Problem Solving

The average age of a group of n people is 15 yrs. One more person aged 39 joins the group and the new average is 17 yrs. What is the value of n?

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

Mo2men wrote:The average age of a group of n people is 15 yrs. One more person aged 39 joins the group and the new average is 17 yrs. What is the value of n?

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

This is a weighted average/mixture question.

Average for the first n people = 15.
Average for added person = 39.
Average for the MIXTURE of all the people = 17.

To determine the value of n, use ALLIGATION.
Let a = the added person.

Step 1: Plot the 3 averages on a number line, with the average for the first n people and that for the added person on the ends and the average for the mixture in the middle.
n 15----------------17----------------39 a

Step 2: Calculate the distances between the averages.
n 15--------2-------17--------22------39 a

Step 3: Determine the ratio in the mixture.
The required ratio of n to a is equal to the RECIPROCAL of the distances in red.
n:a = 22:2 = 11:1.

In the resulting ratio, n=11 and a=1, implying that to 11 original people 1 person was added.
Thus, n=11.

The correct answer is C.

Mo2men wrote:The average age of a group of n people is 15 yrs. One more person aged 39 joins the group and the new average is 17 yrs. What is the value of n?

(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

An alternate approach is to PLUG IN THE ANSWERS, which represent the value of n.
When the correct answer choice is plugged in, the age of the added person will be 39.

B: n=10
Sum of the ages of the 10 original people = (original number of people)(original average) = 10*15 = 150.
Sum of the ages after 1 person is added = (new number of people)(new average age) = 11*17 = 187.
Age of the added person = (new sum) - (old sum) = 187-150 = 37.
Here, the age of the added person is TOO SMALL.

D: n=12
Sum of the ages of the 12 original people = (original number of people)(original average) = 12*15 = 180.
Sum of the ages after 1 person is added = (new number of people)(new average age) = 13*17 = 221.
Age of the added person = (new sum) - (old sum) = 221-180 = 41.
Here, the age of the added person is TOO GREAT.

Since the age of added person is TOO SMALL when n=10 but TOO GREAT when n=12, the correct answer choice must be BETWEEN 10 and 12.

The correct answer is C.