algebra question

Question

In an exhibition hall, a worker arranges 240 chairs in rows. Each row contains the same number of chairs. If there were 30 more chairs in each row, the worker would need 4 fewer rows. How many rows are there?

Answer 1

C = # chairs R = # rows

C x R = 240

If add 30 chairs, need 4 fewer rows.

(C + 30)(R - 4) = 240; C = 240/R Substitute:

((240/R) + 30)(R - 4) = 240 Use FOIL

240 +30R - 120 - 960/R = 240

30R - 120 - 960/R = 0 divide by 30;

R - 4 - 32/R = 0 mult by R to eliminate fraction;

R² - 4R - 32 = 0 factor:

(R - 8)(R + 4) = 0 so either R = 8 or R = -4 (disregard the latter)

R = 8; C = 30 initially

Check: after revising the setup; R - 4 = 4; C + 30 = 60 4 x 60 = 240 chairs.

In retrospect, could just take the two factors of 240 that make sense in the real world for no. of rows and no. of chairs/row. 1 x 240 wouldn't work, but 3 x 80, 4 x 60, 6 x 40, 8 x 30, 10 x 24, etc. would.

Answer 2

Step 1: Set up the equations

Let x represent the number of chairs in each row, and let y represent the number of rows. We know that the total number of chairs is the product of the rows and the number of chairs per row, so we know that 240=x*y.

We also know that if the number of chairs per row were to be increased by 30, then the number of rows would be decreased by 4. However, the product of the rows and the number of chairs per row still has to be the same. This means that (x+30)*(y-4)=240.

Step 2: Solve the equations

We have that 240=x*y and (x+30)*(y-4)=240. We therefore have a system of two equations with two unknowns (x and y), so we can solve! I won’t show the steps here, so I highly encourage you to work through them!

Mathematically, solving this system yields two solutions.

The first is x=30 and y=8, and the second solution is x=-60 and y=-4.

Step 3: Interpret!

We got two solutions from our equations, but we have to be careful!

The second solution had negative values for x and y, but we have to remember what these variables represent: you cannot have a negative number of rows of chairs!

Therefore, only the first solution applies to the physical scenario: there are 8 rows of 30 chairs.

Hope this clarifies,

Sam

Answer 3

8 rows of 30

or

4 rows of 60