Question

If the numerator of a fraction is decreased by 1, its value becomes 2/3, but if the denominater is increased by 5, its value becomes 1/2. Find the fraction? ​

Answer 1

Answer:

⁷/₉

Step-by-step explanation:

As per the provided information in the given question, we have :

• If the numerator of a fraction is decreased by 1, its value becomes ⅔.
• But, if the denominator is increased by 5, its value becomes ½.

We are asked to calculate the fraction.

Let us assume the numerator as x and denominator of the fraction as y.

― According to the question, if the numerator of a fraction is decreased by 1, its value becomes ⅔. Writing it in the form of an equation,

E⠀Q⠀U⠀A⠀T⠀I⠀O⠀N⠀( 1 ) :

$\longrightarrow \sf{\quad { \dfrac{x -1}{y} = \dfrac{2}{3} }}$

Also, if the denominator is increased by 5, its value becomes ½.

E⠀Q⠀U⠀A⠀T⠀I⠀O⠀N⠀( 2 ) :

$\longrightarrow \sf{\quad { \dfrac{x }{y + 5} = \dfrac{1}{2} }}$

Working out in the equation ( 2 ), we'll write the value of x in the terms of y. From the equation ( 2 ) :

$\longrightarrow \sf{\quad { x = \dfrac{1}{2} (y + 5) }}$

Here, we just transposed (x + 5) from LHS to RHS. Now, x can be written as,

$\longrightarrow \quad \boxed{\sf{ x = \dfrac{y + 5}{2} }}$

Now, substitute the value of x in the equation ( 1 ) to find the value of y.

$\longrightarrow \sf{\quad { \dfrac{x -1}{y} = \dfrac{2}{3} }}$

Substitute the value of x.

$\longrightarrow \sf{\quad { \dfrac{ \Bigg ( \cfrac{y+5}{2} \Bigg ) -1}{y} = \dfrac{2}{3} }}$

Remove the brackets.

$\longrightarrow \sf{\quad { \dfrac{ \cfrac{y+5}{2} -1}{y} = \dfrac{2}{3} }}$

Now take the LCM in LHS, and perform subtraction.

$\longrightarrow \sf{\quad { \dfrac{ \cfrac{y+5-2}{2} }{y} = \dfrac{2}{3} }}$

Performing subtraction in LHS.

$\longrightarrow \sf{\quad { \dfrac{ \cfrac{y+3}{2} }{y} = \dfrac{2}{3} }}$

Now, transpose y in the denominator in LHS to RHS.

$\longrightarrow \sf{\quad { \dfrac{y+3}{2} = \dfrac{2}{3}(y) }}$

Performing multiplication in RHS.

$\longrightarrow \sf{\quad { \dfrac{y+3}{2} = \dfrac{2y}{3} }}$

Now by cross multiplication,

$\longrightarrow \sf{\quad { 3(y+3) = 2y(2) }}$

Performing multiplication.

$\longrightarrow \sf{\quad { 3y+9 = 4y}}$

Now, transpose 3y from LHS to RHS.

$\longrightarrow \sf{\quad { 9 = 4y - 3y}}$

Performing subtraction in RHS.

$\longrightarrow \quad { \textbf{\textsf{ 9 = y }}}$

Now, substitute the value of y in the equation equation ( 2 ). We have,

$\longrightarrow \quad \sf{ x = \dfrac{y + 5}{2} }$

Substitute the value of y.

$\longrightarrow \quad \sf{ x = \dfrac{9 + 5}{2} }$

Performing addition in the numerator.

$\longrightarrow \quad \sf{ x = \cancel{\dfrac{14}{2}} }$

Dividing 14 by 2.

$\longrightarrow \quad { \textbf{\textsf{ x = 7 }}}$

Now, as we have assumed x as the numerator and y as the denominator. So,

$\longrightarrow \sf{\quad {Fraction = \dfrac{x}{y} }}$

Substitute the value of x and y.

$\longrightarrow \quad \underline{ \boxed{\textbf{\textsf{ Fraction}} = \dfrac{\textbf{\textsf{7}}}{\textbf{\textsf{9}}} }}$

$\underline{ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$

Therefore,

⠀⠀⠀⠀⠀⠀★ Fraction = ⁷/₉