A math teacher has 30 cards, each of which is in the shape of a geometric figure. Half of the cards are rectangles, and a third of the cards are rhombuses. If 8 cards are squares, what is the maximum possible number of cards that re circles?
A. 9
B. 10
C. 11
D. 12
E. 13
The OA is E
Source: Manhattan Prep
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A math teacher has 30 cards, each of which is in the shape
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swerve wrote:A math teacher has 30 cards, each of which is in the shape of a geometric figure. Half of the cards are rectangles, and a third of the cards are rhombuses. If 8 cards are squares, what is the maximum possible number of cards that re circles?
A. 9
B. 10
C. 11
D. 12
E. 13
A square is also a rhombus and a rectangle.
Thus, we will maximize the number of circles if the 8 squares are included among the 15 rectangles and the 10 rhombuses, implying the following composition for the 30 cards:
8 squares (which also qualify as 8 rectangles and 8 rhombuses)
7 non-square rectangles, for a total of 15 rectangles (including the 8 squares)
2 non-square rhombuses, for a total of 10 rhombuses (including the 8 squares)
C circles
The following equation is yielded:
8 + 7 + 2 + C = 30
17 + C = 30
C = 13
The correct answer is E.
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We see that we have ½ x 30 = 15 rectangles and ⅓ x 30 = 10 rhombuses. Since there are 8 squares (recall that squares are both rectangles and rhombuses), we have 15 + 10 - 8 = 17 rectangles or rhombuses. If the remaining cards are all circles, we have a maximum number of 30 - 17 = 13 circles.swerve wrote:A math teacher has 30 cards, each of which is in the shape of a geometric figure. Half of the cards are rectangles, and a third of the cards are rhombuses. If 8 cards are squares, what is the maximum possible number of cards that re circles?
A. 9
B. 10
C. 11
D. 12
E. 13
The OA is E
Source: Manhattan Prep
Answer: E
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