the numerator of a fraction is one less than its denominator if 3 is added to each of the numerator and denominator the fraction is increased by 3 by 8 find the fraction
Answers
Answer:
3/4
Step-by-step explanation:
Let the denominator be x.
A.T.Q., the numerator will be (x - 1).
Original fraction = {\sf{{\dfrac{x - 1}{x}}}}
x
x−1
Now, as given in the question, 3 is added to both numerator and denominator.
So,
Numerator = x - 1 + 3 = x + 2
Denominator = x + 3
A.T.Q.
{\sf{{\dfrac{x + 2}{x + 3}} = {\dfrac{x - 1}{x}} + {\dfrac{3}{28}} }}
x+3
x+2
=
x
x−1
+
28
3
On further solving, we get
\Rightarrow{\sf{ {\dfrac{x + 2}{x + 3}} = {\dfrac{28(x - 1) + 3(x)}{(28)(x)}} }}⇒
x+3
x+2
=
(28)(x)
28(x−1)+3(x)
\Rightarrow{\sf{ {\dfrac{x + 2}{x + 3}} = {\dfrac{28x - 28 + 3x}{28x}} }}⇒
x+3
x+2
=
28x
28x−28+3x
\Rightarrow{\sf{ {\dfrac{x + 2}{x + 3}} = {\dfrac{31x - 28}{28x}} }}⇒
x+3
x+2
=
28x
31x−28
Cross - multiplication.
\Rightarrow⇒ 28x(x + 2) = (x + 3)(31x - 28)
\Rightarrow⇒ 28x(x + 2) = x(31x - 28) + 3(31x - 28)
\Rightarrow⇒ 28x² + 56x = 31x² - 28x + 93x - 84
\Rightarrow⇒ 28x² + 56x = 31x² + 65x - 84
Transposing the terms.
\Rightarrow⇒ 28x² - 31x² = 65x - 56x - 84
\Rightarrow⇒ - 3x² = 9x - 84
\Rightarrow⇒ 3x² + 9x - 84 = 0
\Rightarrow⇒ 3(x² + 3x - 28) = 0
\Rightarrow⇒ x² + 3x - 28 = 0
Using Middle Term Factorisation, we get
\Rightarrow⇒ x² - 4x + 7x - 28 = 0
\Rightarrow⇒ x(x - 4) + 7(x - 4) = 0
\Rightarrow⇒ (x + 7)(x - 4) = 0
Using zero product rule, we get
\Rightarrow⇒ x + 7 = 0 and x - 4 = 0
\Rightarrow⇒ x = - 7 and x = 4
Here, x is a natural number, where x > 0.
Hence, x = 4
The required fraction is :
\Rightarrow{\sf{ {\dfrac{x - 1}{x}} }}⇒
x
x−1
\Rightarrow{\sf{ {\dfrac{4 - 1}{4}} }}⇒
4
4−1
\Rightarrow{\boxed{\sf{\red{ {\dfrac{3}{4}} }}}}⇒
4
3