Question

Alan purchased pens and pencils at a certain shop, where each pen costs 3 dollars and each pencil cost 2 dollars. What is the total number of pen and pencils Alan purchased?
I. Alan bought pen and pencils for the total cost of 10 dollars.
II. Total cost of the pens which Allan bought is less than 10 dollars.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In Diophantine equation problems, the constraints that variables must be integers (and often positive) drastically reduce the number of possible solutions. Always test small integer values to find all possible combinations.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a Data Sufficiency problem that translates to a linear Diophantine equation (an equation where we are only interested in integer solutions). Let \(p\) be the number of pens and \(n\) be the number of pencils. The total cost is given by the equation \(3p + 2n = \text{Total Cost}\). The question asks for the value of \(p + n\). 
A key ambiguity in such problems is whether the quantities must be positive (\(>0\)) or non-negative (\(\geq 0\)). The phrasing ""purchased pens and pencils"" strongly implies that Alan bought at least one of each, so we will assume \(p \geq 1\) and \(n \geq 1\). 
Step 2: Detailed Explanation: 
Analyze Statement I: ""Alan bought pen and pencils for the total cost of 10 dollars."" 
This gives us the equation: \[ 3p + 2n = 10 \] We need to find integer solutions where \(p \geq 1\) and \(n \geq 1\). 

If \(p = 1\), \(3(1) + 2n = 10 \implies 2n = 7\). This gives \(n = 3.5\), which is not an integer. 
If \(p = 2\), \(3(2) + 2n = 10 \implies 6 + 2n = 10 \implies 2n = 4\). This gives \(n = 2\), which is an integer. So, (p=2, n=2) is a valid solution. 
If \(p = 3\), \(3(3) + 2n = 10 \implies 9 + 2n = 10 \implies 2n = 1\). This gives \(n = 0.5\), not an integer. 
If \(p \geq 4\), \(3p\) would be 12 or more, which is already greater than the total cost of 10. So there are no more solutions. 
Under the assumption that he bought at least one of each, there is only one possible solution: 2 pens and 2 pencils. The total number of items is \(p + n = 2 + 2 = 4\). Since we found a unique value for the total number of items, Statement I alone is sufficient. 
Analyze Statement II: ""Total cost of the pens which Allan bought is less than 10 dollars."" 
This gives the inequality: \[ 3p<10 \] \[ p<10/3 \implies p<3.33... \] Since \(p\) must be an integer and \(p \geq 1\), the possible values for \(p\) are 1, 2, or 3. This statement gives no information about the number of pencils or the total cost. Therefore, Statement II alone is not sufficient. 
Step 3: Final Answer: 
Statement I alone is sufficient to answer the question, but Statement II alone is not. This corresponds to option (A). 
 

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