Question

The $38$ movies in the video store fall into the following three categories: $10$ action, $20$ drama, and $18$ comedy. However, some movies are classified under more than one category: $5$ are both action and drama, $3$ are both action and comedy, and $4$ are both drama and comedy. How many action-drama-comedy movies are there ?

• A. $40$
• B. $42$
• C. $44$
• D. $38$

Answer 1

Answer: (D)

$38$

There are three overlapping sets here; therefore,use a Venn diagram to solve the problem. First, fill in the numbers given in the problem, working from the inside out. Assign the variable x to represent the number of action-drama-comedy movies. Then, create variable expressions, using the totals given in the problem, to represent the number of movies in each of the other categories. You know that there is a total of $38$ movies; therefore, you can write the following equation to represent the total number of movies in the store:

$10-8+x$

$20-9+x$

$18-7+x$

$5-x$

$4-x$

$3-x$

$\underset{–}{+x}$

$36+x=38$

$x=2$

If you are unsure of the algebraic solution, you can also guess a number for x and fill in the rest of the diagram until the total number of movies reaches $38$.