a. The denominator of a fraction is greater than its numerator by 9. If 7 is subtracted
from both, its numerator and denomina

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1. [tex]\huge\bf\underline\mathfrak{Corrected \: Question :}[/tex]

   The denominator of a fraction is greater than it’s numerator by 9. If 7 is subtracted from both it’s numberator and denominator, the new fraction becomes 2/3. Find the original fraction.

   [tex]\huge\bf\underline\mathfrak{Answer :}[/tex]

   • [tex]\text {The required fraction is 25/34 ✓} [/tex].

   [tex]\huge\bf\underline\mathfrak{Concept :}[/tex]

   • Solving simple equations with opening brackets and with the help of cross-multiplication method.

   [tex]\huge\bf\underline\mathfrak{Step \: by \: step \: explanation :}[/tex]

   [tex]\huge\bf\underline\mathfrak{Given :}[/tex]

   • [tex]\text {Denominator is greater than the numerator by 9.}[/tex]
   • The fraction becomes 2/3 if 7 is subtracted from the both numerator and denominator.

   [tex]\huge\bf\underline\mathfrak{To \: find :}[/tex]

   • [tex]\text {Original fraction.}[/tex]

   [tex]\huge\bf\underline\mathfrak{Solution :}[/tex]

   [tex]\underline\text {From the given conditions :-}[/tex]

   [tex]\text {Let the original fraction be in the form of p/q.}[/tex]

   [tex]\text {Let numerator be p.}[/tex]

   [tex]\implies\text {Denominator of the fraction = p+9.}[/tex]

   • Since, it is said that denominator is greater than numerator by 9, we have taken denominator as p+9.

   [tex]\bf\underline {Now, \: according \: to \: question :-}[/tex]

   [tex]\underline\text {It is given that :-}[/tex]

   • When 7 is subtracted from both numerator and denominator, then the fraction becomes 2/3.

   [tex]\implies\bf\ \frac{( \: p – 7 \: )}{( \: p \: + \: 9 \: ) – 7} = \frac{2}{3} [/tex]

   • Opening brackets.

   [tex]\implies\bf\ \frac{p – 7}{p + 9 – 7} = \frac{2}{3} [/tex]

   [tex]\implies\bf\ \frac{p – 7}{p + 2} = \frac{2}{3} [/tex]

   • On cross-multiplying.

   [tex]\implies\bf\ 3(p – 7) = 2(p + 2)[/tex]

   [tex]\implies\bf\ 3p – 21 = 2p + 4[/tex]

   • Moving all the variable in the left hand side and all the variables on the right hand side.

   [tex]\implies\bf\ 3p – 2p = 4 + 21[/tex]

   [tex]\implies\bf\boxed {p = 25}[/tex]

   [tex]\underline\text { Putting the value of p in the fraction :-}[/tex]

   [tex]\bf\text {Numerator = p = 25, and}[/tex]

   [tex]\bf\text {Denominator = p+9 = 25+9 = 34}[/tex]

   [tex]\underline\mathfrak {∴ \: Required \: Fraction = 25/34}[/tex]

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