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Detailed Questions
 
Read the passage given below and solve the questions based on it.
 
In a shooting competition, a person is allowed to shoot at four targets successively, followed by the next shooter.

When all the shooters have finished one such round, the process is repeated. If a target is hit, the shooter gets 2 points and if he misses the target, the other shooters are awarded one point each. The first shooter to get 60 points wins the shooting competition. In a contest among three persons—Akhil, Bharat and Chand, their score at the end is as follows:

Akhil = 60, Bharat = 53 and Chand = 43.
Out of a total of 78 shots being fired, only 43 hit the target.

1. Who was the first to shoot?
  1. Akhil
  2. Bharat
  3. Chand
  4. Cannot be determined
2. Who was the second to shoot?
  1. Akhil
  2. Bharat
  3. Chand
  4. Cannot be determined
3. Who was the third to shoot?
  1. Akhil
  2. Bharat
  3. Chand
  4. Cannot be determined

Answers :-
 
Scenario—The problem set given above involves many facts–the scheme of firing shots, the way the points are awarded, the number of shots fired and the number of shots hitting the target.

Observation—After going through the setup given above, following points should come of the surface:
  1. One round of firing involves 12 shots being fired, four shots by each shooter.
  2. Since 12 shots are being fired in one round, so a total of 72 shots are being fired in six such rounds. Out of the 72 shots, 24 shots have been fired by each of them. Now, in the next six shots, 4 shots must have been fired by the person who was the first person to shoot and rest two must have been fired by the person who was the second person to shoot.
  3. So, total number of shots fired by the 1st person= 28
     
    Total number of shots fired by the 2nd person= 26
     
    Total number of shots fired by the 3rd person= 24
Problem—Besides the problems given in the question set, let us raise some more pertinent points regarding this set:
  1. Since Akhil was the first on to get 60 points, can we assume that Akhil was the first one to shoot? If yes, then why and if no, then why?
  2. Since Bharat was the second ranker, can we assume that Bharat was the second one to shoot? If yes, then why and if no, then why?

Flaw Detector—While operating on the surface only gives us an idea that Akhil was the first one to shoot because he got 60 points first, it also gives us the reason why we get this conclusion–because we were operating on surface only.

 

Understand the points scheme—A person can get points without hitting any target or even without firing shots. As it is given that two points are awarded for a hit and one point is awarded to the opponents in case of a miss. It might be a possibility that Akhil would have got 59 points or 58 points or so in his round, and then other shooters go to shoot, they miss and in turn Akhil gets the point, and thus he gets 60 points.

And otherwise also, a deep thinking tells us that game ends with second shooter (because a total of 78 shots are fired), so Akhil could be at best second shooter and not the first shooter.

Explanation—As we have discussed above also, this question set involves many facts and hence, lets make some equations.
 
 
We are using six variables above, so we need to have six equations to solve this set. Let us make the equations:

Points scored by Akhil = 2a + d + f = 60 ......(i)
Points scored by Bharat = 2c + b + f = 53 ......(ii)
Points scored by Chand = 2e + b + d = 43 ......(iii)
Total hits = a + c + e = 43 ......(iv)
Total misses = b + d + f = 35 ......(v)

Till now we have been able to construct only five equations. Since we have used all the given information, we cannot have a sixth equation directly from the given set. Hence, we will introduce the ‘hypothetical equations’ now that will work as 6th equation.

Assume that Akhil is the first one to start,
so                                              
a + b = 28 ......(vi)
Doing (i) – (v) gives us:              
2a - b = 25 ......(vii)
Adding (vi) and (vii),

3a = 53, since we are not getting the integral value of ‘a’ from here, we would conclude that (vi)th equation a + b = 28 is not a valid equation and so Akhil is not the first one to shoot.

 
Similarly, assuming Bharat to be the first one to shoot gives us: c + d = 28 ......(viii)
Solving equation (ii) and equation (v), 2c - d = 18 ......(ix)

Adding equation (viii) and equation (ix), 3c = 46. Again we are not getting the integral value of c from here, so Bharat is not the first one to shoot.

Obviously, it means Chandan is the first one to shoot. Let us check that also:
If Chandan is the first one to shoot then, e + f = 28 .....(x)
Solving equation (iii) and equation (v), 2e - f = 8 ......(xi)
Adding equation (x) and equation (xi), 3e = 36, so,e = 12.

Similarly, to find out the second shooter, we will insert, one by one, an additional equation besides the above given 5 equations, by assuming that Akhil is the second one to shoot (and hence, a + b = 26). If it does not satisfy the given conditions, then we will construct the additional equation by assuming Chandan to be the second shooter (and hence,
c + d = 26).

Alternative Solution—The more the number of variables, the more difficult the solution will be.
Let us reduce some variables and start directly from the ‘hypothetical equation’ itself.
Assume that the total number of hits by the first one to shoot = N, so total number of misses by him =28 - N.

So, total number of misses by the other two shooters = 35 - (28 - N) = N + 7
[Total number of shots fired by all the shooters = 78 and total hits by all the shooters = 43, so total number of misses by all the shooters = 35]
 
So, the points scored by the first person to shoot = 2 N + N + 7 = 3 N + 7
Now, if Akhil is the first one to shoot, then 3 N + 7 = 60 & 3 N = 53

Since no integral value of N is obtained from here, so Akhil is not the first one to start.
Again, if Bharat is the second one to shoot, then 3 N + 7 = 53 & 3 N = 46

We are not getting integral value of N, so Bharat is not the first one to shoot.
Hence, Chandan is the first one to shoot.

Let us verify that:
If Chandan is the first one to shoot, then 3 N + 7 = 43 & 3 N = 36 & N = 12
Now to find the second person to shoot, assume that the total number of hits by the second shooter = M, so total number of misses by him = 26 - M.

So, total number of misses by the other two shooters = 35 - (26 - M) = M + 9
Or, the points scored by the second person to shoot = 2 M + M + 9 = 3 M + 9
If Akhil is the second one to shoot, then 3 M + 9 = 60 & 3 M = 51 & M = 17

Hence, Bharat is the second one to shoot.
  1. Option (c)
  2. Option (a)
  3. Option (b)




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