Raj went to the theatre to watch a traditional Indian dance performance with his family. The theatre had 1050 seats. There were 50% fewer $50-seats than $30-seats. 90% of the $30-seats and some $50-seats were sold. A total of $34 650 was collected. How many seats were unsold?

Let's break down the problem step by step:

Step 1: Let's assume the number of $30 seats is x. According to the problem, there are 50$50 seats than $30 seats, which means the number of$50 seats is 0.5x.

Step 2: The total number of seats in the theatre is 1050. So, x + 0.5x = 1050. Solving this equation gives us x = 700. So, there are 700 $30 seats and 350$50 seats.

Step 3: We know that 90% of the $30 seats were sold, which is 0.9 * 700 = 630 seats.

Step 4: The total amount collected from the sale of $30 seats is 630 *$30 = $18,900.

Step 5: The total collection was $34,650. If we subtract the amount collected from the$30 seats, we get the amount collected from the $50 seats, which is$34,650 - $18,900 =$15,750.

Step 6: The number of $50 seats sold is$15,750 / $50 = 315 seats.

Step 7: The total number of seats sold is 630 (from $30 seats) + 315 (from$50 seats) = 945 seats.

Step 8: The number of unsold seats is the total number of seats in the theatre minus the number of seats sold, which is 1050 - 945 = 105 seats.

So, there were 105 seats that were unsold.