Math, asked by rishabh1202kumar, 3 months ago

The denominator of a fraction is one more than twice its numerator if the numerator and denominator are both increased by 5 it will become 3/5 find the original fraction

Answers

Answered by CɛƖɛxtríα
50

{\underline{\underline{\bf{Given:}}}}

  • The denominator of a fraction is one more than twice its numerator.
  • If both the numerator and the denominator increases by 5, the fraction will become \large{\sf{\frac{3}{5}}}.

{\underline{\underline{\bf{Need\:to\:find:}}}}

  • The original fraction (the original values of numerator and denominator).

{\underline{\underline{\bf{Solution:}}}}

Let the numerator be '\sf{x}'.

According to the question,

  • The original fraction will be \large{\sf{\frac{x}{2x+1}}}
  • If both the numerator and the denominator increases by 5, we can form an equation,

\:\:\:\:\:\:\:\:\:\:\:\:\small{\boxed{\sf{\purple{ \frac{x + 5}{2x + 1 + 5}=\frac{3}{5} }}}}

Now, solving the equation gives the value of \sf{x}.

\longrightarrow{\sf{\frac{x + 5}{2x + 1 + 5}=\frac{3}{5}}}

\longrightarrow{\sf{\frac{x+5}{2x+6}=\frac{3}{5}}}\sf{\:\:\:\:\:\:\:\:(Cross\: multiplying)}

\longrightarrow{\sf{5(x+5)=3(2x+6)}}

\longrightarrow{\sf{5x+25=6x+18}}

\longrightarrow{\sf{5x-6x=18-25}}

\longrightarrow{\sf{-1x=-7}}

\longrightarrow{\sf{x=\frac{\cancel{-}\:7}{\cancel{-}\:1}}}

\longrightarrow{\underline{\underline{\sf{\red{x=7}}}}}

{\underline{\underline{\bf{Verification:}}}}

\longrightarrow Substitute 7 in places of \sf{x} in the equation formed.

\longrightarrow{\sf{\frac{x + 5}{2x + 1 + 5}=\frac{3}{5}}}

\longrightarrow{\sf{\frac{\red{7}+5}{2\times \red{7} + 1 + 5} = \frac{3}{5}}}

\longrightarrow{\sf{\frac{12}{14+6}=\frac{3}{5}}}

\longrightarrow{\sf{\frac{\cancel{12}}{\cancel{20}}=\frac{3}{5}}}

\longrightarrow{\sf{\frac{3}{5}=\frac{3}{5}}}

\longrightarrow{\sf{L.H.S=R.H.S}}

\longrightarrow Hence, the value of \sf{x} is correct.

Finally, by putting the value of \sf{x} in the expression of original fraction gives the required answer.

\:\:\:\:\:\:\:\:\implies{\sf{\frac{x}{2x+1}}}

\:\:\:\:\:\:\:\:\implies{\sf{\frac{\red{7}}{2\times \red{7}+1}}}

\:\:\:\:\:\:\:\:\implies{\sf{\frac{7}{14+1}}}

\:\:\:\:\:\:\:\:\implies{\boxed{\sf{\purple{\frac{7}{15}}}}}

{\underline{\underline{\bf{Required\: answer:}}}}

  • Therefore, the original fraction is \bf{\frac{7}{15}}

_________________________________________


Anonymous: Awesome Answer !
CɛƖɛxtríα: Thnk uh..!! ^▽^
Anonymous: Adorable :D !
CɛƖɛxtríα: Thnkuu ^^
Answered by Anonymous
4

Correct Question-:

  • The denominator of a fraction is one more than twice its numerator if the numerator and denominator are both increased by 5 it will become 3/5 . Find the original fraction .

AnswEr-:

  • \underline{\boxed{\star{\sf{\blue{ The\:Original \:Fraction= \frac {7}{15}}}}}}

Explanation -:

 \frak{Given\:-:}\begin{cases} \sf{The \:denominator\: of \:a \:fraction\: is\: one\: more\: than\: twice\: its\: numerator.} & \\\\ \sf{ If\: the\: numerator \:and\: denominator\: are \:both\: increased \:by \:5 \:it\: will\: become \:3/5 } \end{cases}\\\\

 \frak{To\:Find-:}\begin{cases} & \sf{The\:Original\: fraction\:with\:numerator \:and\:Denominator  .} \end{cases}\\\\

Solution-:

  • Let the numerator be x .

Then ,

  • The denominator of a fraction is one more than twice its numerator.

Now ,

  • The Denominator = 2x + 1

Then ,

  • The original fraction = x / 2x + 1

 \frak{According \:To\:The\:Question \:-:}\begin{cases} & \sf{  If\: the\: numerator \:and\: denominator\: are \:both\: increased \:by \:5 \:it\: will\: become \:3/5.} \end{cases}\\\\

Then ,

  • x + 5 / 2x + 1 + 5 = 3/5

Now , Solving for the Value of X .

  • \implies{\sf{\large {\frac {x+5}{2x+1+5}=\frac{3}{5}}}}
  • \implies{\sf{\large {\frac {x+5}{2x+6}=\frac{3}{5}}}} _______________[ Cross ✝️ Multiplication]
  • \implies{\sf{\large { 5 (x +5) = 3(2x +6)}}}
  • \implies{\sf{\large { 5x +25 = 6x +18}}}
  • \implies{\sf{\large { 5x -6x = 18- 25}}}
  • \implies{\sf{\large { -x = -7}}}__________[ - Sign will be eliminated from both side ]
  • \implies{\sf{\large { x = 7}}}

Therefore ,

  • \underline{\boxed{\star{\sf{\blue{ x = 7 }}}}}

Now ,

  • Numerator = x = 7
  • Denominator = 2x + 1 = 7×2 + 1 = 14 + 1= 15

Hence ,

  • \underline{\boxed{\star{\sf{\blue{ The\:Original \:Fraction= \frac {7}{15}}}}}}

_____________________________________

♤ Verification ♤

  • The original fraction = x / 2x + 1

 \frak{According \:To\:The\:Question \:-:}\begin{cases} & \sf{  If\: the\: numerator \:and\: denominator\: are \:both\: increased \:by \:5 \:it\: will\: become \:3/5.} \end{cases}\\\\

Then ,

  • x + 5 / 2x + 1 + 5 = 3/5

Now ,

  • \underline{\boxed{\star{\sf{\blue{ x = 7 }}}}}

By Substituting Value -:

  • \implies{\sf{\large {\frac {7+5}{2×7+1+5}=\frac{3}{5}}}}
  • \implies{\sf{\large {\frac {12}{14+1+5}=\frac{3}{5}}}}
  • \implies{\sf{\large {\frac {12}{15+5}=\frac{3}{5}}}}
  • \implies{\sf{\large {\frac {12}{20}=\frac{3}{5}}}} [ 12 /20 is devided by 4 ]
  • \implies{\sf{\large {\frac {3}{5}=\frac{3}{5}}}}

Therefore,

  • \underline{\boxed{\star{\sf{\blue{ LHS = RHS  }}}}}
  • \underline{\boxed{\star{\sf{\blue{ Hence , \: Verified }}}}}

_________________♡_______________


Anonymous: Impressive ! :D
Anonymous: Thankiew!
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