Math
posted by Surya on .
There is a ratio of number of boys to girls 3:5. After adding 5 boys and 3 girls, the ratio become 5:7. How many girls were there at first?

b = numbers of boys
g = numbers of girls
Ratio of boys to girls:
b / g = 3 / 5 Multiply both sides by 5
5 b / g = 3 Multiply both sides by g
5 b = 3 g Divide both sides by 5
b = 3 g / 5
After adding ratio is :
( b + 5 ) / ( g + 3 ) = 5 / 7
[ ( 3 g / 5 ) + 5 ] / ( g + 3 ) = 5 / 7 Multiply both sides by 7
7 [ ( 3 g / 5 ) + 5 ] / ( g + 3 ) = 5 Multiply both sides by ( g + 3 )
7 [ ( 3 g / 5 ) + 5 ] = 5 ( g + 3 )
7 * 3 g / 5 + 7 * 5 = 5 ( g + 3 )
21 g / 5 + 35 = 5 ( g + 3 ) Multiply both sides by 5
21 g + 35 * 5 = 5 * 5 ( g + 3 )
21 g + 175 = 25 ( g + 3 )
21 g + 175 = 25 g + 25 * 3
21 g + 175 = 25 g + 75
175  75 = 25 g  21 g
100 = 4 g
4 g = 100 Divide both sides by 4
g = 100 / 4
g = 25
b = 3 g / 5
b = 3 * 25 / 5
b = 75 / 5
b = 15
Checking :
b / g = 15 / 25 =
( 5 * 3 ) / ( 5 * 5 ) = 3 / 5
( b + 5 ) / ( g + 3 ) =
( 15 + 5 ) / ( 25 + 3 ) =
20 / 28 =
( 4 * 5 ) / ( 4 * 7 ) = 5 / 7 
Initial:
number of boys to girls 3:5
=3U:5U
U means unit
Later (3U +5): (5U +3) = 5:7
3U+5 5
 = 
5U+3 7
Cross multiply
21U+35 = 25U+15
3515 = 25U21U
20 = 4U
Therefore 1U =5
Initially girls are 5Units.
So the answer is 25. 
My answer is also numbers of girls
g = 25 
Thanks a lot Bosnian.
Answer is exactly correct. Just wanted to provide the alternate method which got through Maths olympiad book.