Question

A fraction become \[\dfrac{1}{3}\]when 2 is subtracted from the numerator and it becomes \[\dfrac{1}{2}\]when 1 is subtracted from the denominator. Find the fraction.

Answer 1

Hint: Assume any general fraction by yourself (i.e. Let the fraction be \[\dfrac{a}{b}\]) and try to convert the above problem statement into numerical form and find the value of a, b.

Complete step-by-step solution:
Let us assume the fraction to be \[\dfrac{a}{b}\] where a and b are real numbers.
where a is called the numerator of the fraction
and b is called the denominator of the fraction.
According to the question when we subtract 2 from the numerator of the fraction \[\dfrac{a}{b}\], that is from a, the fraction becomes \[\dfrac{1}{3}\].
So, it can also be written as:
\[\dfrac{a-2}{b}=\dfrac{1}{3}\]
After cross multiplying the above equation will become:
\[\Rightarrow 3\left( a-2 \right)=b\] (Let it be our equation (1)).
Now, again according to the question when we subtract 1 from the denominator of the fraction \[\dfrac{a}{b}\], that is from b, the fraction become \[\dfrac{1}{2}\]So, it can also be written as:
\[\dfrac{a}{b-1}=\dfrac{1}{2}\]
After cross multiplying the above equation will become:
\[\Rightarrow 2a=b-1\] (Let it be our equation (2)).
Now, by substituting the value of ‘b’ from equation (1) to equation (2), we will get:
\[\Rightarrow 2a=\left( 3a-6 \right)-1\]
By taking all the variable one side and constant other side, then we will get:
\[\Rightarrow -a=-7\]
\[\therefore a=7\]
Now, by putting the value ‘a’ in equation (1) we will get:
\[\Rightarrow 3\left( 7 \right)-6=b\]
\[\Rightarrow b=21-6\]
\[\therefore b=15\]
Since, we have assumed our fraction to be \[\dfrac{a}{b}\] and we have calculated the value ‘a’ and ‘b’.
Here, \[a=7\] and \[b=15\]
Hence, the required fraction is \[\dfrac{7}{15}\].

Note: The above question is solved by using the substitution method of solving the equation of two variables. It can also be solved by using the elimination method of solving two equations. So to eliminate the variable a, we will have to multiply the first equation with 2 and the second equation with 3 and then subtract them. This will give us the value of b. Then, substituting b in any equation, we can get the value of a.