0.2
Therefore, balance work = 1 – 0.2 = 0.8

Next 2hrs, work completed by A +B = \(0.2+\frac{1}{5}*2\) = 0.6
Therefore, balance work left = 0.8 – 0.6 = 0.2

Therefore for next 15min, we get:

\(\frac{1}{10} *0.25 +\frac{1}{5}*0.25+\frac{1}{x}*0.25 = 0.2\)

=> \(0.25 [0.1+ 0.2 + \frac{1}{x}] = 0.2\)
=> \(0.3 + \frac{1}{x} = \frac{0.2}{0.25}\)
=>\( 0.3 + \frac{1}{x} = \frac{20}{25}\)
=>\( 0.3 + \frac{1}{x} = \frac{4}{5}\)
=> \(0.3 + \frac{1}{x} = 0.8\)
=> \(\frac{1}{x} = 0.8 – 0.3\)
=>\( \frac{1}{x} = 0.5\)
=> \(x = \frac{1}{0.5}\)
=> \(x = \frac{10}{5}\)
=> x = 2

Therefore, the value of x = 2 hours.

“Three workers, A, B, and C, can complete a certain task” - is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Prep Plus 2021”. To solve the GMAT Problem Solving questions, the candidate must possess concrete knowledge of arithmetic, algebra and geometry. The candidate can follow the GMAT Quant practice papers to analyze several sorts of questions that will help them to improve their mathematical skills.

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