Part of the seating arrangement in a theatre is shown in the diagram below - Leaving Cert Mathematics - Question 7 - 2018

To find the total number of rows, we know that the last row has 50 seats. The relationship between the rows can be expressed as:

$28 + (n - 1) = 50$

Solving for n:

$n - 1 = 50 - 28$ $n - 1 = 22$ $n = 23$

Thus, there are 23 rows of seats in the theatre.

The total number of seats can be found by summing the seats in each row. We can use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} (a + l)$

Where:

• n is the number of terms (rows),
• a is the first term (seats in row 1),
• l is the last term (seats in the last row).

Plugging in the values:

• n = 23,
• a = 28,
• l = 50,

$S_{23} = \frac{23}{2} (28 + 50) = \frac{23}{2} (78) = 897$

Therefore, the total number of seats in the theatre is 897.

Given that 600 people are seated:

In the first n rows, the total seating is given by:

$\frac{n}{2} (a + l)$

Where: a = 28, l = 28 + (n-1), n = number of rows occupied. Thus, we set up the equation:

$\frac{n}{2} (56 + n) = 600$

Solving for n gives us:

$n^2 + 55n - 1200 = 0$

Using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where a = 1, b = 55, c = -1200.

Solving this yields:

$n = 32$

The number of people seated in the next row (row 33) is:

$l = 28 + 32 = 60$

So, there are 60 people seated in the next row.

The total income can be computed using:

$Income = (number\ of\ adult\ tickets \times Price\ of\ adult\ ticket) + (number\ of\ children’s\ tickets \times Price\ of\ children’s\ ticket)$

Calculating:

$Income = 276 imes 25 + 212 imes 12$

This results in:

$Income = 6900 + 2544 = 9444\€$

Thus, the total income is €9,444.

Let x represent the number of children's tickets sold. According to the ratio of adult to children's tickets:

$A = 3x$

The total number of tickets sold is given by:

$A + x = 752$

Substituting for A:

$3x + x = 752\rightarrow 4x = 752\rightarrow x = 188$

Therefore, the number of adult tickets sold is:

$A = 3(188) = 564$

So, 564 adult tickets and 188 children's tickets were sold.

Let the cost of a children's ticket be x. The cost of an adult ticket is given as:

$A = \frac{22}{5}x$

Using the total income generated:

$188A + 564x = 17578$

Substituting in the expression for A:

$188 \left( \frac{22}{5}x \right) + 564x = 17578\rightarrow 759.2x + 564x = 17578$

Simplifying:

$1323.2x = 17578\rightarrow x = 13.3\€$

Thus, the cost of a children's ticket is approximately €11.