Overlapping Sets In An Interesting GMAT, GRE Quant Problem

There are 750 male and female participants in a meeting. Half of the female participants and one-quarter of the male participants are Democrats. One-third of all the participants are Democrats. How many of the Democrats are female?

(A) 75
(B) 100
(C) 125
(D) 175
(E) 225





Lets make an equation to represent this.

We'll make m the number of male participants and f the number of female participants in the meeting. The total number of participants is 750, although it's phrased in an obscure way. 

m + f = 750

It's a very simple equation, but you'll notice that there are two variables. When we have two variables we need two equations. 

Also, we have males and females for each group listed. So we have that half the total female participants and one-quarter of the total male participants are Democrats. Let

Let's let d equal the number of the Democrats. Then we have the equation f/2 + m/4 = d

With a third variable, we simply need a third equation. We have that one-third of the total participants are Democrats. Hence, we have the equation d = 750/3 = 250

Solving the three equations yields the solution f = 250, m = 500, and d = 250. The number of female democratic participants equals half the female participants equals 250/2 = 125. 

C.

Refresher on "Counting" Sets

How many integers are there between 49 and 101, inclusive?

(A) 50 

(B) 51 

(C) 52 

(D) 53 

(E) 54

If the number of integers between 49 and 101 inclusive is (101 – 49) + 1 = 53. (Add one when they mention "inclusive" to add in the number we are starting with). 

Let's look at this another way. Let's choose smaller numbers: 9 and 11. The difference between 9 and 11 is 2. But there are three numbers between them inclusively—9, 10, and 11—one more than their difference.

D.  

A drum contains 3 to 5 jars each of which contains 30 to 40 marbles. If 10 percent of the marbles are flawed, then what is the greatest possible number of flawed marbles in the drum?

(A) 51
(B) 40 

(C) 30 

(D) 20 

(E) 12



There are, at most, 5 jars each of which contains at most 40 marbles. Then by counting, there are at most 5 40 = 200 marbles in the drum. Since 10 percent of the marbles are flawed, there is at most 20 = 10% 200 flawed marbles.

D.