A hall has 1325 seats. These are arranged in rows of 35 seats or 40 seats. The ratio of

35-seat rows to 40-seat rows is 3 : 4. How many rows are there?
Simultaneous equation please

To find out how many rows there are, we can set up a system of simultaneous equations based on the given information.

Let's assume the number of 35-seat rows is represented by a variable, x, and the number of 40-seat rows is represented by a variable, y.

From the given information, we can establish the following two equations:

Equation 1: 35x = total number of seats in 35-seat rows
Equation 2: 40y = total number of seats in 40-seat rows

We also know that the ratio of 35-seat rows to 40-seat rows is 3:4. Therefore, we can set up another equation to represent this ratio:

Equation 3: x/y = 3/4

Now, we can solve this system of simultaneous equations for x and y.

First, rewrite Equation 3 in terms of one variable, either x or y:
x = (3/4)y

Substitute this value of x in Equation 1:
35((3/4)y) = total number of seats in 35-seat rows
or (105/4)y = total number of seats in 35-seat rows

Substitute this value of x in Equation 2:
40y = total number of seats in 40-seat rows

Now, we know that the total number of seats in the hall is 1325. Thus, we can set up another equation:

Equation 4: (105/4)y + 40y = 1325

Combine like terms:
(105y/4) + (160y/4) = 1325
(265y/4) = 1325

Multiply both sides of the equation by 4 to eliminate the denominator:
265y = 5300

Now, divide both sides of the equation by 265 to solve for y:
y = 5300/265
y = 20

Substitute this value of y in Equation 1:
(105/4)(20) = total number of seats in 35-seat rows
or (35/4)(20) = total number of seats in 35-seat rows
or 700/4 = total number of seats in 35-seat rows
or 175 = total number of seats in 35-seat rows

Since there are 35 seats in each 35-seat row, we can find the number of 35-seat rows by dividing the total number of seats in 35-seat rows by the number of seats in each row:
Number of 35-seat rows = 175/35
Number of 35-seat rows = 5

Similarly, the number of 40-seat rows can be found by dividing the total number of seats in the hall by the number of seats in each 40-seat row:
Number of 40-seat rows = 1325/40
Number of 40-seat rows = 33

Therefore, there are 5 rows of 35 seats and 33 rows of 40 seats in the hall.