Question

Material $A$ costs $\$3$ per kilogram, and Material $B$ costs $\$5$ per kilogram. If $10$ kilograms of Material $K$ consists of $x$ kilograms of Material $A$ and $y$ kilograms of Material $B$, is $x>y$?
(1) $y>4$
(2) The cost of the $10$ kilograms of Material $K$ is less than $\$40$.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is B Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Since x + y = 10x, the relation x > y is equivalent to x > 10 - x, or x > 5.

1. The given information is consistent with x = 5.5 and y = 4.5, and the given information is also consistent with x = y = 5. Therefore, it is possible for x > y to be true and it is possible for x > y to be false; NOT sufficient.
2. Given that 3x + 5y < 40, or 3x + 5(10 - x) < 40, then 3x - 5x < 40 - 50. It follows that -2x < -10, or x > 5; SUFFICIENT.

• The correct answer is B; statement 2 alone is sufficient.