To find the number of students in each grade, let's represent the number of students in each grade as follows:
- 1st grade: \( G_1 \)
- 2nd grade: \( G_2 \)
- 3rd grade: \( G_3 \)
- 4th grade: \( G_4 \)
- 5th grade: \( G_5 \)
From the problem, we know:
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The total number of students is \( 500 \).
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\( 17% \) of the students are in 1st grade: \[ G_1 = 0.17 \times 500 = 85 \]
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\( 19% \) of the students are in 4th grade: \[ G_4 = 0.19 \times 500 = 95 \]
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The number of 3rd-grade students is \( 9 \) less than the number of 4th-grade students: \[ G_3 = G_4 - 9 = 95 - 9 = 86 \]
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The number of 2nd-grade students is \( 10 \) less than the number of 5th-grade students: \[ G_2 = G_5 - 10 \]
Now, we can create the equation for the total number of students: \[ G_1 + G_2 + G_3 + G_4 + G_5 = 500 \] Substituting the known values: \[ 85 + G_2 + 86 + 95 + G_5 = 500 \] Simplifying: \[ 266 + G_2 + G_5 = 500 \] This leads to: \[ G_2 + G_5 = 500 - 266 = 234 \]
Now, substituting \( G_5 \) in terms of \( G_2 \): From the previous equation, we have: \[ G_5 = G_2 + 10 \]
Substituting \( G_5 \) in our equation: \[ G_2 + (G_2 + 10) = 234 \] This simplifies to: \[ 2G_2 + 10 = 234 \] \[ 2G_2 = 234 - 10 = 224 \] \[ G_2 = \frac{224}{2} = 112 \]
Now, substituting \( G_2 \) back to find \( G_5 \): \[ G_5 = G_2 + 10 = 112 + 10 = 122 \]
Now, we can summarize the number of students in each grade:
- \( G_1 = 85 \) (1st grade)
- \( G_2 = 112 \) (2nd grade)
- \( G_3 = 86 \) (3rd grade)
- \( G_4 = 95 \) (4th grade)
- \( G_5 = 122 \) (5th grade)
Here's the final table:
| Grade | Number of Students | |-------|--------------------| | 1st | 85 | | 2nd | 112 | | 3rd | 86 | | 4th | 95 | | 5th | 122 | | Total | 500 |