Question : In a party hall, there are people in blue and red dresses. The ratio of the number of men in blue to the number of women in red is 3 : 7. The ratio of men in red to the number of women in blue is 2 : 1. If the ratio of the number of people in blue to the number of people in red is 35 : 76, then what is the ratio of number of men to the number of women?
Option 1: 47 : 51
Option 2: 40 : 43
Option 3: 62 : 57
Option 4: 52 : 59
Correct Answer: 52 : 59
Solution :
Given: (Men in blue) : (Women in red) = 3 : 7
(Men in red) : (Women in blue) = 2 : 1
(People in blue) : (People in red) = 35 : 76
Let the men in blue and women in red be 3$x$ and 7$x$, respectively.
Let the men in red and women in blue be 2$y$ and $y$, respectively.
According to the question,
$\frac{3x+y}{7x+2y}=\frac{35}{76}$
⇒ $245x+70y=228x+76y$
⇒ $\frac{x}{y}=\frac{6}{17}$
Let $x$ = 6k and $y$ = 17k
Number of men = 3$x$ + 2$y$ = (3 × 6k) + (2 × 17k) = 18k + 34k = 52k
Number of women = 7$x$ + $y$ = (7 × 6k) + (17k) = 59k
So, the ratio of the number of men to the number of women = 52 : 59
Hence, the correct answer is 52 : 59.
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