Question

The ratio of the number of men to the number of women in a sports club is 5:2.

After 63 men and 12 women quit the club, the ratio of the number of men to the number of women became 7:4.
How many members did the club have in the beginning?

Answer 1

Let the number of men and the number of women in the beginning be x and y

Now,
`\sf \frac{x} {y} =(5)/(2) \\\\\sf</p><p>=2x=5y \\\\\sf</p><p>=x=(5y)/(2)`

Now, In 2nd case,

No. Of men=x-63 and no. Of women =y-12

Now,the ratio becomes
`(x-63)/(y-12)=(7)/(4) \\\\\sf</p><p>=4(x-63)=7(y-12) \\\\\sf</p><p>=4x-252=7y-84 \\\\\sf</p><p>Putting,\:x=(5y)/(2)\:in\: 4x-252=7y-84\:we\: get, \\\\\sf</p><p>10y-252=7y-84 \\\\\sf</p><p>=10y-7y=252-84 \\\\\sf</p><p>=3y=168 \\\\\sf</p><p>=y=56 \\\\\sf</p><p>=\boxed{Y=56}</p><p>`

Now,we have, 2x=5y

Putting the value of Y=56 in 2x=5y,we get,

`\sf 2x=5\:x\:56 \\\\\sf</p><p>x=(5\:x\:56)/(2) \\\\\sf</p><p>x=\frac{5\:x\: \cancel{56}\:\:28}{ \cancel{2}} \\\\\sf</p><p>=x=5\: x\: 28 \\\\\sf</p><p>x=140 \\\\\sf</p><p>\boxed{x=140}</p><p>`

Hence, the number of men was 140 and the no. of women was 56

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