For Billy's birthday party his mother has baked 80 cookies. 40 cookies contain vanilla and 45 contains nuts, and 15 contain both. One of Billy's friends, Jim, is allergic to both nuts and vanilla, but he loves cookies! What is the largest number of cookies that Jim can eat on Billy's birthday? (Note that it is fine for Jim to smell nuts and vanilla as long as he is not eating anything that contains them.)

We are given:

85 baked cookies where:

40 cookies contain vanilla;

and 45 contains nuts;

and 15 contain both.

Jim is allergic to both nuts and vanilla, but he loves cookies!

The largest number of cookies that Jim can eat on Billy’s birthday can be determined by using a Venn Diagram:

The total on one circle (A) is 40 and the total on the other side (B) is 45. At the center, there are 15 cookies, so to get the amount of A:

A = 40 - 15 = 25

B = 45 - 15 = 30

Therefore the largest amount that Jim can eat is 30 cookies.