Susan’s business makes handmade dresses. The business sells three different kinds of dresses: evening gowns, sundresses, and cocktail dresses. To make an evening gown, it takes 3 hours to cut the material, 2 hours to pin the pieces together, and 5 hours to sew the dress. To make a sundress, it takes 1 hour to cut the material, 1.5 hours to pin the pieces together, and 3 hours to sew the dress. To make a cocktail dress, it takes 2 hours to cut the material, 1 hour to pin the pieces together, and 2.5 hours to sew the dress. Susan’s employees spent a total of 54 hours cutting, 38.5 hours pinning, and 91 hours sewing in a particular week. How many of each type of dress did they produce? That week, the employees produced evening gowns, sundresses, and cocktail dresses.

An equation is a mathematical statement which equate two algebraic expressions. An equation has an equal to (=) sign in between the expression.

How to find how many of each type of dress did they produce?

Let the number of evening dresses produced be x , sundresses produced be y and cocktail dresses produced be z.

According to the problem,

To produce a evening dress 3 hours of cut is required.
To produce a sundress 1 hour of cut is required.
To produce a cocktail dress 2 hours of cut is required.
Total tie spent in cutting is 54 hours

∴ We can write 3x + y + 2z = 54........(1)

Again, according to the problem,

To produce a evening dress 2 hours of pin is required.
To produce a sundress 1.5 hours of pin is required.
To produce a cocktail dress 1 hour of pin is required.
Total tie spent in pinning is 38.5 hours

∴ We can write 2x + 1.5y + z = 38.5........(2)

Again,

To produce a evening dress 5 hours of sew is required.
To produce a sundress 3 hours of sew is required.
To produce a cocktail dress 2.5 hours of sew is required.
Total tie spent in sewing is 91 hours

∴ We can write, 5x + 3y + 2.5z = 91......(3)

We have to solve the three equations.

From equation (1) we can write , y = 54 - 3x - 2z

Putting the value of y in equation (2), we get,

2x + 1.5(54- 3x - 2z) + z = 38.5

⇒ 2x + 81 - 4.5x - 3z + z = 38.5

⇒ 2z + 2.5x = 42.5.....(4)

Again, putting the value of y from equation (1) in equation (3), we get,

5x + 3(54- 3x - 2z) + 2.5z = 91

⇒ 5x + 162 - 9x - 6z + 2.5z = 91

⇒ 4x +3.5z = 71

⇒ x = (71 - 3.5z)/4

Putting the value of x in equation (4)

2z + 2.5{(71 - 3.5z)/4} = 42.5

⇒ 8z + 177.5 - 8.75z = 170

⇒ -0.75z = -7.5

⇒ z= 10

Again putting the value of z in equation (4)

x = (71 - 35)/4 = 9

Putting the value of x and z in equation (1)

y = 54 - 27 - 20

⇒ y = 7

Number of evening dress = 9

Number of sundresses = 7

Number of cocktail dresses= 10

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