Answer :
To determine the truth of each statement regarding the transportation of 80 students with cars and vans, let's analyze each part individually using the given condition [tex]\(4c + 6v = 80\)[/tex]:
Statement A: If 12 cars go, then 2 vans are needed.
To verify this, we should substitute [tex]\(c = 12\)[/tex] and [tex]\(v = 2\)[/tex] into the equation:
[tex]\[4 \cdot 12 + 6 \cdot 2 = 48 + 12 = 60 \neq 80\][/tex]
This statement is false, as 60 does not equal 80.
Statement B: The pair [tex]\(c = 14\)[/tex] and [tex]\(v = 4\)[/tex] is a solution to the equation.
Substitute [tex]\(c = 14\)[/tex] and [tex]\(v = 4\)[/tex] into the equation:
[tex]\[4 \cdot 14 + 6 \cdot 4 = 56 + 24 = 80\][/tex]
This statement is true, as 80 equals 80.
Statement C: If 6 cars go and 11 vans go, there will be extra space.
Substitute [tex]\(c = 6\)[/tex] and [tex]\(v = 11\)[/tex] into the equation:
[tex]\[4 \cdot 6 + 6 \cdot 11 = 24 + 66 = 90 > 80\][/tex]
This statement is true, as 90 is greater than 80, indicating extra space.
Statement D: 10 cars and 8 vans isn't enough to transport all the students.
Substitute [tex]\(c = 10\)[/tex] and [tex]\(v = 8\)[/tex] into the equation:
[tex]\[4 \cdot 10 + 6 \cdot 8 = 40 + 48 = 88 \neq 80\][/tex]
This statement is false, as 88 is more than 80 so it would be enough space (and even extra).
Statement E: If 20 cars go, no vans are needed.
Substitute [tex]\(c = 20\)[/tex] and [tex]\(v = 0\)[/tex] into the equation:
[tex]\[4 \cdot 20 + 6 \cdot 0 = 80 + 0 = 80\][/tex]
This statement is true, as 80 equals 80.
Statement F: 8 vans and 8 cars are numbers that meet the constraints in this situation.
Substitute [tex]\(c = 8\)[/tex] and [tex]\(v = 8\)[/tex] into the equation:
[tex]\[4 \cdot 8 + 6 \cdot 8 = 32 + 48 = 80\][/tex]
This statement is true, as 80 equals 80.
So, the true statements about the situation are:
- Statement B
- Statement C
- Statement E
- Statement F
Statement A: If 12 cars go, then 2 vans are needed.
To verify this, we should substitute [tex]\(c = 12\)[/tex] and [tex]\(v = 2\)[/tex] into the equation:
[tex]\[4 \cdot 12 + 6 \cdot 2 = 48 + 12 = 60 \neq 80\][/tex]
This statement is false, as 60 does not equal 80.
Statement B: The pair [tex]\(c = 14\)[/tex] and [tex]\(v = 4\)[/tex] is a solution to the equation.
Substitute [tex]\(c = 14\)[/tex] and [tex]\(v = 4\)[/tex] into the equation:
[tex]\[4 \cdot 14 + 6 \cdot 4 = 56 + 24 = 80\][/tex]
This statement is true, as 80 equals 80.
Statement C: If 6 cars go and 11 vans go, there will be extra space.
Substitute [tex]\(c = 6\)[/tex] and [tex]\(v = 11\)[/tex] into the equation:
[tex]\[4 \cdot 6 + 6 \cdot 11 = 24 + 66 = 90 > 80\][/tex]
This statement is true, as 90 is greater than 80, indicating extra space.
Statement D: 10 cars and 8 vans isn't enough to transport all the students.
Substitute [tex]\(c = 10\)[/tex] and [tex]\(v = 8\)[/tex] into the equation:
[tex]\[4 \cdot 10 + 6 \cdot 8 = 40 + 48 = 88 \neq 80\][/tex]
This statement is false, as 88 is more than 80 so it would be enough space (and even extra).
Statement E: If 20 cars go, no vans are needed.
Substitute [tex]\(c = 20\)[/tex] and [tex]\(v = 0\)[/tex] into the equation:
[tex]\[4 \cdot 20 + 6 \cdot 0 = 80 + 0 = 80\][/tex]
This statement is true, as 80 equals 80.
Statement F: 8 vans and 8 cars are numbers that meet the constraints in this situation.
Substitute [tex]\(c = 8\)[/tex] and [tex]\(v = 8\)[/tex] into the equation:
[tex]\[4 \cdot 8 + 6 \cdot 8 = 32 + 48 = 80\][/tex]
This statement is true, as 80 equals 80.
So, the true statements about the situation are:
- Statement B
- Statement C
- Statement E
- Statement F
