To determine the number of seats in each section of the 500-seat auditorium, let's break the problem down step-by-step:
1. Identify the Known Values:
- Total number of seats in the auditorium: 500
- Number of seats in the Blue section: 170
- The Red section has 50 more seats than the White section.
2. Define Variables:
- Let [tex]\( W \)[/tex] represent the number of seats in the White section.
- Consequently, the number of seats in the Red section will be [tex]\( W + 50 \)[/tex].
3. Set Up the Equation:
- We know that the total number of seats is the sum of the seats in the Blue, Red, and White sections.
- Hence, the equation based on the total number of seats is:
[tex]\[ 170 + (W + 50) + W = 500 \][/tex]
4. Simplify the Equation:
- Combine like terms in the equation:
[tex]\[ 170 + W + 50 + W = 500 \][/tex]
[tex]\[ 170 + 50 + W + W = 500 \][/tex]
[tex]\[ 220 + 2W = 500 \][/tex]
5. Solve for [tex]\( W \)[/tex] (White seats):
- Isolate [tex]\( 2W \)[/tex] by subtracting 220 from both sides of the equation:
[tex]\[ 2W = 500 - 220 \][/tex]
[tex]\[ 2W = 280 \][/tex]
- Divide both sides by 2 to find [tex]\( W \)[/tex]:
[tex]\[ W = \frac{280}{2} \][/tex]
[tex]\[ W = 140 \][/tex]
6. Calculate the Number of Seats in the Red Section:
- Recall that the Red section has 50 more seats than the White section:
[tex]\[ R = W + 50 \][/tex]
[tex]\[ R = 140 + 50 \][/tex]
[tex]\[ R = 190 \][/tex]
7. Summarize the Results:
- Number of seats in the Blue section: 170
- Number of seats in the White section: 140
- Number of seats in the Red section: 190
Therefore, the auditorium has:
- 170 seats in the Blue section,
- 140 seats in the White section,
- 190 seats in the Red section.
This completes the solution.
