# SAT Math: Arithmetic

## 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.