Question

A fraction becomes 9 / 11 if 2 is added to both numerator and denominator . It becomes 5 /3 if 3 is added to both numerator and denominator . What is the fraction

Answer 1

Answer:

$\large Fraction=\dfrac{102}{243}$

Step-by-step explanation:

Let the numerator and denominator of  fraction be x and y respectively.

Fraction = $\dfrac{x}{y}$

In first case when 2 is added to both it become $\dfrac{9}{11}$

$\large \dfrac{x+2}{y+2}=\dfrac{9}{11} \\\\\\\large 11x+22=9y+18\\\\\\\large 11x-9y=-4 ...( i)$

In second case when 3 is added from both it become $\dfrac{5}{3}$

$\large \dfrac{x+3}{y+3}=\dfrac{5}{3}\\\\\\\large 3x+9=5y+15\\\\\\\large 3x-5y=6 \ ...( ii)$

Now multiply by 9 in ( ii )

27 x - 45 y = 54

45 y = 27 x  + 54 .....( iii )

Now multiply by 5 in ( i )

55 x - 45 y = - 20

45 y = 55 x + 20 ....( iv )

From ( iii ) and  ( iv ) we have

27 x + 54 = 55 x + 20

55 x - 27 x = 54 - 20

28 x = 34

$\large x=\dfrac{17}{14}$

Now put   $\large x=\dfrac{17}{14}$  in ( i )

11 x - 9 y = - 4

9 y = 11 x + 4

$9 y = 9y=11\times\dfrac{17}{14}+4\\\\\\9y=\dfrac{187+56}{2}\\\\\\9y=\dfrac{243}{14}\\\\\\y=\dfrac{243}{126}$

Thus we get  $\large x=\dfrac{17}{14} \ and \ \dfrac{243}{126}$

Decimal values of x = 1.214 and y = 1.929 that shown in graph.

It show that values of x and y is correct.

Thus we get

 $\large \text{ Fraction=\dfrac{\frac{17}{14} }{\frac{243}{126} } }\\\\\\\large Fraction=\dfrac{17\times126}{14\times243} \\\\\\\large Fraction=\dfrac{17\times9}{243}\\\\\\\large Fraction=\dfrac{102}{243}$