GATE ECE 2016 Set 1 | GA | Question: 10

Question

P, Q, R and S are working on a project. Q can finish the task in $25$ days, working alone for $12$ hours a day. R can finish the task in $50$ days, working alone for $12$ hours per day. Q worked $12$ hours a day but took sick leave in the beginning for two days. R worked $18$ hours a day on all days. What is the ratio of work done by Q and R after $7$ days from the start of the project?

• A. $10:11$
• B. $11: 10$
• C. $20:21$
• D. $21:20$

Comments

lets total amount of work in unit lcm(25,50)=50 unit

unit of work done by P in 1 day=50/25=2 unit

unit of work done by Q in 1 day=50/50=1unit

unit of work done by Q in 1hr=1/12 unit

after 7 days

unit of work done by P  5*2=10 unit

unit of work done by Q  18*7/12=21/2

P/Q=10/(21/2)

P/Q=20/21

Option C

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 Q can finish the task in  25 days, working alone for  12 hours a day. R can finish the task in 50  days, working alone for 12  hours per day

from this we can say Q efficiency is double that of R . lets say R does 1 work/hr then Q does 2 work/hr

in the given time of 5 days(12hr /day) Q work = total time * workdone/hr = 5*12*2  and for R (7 days 18hr/day)= 7*18*1.

hence the ratio of work done = 5*12*2 : 7*18*1 = 20:21.

Answer 1

Correct answer is C.

Q can finish the task in $25$ days, working alone for $12$ hours a day.

What can we learn from this line??

That Q, in $25\times 12$ hours can complete the work alone.

That is, his rate of doing work per hour is $\frac{1}{25\times 12}=\frac{1}{300}.$

R can finish the task in $50$ days, working alone for $12$ hours per day.

Now, similarly, R's per hour work is $\frac{1}{600}.$

Now, Q has worked for $5$ days of $12$ hours ($60$ hours) and R for $7$ days of $18$ hours ($126$ hours).

We finally know, how many hours both worked, and their capacity for an hour.

Ratio of work done by Q and R after $7$ days from the start of the project

$\qquad\qquad=\frac{\text{Work done by Q}}{\text{Work done by R}} = \frac{60\times \frac{1}{300}}{126\times  \frac{1}{600}}=\frac{20}{21}.$

Comments

$time \propto$ $\dfrac{work}{rate}$

$work \propto time \times\ rate $

Answer 2

When both working 12 Hours a day

Q’s 1 day work→ x /25

R’s 1 day work→ x/50

After 7 Days 

Q could only work for 5 days as he took 2 days leave  hence = 5x/25..

R worked 18Hours per day in that week...18*7 = 126 = which is for him Working 10 days and 1 half day.

So 10x/50 + x/(50*2) = 21x/100

In short Q worked for 5x/25 in a week

R worked for 21x/100

Ratio is 20:21...C is the answer
 

Answer 3

Q does W work in 25 days or W/25 work in a day (12 hrs)

R does W work in 50 days or W/50 work in a day (12 hrs).

For R, 12 hrs = W/50 work

18 hrs =3W/100 work.

 

Now Q worked only 5 days from the start of project while R worked 7 days

Work by Q= 5* W/25

Work by R= 7*(3W/100)

Q:R =20:21