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A fraction becomes $\frac{5}{6}$ when $3$ is added to both the numerator and the denominator. If $2$ is added to both the numerator and the denominator, the fraction becomes $\frac{9}{11}$. Express the given information algebraically as a system of linear equations in two variables. Hence, find the original fraction.
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Solution
Let numerator be $x$ and denominator be $y$
$\therefore$ fraction is $\frac{x}{y}$ (1/2 Mark)
$\frac{x+3}{y+3} = \frac{5}{6}$ (1 Mark)
$\Rightarrow 6x – 5y = -3$ ---------- (i) (1/2 Mark)
Also $\frac{x+2}{y+2} = \frac{9}{11}$ (1 Mark)
$\Rightarrow 11x – 9y = -4$ ---------- (ii) (1/2 Mark)
Solving (i) and (ii) we get
$x = 7$ and $y = 9$ (1/2+1/2 Mark)
Fraction is $\frac{7}{9}$ (1/2 Mark)
Let numerator be $x$ and denominator be $y$
$\therefore$ fraction is $\frac{x}{y}$ (1/2 Mark)
$\frac{x+3}{y+3} = \frac{5}{6}$ (1 Mark)
$\Rightarrow 6x – 5y = -3$ ---------- (i) (1/2 Mark)
Also $\frac{x+2}{y+2} = \frac{9}{11}$ (1 Mark)
$\Rightarrow 11x – 9y = -4$ ---------- (ii) (1/2 Mark)
Solving (i) and (ii) we get
$x = 7$ and $y = 9$ (1/2+1/2 Mark)
Fraction is $\frac{7}{9}$ (1/2 Mark)