Data Sufficiency

If the average (arithmetic mean) annual income for a group of people is x and the annual income of one of the members of the group, Joe, increases by 25%, how much does the average income of the whole group increase?

(A) There are 8 people in the group.
(B) The total initial income of the group is 9 times Joes initial income.

[spoiler]oa B[/spoiler]

mariah wrote:If the average (arithmetic mean) annual income for a group of people is x and the annual income of one of the members of the group, Joe, increases by 25%, how much does the average income of the whole group increase?

(A) There are 8 people in the group.
(B) The total initial income of the group is 9 times Joes initial income.

Hi!

We see we're being tested on averages, so let's start by jotting down the average formula:

average = (sum of terms)/(# of terms)

We know that the average is x and that the income of one person increases by 25%. We don't know the number of people in the group or the sum of their incomes.

On to the statements!

(1) # of terms = 8

We don't know how Joe relates to the rest of the group, so this doesn't help us, since we still don't know how much overall impact that 25% increase to Joe will have on the total sum: insufficient.

(2) thinking about the sum of the terms, we can determine that:

9J = J + everyone else

or

8J = sum of everyone except J

Knowing that J went up by 25%, we now know that

sum = 8J + 1.25J
sum = 9.25J

Since the sum went from 9J to 9.25J, we can calculate the % increase in the sum of the terms. However, without the number of terms or the original annual income, there's no way to determine the increase in the group average: insufficient.

We could also set up the following two equations:

x = 9J/#
x + .25J = (9.25)J/#

3 variables, 2 equations, no way to solve for any of the variables.

Together: with the number of people we can determine the percent increase in overall average, but there's no way to determine "how much" the average income of the whole group increases: insufficient, choose (E).

What's the source of this question? There are a number of problems with it:

1) nowhere does it say that the annual incomes of the rest of the group remain constant, and we can't make that assumption. If you notice that fact, you can automatically choose (E).

2) it asks for how much the average income increases, but nowhere is there an actual number. Here's a great tip for DS: if a question asks you to solve for an actual value, you need at least one actual value to solve (i.e. a percent or ratio won't cut it). If you notice that fact, you can automatically choose (E).

3 the OA is listed as (b), but even taking into account the flaws and rewriting the question, there's no way that (b) is sufficient alone.

Tried to think in terms of equations.. which will result in avg2/avg1 = 10.25/9

my solving:

Avg1 = ( X+ J ) / n ... (1)
Avg2 = (X + J + 1.25 J ) / n ... (2)

1) Clearly insufficient.
2)X= 8J

Re-derive 1 & 2
Avg1 = ( 8J + J )/ n => 9J/n
Avg2 = ( 8J + J + 1.25 J) /n => 10.25J/n

Avg2/ Avg1 = 10.25/9
Avg2 = 10.25/9 * Avg1 => Avg1 + 0.13 Avg1 = 13 % increase..

if question is about percentage increase .. B is sufficient.

Combining 1,2 will be the same as above as 1 choice is irrelevant and it cancels out in equation.