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 1

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

• 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 $\bf{\frac{7}{15}}$

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Answer 2

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 ,

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

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♤ 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,

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

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