Question

The sum of numerator and denominator of fraction is 4 more than twice the numerator if numerator and denominator are increase by 3 they are in ratio 2:3 determine the fraction

Answer 1

$\bold{let : } \\ \\ numerator \: = x \\ \\ denominator \: = \: y \\ \\ the \: fraction : \bold{ \frac{x}{y}} \\ \\ \\ \\ \bold{by \: given \: conditions : }$$\\ \\ x \: + \: y \: = 4 + 2x \\ \\ x - 2x + y = 4 \\ \\ - x + y = 4 \\ \\\bold{y = 4 + x}\\ \\ \\ \\\bold{ by \: second \: condition : } \\ \\ \frac{x + 3}{y + 3} = \frac{2}{3} \\ \\ (x + 3) \times 3 = (y + 3) \times 2$$3x + 3 \times 3 = 2y + 3 \times 2 \\ \\ 3x + 9 = 2y + 6 \\ \\ 3x - 2y = 6 - 9 \\ \\ \bold{3x - 2y = - 3} \\ \\ \\ \\ \bold{put \: y \: = 4 + x \: in \: this \: equation : }$$\\ \\ 3x - 2(4 + x) = - 3 \\ \\ 3x - 8 - 2x = - 3 \\ \\ x - 8 = - 3 \\ \\ x = - 3 + 8 \\ \\ \bold{x = 5 }\\ \\ \\ \\ \bold{put \: x = 5 \: in \: any \: equation : }$$\\ \\ y = 4 + x \\ \\ y = 4 + 5 \\ \\ \bold{y = 9} \\ \\ \\ \\ \bold {the\: fraction\: is\: : }\\ \\ \bold {\: 5\: /\: 9}$

Answer 2

Let Numerator=x
Let Denominator=y
Sum of Numerator and Denominator=2×numerator+4

x+y=2x+4
y=2x-x+4
y=x+4 (Equation 1)

Now if
Numerator increase by 3
Now Numerator=x+3

Now if
Denominator increase by 3
Now Denominator=y+3

Then Fraction will be=2:3=2/3
A/Q
$\frac{x + 3}{y + 3} = \frac{2}{3}$
Now y=x+4 put value .Now Fraction
$\frac{x + 3}{x + 4 + 3} = \frac{2}{3} \\ = \frac{x + 3}{x + 7} = \frac{2}{3} \\ = 3(x + 3) = 2(x + 7) \\ = 3x + 9 = 2x + 14 \\ 3x - 2x = 14 - 9 \\ x = 5$
Now We get x=5
And We know
y=x+4
y=5+4
y=9

Now Fraction=x/y
=5/9
$\boxed{\mathbf{Fraction=\frac{5}{9}}}$