Question

If we add 1 to the numerator and subtract 1 in the denominator a fraction becomes 1. It also becomes \[\dfrac{1}{2}\] if we only add 1 in the denominator, what is the numerator of the fraction?
A) 2
B) 4
C) 5
D) 3

Answer 1

Hint: Take any fraction of your choice where denominator and numerator must be different variables. Now use the two parameters given to construct two different equations and now you have two separate equations for two different variables. Solve them to get the desired value.
Complete step by step answer:
 Let us assume a fraction \[\dfrac{x}{y}\] where x is the numerator and y is the denominator in this fraction. Now according to the given question we have to find the value of x. Now try to form the equation using the given conditions, i.e., 1 must be added in the numerator so the numerator becomes \[x + 1\] and at the sametime 1 must be subtracted from y to produce the fraction as 1.
Thus the equation will somehow look like \[\dfrac{{x + 1}}{{y - 1}} = 1\] after solving a bit we can get it as
\[\begin{array}{l}
\dfrac{{x + 1}}{{y - 1}} = 1\\
 \Rightarrow x + 1 = y - 1\\
 \Rightarrow x + 2 = y.......................(i)
\end{array}\]
For the second equation it is given that the numerator (x) is not changing just the denominator (y) is changing i.e., 1 is added to y for producing \[\dfrac{1}{2}\] . Thus the equation becomes
 \[\begin{array}{l}
 \Rightarrow \dfrac{x}{{y + 1}} = \dfrac{1}{2}\\
 \Rightarrow 2x = y + 1
\end{array}\]
Now putting the value of y from equation (i) we get
\[\begin{array}{l}
 \Rightarrow 2x = y + 1\\
 \Rightarrow 2x = x + 2 + 1\\
 \Rightarrow x = 3
\end{array}\]
So x was our numerator and we got it as 3. Therefore option D is correct.

Note: We could also substitute x in place of y but it would only stretch the problem and we will get the value of y and then again by substitution we will get x. So keep in mind to substitute the value which you don't want in your answer.