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 23×10=230. Let x denote the age of the new person that enters the group, then the new mean is 10+1230+x=23−1. Solving for x 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 x denote the ages of the 6 people, and y denote the age of the 7th person that doesn't share the same age. Using the definition of mean, we have 76x+y=2⋅66x. Upon simplication, we see that 8x=y. So the ratio in question is simply x:y=1:8□.
I am thinking of 4 numbers.
The average of the 4 numbers is 44.
The average of the first 3 numbers is 33.
What is the value of the 4th number?
a, b, c, d and e are five consecutive integers in increasing order. When we delete one of the 5 from the set, then the sum of the numbers would have decreased by 20%
Which one of the numbers was deleted from the set?
Problem Solving - Intermediate
What is the average (arithmetic mean) of 315,320, and 325?
We have a five digit positive integer N. We select every pair of digits of N (still retaining their relative order) and arrange them in numerical order to obtain 10 numbers: 33,37,37,37,38,73,77,78,83,87. Find N.
I have a list of twelve numbers where the first number is 1, the last number is 12 and each of the other numbers is one more than the average of its two neighbors. What is the largest number in the list?
Problem Solving - Advanced
II and III only
I and II only
I, II, and III
The sum of N real numbers (not necessarily unique) is 20. The sum of the 3 smallest of these numbers is 5. The sum of the 3 largest is 7. Which of the following are possible values for N?
Calvin is thinking of a sequence of distinct positive integers, which includes the number 35. The average of the numbers is 53. If he removes the number 35 from the list, the average increases to 54.
What is the largest possible number in Calvin's list?