Problems based on analytical reasoning are essentially problems where one is expected to first refer all the facts given in the problem, and then arrange the facts, so that all the conditions given in the problem are satisfied. Let few examples to illustrate this.

**Question**

5 horses, Red, white, Grey, Black and spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs.3000, Rs. 2000, Rs.1000 on Red, White and Black horses respectively and ends up with up no profit and no loss.

Which of the following cannot be true?

(i) At least 2 horses finished before spotted.

(ii)Red finished last

(iii)There were 3 horses between Black and Spotted..

(iv) There were 3 horses between White and Red.

(v) Grey came in second.

**Solution**

Let us first understand the constraint of the problem, Raju betted Rs. 3000, 2000 and 1000 and ended up with no profit or loss, that means he betted total amount of 6000 and get back Rs.6000.

Let us next observe how the money is returned.

Winning horse – 4 times the bet amount.

Horse coming second- 3 times the bet amount

Horse coming third- The bet amount is returned.

Horse coming third- The bet amount is returned.

So if Raju got back 6000 then let us see what are the possibilities.

Position of horses Amount betted Amount received

I 3000 12000 -----------(i)

I 2000 8000 ------------(ii)

I 1000 4000 ------------(iii)

II 3000 9000 -----------(iv)

II 2000 6000 -----------(v)

II 1000 3000 -----------(vi)

III 3000 3000 ----------(vii)

III 2000 2000 ----------(viii)

III 1000 1000 ----------(ix)

Since Raju received Rs.6000 we need to observe from the above possibilities which combinations can give us Rs. 6000.

Possibilities (I) and (ii),(iv) are more than 6000 so those cannot be the possibilities.Combination (iii) and (viii), (v); (vi) and (vii); and no other possibilities exists.Hence we get following possibilities for the Rank of horses.

Rank I II III IV V R-Red

Horse Black Grey/Spotted White R/G/S R/G/S G-Grey

amt. Received 4000 2000 0 0 S- Spotted

Horse G/S White G/S R/G/S/B R/G/S/B

amt Received 6000

Horse G/S Black Red W/G/S W/G/S

amt received 3000 3000

Looking at the table above we find

(i)can be true

(ii)can be true as R can be the last horse in each of the 3 possibilities.

(iii)Can be true.

(iv)Cannot be true as in none of the options White is first and Red is last.

Hence option(iv) is the answer.

**Question**

Suppose, in addition, it is known that Grey came in fourth, Then which of the following cannot be true?

(i) Spotted came in first.

(ii) Red finished last.

(iii) White came in second.

(iv) Black came in second.

(v) There was one horse between Black and White.

**Solution**

If Grey came in 4 ^{ th} then we get possible rank of horses

I II III IV V

Black Spotted White Grey

RedWhite Grey Not possible

Spotted Black Red Grey White

So we have only 2 rank orders possible for the horses

(i) can be true spotted in either first or second.

(ii) can be true as Red is last in one of the possibilities.

(iii) cannot be true as White is either third or fifth.

Hence correct option is (iii).

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