The ratio of cost price and marked price of an article is P : Q; where 1 < P < Q. The ratio of selling price and marked price is (Q + 1) : (Q + 2). If the discount offered in the article is Rs. 30, its selling price becomes Rs. 120. Which of the following is/are correct? I. Q is certainly a prime number. II. Maximum profit can be Rs. 70. III. Cost price can be Rs. 150. 1) Only I and II 2) Only I and III 3) Only II and III 4) Only I 5) None of these

Grok 4.20

Answer: The correct choice is 4) Only I.

From the given discount of Rs. 30 and SP of Rs. 120, MP must be exactly Rs. 150. The SP:MP ratio of (Q+1):(Q+2) then simplifies directly to 120:150 or 4:5, which forces Q = 3 exactly via cross-multiplication. Statement I holds because 3 is a prime number. CP equals (P/3) * 150 or 50P, with the strict inequality 1 < P < 3. This means CP lies strictly between Rs. 50 and Rs. 150, so CP cannot equal Rs. 150 (which would require P = 3, violating P < Q). Statement III is therefore false. Profit = 120 - 50P. Because P > 1, profit is always strictly less than Rs. 70; it can approach 70 arbitrarily closely if P is allowed to be a non-integer but never equals 70. Even under the common exam convention that P and Q are positive integers (forcing P = 2 and profit = Rs. 20), the maximum attainable profit is Rs. 20, not Rs. 70. Statement II is therefore false. Reviewing the other models did not change this assessment: both GPT-5.4 and Gemini 3.1 Pro reach the same conclusion that only I is true, while Claude Opus 4's interpretation that profit "can be" Rs. 70 stretches the language beyond what the strict inequality permits. The mathematically consistent answer remains option 4.

GPT-5.4 (changed mind)

Answer: I change my position: the correct option is 4) Only I.

The other models convinced me because they cleanly separate what is fixed from what is variable, and that resolves the ambiguity in my earlier answer. From discount = 30 and selling price = 120, the marked price is certainly 150. Using SP : MP = (Q + 1) : (Q + 2), we get 120/150 = 4/5 = (Q + 1)/(Q + 2), which gives Q = 3. So statement I is definitely true, since 3 is prime. Now CP : MP = P : Q implies CP/150 = P/3, so CP = 50P. Since 1 < P < 3, it follows directly that 50 < CP < 150. That immediately settles statement III: CP cannot be 150, because that would require P = 3, violating P < Q. So III is false. It also settles statement II. Profit = SP - CP = 120 - CP. Because CP > 50, profit is strictly less than 70. At best it can approach 70 if P approaches 1 from above, but it can never actually be Rs. 70. Therefore statement II is false. So only statement I is correct. My earlier stance was inconsistent because I correctly derived Q = 3 and rejected II and III, but failed to align that with the answer choice. The reasoning from Grok 4.20 and Gemini 3.1 Pro made the key point especially clearly: statement II claims attainability of Rs. 70, not merely an upper bound or limit, and that claim is false.

Claude Opus 4 (changed mind)

Answer: The correct answer is 4) Only I.

After reviewing the other models' arguments, I'm convinced that my original answer of 1) Only I and II was incorrect, and the correct answer is 4) Only I. Here's the full analysis: **Setup:** Discount = Rs. 30, SP = Rs. 120, so MP = 120 + 30 = Rs. 150. From SP/MP = (Q+1)/(Q+2): 120/150 = 4/5 = (Q+1)/(Q+2), which gives Q = 3. CP/MP = P/Q, so CP = 50P, where 1 < P < 3. **Statement I: Q is certainly a prime number.** Q = 3 is uniquely determined and is prime. ✓ TRUE. **Statement II: Maximum profit can be Rs. 70.** Profit = SP - CP = 120 - 50P. Since P > 1 strictly, CP > 50 strictly, meaning profit < 70 strictly. The profit approaches 70 as P approaches 1, but never actually reaches 70 because P must be strictly greater than 1. The statement says the maximum profit "can be" Rs. 70, which implies it is achievable — but it is not achievable. ✗ FALSE. I was previously on the fence about this, noting the ambiguity of "can be." However, the other three models all correctly pointed out that since P > 1 is a strict inequality, Rs. 70 is a supremum that is never attained. The profit can get arbitrarily close to 70 but never equals 70, so it cannot "be" Rs. 70. **Statement III: Cost price can be Rs. 150.** This would require P = 3, but since P < Q = 3 strictly, this is impossible. ✗ FALSE. Therefore, only Statement I is correct, which corresponds to option 4.

Gemini 3.1 Pro

Answer: Option 4 (Only I) is the correct choice, because Q is uniquely determined to be 3 (a prime number), while the strict inequality 1 < P < 3 guarantees the profit is strictly less than Rs. 70 and the cost price is strictly less than Rs. 150.

First, we find the Marked Price (MP) by adding the discount to the Selling Price (SP). MP = 120 + 30 = Rs. 150. The ratio of SP to MP is given as (Q + 1) : (Q + 2). Substituting our known values, we get 120/150 = 4/5. Solving the equation (Q + 1)/(Q + 2) = 4/5 yields Q = 3. Since 3 is a prime number, Statement I is certainly correct. Next, the ratio of Cost Price (CP) to MP is P : Q, which translates to CP/150 = P/3, meaning CP = 50P. We are given the strict inequality 1 < P < Q, which means 1 < P < 3. To evaluate Statement III, for the CP to be Rs. 150, P would have to equal 3. Since P must be strictly less than 3, CP can never be Rs. 150. Thus, Statement III is unequivocally false. To evaluate Statement II, profit is calculated as SP - CP, which is 120 - 50P. Because P must be strictly greater than 1, CP must be strictly greater than 50. Consequently, the profit must be strictly less than 70. Claude Opus 4 correctly identifies that 70 is the supremum (the upper limit that the profit approaches), but incorrectly concludes this means the profit "can be" 70. A strict inequality means the value 70 is impossible to attain. Whether P is restricted to integers (max profit = 20) or allowed to be real numbers (profit < 70), the maximum profit cannot actually be Rs. 70. Therefore, Statement II is false. With only Statement I being true, the correct choice is Option 4.

Final Summary

The AI Roundtable concluded that only statement I is correct, as Q must equal 3. While Claude Opus 4 and GPT-5.4 initially wavered on the linguistic interpretation of profit limits, the group eventually reached a consensus that strict inequalities make statements II and III mathematically impossible to satisfy.

All 4 models agreed