## Groups

At a certain restaurant, guests have a choice of either mustard or mayonnaise on their sandwiches. At lunch yesterday, the restaurant served a total of 100 sandwiches. 55 of the sandwiches had mustard, 37 mayonnaise, and 18 had neither. How many of the sandwiches had both mustard and mayonnaise?

A. 0

B. 2

C. 8

D. 10

E. 12

## Knowsys Method

**Read the problem carefully.** Be sure to keep the numbers in this problem straight so that you do not make any careless mistakes in your work.

**Identify the bottom line. **# of sand. with must. and mayo = ?

**Assess your options. **You could solve this problem by creating a Venn diagram, but there is a faster method. If you memorize "the group formula," you can solve these problems quickly and easily without drawing out a diagram.

**The Group Formula**

2 groups:

Total = Group 1 + Group 2 + Neither - Both

3 groups:

Total = Group 1 + Group 2 + Group 3 + Neither - Both - 2(All)

**What do all these terms mean? **

Total = Total number of items. In this problem, it is the total number of sandwiches.

Group 1, 2, 3 = The number of items in each group. In this problem, the two groups are the sandwiches with mustard and the sandwiches with mayo.

Neither = The number of items that do not fall into one of the 2 (or 3) groups. In this problem, it is the sandwiches that have neither mustard nor mayo on them.

Both = The number of items that fall into 2 groups. In this problem, it is the sandwiches that have both mustard and mayo.

All = (Used for 3 group problems ONLY!) This is the number of items that fall into all 3 groups (as opposed to the number of items that fall into only 2 out of the three groups). There are only 2 groups in the problem at hand, so this term does not apply.

**Attack the problem.** Sub the numbers from the problem into the group formula to figure out how many sandwiches have both mustard and mayo.

Total = Group 1 + Group 2 + Neither - Both

100 = 55 + 37 + 18 - Both

100 = 110 - Both

10 = Both

So, 10 sandwiches have both mustard and mayo.

**Loop back. **Verify that you solved for the bottom line.

The correct answer is D.

This is a medium level problem.

**Want some help reviewing the math concepts you need to master? **Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.