Question

The denominator of a fraction is 1 more than twice the numerator. If the numerator & the denominator are both increased by 5, it becomes 3/5. Find the original fraction.​

Answer 1

• The original fraction = $\bf\dfrac{7}{15}$

Given :

• The denominator of a fraction is 1 more than twice the numerator.

• If 5 is added to both the numerator and the denominator of the fraction. It becomes $\bf\dfrac{3}{5}$

To Find :

• The original fraction.

Solution :

Let,

The numerator be x.

The denominator be 2x + 1 because according to the question, the denominator of a fraction is 1 more than twice the numerator.

Therefore,

The original fraction = $\bf\dfrac{x}{2x+1}$ ––––(1)

According to the condition,

If 5 is added to both the numerator and the denominator. It becomes $\bf\dfrac{3}{5}$

After adding 5 to both the numerator and the denominator of the fraction.

• The numerator = x + 5.
• The denominator = 2x + 1 + 5 = 2x + 6.

Now,

$\bf \implies \dfrac{x + 5}{2x + 6} = \dfrac{3}{5} \\ \\ \\\bf \implies 5(x + 5) = 3(2x + 6) \\ \\ \\ \bf \implies 5x + 25 = 6x + 18 \\ \\ \\ \bf \implies 25 - 18 = 6x - 5x \\ \\ \\ \bf \implies 7 = 1x \\ \\ \\ \: \: \: \bf \therefore \: \: x = 7$

Now, substitute the value of x in the equation (1),

$\bf \implies \dfrac{x}{2x + 1} \\ \\ \\ \bf \implies \dfrac{7}{2(7) + 1} \\ \\ \\ \bf \implies \dfrac{7}{(2 \times 7) + 1} \\ \\ \\ \bf \implies \dfrac{7}{14 + 1} \\ \\ \\ \bf \implies \dfrac{7}{15}$

Hence,

The original fraction is $\bf\dfrac{7}{15}$

Answer 2

Answer:

✯ Given :-

• The denominator of a fraction is 1 more than twice the numerator.
• If the numerator and the denominator are both increased by 5, it becomes ⅗.

✯ To Find :-

• What is the original fraction.

✯ Solution :-

» Let, the numerator be x

» And, the denominator be 2x + 1

» So, the original fraction will be = $\dfrac{x}{2x + 1}$

➙ If the numerator and the denominator are both increased by 5 it becomes ⅗.

➣ According to the question,

⇒ $\dfrac{x + 1}{2x + 1 + 5}$ = $\dfrac{3}{5}$

⇒ $\dfrac{x + 5}{2x + 6}$ = $\dfrac{3}{5}$

By doing cross multiplication we get,

⇒ 5(x + 5) = 3(2x + 6)

⇒ 5x + 25 = 6x + 18

⇒ 25 - 18 = 6x - 5x

⇒ 7 = x

➠ $\large\bf{\underbrace{\green{x\: =\: 7}}}$

Hence, the required original fraction,

↦ $\dfrac{7}{2 × 7 + 1}$

$\implies$ $\dfrac{7}{15}$

$\therefore$ The original fraction is $\boxed{\bold{\large{\dfrac{7}{15}}}}$

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