Let's solve the following problem:
Step 1
After reading the task we draw two circles that overlap in the middle. On one side, we record the number of Alex's friends, which includes the guests who are only his friends plus the guests who are mutual friends. In the condition of the problem it is said that they are 15. On the other side, in the same way, we write down Bobby's friends, which are 17 in total.
Step 1
Let's start solving the task:
1. To find how many of the guests are ONLY Alex's friends: From a total of 25 guests, we subtract 15 (Alex's guests) and get the number of friends that are only his. (25 - 15 = 10). We write 10 in Alex's purple circle.
2. Similarly, we find how many of the guests are ONLY Bobby's friends: (25 - 17 = 8). We write 8 in Bobby's green circle.
3. To find how many guests are mutual friends of Alex and Bobby, we subtract the number of friends of Alexa and Bobby from the total number of guests (25). (25 - 10 - 8 = 7). We write 7 in the middle where the two circles overlap - this is the number of friends Alex and Bobby have in common.
Check: 10 + 7 + 8 = 25
Answer: 7
* Problems with Euler circles can be found in the course Competitive Mathematics - Level 3