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The numerator of a fraction is $4$less than the denominator. If the numerator is decreased by $2$ and the denominator is increased by $1$, then the denominator is eight times the numerator. The fraction is also written as $\dfrac{m}{{14}}$. What is the value of $m$?

Last updated date: 19th Mar 2023
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Hint: First make linear equations from the given information and then solve those equations in order to find the value of two numbers.

Let the numerator be $x$and denominator be$y$.
According to the question:
${\text{Case1: }}x = y - 4{\text{ }} \ldots \ldots \left( 1 \right) \\ {\text{Case2: }}8\left( {x - 2} \right) = y + 1 \\ \Rightarrow 8x - 16 = y + 1 \\ \Rightarrow 8x - y = 17{\text{ }} \ldots \ldots \left( 2 \right) \\$
Put the value of $x$from equation $\left( 1 \right)$in equation $\left( 2 \right)$and solve for$y$, we get:
$\Rightarrow 8\left( {y - 4} \right) - y = 17 \\ \Rightarrow 8y - 32 - y = 17 \\ \Rightarrow 7y = 49 \\ \Rightarrow y = 7 \\$
Now, put the value of$y$in equation$\left( 1 \right)$and solve for$x$, we get:
$\Rightarrow x = y - 4 \\ \Rightarrow x = 7 - 4 \\ \Rightarrow x = 3 \\$
Hence, the fraction $\dfrac{x}{y}$is equal to $\dfrac{3}{7}.$
Now, the given fraction is of the form $\dfrac{m}{{14}}$. So, we will also convert our fraction of the same form.
Therefore, multiplying and dividing the fraction by$2$, we get:
$\dfrac{x}{y}{\text{ = }}\dfrac{3}{7} \times \dfrac{2}{2} \\ \dfrac{x}{y} = \dfrac{6}{{14}}{\text{ }} \ldots \ldots \left( 3 \right) \\$
Now, the equation$\left( 3 \right)$is of the form $\dfrac{m}{{14}}$. By comparison, we get:
$m = 6$

Note- Whenever you see a problem like this, always try to identify the number of variables and try to make that many equations. Also, in order to compare two fractions, always make their denominator or numerator equal.