Given that three students with weights of [tex]\( 72 \)[/tex] kg, [tex]\( 83 \)[/tex] kg, and [tex]\( 60 \)[/tex] kg leave the class, and they are replaced by new students weighing [tex]\( 51 \)[/tex] kg, [tex]\( 45 \)[/tex] kg, and [tex]\( 69 \)[/tex] kg respectively, we need to find out how many students are there in the class if the average weight decreases by [tex]\( 1.25 \)[/tex] kg.
Let's denote:
- The initial weights of the students leaving as [tex]\( 72 \)[/tex] kg, [tex]\( 83 \)[/tex] kg, and [tex]\( 60 \)[/tex] kg.
- The weights of the new students as [tex]\( 51 \)[/tex] kg, [tex]\( 45 \)[/tex] kg, and [tex]\( 69 \)[/tex] kg.
- The decrease in average weight as [tex]\( 1.25 \)[/tex] kg.
First, we calculate the total weight of the students who left:
[tex]\[
72 \, \text{kg} + 83 \, \text{kg} + 60 \, \text{kg} = 215 \, \text{kg}
\][/tex]
Next, we calculate the total weight of the new students:
[tex]\[
51 \, \text{kg} + 45 \, \text{kg} + 69 \, \text{kg} = 165 \, \text{kg}
\][/tex]
The change in total weight due to the exchange of students can be found by subtracting the total new weight from the total old weight:
[tex]\[
215 \, \text{kg} - 165 \, \text{kg} = 50 \, \text{kg}
\][/tex]
This [tex]\( 50 \, \text{kg} \)[/tex] decrease in total weight caused the average weight to decrease by [tex]\( 1.25 \, \text{kg} \)[/tex]. To find the number of students in the class, we need to set up the following relation:
[tex]\[
\Delta \text{total weight} = \Delta \text{average weight} \times \text{number of students}
\][/tex]
Where:
[tex]\[
\Delta \text{total weight} = 50 \, \text{kg}
\][/tex]
[tex]\[
\Delta \text{average weight} = 1.25 \, \text{kg}
\][/tex]
Let [tex]\( n \)[/tex] be the total number of students in the class. So, we have:
[tex]\[
50 = 1.25 \times n
\][/tex]
Solving for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{50}{1.25} = 40
\][/tex]
Thus, the total number of students in the class is [tex]\( 40 \)[/tex].
