To analyze the situation, we have a ratio of children to adults of 2:5. This means that for every 2 children, there are 5 adults.
First, we'll start by letting \(2x\) represent the number of children and \(5x\) represent the number of adults. The total number of people watching the movie can be expressed as:
\[ 2x + 5x = 7x \]
Given that there are 280 people in total, we can set up the equation:
\[ 7x = 280 \]
Solving for \(x\):
\[ x = \frac{280}{7} = 40 \]
Now we can find the number of children and adults:
- Number of children: \[ 2x = 2(40) = 80 \]
- Number of adults: \[ 5x = 5(40) = 200 \]
Now we can evaluate each of the statements:
-
There are exactly 120 more adults than children. \[ 200 - 80 = 120 \quad \text{(True)} \]
-
There are exactly 120 children watching the movie. \[ \text{(False: There are actually 80 children)} \]
-
Two-fifths of the people watching the movie are children. \[ \text{Fraction of children} = \frac{80}{280} = \frac{2}{7} \quad \text{(False: It is not two-fifths)} \]
-
The number of children is \(\frac{2}{5}\) of the number of adults. \[ \text{Fraction of children to adults} = \frac{80}{200} = \frac{2}{5} \quad \text{(True)} \]
-
Five-sevenths of the people watching the movie are adults. \[ \text{Fraction of adults} = \frac{200}{280} = \frac{5}{7} \quad \text{(True)} \]
The correct statements are:
- There are exactly 120 more adults than children.
- The number of children is \(\frac{2}{5}\) of the number of adults.
- Five-sevenths of the people watching the movie are adults.
So the true statements are: 1, 4, and 5.
1 answer