Question

A citrus fruit grower receives $15foreachcrateoforangesshippedand$18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?
(1) Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped.
(2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped.

• 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 C Both statements together are sufficient, but neither statement alone is sufficient.

Algebra Simultaneous equations

If $x$ represents the number of crates of oranges and $y$ represents the number of crates of grapefruit, find a unique value for $x$.

(1) Translating from words into symbols gives $x=2y+20$, but there is no information about $y$ and no way to find a unique value for $x$ from this equation. For example, if $y=10$, then $x=40$, but if $y=100$, then $x=220$; NOT sufficient.

(2) Translating from words to symbols gives $15x+18y=38,700$, but there is no way to find a unique value for $x$ from this equation. For example, if $y=2,150$, then $x=0$ and if $y=0$, then $x=2,580$; NOT sufficient.

Taking (1) and (2) together gives a system of two equations in two unknowns. Substituting the equation from (1) into the equation from (2) gives a single equation in the variable $y$. This equation can be solved for a unique value of $y$ from which a unique value of $x$ can be determined.

The correct answer is C; both statements together are sufficient.