Sure, I will explain the solution step-by-step.
1. Understanding the Problem:
Mrs. Hopkins has 120 seventh-grade students that she needs to divide into three groups. There are specific conditions given:
- The second group has 1.5 times the number of students as the first group.
- The third group has 2.5 times the number of students as the first group.
2. Define Variables:
- Let [tex]\( x \)[/tex] be the number of students in the first group.
3. Express the Other Groups in Terms of [tex]\( x \)[/tex]:
- The second group has [tex]\( 1.5 \times x \)[/tex] students.
- The third group has [tex]\( 2.5 \times x \)[/tex] students.
4. Set Up the Equation:
Since the total number of students is 120, we can set up the following equation:
[tex]\[
x + 1.5x + 2.5x = 120
\][/tex]
5. Combine Like Terms:
Combine all terms involving [tex]\( x \)[/tex]:
[tex]\[
(1 + 1.5 + 2.5)x = 120
\][/tex]
Simplifying inside the parentheses gives:
[tex]\[
5x = 120
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], divide both sides by 5:
[tex]\[
x = \frac{120}{5} = 24
\][/tex]
7. Determine the Number of Students in Each Group:
- First group: [tex]\( x = 24 \)[/tex] students
- Second group: [tex]\( 1.5 \times x = 1.5 \times 24 = 36 \)[/tex] students
- Third group: [tex]\( 2.5 \times x = 2.5 \times 24 = 60 \)[/tex] students
Therefore, the number of students in each group is:
- First Group: 24 students
- Second Group: 36 students
- Third Group: 60 students
