Question

The cost of 10 pounds of apples and 2 pounds of grapes was \$12. What was the cost per pound of apples?
(1) The cost per pound of grapes was \$2.
(2) The cost of 2 pounds of apples was less than the cost of 1 pound of grapes.

Hint:

An equation generally provides more specific information than an inequality. If a statement gives you a definite value for one variable in a two-variable linear equation, it will almost always be sufficient to solve for the other variable.

Answer 1

Step 1: Understanding the Concept:
This is a Data Sufficiency question involving a linear equation. We need to determine if the statements provide enough information to find a unique value for the cost per pound of apples.
Let \(A\) be the cost per pound of apples and \(G\) be the cost per pound of grapes.
From the question stem, we can form the equation: \(10A + 2G = 12\).
This can be simplified by dividing by 2: \(5A + G = 6\).
The question asks for the value of \(A\).
Step 2: Detailed Explanation:
Evaluating Statement (1) Alone:
"The cost per pound of grapes was \$2."
This gives us a specific value for \(G\): \(G = 2\).
We can substitute this value into our equation:
\[ 5A + 2 = 6 \] \[ 5A = 4 \] \[ A = \frac{4}{5} = 0.80 \] Since we can find a single, unique value for \(A\), this statement is sufficient.
Evaluating Statement (2) Alone:
"The cost of 2 pounds of apples was less than the cost of 1 pound of grapes."
This gives us an inequality: \(2A<G\).
We have a system of one equation (\(5A + G = 6\)) and one inequality (\(2A<G\)).
We can substitute \(G = 6 - 5A\) from the equation into the inequality:
\[ 2A<6 - 5A \] \[ 7A<6 \] \[ A<\frac{6}{7} \] This tells us that the cost per pound of apples is less than approximately \$0.86, but it does not give a specific value. Therefore, this statement is not sufficient.
Step 3: Final Answer:
Statement (1) alone is sufficient, but statement (2) alone is not sufficient. This corresponds to option (A).