Question

The sum of numerator & denominator at
a fraction is 18. If the denominator increased by 2
the fraction reduces to 1/3.Find the Fraction​

Answer 1

Answer:

$\boxed{\red{\bf Fraction = \dfrac{5}{13}}}$

Step-by-step explanation:

Given that :

• The sum of numerator and denominator of a fraction is 18.
• If the denominator increased by 2
• the fraction reduces to 1/3.

To find :

• The fraction.

Solution :

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

★ $\boxed{\bf Original \: fraction = \dfrac{x}{y}}$

According to the question,

The sum of numerator and denominator of the fraction is 18.

$\implies \sf x + y = 18 \\\\\implies \sf x = 18 - y \: \: \: \dots (i)$

Also,

If the denominator increased by 2

the fraction reduces to 1/3.

Equation :

$\implies \sf \frac{x}{y+2} = \frac{1}{3} \\\\\implies \sf 3x = y + 2 \\\\\implies \sf 3(18-y) = y + 2$

[ Using equation (i) ]

$\sf \implies 54 - 3y = y + 2\\\\\implies \sf - 3y - y = 2 - 54 \\\\\implies \sf - 4y = - 52 \\\\\implies \sf y = \frac{-52}{-4} \\\\\implies \sf y = 13$

Now, substituting the value of y in (i),

$\sf \implies x = 18 - y \\\\\implies \sf x = 18 - 13 \\\\\implies \sf x = 5$

Thus, original fraction = $\sf \dfrac{x}{y}$

                                    = $\sf \dfrac{5}{13}$

Hence, the original fraction is 5/13.

Answer 2

${\sf{\underline{\underline{\pink{Solution:}}}}}$

$\boxed{\pink{\bf Fraction = \dfrac{5}{13}}}$

Step-by-step explanation:

$\sf\purple{Given \ that:}$

• $\sf\orange{The \ sum \ of \ numerator \ and \ denominator \ of \ a \ fraction \ is \ 18.}$

• $\sf\green{If \ the \ denominator\ increased \ by \ 2}$

• $\sf\green{The \ fraction \ reduces \ to}$ $\frac{1}{3}$

$\sf\pink{To \ find :}$

$\sf\green{The \ fraction}$

${\sf{\underline{\underline{\pink{Answer:-}}}}}$

$\sf\pink{Let \ the \ numerator \ and \ denominator \ of \ the \ fraction \ be x \ and \ y \ respectively.}$

$★ \boxed{\bf Original \:fraction = \dfrac{x}{y}}$

$\sf\green{According \ to \ the \ question,}$

$\sf\red{The \ sum \ of \ numerator \ and \ denominator \ of \ the \ fraction \ is \ 18.}$

$\begin{gathered}\implies \sf x + y = 18 \\\\\implies \sf x = 18 - y \: \: \: \dots (i)\end{gathered}$

$\sf\green{Also,}$

$\sf\green{If \ the \ denominator \ increased \ by \ 2}$

$\sf\pink{the \ fraction \ reduces \ to}$ $\frac{1}{3}$ .

$\sf\green{Equation :}$

$\begin{gathered}\implies \sf \frac{x}{y+2} = \frac{1}{3} \\\\\implies \sf 3x = y + 2 \\\\\implies \sf 3(18-y) = y + 2\end{gathered}$

$\sf\green{[ Using \ equation \ (i) ]}$

$\begin{gathered}\sf \implies 54 - 3y = y + 2\\\\\implies \sf - 3y - y = 2 - 54 \\\\\implies \sf - 4y = - 52 \\\\\implies \sf y = \frac{-52}{-4} \\\\\implies \sf y = 13\end{gathered}$

$\sf\green{Now, \ substituting \ the \ value \ of \ y \ in \ (i),}$

$\begin{gathered}\sf \implies x = 18 - y \\\\\implies \sf x = 18 - 13 \\\\\implies \sf x = 5\end{gathered}$

$\sf\green{Thus, \ original \ fraction \ =}$ $\sf \dfrac{x}{y}$

= $\sf \dfrac{5}{13}$

$\sf\pink{Hence, \ the \ original \ fraction \ is}$ $\frac{5}{13.}$