A, B and C work on a task. To complete the task alone, B takes twice the time that A would take to complete the task alone and 2/3rd the time that C would take to complete the task alone. If B actually worked for half the number of days that A worked and 3/2 times the number of days that C worked, what proportion of the total work was completed by B?

A. 1/3
B. 2/9
C. 9/49
D. 7/19
E. 1/6
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This is a variable question, so we can just assign arbitrary values to simplify the problem greatly.

To complete the task alone, B takes twice the time that A would take to complete the task alone and 2/3rd the time that C would take to complete the task alone. If it takes 1 day for A to do the project, then it takes 2 days for B and 3 days for C.

If B actually worked for half the number of days that A worked and 3/2 times the number of days that C worked, what proportion of the total work was completed by B?

Let's say A worked 1 day. Then A completed 1 project. B worked for 1/2 a day, and since it takes him 2 days, he completed \(\frac{1/2}{2} = \frac{1}{4}\) of a project. C then worked for 1/3 of a day, and completed \(\frac{1/3}{3} = \frac{1}{9}\) of a project.

Then to find the proportion, we take \(\frac{1/4}{1+1/4+1/9} = \frac{9}{49}\).

Answer: C

How did you arrive at ...C does 2/3 of the work??? I did not understand ...B works for 3/2 times the number of days C worked???

A=A
B=2A
B=\(\frac{2}{3}\)C

Put the value of B here, so that we can get everything in term A
so, C=3A



A=x
B=\(\frac{1}{2}\)x
B=\(\frac{3}{2}\)C

In this also we put value of B so that we can get everything in term of x
C=\(\frac{1}{3}\)x

Now for suppose the x=6.
A=6
B=3
C=2

Let,total work =1

work is done by A, \(\frac{1}{A}\)=\(\frac{W(A)}{6}\)= W(A)=\(\frac{6}{A}\)

work is done by B, \(\frac{1}{2A}\)=\(\frac{W(B)}{3}\)= W(B)=\(\frac{3}{2A}\)

work is done by C, \(\frac{1}{3A}\)=\(\frac{W(C)}{2}\)= W(C)=\(\frac{2}{3A}\)

now the proportion of the total work was completed by B

\(\frac{(3|2A)}{(6/A)+(3/2A)+(2/3A)}\)
=\(\frac{9A}{49A}\)
=9/49

IMO C

yes, I have done it wrong on the last part calculating the proportion of the total work completed by B.


Thanks, Rachit4126 now, I have corrected it.