manik11 wrote:If the average (arithmetic mean) annual income for a group of people is x, where x >0, and the annual income of one of the members of the group, Joe, increases by 25%, by what percent does the average income of the whole group increase?

(1) There are 8 people in the group.
(2) The total initial income of the group is 9 times Joe's initial income.

OA : B

Let Joe's initial income = 100.
Increase in Joe's new income = 25% of 100 = 25.

Question stem, rephrased:
What is the value of (new average)/(old average)?

Statement 1:
No way to determine the new average or the old.
INSUFFICIENT.

Statement 2:
Total old income = 9*100 = 900.
After Joe's income increases, new total income = 900+25 = 925.

Case 1: 10 people in the group
Old average = 900/10 = 90.
New average = 925/10 = 92.5
Resulting ratio:
(new average)/(old average) = 92.5/90 = 185/180.

Case 2: 5 people in the group
Old average = 900/5 = 180.
New average = 925/5= 185.
Resulting ratio:
(new average)/(old average) = 185/180.

Since the ratio is THE SAME in each case, SUFFICIENT.

The correct answer is B.

manik11 wrote:If the average (arithmetic mean) annual income for a group of people is x, where x > 0, and the annual income of one of the members of the group, Joe, increases by 25%, by what percent does the average income of the whole group increase?

(1) There are 8 people in the group.
(2) The total initial income of the group is 9 times Joe's initial income.

OA : B

Average Income People In The Group = (Total Income)/(Number Of People)

Statement 1:

Statement 1 tells us the number of members, but nothing about the total income of the members or the relationship between Joe's income and the income of other members.

Joe's income could be 1% of Total Income, in which case a 25% increase in Joe's income would have only a tiny effect on Total Income and Average Income.

Joe's income could be 80% of Total Income, in which case a 25% increase in Joe's income would have a large effect on Total Income and Average Income.

Insufficient.

Statement 2:

Given the information that Joe's initial income represents 1/9 of the total income, we can calculate that a 25% increase in Joe's income will cause Total Income to increase by .25 x 1/9 x Original Total Income.

Original Average Income = Original Total Income/Number Of People

New Average Income = (Original Total Income)(1 + (.25 x 1/9)/Number Of People.

There's no need to actually do the math. We know that we could use the ratio of New Average Income to Original Average Income to calculate the percentage change.

Sufficient.

The correct answer is B.

Another way of thinking about this:

Joe's income = j
Average income = m
Number of people = n

Average = (Sum of incomes)/# of people
m = (Sum)/n
nm = Sum

So the sum of everybody's income is nm.

When Joe's income goes up by 25%, we have a new sum of nm + .25j, and a new average of (nm + .25j)/n.

To find the % increase, we start with (New - Old)/Old:

((nm + .25j)/n - m) =

.25j/(m*n)

S1: n = 8. Now we have .25j/8m, not sufficient.

S2: n*m = 9j. So we have .25j/9j, sufficient!