Question

The numerator of a fraction is one less than its denominator. If 3 is added to each of its
numerator and denominator then the fraction is increased by 3/28.Find the fraction.​

Answer 1

Answer :

The fraction = 3/4

Step-by-step explanation :

Let the denominator be x

The numerator of a fraction is one less than its denominator.

So, the numerator = (x - 1)

The fraction = (x - 1)/x

• If 3 is added to each of its  numerator and denominator then the fraction is increased by 3/28.

     $\sf \frac{x-1+3}{x+3} =\frac{x-1}{x}+ \frac{3}{28} \\\\ \frac{x+2}{x+3} =\frac{28(x-1)+3x}{28x} \\\\ \frac{x+2}{x+3} =\frac{28x-28+3x}{28x} \\\\ \frac{x+2}{x+3}=\frac{31x-28}{28x} \\\\ 28x(x+2)=(x+3)(31x-28) \\\\ 28x^2+56x=x(31x-28) +3(31x-28) \\\\ 28x^2+56x=31x^2-28x+93x-84 \\\\ 28x^2+56x=31x^2+65x-84 \\\\ 31x^2-28x^2+65x-56x-84=0 \\\\ 3x^2+9x-84=0 \\\\ 3(x^2+3x-28)=0 \\\\ x^2+3x-28=0/3 \\\\ x^2+3x-28=0 \\\\ x^2+7x-4x-28=0 \\\\ x(x+7)-4(x+7)=0 \\\\ (x+7)(x-4)=0 \\\\ \longrightarrow x=4 \ (or) -7$

x > 0 , x is a natural number.

• If x = 4,

denominator = 4

numerator = 4 - 1 = 3

⇒ The fraction = 3/4

Verification :

• The fraction = 3/4

 $\sf \frac{3+3}{4+3}= \frac{3}{4} + \frac{3}{28} \\\\ \frac{6}{7}=\frac{3\times7}{4\times 7} +\frac{3}{28} \\\\ \frac{6}{7} =\frac{21}{28}+\frac{3}{28} \\\\ \frac{6}{7} =\frac{21+3}{28} \\\\ \frac{6}{7} =\frac{24}{28} \\\\ \frac{6}{7} =\frac{\not{4}\times6}{\not{4}\times7} \\\\ \frac{6}{7} = \frac{6}{7} \\\\ LHS=RHS$

Hence verified!