Question

in a rational number,twice the numerator is 2 more than the denomater. if 3 is added to each ,the numator and the denomater ,the new fraction 2/3.find the original number




pls help me​

Answer 1

Correct Question

In a rational number, twice the numerator is 2 more than the denominator. If 3 is added to each, the numerator and the denomater, the new fraction 2/3. Find the original number.

Answer

Let the original rational number be a/b.

Given that,

Twice the numerator is 2 more than the denominator. So,

$\sf \dashrightarrow 2a - 2 = b \: \: --- (i)$

If 3 is added to each, the numerator and the denominator, the new fraction 2/3. So,

$\sf \dashrightarrow \dfrac{a + 3}{b + 3} = \dfrac{2}{3}$

$\sf \dashrightarrow 3(a + 3) = 2(b + 3)$

$\sf \dashrightarrow 3a + 9 = 2b + 6$

$\sf \dashrightarrow 3a - 2b = 6 - 9$

$\sf \dashrightarrow 3a - 2b = -3 \: \: --- (ii)$

By first equation,

$\sf \dashrightarrow 2a - 2 = b$

$\sf \dashrightarrow 2a = b + 2$

$\sf \dashrightarrow a = \dfrac{b + 2}{2}$

Now, let's find the value of b by second equation.

$\sf \dashrightarrow 3a - 2b = -3$

$\sf \dashrightarrow 3 \bigg( \dfrac{b + 2}{2} \bigg) - 2b = -3$

$\sf \dashrightarrow \dfrac{3b + 6}{2} - 2b = -3$

$\sf \dashrightarrow \dfrac{3b + 6 - 4b}{2} = -3$

$\sf \dashrightarrow \dfrac{-1b + 6}{2} = -3$

$\sf \dashrightarrow -1b + 6 = -3 \times 2$

$\sf \dashrightarrow -1b + 6 = -6$

$\sf \dashrightarrow -1b = -6 - 6$

$\sf \dashrightarrow -1b = -12$

$\sf \dashrightarrow b = 12$

Now, let's find the value of a by first equation.

$\sf \dashrightarrow 2a - 2 = b$

$\sf \dashrightarrow 2a - 2 = 12$

$\sf \dashrightarrow 2a = 12 + 2$

$\sf \dashrightarrow 2a = 14$

$\sf \dashrightarrow a = \dfrac{14}{2}$

$\sf \dashrightarrow a = 7$

We know that, the values of a and b are 7 and 12 respectively. So,

$\sf \dashrightarrow \dfrac{a}{b} = \dfrac{7}{12}$

Hence, the original rational number is 7/12.