Five students – Anu, Bala, Chetan, Divya and Esha – took three quizzes: Quiz 1, Quiz 2 and Quiz 3. In each quiz, the five students received five different scores from the set {1, 2, 3, 4, 5} (so each of these scores was received by exactly one student in each quiz). A student's total is the sum of her or his three quiz scores. The following is known:
1. Divya scored 4 in Quiz 1, and her total was 7.
2. Chetan scored 1 in Quiz 1, and his score strictly increased from each quiz to the next.
3. Esha never scored more than 3 in any quiz. She scored more in Quiz 2 than in Quiz 3, and her total was strictly lower than everyone else's.
4. Bala scored 5 in Quiz 2, and his total was strictly higher than everyone else's.
5. Anu and Chetan had the same total. Anu's score in Quiz 1 was higher than each of her scores in Quiz 2 and Quiz 3.
1. Divya scored 4 in Quiz 1, and her total was 7.
2. Chetan scored 1 in Quiz 1, and his score strictly increased from each quiz to the next.
3. Esha never scored more than 3 in any quiz. She scored more in Quiz 2 than in Quiz 3, and her total was strictly lower than everyone else's.
4. Bala scored 5 in Quiz 2, and his total was strictly higher than everyone else's.
5. Anu and Chetan had the same total. Anu's score in Quiz 1 was higher than each of her scores in Quiz 2 and Quiz 3.
Question 1
Numerical answer
What was Bala's total score?
Show answer and solution
Answer: 12
Deduction: Divya's Quiz 2 + Quiz 3 = 7 − 4 = 3, so she scored 1 and 2 in some order. In Quiz 1 the scores 2, 3, 5 remain for Anu, Bala and Esha.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Bala: 3 + 5 + 4 = 12.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Bala: 3 + 5 + 4 = 12.
Question 2
Single correct option
Who scored 3 in Quiz 3?
- A. Anu
- B. Bala
- C. Divya
- D. Esha
Show answer and solution
Answer: (A) Anu
Deduction: Divya's Quiz 2 + Quiz 3 = 7 − 4 = 3, so she scored 1 and 2 in some order. In Quiz 1 the scores 2, 3, 5 remain for Anu, Bala and Esha.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Quiz 3 scores: Anu 3, Bala 4, Chetan 5, Divya 2, Esha 1. So Anu scored 3.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Quiz 3 scores: Anu 3, Bala 4, Chetan 5, Divya 2, Esha 1. So Anu scored 3.
Question 3
Single correct option
What was Esha's score in Quiz 1?
- A. 1
- B. 2
- C. 3
- D. Cannot be determined
Show answer and solution
Answer: (B) 2
Deduction: Divya's Quiz 2 + Quiz 3 = 7 − 4 = 3, so she scored 1 and 2 in some order. In Quiz 1 the scores 2, 3, 5 remain for Anu, Bala and Esha.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Esha scored 2 in Quiz 1 (and 3, 1 in Quizzes 2 and 3).
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Esha scored 2 in Quiz 1 (and 3, 1 in Quizzes 2 and 3).
Question 4
Single correct option
How many students scored more in Quiz 3 than in Quiz 1?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: (B) 2
Deduction: Divya's Quiz 2 + Quiz 3 = 7 − 4 = 3, so she scored 1 and 2 in some order. In Quiz 1 the scores 2, 3, 5 remain for Anu, Bala and Esha.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Quiz 3 vs Quiz 1: Anu 3 < 5; Bala 4 > 3 ✓; Chetan 5 > 1 ✓; Divya 2 < 4; Esha 1 < 2. So 2 students.
*Anu's Quiz 1 score is 5:* if it were 2, she would need 1 in both later quizzes, clashing with Divya's 1. If it were 3, Anu and Divya would take 1 and 2 in both Quiz 2 and Quiz 3, forcing Esha to 3 in both – contradicting Esha's Quiz 2 > Quiz 3.
*Quiz 3:* Esha's Quiz 3 < Quiz 2 ≤ 3, so she has 1 or 2; with Divya, they take {1, 2}. If Divya had 1 in Quiz 3, Esha would have 2 there and 3 in Quiz 2, total ≥ 2 + 3 + 2 = 7, not below Divya's 7 ✗. So Divya: Quiz 2 = 1, Quiz 3 = 2; Esha Quiz 3 = 1.
*Anu = Chetan:* Quiz 2 leaves {2, 3, 4} for Anu, Chetan, Esha; Quiz 3 leaves {3, 4, 5} for Anu, Bala, Chetan. 5 + a₂ + a₃ = 1 + c₂ + c₃ needs c₂ + c₃ ≥ 4 + (2 + 3) = 9, so Chetan has 4 and 5, Anu has 2 and 3. Then Esha's Quiz 2 = 3, Bala's Quiz 3 = 4, and Esha's total ≤ 6 forces Esha's Quiz 1 = 2, Bala's = 3.
| Student | Quiz 1 | Quiz 2 | Quiz 3 | Total |
|---|---|---|---|---|
| Anu | 5 | 2 | 3 | 10 |
| Bala | 3 | 5 | 4 | 12 |
| Chetan | 1 | 4 | 5 | 10 |
| Divya | 4 | 1 | 2 | 7 |
| Esha | 2 | 3 | 1 | 6 |
Check: Bala's 12 is the strict maximum; Esha's 6 is the strict minimum; Anu = Chetan = 10 ✓.
Quiz 3 vs Quiz 1: Anu 3 < 5; Bala 4 > 3 ✓; Chetan 5 > 1 ✓; Divya 2 < 4; Esha 1 < 2. So 2 students.
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