Question: A and B working separately can do a piece of work in 9 and 12 days respectively.If they work for a day alternately, A beginning, in how many days, the work will be completed?

1. 10 ⅕
2. 10 ¼
3. 10 ⅓
4. 10 ½
5. 10 ⅔

Correct Answer: B
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• A and B working separately can do a piece of work in 9 and 12 days respectively.
• They work for a day alternately
• A begins first.

Find out:

• The number of days needed to complete the work.

Whole work = 1
A does the whole work in 9 days
In 1 day, A does = 1/9 work
B does the whole work in 12 days
In 1 day B does= 1/12 work
A and B together do the work in 1 day = 1/9+1/12= 7/36
In 2 days the work=7/36
In 1 day the work= 7/72
In 10 days the work= 7×10/72 = 35/36
We have to take 10 because this gives the nearest multiplication of 72.
Remaining work = 1- 35/36 = 1/36
A did the whole work in 9 days
A did the 1/36 work in 9 × 1/36 = ¼ days
Thus, the time needed to complete the work = 10 + ¼ = 10 ¼ days

Approach Solution 2:
The problem statement suggests that:
Given:

• A and B working separately can do a piece of work in 9 and 12 days respectively.
• They work for a day alternately
• A begins first.

Find out:

• The number of days needed to complete the work.

Combining both works of A and B:
Given rate 1/9 and 1/12; LCM = 36
Therefore, days taken by A = 36/9 = 4
Therefore, days taken by B = 36/12 = 3 days
Average time = (4+3)/2 = 3.5
So in 10 days 3.5×10 = 35 days work done
Remaining = 36-35= 1 day work
which A can do as its rate is ¼
Thus, the time needed to complete the work = 10 + ¼ = 10 ¼ days

Approach Solution 3:
The problem statement suggests that:
Given:

• A and B working separately can do a piece of work in 9 and 12 days respectively.
• They work for a day alternately
• A begins first.

Find out:

• The number of days needed to complete the work.

Rate of A= 1/9, and
Rate of B= 1/12.
Since A starts first, A works one more day than B.
So if we let x = the number of days B works,
then x + 1 = the number of days A works.
Let’s create the equation:
(1/9)(x + 1) + (1/12)x = 1
Multiplying both sides by 36, we get:
4(x + 1) + 3x = 36
4x + 4 + 3x = 36
7x = 32
x = 32/7 = \(4\frac{4}{7}\)
We see that x is not a whole number.
If we round x up to 5, then certainly the two workers will be “overworked.” That is, the amount they work together will exceed 1 job.
Therefore, we have to round x down to 4. That is, B works 4 days and A works 5 days.
They will now be “underworked,” but we can figure out how much time will be needed for the remainder of the job.
So if B works 4 days and A works 5 days, together they have completed 4(1/12) + 5(1/9) = 1/3 + 5/9 = 8/9 of the job.
If the remaining 1/9 of the job will be completed by B for one more day, then 1/9 - 1/12 = 1/36 of the job still needs to be completed and that fraction of the job can be completed by A in (1/36)/(1/9) = 9/36 = 1/4 of a day.
Therefore, in total, they work 4 + 5 + 1 + ¼ = 10 ¼ days.

“A and B working separately can do a piece of work in 9 and 12 days respectively”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Prep Plus 2021”. To solve the GMAT Problem Solving questions, the candidates must have the basic concept of mathematics. The candidates can analyse GMAT Quant practice papers to practise varieties of questions that will enable them to improve their mathematical knowledge.

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