A total of $60, 000 w a s i n v e s t e d f o r o n e y e a r . P a r t o f t h i s a m o u n t e a r n e d s i m p l e a n n u a l i n t e r e s t a t t h e r a t e o f x p e r c e n t p e r y e a r, a n d t h e r e s t e a r n e d s i m p l e a n n u a l i n t e r e s t a t t h e r a t e o f y p e r c e n t p e r y e a r . I f t h e t o t a l i n t e r e s t e a r n e d b y t h e$ 60,000 for that year was $4, 080, w h a t i s t h e v a l u e o f x ? (1)$ x=\cfrac{3y}{4}$
(2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.

Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

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Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

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

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Each statement alone is sufficient.

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Statements (1) and (2) together are not sufficient.

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Solution

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.
Let X be the amount invested at x percent and let Y be the amount invested at y percent, where X + Y = 60000. If $\frac{x}{100}X+\frac{y}{100}Y= 4080$ determine the value of x.
Given that $x= \frac{3}{4}y$ , then $y= \frac{4}{3}x$ . Since, X + Y = 60000, then Y = 60000 - X. By substitution, $\frac{x}{100}X+\frac{y}{100}Y= 4080$ becomes $\frac{x}{100}X+\frac{3}{100}\left ( 60000-X \right)= 4080$ . However, because the value of X is unknown, it is impossible to determine a unique value for x; NOT sufficient.
Given that $\frac{X}{Y}= \frac{3}{2}$ ,then $Y= \frac{2}{3}X$ and $Y= \frac{2}{3}\left ( 60000-Y \right )$ , from which the values of Y and, in turn, X can be determined. However, since the value of y in the equation $\frac{x}{100}X+\frac{y}{100}Y= 4080$ is still unknown, it is impossible to determine a unique value for x; NOT sufficient.
Taking (1) and (2) together, $\frac{x}{100}X+\frac{y}{100}Y= 4080$ from the stem becomes $\frac{x}{100}X+\frac{\frac{4}{3}x}{100}Y= 4080$ , using the information from (1). Then, using the values of X and Y obtained from the information in (2) leaves an equation with x as the only unknown. This equation can be solved for a unique value of x.
The correct answer is C; both statements together are sufficient.

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