To find the number of students in each grade (1st through 5th) based on the information provided, we can set up a system of equations.
Given:
- Total number of students = 500
- \( x_1 \): number of 1st-grade students
- \( x_2 \): number of 2nd-grade students
- \( x_3 \): number of 3rd-grade students
- \( x_4 \): number of 4th-grade students
- \( x_5 \): number of 5th-grade students
From the information given, we can create the following equations:
-
Total students equation:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 500 \] -
1st grade students (17% of total):
\[ x_1 = 0.17 \times 500 = 85 \] -
4th grade students (19% of total):
\[ x_4 = 0.19 \times 500 = 95 \] -
3rd grade students (9 less than 4th grade):
\[ x_3 = x_4 - 9 = 95 - 9 = 86 \] -
2nd grade students (10 less than 5th grade):
\[ x_2 = x_5 - 10 \]
Now, substituting \( x_1 \), \( x_3 \), and \( x_4 \) into the total students equation:
\[ 85 + x_2 + 86 + 95 + x_5 = 500 \]
Combining the known numbers:
\[ 266 + x_2 + x_5 = 500 \]
Subtracting 266 from both sides gives:
\[ x_2 + x_5 = 234 \]
Now, substitute \( x_2 \) from the equation related to \( x_5 \):
\[ (x_5 - 10) + x_5 = 234 \]
Combine the terms:
\[ 2x_5 - 10 = 234 \]
Adding 10 to both sides:
\[ 2x_5 = 244 \]
Dividing by 2:
\[ x_5 = 122 \]
Now substitute \( x_5 \) back to find \( x_2 \):
\[ x_2 = 122 - 10 = 112 \]
Now we have all the values for each grade:
- \( x_1 = 85 \) (1st grade)
- \( x_2 = 112 \) (2nd grade)
- \( x_3 = 86 \) (3rd grade)
- \( x_4 = 95 \) (4th grade)
- \( x_5 = 122 \) (5th grade)
Summary:
- 1st grade: 85 students
- 2nd grade: 112 students
- 3rd grade: 86 students
- 4th grade: 95 students
- 5th grade: 122 students
All student counts add up to the total of 500: \[ 85 + 112 + 86 + 95 + 122 = 500 \]