Five students — Arnav, Bhavna, Chirag, Divya and Eshan — took examinations in Mathematics, Physics and Chemistry. In each subject they were ranked 1 to 5 with no ties (rank 1 is the best). A student's rank sum is the sum of the three subject ranks. The overall ranking is by rank sum (lower is better); if two students have the same rank sum, the one with the better Mathematics rank is placed higher.
1. Arnav was ranked 1st in Mathematics but 4th in Physics.
2. Eshan was last in Physics, and his rank in Chemistry was the same as his rank in Mathematics.
3. Divya was ranked 1st in Chemistry, and her Physics rank was better than Chirag's Physics rank.
4. Chirag's Chemistry rank was worse than both his Mathematics rank and his Physics rank.
5. Chirag had a better Mathematics rank than Bhavna.
6. Bhavna had the lowest rank sum, and no one else had the same rank sum as her.
7. Arnav and Divya had the same rank sum.
1. Arnav was ranked 1st in Mathematics but 4th in Physics.
2. Eshan was last in Physics, and his rank in Chemistry was the same as his rank in Mathematics.
3. Divya was ranked 1st in Chemistry, and her Physics rank was better than Chirag's Physics rank.
4. Chirag's Chemistry rank was worse than both his Mathematics rank and his Physics rank.
5. Chirag had a better Mathematics rank than Bhavna.
6. Bhavna had the lowest rank sum, and no one else had the same rank sum as her.
7. Arnav and Divya had the same rank sum.
Question 1
Single correct option
What was Divya's rank in Mathematics?
- A. 2
- B. 3
- C. 4
- D. 5
Show answer and solution
Answer: (D) 5
Key deductions:
1. Physics ranks 1–3 go to Bhavna, Chirag, Divya, with Divya ahead of Chirag, so Divya's Physics rank is 1 or 2.
2. Arnav's sum = 1 + 4 + (his Chemistry) = 5 + ChA. Divya's sum = MD + PhD + 1. Equal sums → MD + PhD = 4 + ChA. Since MD ≤ 5 and PhD ≤ 2, ChA ≤ 3.
3. If ChA = 2, Arnav's sum is 7. Bhavna's Maths rank is at least 3 (Chirag beats her and rank 1 is Arnav's) and her Chemistry at least 3, so her sum ≥ 3 + 1 + 3 = 7 — not strictly lowest. Hence ChA = 3, sums of Arnav and Divya = 8, so MD = 5 and PhD = 2 → Chirag Physics 3, Bhavna Physics 1.
4. Eshan's Maths = Chemistry = 2 or 4. If 2, Bhavna's Maths is 4 and her sum ≥ 9 > 8 — impossible. So Eshan = 4, 4; Chirag Maths 2, Bhavna 3; Chirag Chemistry 5, Bhavna 2.
| | Maths | Physics | Chemistry | Sum |
|---|---|---|---|---|
| Arnav | 1 | 4 | 3 | 8 |
| Bhavna | 3 | 1 | 2 | 6 |
| Chirag | 2 | 3 | 5 | 10 |
| Divya | 5 | 2 | 1 | 8 |
| Eshan | 4 | 5 | 4 | 13 |
Divya's Maths rank = 5.
1. Physics ranks 1–3 go to Bhavna, Chirag, Divya, with Divya ahead of Chirag, so Divya's Physics rank is 1 or 2.
2. Arnav's sum = 1 + 4 + (his Chemistry) = 5 + ChA. Divya's sum = MD + PhD + 1. Equal sums → MD + PhD = 4 + ChA. Since MD ≤ 5 and PhD ≤ 2, ChA ≤ 3.
3. If ChA = 2, Arnav's sum is 7. Bhavna's Maths rank is at least 3 (Chirag beats her and rank 1 is Arnav's) and her Chemistry at least 3, so her sum ≥ 3 + 1 + 3 = 7 — not strictly lowest. Hence ChA = 3, sums of Arnav and Divya = 8, so MD = 5 and PhD = 2 → Chirag Physics 3, Bhavna Physics 1.
