If 45 chairs are​ manufactured, how many ottomans and sofas can be​ manufactured?

Question

A furniture manufacturer makes​ sofas, chairs, and ottomans. The accompanying table gives the number of hours of labor required for the carpentry and upholstery that goes into each item. Suppose that each​ day,

401

​labor-hours are available for carpentry and

465

​labor-hours are available for upholstery. Give three different combinations for the number of each type of furniture that can be manufactured each day.

	Ottoman	Sofa	Chair
Carpentry	1 hours	3 hours	6 hours
Upholstery	3 hours	6 hours	3 hours

If 45 chairs are​ manufactured, how many ottomans and sofas can be​ manufactured?

Answer 1

Firstly express the information given as mathematical expressions:

Let the number of ottomans be X, the number of sofas be Y and the number of chairs be Z

The total number of hours available for carpentry and upholstery is given therefore the number of hours available for each piece of furniture can be expressed:

(1) 401 = 1*X + 3*Y + 6*Z

(2) 465 = 3*X + 6*Y + 3*Z

However, the problem tells us that 45 chairs are made, therefore 45 can be substituted for Z in both equations:

(1) 401 = X + 3Y + 6*45

401 = X + 3Y + 270 and taking 270 from each side gives

131 = X + 3Y

(2) 465 = 3X +6Y + 3*45

465 = 3X + 6Y + 135 and taking 135 away fro each side gives

330 = 3X + 6Y

If equation (1) is then multiplied by 2, this gives

262 = 2X + 6Y and then subtracting this from equation (2) gives

330 - 262 = 3X - 2X +6Y -6Y . the Y's cancel out and this leaves

68 = X

Substituting for X in equation (1) gives:

131 = 68 + 3Y and Y is therefore 21

Therefore 68 ottomans and 21 sofas can be produced