Problem Solving

When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Source: Veritas

Mo2men wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Source: Veritas

Hello Mo2men.

Let's see your question.

In the first case we have the following: $$\frac{39+x}{n+1}=\frac{x}{n}+2\ -----\left(1\right)$$ and in the second case we have: $$\frac{15+x}{n+1}=\frac{x}{n}-1\ -----\left(2\right)$$ where x represents the sum of the ages of all n people in the group.

Now, if we compute (1)-(2) we will get: $$\frac{39+x}{n+1}-\frac{15+x}{n+1}=2-\left(-1\right)$$ $$\Leftrightarrow\ \frac{24}{n+1}=3$$ $$\Leftrightarrow\ 8=n+1$$ $$\Leftrightarrow\ n=7.$$ Therefore, the correct answer is the option A.

I hope it helps you.

Feel free to ask me again if you have a doubt.

Regards.

Mo2men wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Alternate approach:

Let s = the original sum.
New sum when a person of 39 years is added = s+39.
New sum when a person of 15 years is added = s+15.
Difference between the two new sums = (s+39) - (s+15) = 24.

We can PLUG IN THE ANSWERS, which represent the current number of people.
Let a = the original average.
When the correct answer is plugged in, the difference between the two new sums = 24.

D: 10 people
Since a person of 39 years increases the average by 2, the new sum when a 39-year-old is added = (new number of people)(new average) = 11(a+2) = 11a+22.
Since a person of 15 years decreases the average by 1, the new sum when a 15-year-old is added = (new number of people)(new average) = 11(a-1) = 11a-11.
Difference between the two new sums = (11a+22) - (11a-11) = 33.
The difference is TOO BIG.
Eliminate D.

B: 8 people
Since a person of 39 years increases the average by 2, the new sum when a 39-year-old is added = (new number of people)(new average) = 9(a+2) = 9a+18.
Since a person of 15 years decreases the average by 1, the new sum when a 15-year-old is added = (new number of people)(new average) = 9(a-1) = 9a-9.
Difference between the two new sums = (9a+18) - (9a-9) = 27.
The difference is still TOO BIG.
Eliminate B.

Notice the PATTERN.
As the number of people gets SMALLER, the difference between the two new sums also gets smaller.
Implication:
For the difference between the two new sums to decrease to 24, a smaller answer choice is needed.

The correct answer is A.

A: 7 people
Since a person of 39 years increases the average by 2, the new sum when a 39-year-old is added = (new number of people)(new average) = 8(a+2) = 8a+16.
Since a person of 15 years decreases the average by 1, the new sum when a 15-year-old is added = (new number of people)(new average) = 8(a-1) = 8a-8.
Difference between the two new sums = (8a+16) - (8a-8) = 24.
Success!

Mo2men wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

I received a PM requesting that I explain how the problem can be solved with alligation.
Let A = the one added person.

Case 1:
Average age for the first N people = x.
Average age for A = 39.
Average for the MIXTURE of all the people = x+2.

Step 1: Plot the 3 averages on a number line, with the averages for N and A on the ends and the average for the mixture in the middle.
N x----------------x+2----------------39 A

Step 2: Calculate the distances between the averages.
N x--------2-------x+2--------37-x------39 A

Step 3: Determine the ratio in the mixture.
The ratio of N to A is equal to the RECIPROCAL of the distances in red.
N/A = (37-x)/ 2

Since there is only ONE added person, A=1.
Substituting A=1 into N/A = (37-x)/2, we get:
N/1 = (37-x)/2
2N = 37-x
x = 37-2N.

Case 2:
Average age for the first N people = x.
Average age for A = 15.
Average for the MIXTURE of all the people = x-1.

Step 1: Plot the 3 averages on a number line, with the averages for A and N on the ends and the average for the mixture in the middle.
A 15----------------x-1----------------x N

Step 2: Calculate the distances between the averages.
A 15--------x-16-------x-1--------1------x N

Step 3: Determine the ratio in the mixture.
The ratio of A to N is equal to the RECIPROCAL of the distances in red.
A/N = 1/(x-16).
N/A = x-16.

Since there is only ONE added person, A=1.
Substituting A=1 into N/A = x-16, we get:
N/1 = x-16
x = N+16.

Since x=N+16 and x=37-2N, the expressions in blue are EQUAL:
N+16 = 37-2N
3N = 21
N = 7.

The correct answer is A.

Mo2men wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Source: Veritas

Not a whole lot different than the other answers, but:

A= original average, therefore nA = total of ages

Adding age 39 person increases average by two means: nA/(n+1) + 39/(n+1) = A+2

Adding age 15 person decreases average by 1 means nA/(n+1) + 15/(n+1) = A-1

Subtract equation two from equation one: 24/(n+1) = 3

Rearrrange: 3n+3 = 24, 3n=21, n=7, A