4. Eshan's Maths = Chemistry = 2 or 4. If 2, Bhavna's Maths is 4 and her sum ≥ 9 > 8 — impossible. So Eshan = 4, 4; Chirag Maths 2, Bhavna 3; Chirag Chemistry 5, Bhavna 2.
| | Maths | Physics | Chemistry | Sum |
|---|---|---|---|---|
| Arnav | 1 | 4 | 3 | 8 |
| Bhavna | 3 | 1 | 2 | 6 |
| Chirag | 2 | 3 | 5 | 10 |
| Divya | 5 | 2 | 1 | 8 |
| Eshan | 4 | 5 | 4 | 13 |
Divya's Maths rank = 5.
Question 2
Single correct option
Who was placed third in the overall ranking?
- A. Arnav
- B. Chirag
- C. Divya
- D. Eshan
Show answer and solution
Answer: (C) Divya
Rank sums: Bhavna 6, Arnav 8, Divya 8, Chirag 10, Eshan 13.
Arnav and Divya tie at 8; the tie-break (better Maths rank) favours Arnav (Maths 1) over Divya (Maths 5).
Overall order: Bhavna, Arnav, Divya, Chirag, Eshan.
So Divya is third. Note that Bhavna tops overall without topping Mathematics — she is 1st in Physics and 2nd in Chemistry.
Arnav and Divya tie at 8; the tie-break (better Maths rank) favours Arnav (Maths 1) over Divya (Maths 5).
Overall order: Bhavna, Arnav, Divya, Chirag, Eshan.
So Divya is third. Note that Bhavna tops overall without topping Mathematics — she is 1st in Physics and 2nd in Chemistry.
Question 3
Numerical answer
What was Chirag's rank sum?
Show answer and solution
Answer: 10
Chirag: Mathematics 2, Physics 3, Chemistry 5 → rank sum 10.
How Chirag's ranks are fixed: Arnav's Chemistry rank must be 3 (a rank of 2 would stop Bhavna from having the strictly lowest sum), which forces Divya to Maths 5 and Physics 2, leaving Chirag Physics 3. Eshan's equal Maths/Chemistry rank must be 4 (a 2 would push Bhavna's sum above 8), so Maths ranks 2 and 3 go to Chirag and Bhavna — Chirag 2 by fact 5. Chirag's Chemistry must be worse than 3 and 4 is Eshan's, so it is 5.
How Chirag's ranks are fixed: Arnav's Chemistry rank must be 3 (a rank of 2 would stop Bhavna from having the strictly lowest sum), which forces Divya to Maths 5 and Physics 2, leaving Chirag Physics 3. Eshan's equal Maths/Chemistry rank must be 4 (a 2 would push Bhavna's sum above 8), so Maths ranks 2 and 3 go to Chirag and Bhavna — Chirag 2 by fact 5. Chirag's Chemistry must be worse than 3 and 4 is Eshan's, so it is 5.
Question 4
Single correct option
In how many of the three subjects did Bhavna have a better rank than Arnav?
- A. 1
- B. 2
- C. 3
- D. 0
Show answer and solution
Answer: (B) 2
Compare subject by subject:
- Mathematics: Arnav 1, Bhavna 3 → Arnav better
- Physics: Arnav 4, Bhavna 1 → Bhavna better
- Chemistry: Arnav 3, Bhavna 2 → Bhavna better
Bhavna was better in 2 subjects, which is why her rank sum (6) beats Arnav's (8) even though Arnav topped Mathematics.
- Mathematics: Arnav 1, Bhavna 3 → Arnav better
- Physics: Arnav 4, Bhavna 1 → Bhavna better
- Chemistry: Arnav 3, Bhavna 2 → Bhavna better
Bhavna was better in 2 subjects, which is why her rank sum (6) beats Arnav's (8) even though Arnav topped Mathematics.
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