Given that there are 10 people in a group such that the mean of their ages is 23, if another person enters the group, the new mean is decreased by 1. Find the age of the new person that enters the group.
Solution: The sum of ages of the 10 people equals to . Let denote the age of the new person that enters the group, then the new mean is . Solving for yields 12. Hence the age of the new person that enters the group is 12. .
Given that in a group of 7 people, there are 6 people share the same age. If the mean of the age of these 7 people is twice the mean of the the 6 people who share the same age. What is the ratio of age between the person who don't share the same age versus the rest of the people in the group?
Solution: Let denote the ages of the 6 people, and denote the age of the 7th person that doesn't share the same age. Using the definition of mean, we have . Upon simplication, we see that . So the ratio in question is simply .