F.2 mathz

1. The denominator of a fraction is 7 more than the numerator. if 5 is added to both the numerator and the denominator, the resulting fraction is 1/2. find the original fraction.
Let x be the numerator, y be the denominator
x + 7 = y..............................(1)
( x+5 ) / ( y+5 ) = 1 / 2.........(2)
From (2),
2( x+5 ) = 1( y+5 )
2x + 10 = y + 5
2x + 5 = y..............................(3)
Substitute (3) into (1)
x + 7 = 2x +5
x = 2 //
Substitute x = 2 into (1)
y = 2 + 7
y = 9 //
The original fraction is 2/9.
2. If the numerator and the denominator of a fraction are both decreased by 1, the resulting fraction is 1/2. 2/3 will be obtained if both the numerator and denominator are increased by 2, find the original number
Let x be the numerator, y be the denominator
( x-1 ) / ( y-1 ) = 1/2....................(1)
( x+2 ) / ( y+2 ) = 2/3.................(2)
From (1),
2( x-1 ) = 1( y-1 )
2x - 2 = y - 1
2x - 1 = y
4x - 2 = 2y......................................(3)
From (2),
3( x+2 ) = 2( y+2 )
3x + 6 = 2y + 4
3x + 2 = 2y......................................(4)
Substitute (3) into (4),
4x - 2 = 3x + 2
4x = 3x + 4
x = 4 //
Substitute x = 4 into (1)
2y = 12 + 2
2y = 14
y = 7 //
The original fraction is 4/7