**> Ranking and Ordering: Introduction**

**> Ranking and Ordering: Solved Examples**

**> Ranking and Ordering: Solved Examples – 2**

**> Ranking and Ordering: Solved Examples – 3**

**> Ranking and Ordering: Solved Examples – 4**

**> Ranking and Ordering: Solved Examples – 5**

**> Ranking and Ordering: Solved Examples – 6**

**> Ranking and Ordering: Solved Examples – 7**

**> Ranking and Ordering: Solved Examples – 8**

**> Ranking and Ordering: Solved Examples – 11**

**> Ranking and Ordering: Solved Examples – 12**

**> Ranking and Ordering: Solved Examples – 13**

**> Ranking and Ordering: Solved Examples – 14**

**> Ranking and Ordering: Solved Examples – 15**

**> Ranking and Ordering: Solved Examples – 16**

**> Ranking and Ordering: Solved Examples – 17**

**Practice Exercise 1: Four friends Ashok, Bashir, Chirag, and Deepak are out shopping. Ashok has less money than three times the amount that Bashir has. Chirag has more money than Bashir. Deepak has an amount equal to the difference of amounts with Bashir and Chirag. Ashok has three times the money with Deepak. They each have to buy at least one shirt, or one shawl, or one sweater, or one Jacket, that are priced 200, 400, 600 and 1000 a piece, respectively. Chirag borrows 300 from Ashok and buys a jacket. Bashir buys a sweater after borrowing 100 from Ashok and is left with no money. Ashok buys three shirts. What is the costliest item that Deepak could buy with his own money?**

A. A shirt

B. A shawl

C. A Sweater

D. A Jacket

**> Chirag borrows 300 from Ashok and buys a jacket.**

This implies Chirag has at least 700 units of money.

**> Bashir buys a sweater after borrowing 100 from Ashok and is left with no money.**

The cost of a sweater is 400. This implies Bashir had 300 units of money.

**> Ashok has less money than three times the amount that Bashir has.**

Therefore, Ashok had less than 1200 units of money.

**> Deepak has an amount equal to the difference of amounts with Bashir and Chirag.**

This implies Deepak has at least 300 units of money.

**> Ashok has three times the money with Deepak.**

This implies Deepak has less than 400 units of money.

Therefore, the costliest item that Deepak can buy with his money is a shirt. **So, A is the correct option.**

**Practice Exercise 2: Answer the questions on the basis of the information given below.**

**Some children were taking free throws at the basketball court in school during lunch break. Below are some facts about how many baskets these children shot.**

**(i) Ganesh shot 8 baskets less than Ashish.**

**(ii) Dhanraj and Ramesh together shot 37 baskets.**

**(iii) Jugraj shot 8 baskets more than Dhanraj.**

**(iv) Ashish and Ganesh together shot 40 baskets.**

**(v) Ashish shot 5 baskets more than Dhanraj.**

**1. **Which of the following statements is true?

A. Ramesh shot 18 baskets and Dhanraj shot 19 baskets.

B. Ganesh shot 24 baskets and Ashish shot 16 baskets.

C. Jugraj shot 19 baskets and Dhanraj shot 27 baskets.

D. Dhanraj shot 11 baskets and Ashish shot 16 baskets.

**2. **Which of the following statements is true?

A. Dhanraj and Jugraj shot 46 baskets.

B. Ganesh shot 18 baskets and Ramesh shot 21 baskets.

C. Dhanraj shot 3 more baskets than Ramesh.

D. Ramesh and Jugraj together shot 29 baskets.

**> Ganesh shot 8 baskets less than Ashish. Ashish and Ganesh together shot 40 baskets.**

This implies Ganesh shot 16 baskets and Ashish shot 24 baskets.

**> Ashish shot 5 baskets more than Dhanraj.**

This implies Dhanraj shot 19 baskets.

**> Jugraj shot 8 baskets more than Dhanraj.**

This implies Jugraj shot 27 baskets.

**> Dhanraj and Ramesh together shot 37 baskets.**

This implies Ramesh shot 18 baskets.

