Question

There are a total of
500
500 students in grades
1
1 through
5
5 in an elementary school.

​

17
%
17% of the total number of students are in
1
1st grade.
19
%
19% of the total number of students are in
4
4th grade.
The number of
3
3rd-grade students is
9
9 less than the number of
4
4th-grade students.
The number of
2
2nd-grade students is
10
10 less than the number of
5
5th-grade students.
​

Answer 1

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:

1. **Total students equation**:
\[
x_1 + x_2 + x_3 + x_4 + x_5 = 500
\]

2. **1st grade students (17% of total)**:
\[
x_1 = 0.17 \times 500 = 85
\]

3. **4th grade students (19% of total)**:
\[
x_4 = 0.19 \times 500 = 95
\]

4. **3rd grade students (9 less than 4th grade)**:
\[
x_3 = x_4 - 9 = 95 - 9 = 86
\]

5. **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:
\[
85 + 112 + 86 + 95 + 122 = 500
\]