70% of stadium visitors are men. 1/7 of men and 1/5 of women at the stadium are fans of visiting team. If every person at the stadium is a fan of one of the playing teams and there are 6300 fans of home team overall then how many women are there?
A. 450 B. 1800 C. 2250 D. 5250 E. 11800
(C) Let's count the fraction of people supporting the visiting team. 70% of the stadium visitors are men, so 30% are women. We know what part of the men (1/7) and what part of women (1/5) support the visiting team, so we can find the fraction of the visiting team's fans out of the total: (7/10) × (1/7) + (3/10) × (1/5) = 1/10 + 3/50 = 5/50 + 3/50 = 8/50 = 4/25. If 4/25 of the fans support the visiting team, then 1 – 4/25 = 21/25 support the home team. (21/25) × (people at the stadium) = 6300 (people at the stadium) = 6300 × 25/21 = 7500 Women are 30% of the visitors, 7500 × 30% = 2250.
The correct answer is choice (C).
The alternative explanation: Instead of dealing with fractions, we can deal with percentages.
(C) Let's count the percentage of people supporting the visiting team. 70% of the stadium visitors are men, so 30% are women. We know what part of the men (1/7) and what part of women (1/5) support the visiting team, so we can find the percentage of the visiting team's fans out of the total: 70% × (1/7) + 30% × (1/5) = 10% + 6% = 16%. If 16% of the fans support the visiting team, then 100% – 16% = 84% support the home team. (84/100) × (people at the stadium) = 6300 (people at the stadium) = 6300 × 100 / 84 = 7500 30% are women: 7500 × 30 / 100 = 2250. The correct answer is choice (C).
The alternative explanation #2: You can consider 100% to be 100 persons.
Suppose there were 100 visitors, in which case there are 70 men and 30 women. 10 men support the visiting team and 60 men support the home team. Following through, 6 women support the visiting team and 24 women support the home team. This implies that 84% of the total number visiting the stadium are 6300 which sets up the equation: (84/100) × P =6300, where P is the actual total number visiting the stadium. Solving this equation gives P = (6300 x 100)/84 =7500 and 30% of this value is the number of women in the stadium which works out to 2250.