**1. Option (A) is correct.**

**2. Option (A) is correct.**

**Practice Exercise 3: Answer the questions based on the following information.**

**Sixteen teams have been invited to participate in the ABC Gold Cup Cricket tournament. The tournament was conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprised of several rounds. A round involves one match for each team. The winner of the match advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.**

**The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage, teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.**

**1. **What is the total number of matches played in the tournament?

A. 28

B. 55

C. 63

D. 35

**2. **The minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage?

A. 5

B. 6

C. 7

D. 4

**3. **What is the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of the first stage?

A. 1

B. 2

C. 3

D. 4

**4. **What is the number of rounds in the second stage of the tournament?

A. 1

B. 2

C. 3

D. 4

**5. **Which of the following statements is true?

A. The winner will have more wins than any other team in the tournament.

B. At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.

C. It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the tournament.

D. The number of teams in the tournament with exactly one win in the second stage of the tournament is 4.

**First Stage: ****The teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once.**

Total number of matches played in each group = Total number of ways of selecting two times from eight items = 8C2 = 28

Therefore, the total number of matches played in the first stage = 28 + 28 = 56

**The top four teams from each group advance to the second stage while the rest are eliminated. **

**Second Stage: ****The second stage comprised of several rounds. A round involves one match for each team. The winner of the match advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.**

**Round-1: **This round consists of four matches, with each team playing one match. The team that wins advance to the next round while the losing team gets eliminated. Therefore, only four teams advance to the next round.

**Round-2: **Similar to round-1, this round consists of two matches, with each team playing one match. Only two teams advance to the next round.

**Round-3: **There would be two teams in this round. Both the teams play a match, and the winner would claim the Gold Cup.

**1. **Total number of matches played in the tournament = 56 + 4 + 2 + 1 = 63. **Therefore, C is the correct option.**

**2. **The total number of matches played in each group in the first stage = 28.

**> **Suppose that a team wins 4 matches. The remaining matches would be 24. To eliminate this team, any other four teams in this group should win at least 4 matches each, which is easily possible.

**> **Suppose that a team wins 5 matches. The remaining matches would be 23. To eliminate this team, any other four teams in this group should win at least 5 matches each, which is also possible.

**>** Suppose that a team wins 6 matches. The remaining matches would be 22. To eliminate this team, any other four teams in this group should win at least 6 matches each, which is not possible.

Therefore, the minimum number of wins needed for a team in the first stage to guarantee its advancement to the next stage is 6. **So, B is the correct option.**

**3. **First, we need to find the minimum number of wins with which a team can advance to the second stage of the tournament. This number minus one would be the highest number of wins in spite of which the team would get eliminated at the end of the first stage.

In the first stage, each team in a group plays eight matches. For the minimum number of wins by a team with which it can advance to the next stage, the first three teams win the maximum number of matches. Therefore, the first team wins 7 matches, the second team wins 6 matches (one match lost to the first team), the third team wins 5 matches, and the remaining teams win 2 matches each. Therefore, a team can qualify for the second stage by just winning 2 matches.

Hence, the highest number of wins for a team in the first stage in spite of which it would be eliminated at the end of the first stage is 1. **So, A is the correct option.**

**4. **There are three rounds in the second stage of the tournament.** So, C is the correct option.**

**5. **Let’s analyze each statement.

**A. The winner will have more wins than any other team in the tournament.**

Earlier we saw that a team with 2 points can also advance to the second stage. If this team wins the tournament, it will have 5 wins in all. On the other hand, a team with 6 or 7 wins in the first stage can get eliminated in the first round of the second stage. So, the winner doesn’t need to have more wins than any other team in the tournament.

**B. At the end of the first stage, no team eliminated from the tournament will have more wins than any of the teams qualifying for the second stage.**

This statement is also false as a team with 2 or 3 wins in one group may qualify for the second stage and a team with 4 or 5 wins in the other group may get eliminated at the end of the first stage.

**C. It is possible that the winner will have the same number of wins in the entire tournament as a team eliminated at the end of the tournament.**

This statement is true. One such scenario can be two teams qualifying to the second stage with 7 and 4 wins in the first stage. The team with 7 wins gets eliminated in the first round of the second stage while the team with 4 wins ends up winning the tournament and hence, it has a total of 7 wins in the entire tournament.

**D. The number of teams in the tournament with exactly one win in the second stage of the tournament is 4.**

The number of teams with exactly one win in the second stage of the tournament is 2. These would be the teams that get eliminated in the round-2 of the second stage.

