Question

The denominator of a fraction is greater than its numerator by 12, if the numerator is decreased by 2 and the denominator is increased by 7, the new fraction is equivalent with \[\dfrac{1}{2}\]. Find the fraction.

Answer 1

Hint: In this problem, we need to consider the numerator of the fraction as \[x\] and denominator of the fraction as \[y\]. Now, apply the given condition over the fraction to obtain the value of \[x\] and \[y\].

Complete step-by-step answer:
Consider the fraction as \[\dfrac{x}{y}\].
Since, the denominator of the fraction is greater than its numerator by 12, it can be written as follows:
\[y = x + 12\]
Now, the fraction becomes \[\dfrac{x}{{x + 12}}\].
Since, the numerator of the fraction is decreased by 2 and the denominator is increased by 7, the new fraction is equivalent to \[\dfrac{1}{2}\], it can be written as shown below.

\[
  \,\,\,\,\dfrac{{x - 2}}{{\left( {x + 12} \right) + 7}} = \dfrac{1}{2} \\
   \Rightarrow \dfrac{{x - 2}}{{x + 19}} = \dfrac{1}{2} \\
   \Rightarrow 2\left( {x - 2} \right) = x + 19 \\
   \Rightarrow 2x - 4 = x + 19 \\
 \]
Further, simplify the above expression.
\[
  \,\,\,\,2x - x = 19 + 4 \\
   \Rightarrow x = 23 \\
  \]

Substitute 23 for \[x\] in expression \[\dfrac{x}{{x + 12}}\] to obtain the old fraction.
\[
  \,\,\,\,\dfrac{{23}}{{23 + 12}} \\
   \Rightarrow \dfrac{{23}}{{35}} \\
\]
Substitute 23 for \[x\] in expression \[\dfrac{{x - 2}}{{\left( {x + 12} \right) + 7}}\] to obtain the new fraction.

\[
  \,\,\,\,\dfrac{{23 - 2}}{{\left( {23 + 12} \right) + 7}} \\
   \Rightarrow \dfrac{{21}}{{35 + 7}} \\
   \Rightarrow \dfrac{{21}}{{42}} \\
   \Rightarrow \dfrac{1}{2} \\
\]
Thus, the old fraction is \[\dfrac{{23}}{{35}} \] and the new fraction is \[\dfrac{{1}}{{2}} \].

Note: Always, try to consider the fraction in one variable. Apply the given conditions over the old fraction to obtain the value of the variable.