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Sample Question 20:
Step 1: Start by working backwards. If three-fourths of the chairs are occupied and there are 6 chairs that are empty, this implies that the number of chairs occupied by people is three times the number of empty chairs.
Step 2: To find the number of occupied chairs, multiply the number of empty chairs - which is 6 - by three. Therefore, \(3 \times 6 = 18\).
Step 3: Next, if we let \(x\) represent the total number of people in the room, and two-thirds of these people are seated, it follows that \(\frac{2}{3}x = 18\).
Step 4: Solve for \(x\) in the equation above to determine the total number of people in the room. \(x = \frac{18 \times 3}{2} = 27\).
Therefore, the total number of people in the room is 27 (\(\textbf{(D)}\)).
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are \(6\) empty chairs, how many people are in the room?
\(\textbf{(A)}\ 12\qquad \textbf{(B)}\ 18\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36\)
Answer Keys
Question 20: DSolutions
Question 20Step 1: Start by working backwards. If three-fourths of the chairs are occupied and there are 6 chairs that are empty, this implies that the number of chairs occupied by people is three times the number of empty chairs.
Step 2: To find the number of occupied chairs, multiply the number of empty chairs - which is 6 - by three. Therefore, \(3 \times 6 = 18\).
Step 3: Next, if we let \(x\) represent the total number of people in the room, and two-thirds of these people are seated, it follows that \(\frac{2}{3}x = 18\).
Step 4: Solve for \(x\) in the equation above to determine the total number of people in the room. \(x = \frac{18 \times 3}{2} = 27\).
Therefore, the total number of people in the room is 27 (\(\textbf{(D)}\)).