In a recent survey of $100$ women, the following information was gathered: $56$ use shampoo A. $51$ use shampoo B. $35$ use shampoo C. $23$ use shampoos A and B. $19$ use shampoos A and C. $14$ use shampoos B and C. $12$ use all three. How many are using all three shampoos (Region VII)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question.
A) 23 women
B) 19 women
C) 12 women
D) 14 women
E) 56 women

Question

In a recent survey of $100$ women, the following information was gathered: $56$ use shampoo A. $51$ use shampoo B. $35$ use shampoo C. $23$ use shampoos A and B. $19$ use shampoos A and C. $14$ use shampoos B and C. $12$ use all three. How many are using all three shampoos (Region VII)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question. 
A) 23 women 
B) 19 women 
C) 12 women 
D) 14 women 
E) 56 women

Quizplus · Accepted Answer

To find the number of women who use all three shampoos (Region VII), we can add up the number of women in the overlapping regions of A, B, and C: 
- region II (women who use shampoos A and B but not C): 23 
- region V (women who use shampoos A and C but not B): 19 
- region VI (women who use shampoos B and C but not A): 14 
- region VII (women who use shampoos A, B, and C): x (what we're solving for)

We know that the total number of women surveyed is 100. So we can set up an equation:

56 (women who use shampoo A) + 51 (women who use shampoo B) + 35 (women who use shampoo C) - region II - region V - region VI - region VII + x = 100

Simplifying:

56 + 51 + 35 - 23 - 19 - 14 - x + x = 100

We can simplify further:

86 - 56 = x

30 = x

Therefore, 30 women use shampoos A, B, and C. But we're only interested in the number of women in Region VII, which is the intersection of A, B, and C. Looking at the figure, we see that this region is equal in size to the intersection of II, V, and VI. So, the answer is the same as the size of the smallest overlapping region, which is Region VII:

Answer: C) 12 women

In a Recent Survey Of 100100100 Women, the Following Information Was Gathered

In a Recent Survey Of $100$ Women, the Following Information Was Gathered