Question on sets

Question

Each student in a cooking class of 50 students is assigned to create a dessert, an appetizer, or both. The total number of students creating an appetizer is seven more than the number of students creating a dessert. If the number of students that create two dishes is the same as the number of students that create exactly one dish, how many students only created a dessert?

Answer 1

Let's try a Venn Diagram (although the Answers Forum doesn't allow graphics).  Imagine 2 intersecting circles:

 

     50 total students

           ------------------------------------------------------------------------------

 

                 Dessert                           Both                              Appetizer

 

 

                     D                                 B                                    A

 

 

Note:  B is inside both circles.

 

Translate:

  "total number of students creating an appetizer is seven more than the number of students creating a dessert"

      means        B+A = 7 + D+B

                         A = 7+D

 

  "the number of students that create two dishes is the same as the number of students that create exactly one dish"

               means                               B                           =                           D      +       A

 

"how many students only created a dessert"     means    "What is D?

 

The math:

 

   Solve for D using these two equations and the fact that A+B+D=50.

 

      A   + B      + D = 50

      A + (D+A) + D = 50

   (7+D) + (D+(7+D)) + D = 50

    14 + 4D = 50                                  [collect terms]

      4D = 36                                  [subtract 14 from both sides]

         D = 9

 

9 students made a dessert.

 

Check:    A=7+D, so 16 students made an appetizer.

    T = 16+B+9 = 50, so 15 students made both.

Answer 2

Hey Sarika,

 

GREAT QUESTION!

 

Let's define a few variables first:

 

-# of students making appetizer = A
-# of students making dessert = D
-# of students making BOTH = B

-Total # of students = 50 = A + D

 

From the question we have:

B = A + D = 50/2 = 25
A = D + 7

 

Okay so we can substitute (D + 7) for A, here we go!

 

D + (D + 7) = 25

2D + 7 = 25

2D = 18

D = 9 students who made dessert only!