Question

The denominator of a fraction is greater than its numerator by 2. If 4 is added to the numerator and 4 to the denominator, the new fraction is equivalent to \[\dfrac{1}{2}\]. Find the fraction.

Answer 1

Hint:
Here, we need to find the fraction. We will assume the numerator and the denominator of the fraction to be \[x\] and \[y\] respectively. We will use the given information to form two linear equations in two variables. We will solve these equations to find the values of \[x\] and \[y\], and use these values to find the fraction.

Complete step by step solution:
Let us assume the numerator and the denominator of the fraction be \[x\] and \[y\] respectively.
Therefore, we get the fraction as \[\dfrac{x}{y}\].
Now, it is given that the denominator is 2 more than its numerator.
Thus, we get
\[ \Rightarrow y = x + 2 \ldots \ldots \ldots \left( 1 \right)\]
The fraction obtained by increasing the numerator and denominator by 4 is equivalent to \[\dfrac{1}{2}\].
Thus, we get
\[ \Rightarrow \dfrac{{x + 4}}{{y + 4}} = \dfrac{1}{2}\]
Multiplying both sides by \[y + 4\], we get
\[\begin{array}{l} \Rightarrow \left( {\dfrac{{x + 4}}{{y + 4}}} \right)\left( {y + 4} \right) = \dfrac{1}{2}\left( {y + 4} \right)\\ \Rightarrow x + 4 = \dfrac{1}{2}y + 2\end{array}\]
Rewriting the equation, we get
\[ \Rightarrow x - \dfrac{1}{2}y = 2 - 4\]
Subtracting the terms, we get
\[ \Rightarrow x - \dfrac{1}{2}y = - 2 \ldots \ldots \ldots \left( 2 \right)\]
We can observe that the equations \[\left( 1 \right)\] and \[\left( 2 \right)\] are a pair of linear equations in two variables.
We will solve the equations to find the values of \[x\] and \[y\].
Substituting \[y = x + 2\] from equation \[\left( 1 \right)\] in equation \[\left( 2 \right)\], we get
\[ \Rightarrow x - \dfrac{1}{2}\left( {x + 2} \right) = - 2\]
Multiplying the terms, we get
\[ \Rightarrow x - \dfrac{1}{2}x - 1 = - 2\]
Subtracting the like terms, we get
\[ \Rightarrow \dfrac{1}{2}x - 1 = - 2\]
Adding 1 to both sides, we get
\[\begin{array}{l} \Rightarrow \dfrac{1}{2}x - 1 + 1 = - 2 + 1\\ \Rightarrow \dfrac{1}{2}x = - 1\end{array}\]
Multiplying both sides by 2, we get
\[\begin{array}{l} \Rightarrow \dfrac{1}{2}x \times 2 = - 1 \times 2\\ \Rightarrow x = - 2\end{array}\]
Substituting \[x = - 2\] in the equation \[\left( 1 \right)\], we get
\[ \Rightarrow y = - 2 + 2\]
Thus, we get
\[ \Rightarrow y = 0\]
Now, we will use the values of \[x\] and \[y\] to find the fraction.
Substituting \[x = - 2\] and \[y = 0\] in the expression \[\dfrac{x}{y}\], we get
\[ \Rightarrow \]Fraction \[ = \dfrac{{ - 2}}{0}\]

Therefore, the fraction is \[\dfrac{{ - 2}}{0}\].

Note:
We have formed two linear equations in two variables and simplified them to find the fraction. A linear equation in two variables is an equation of the form \[ax + by + c = 0\], where \[a\] and \[b\]are not equal to 0. For example, \[2x - 7y = 4\] is a linear equation in two variables.
We can verify our answer by using the information given in the question.
The numerator of the fraction \[\dfrac{{ - 2}}{0}\] is \[ - 2\], and the denominator of the fraction \[\dfrac{{ - 2}}{0}\] is 0.
Thus, the denominator is 2 more than the numerator.
If 4 is added to both the numerator and the denominator of the fraction \[\dfrac{{ - 2}}{0}\], the fraction becomes \[\dfrac{{ - 2 + 4}}{{0 + 4}} = \dfrac{2}{4}\], which is equivalent to \[\dfrac{1}{2}\].
Hence, we have verified our answer.