**Therefore, C is the correct option.**

**Practice Exercise 4: Answer the questions on the basis of the information given below.**

**Five friends meet every morning at Sree Sagar restaurant for an idli-vada breakfast. Each consumes a different number of idlis and vadas. The number of idlis consumed are 1, 4, 5, 6 and 8, while the number of vadas consumed are 0, 1, 2, 4, and 6. Below are some more facts about who eats what and how much.**

**(i) The number of vadas eaten by Ignesh is three times the number of vadas consumed by the person who eats four idlis.**

**(ii) Three persons, including the one who eats four vadas, eat without chutney.**

**(iii) Sandeep does not take any chutney.**

**(iv) ****The one who eats one idli a day does not eat any vadas or chutney. Further, he is not Mukesh.**

**(v) Daljit eats idli with chutney and also eats vada.**

**(vi) Mukesh, who does not take chutney, eats half as many vadas as the person who eats twice as many idlis as he does.**

**(vii) Bimal eats two more idlis than Ignesh, but Ignesh eats two more vadas than Bimal.**

**1. **Which of the following statements is true?

A. Sandeep eats 2 vadas.

B. Mukesh eats 4 vadas.

C. Ignesh eats 6 vadas.

D. Bimal eats 4 vadas.

**2. **Which one of the following statements is true?

A. Daljit eats 5 idlis.

B. Ignesh eats 8 idlis.

C. Bimal eats 1 idli.

D. Bimal eats 6 idlis.

**3. **Which of the following statements is true?

A. Mukesh eats 8 idlis and 4 vadas but no chutney.

B. The person who eats 5 idlis and 1 vada does not take chutney.

C. The person who eats equal number of vadas and idlis also takes chutney.

D. The person who eats 4 idlis and 2 vadas also takes chutney.

**> Clue (i): The number of vadas eaten by Ignesh is three times the number of vadas consumed by the person who eats four idlis.**

This implies Ignesh eats 6 vadas, and the person who eats 4 idlis eats 2 vadas.

**> Clue (iv): The one who eats one idli a day does not eat any vadas or chutney. Further, he is not Mukesh.**

**> Clue (vi): Mukesh, who does not take chutney, eats half as many vadas as the person who eats twice as many idlis as he does.**

This is possible if Mukesh eats 2 vadas and 4 idlis, and the other person eats 4 vadas and 8 idlis.

**> Clue (vii): Bimal eats two more idlis than Ignesh, but Ignesh eats two more vadas than Bimal.**

**> Clues (ii), (iii) and (v): ****Three persons, including the one who eats four vadas, eat without chutney. ****Sandeep does not take any chutney. Daljit eats idli with chutney and also eats vada.**

Now let’s answer the questions:

**1. ****Option (C) is correct.**

**2. ****Option (A) is correct.**

**3. ****Option (C) is correct.**

**Practice Exercise 5: Answer the following questions based on the information given below.**

**Ten coins are distributed among four people P, Q, R, S such that one of them gets one coin, another gets two coins, the third gets three coins, and the fourth gets four coins. It is known that Q gets more coins than P, and S gets fewer coins than R.**

**1. **If the number of coins distributed to Q is twice the number distributed to P, then which one of the following is necessarily true?

A. R gets an even number of coins

B. R gets an odd number of coins

C. S gets an even number of coins

D. S gets an odd number of coins

**2. **If R gets at least two more coins than S, then which one of the following is necessarily true?

A. Q gets at least two more coins than S

B. Q gets more coins than P

C. P gets more coins than S

D. P and Q together get at least five coins

**3. **If Q gets fewer coins than R, then which one of the following is not necessarily true?

A. P and Q together get at least four coins

B. Q and S together get at least four coins

C. R and S together get at least five coins

D. P and R together get at least five coins

**Among P, Q, R and S, one gets one coin, another gets two coins, the third gets three coins, and the fourth gets four coins. It is known that Q gets more coins than P, and S gets fewer coins than R.**

**1. Q gets twice the number of coins that P gets.**

There are two possibilities:

S gets an odd number of coins. **So, D is the correct option.**

**2. R gets at least two more coins than S.**

There are three possibilities:

Q gets more coins than P.** So, B is the correct option.**

**3. Q gets fewer coins than R.**

Since Q gets more coins than P and R gets more coins than S, therefore R gets the highest number of coins.

**> **P and Q together may get 5, 4 or 3 coins.

**> **Q and S together may get 4 or 5 coins.

**> **R and S together may get 5, 6 or 7 coins.

**> **P and R together may get 6 or 5 coins.

We can see that it is not necessary for P and Q to get at least 4 coins together. **So, A is the correct option.**