Question

The denominator of a fraction is 4 more than twice the numerator. When both the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator. Determine the fraction.

Answer 1

Hint: In this particular type of question we need to express the fraction in the form of $\dfrac{x}{y}$. Then we need to form a system of equations using the given information and solve the equations using subtraction and substitution.

Complete Step-by-step answer:
Let the numerator and denominator of the fraction be x and y respectively. Then,
Fraction =$\dfrac{x}{y}$
It is given that
Denominator =2 (Numerator) +4
$ \Rightarrow $ y=2x+4
$ \Rightarrow $2x−y+4=0
Also according to the second condition, we have
y−6=12(x−6)
$ \Rightarrow $y−6=12x−72
$ \Rightarrow $12x−y−66=0
Thus, we have the following system of equations
2x−y+4=0 (i)
12x−y−66=0 (ii)
Subtracting equation (i) from equation (ii), we get
2x-y+4-(12x-y-66)=0
$ \Rightarrow $2x-y+4-12x+y+66=0
$ \Rightarrow $10x−70=0
$ \Rightarrow $x=7
Putting x=7 in equation (i), we get
14−y+4=0
$ \Rightarrow $y=18
Hence, required Fraction =$\dfrac{7}{18}$

Note: Remember to recall the process of forming quadratic equations using the information provided in the question. Note that the equations could also be solved using substitution of x or y from the first equation and putting it in the second equation.