The value of numerators are 3 and 5
Solution:
Let the numerators of two fractions be x and y
Two fractions have denominators 4 and 6. Their sum is 19/12
Therefore,
[tex]\frac{x}{4} + \frac{y}{6} = \frac{19}{12}[/tex]
Cross multiply L.H.S to get simplified
[tex]\frac{6x+4y}{24} = \frac{19}{12}\\\\6x + 4y = 38 ------- eqn 1[/tex]
If the numerators are switched their sum is 7/4
Therefore,
[tex]\frac{y}{4} + \frac{x}{6} = \frac{7}{4}[/tex]
On simplification, we get
[tex]\frac{6y + 4x}{24} = \frac{7}{4}\\\\4x + 6y = 42 -------- eqn 2[/tex]
Solve eqn 1 and eqn 2
Multiply eqn 1 by 4
24x + 16y = 152 --------- eqn 3
Multiply eqn 2 by 6
24x + 36y = 252 ---------- eqn 4
Subtract eqn 3 from eqn 4
24x + 36y = 252
24x + 16y = 152
( - ) -------------------
20y = 100
Substitute y = 5 in eqn 1
6x + 4(5) = 38
6x = 38 - 20
6x = 18
Thus the value of numerators are 3 and 5
