Answer:
original fraction = [tex]\frac{7}{9}[/tex]
Step-by-step explanation:
let x be the numerator and y be the denominator of the original fraction, then
[tex]\frac{x+1}{y+1}[/tex] = [tex]\frac{4}{5}[/tex] ( cross- multiply )
5(x + 1) = 4(y + 1) ← distribute parenthesis on both sides
5x + 5 = 4y + 4 ( subtract 4y from both sides )
5x - 4y + 5 = 4 ( subtract 5 from both sides )
5x - 4y = - 1 → (1)
Also
[tex]\frac{x-5}{y-5}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2(x - 5) = y - 5
2x - 10 = y - 5 ( subtract y from both sides )
2x - y - 10 = - 5 ( add 10 to both sides )
2x - y = 5 → (2)
Solve the 2 equations simultaneously for x and y
Multiply (2) by - 4
- 8x + 4y = - 20 → (3)
Add (1) and (3) term by term to eliminate y , that is
- 3x = - 21 ( divide both sides by - 3 )
x = 7
Substitute x = 7 into either of the 2 equations and solve for y
Substituting into (1)
5(7) - 4y = - 1
35 - 4y = - 1 ( subtract 35 from both sides )
- 4y = - 36 ( divide both sides by - 4 )
y = 9
Thus the original fraction = [tex]\frac{7}{9}[/tex]
