Problem Solving

The time taken by P to complete a piece of work is four times the time taken by Q. Q takes four times the time taken by R and R takes four times the time taken by S. When these four persons are divided into 2 pairs, the ratio of the times taken by the two pairs is 13 : 4. Find the pair, which takes less time to complete the work.



P, S

P, R

Q, R

Q, S

P, Q

P = 4Q , Q = 4R , R = 4S , S
Bring each term in terms of S.

P = 4*4*4S , Q = 4*4S , R = 4S , S
64S , 16S , 4S , S

The ratio 13:4 will come when Q&R are chosen (16S+4S = 20S) and P&S are chosen.
Ratio P&S/Q&R = 65S/20S which is equal to 13/4
Comparing time taken by P&S and Q&R, Q&R take 20S while P&S take 65S
Thus Q&R pair will take less time.
IMO C

The answer is Q, R [C].

The explaination goes as follows :-

Let "x" be the time taken by S to completet the piece of work.

S = x , R = 4x , Q = 4*4x = 16x , P = 4*16x = 64x

time taken by [P + S] / time taken by [R +Q] = 65x / 20x = 13/4

Obviously , the term 20x < 65x , hence 20x is the least time taken to complete the work by the pair Q , R.

goyalsau wrote:The time taken by P to complete a piece of work is four times the time taken by Q. Q takes four times the time taken by R and R takes four times the time taken by S. When these four persons are divided into 2 pairs, the ratio of the times taken by the two pairs is 13 : 4. Find the pair, which takes less time to complete the work.



P, S

P, R

Q, R

Q, S

P, Q

I have a small Surprise for you Guys, Answer is P,S ( option A ) not Q,R i.e. ( option C )


Let the Work is Constant 64 W , Time is T Days.

Time Take by P is 64 Days , Work done by 64 W , Per Day W work

Time Take by Q is 16 Days , Work done is 64 W , Per Day 4 W work

Time Take by R is 4 Days , Work Done is 16 W , Per Day 16 W work

Time Take by S is 1 Day , Work Done is 64 W , Per Day 64 W Work



Now When PS is combined they will do 65 W work & QR will do 20 W work.

65 : 20 , ==== 13: 4 is the ratio of work done by PS and QR in one day , It is not the ratio of time,

I don't understand How they relate 13:4 work ratio to time........ : : : :

Please Help Guys.

I am also posting the Official explanation,

Guys Help......................

Anurag@Gurome wrote:

goyalsau wrote:The time taken by P to complete a piece of work is four times the time taken by Q. Q takes four times the time taken by R and R takes four times the time taken by S. When these four persons are divided into 2 pairs, the ratio of the times taken by the two pairs is 13 : 4. Find the pair, which takes less time to complete the work.

P, S
P, R
Q, R
Q, S
P, Q

I guess previous posters have identified their mistakes.
I'm providing only the solution here.

Say S takes 1 hour to complete the task
=> R takes 4 hrs
=> Q takes 16 hrs
=> P takes 64 hrs

In 1 hour,

• P does 1/64 of the total work
  Q does 1/16 of the total work
  R does 1/4 of the total work
  S does whole of the total work

For option A, the 2 pairs are (P, S) and (Q, R)

• (P, S) does (1 + 1/64) = 65/64 of the total work in 1 hour
  => (P, S) takes 64/64 hrs to complete the work
  
  (Q, R) does (1/4 + 1/16) = 5/16 of the total work in 1 hour
  => (Q, R) takes 16/5 hrs to complete the work
  
  Ratio of times taken by the two pairs = (16/5)/(65/64) = 13/4

Thus our pair selection is correct and pair (P, S) takes the less time to complete the work.

The correct answer is A.

Note: In this case we didn't check the other options because while analyzing option A we got the correct answer. Otherwise we have to check till we get the correct one.

Nicely Explained Anurag !

I would just like to add few observation(s) here (which can potentially help to minimize the choices, in similar problems) :

S is the fastest member. I am listing the rates :

S-> 1
R -> 1/4
Q -> 1/4^2
S -> 1/4^3

Now, even if S does the work alone, and all the rest do it combined, S will be more than 3 times faster.

How did I get 3 ?
Here is the formula you can use : - >The ratio of rate of S vs all the Others combined will always be greater than (1-r)/r [r = 1/4 here]

How can you deduce that ?
Consider a GP : a, ar, ar^2, ar^3,.......to infinity

a/(ar+ar^2+ar^3+...to infinity) = a/[ar*(1+r+r^2+....to infinity)] = a/[ar*1/(1-r)] = (1-r)/r

Now here since the time taken 13:4 or 3.25:1, that means one group is 3.25 times faster than the others.
If S is the fastest, you have to select him to get you this ratio, So only choices : P,S and Q,S makes sense.
Since it is just a little more than 3, my gut feeling says, combine the fastest with the slowest as my my first choice (luckily that is true here).