A roller coaster has 10 carts that will seat up to 3 people each. If 24 people are seated in the carts ant NO carts are empty, what is the greatest possible number of carts that could be filled with 3 people? A. 4 B. 5 C. 6 D. 7

To solve the problem, we need to determine how many carts can be filled with 3 people while ensuring that all 10 carts are used and a total of 24 people are seated. 1. **Identify the total seating capacity**: Each cart can seat 3 people, and there are 10 carts. Therefore, the total seating capacity is: \[ \text{Total capacity} = 10 \text{ carts} \times 3 \text{ people/cart} = 30 \text{ people} \] 2. **Determine the number of people seated**: We know that 24 people are seated in the carts. 3. **Calculate the number of empty seats**: Since the total capacity is 30 and 24 people are seated, the number of empty seats is: \[ \text{Empty seats} = 30 - 24 = 6 \text{ seats} \] 4. **Determine the number of carts filled with 3 people**: If we want to maximize the number of carts filled with 3 people, we need to minimize the number of carts that have fewer than 3 people. Let \( x \) be the number of carts filled with 3 people. The number of people seated in these carts is: \[ 3x \] The remaining people seated in the other carts (which must have at least 1 person each) can be expressed as: \[ 24 - 3x \] The number of carts that are not filled with 3 people is: \[ 10 - x \] Since each of these remaining carts must have at least 1 person, we have: \[ 24 - 3x \geq 10 - x \] 5. **Solve the inequality**: Rearranging the inequality gives: \[ 24 - 3x + x \geq 10 \] \[ 24 - 2x \geq 10 \] \[ 14 \geq 2x \] \[ 7 \geq x \] This means that the maximum number of carts that can be filled with 3 people is 7. 6. **Check if 7 carts can be filled**: If \( x = 7 \): - People in 7 carts: \( 3 \times 7 = 21 \) - Remaining people: \( 24 - 21 = 3 \) - Carts left: \( 10 - 7 = 3 \) - Each of the remaining 3 carts can have 1 person, which satisfies the condition. Thus, the greatest possible number of carts that could be filled with 3 people is \( \boxed{7} \).