Answer: The numerator of the given fraction is 9.
Step-by-step explanation: Given that if the numerator of a fraction is increased by 3, the fraction becomes [tex]\dfrac{3}{4}[/tex] and if the denominator is decreased by 7, the fraction becomes 1.
We are to find the numerator of the function.
Let x and y represents the numerator and denominator of the given function.
Then, according to the given information, we have
[tex]\dfrac{x+3}{y}=\dfrac{3}{4}\\\\\Rightarrow 4(x+3)=3y\\\\\Rightarrow 4x+12=3y\\\\\\\Rightarrow y=\dfrac{4x+12}{3}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
and
[tex]\dfrac{x}{y-7}=1\\\\\Rightarrow x=y-7\\\\\Rightarrow y=x+7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Equating the values of y from equations (i) and (ii), we get
[tex]\dfrac{4x+12}{3}=x+7\\\\\Rightarrow 4x+12=3(x+7)\\\\\Rightarrow 4x+12=3x+21\\\\\Rightarrow 4x-3x=21-12\\\\\Rightarrow x=9.[/tex]
Thus, the numerator of the given fraction is 9.
