Question

the denominator of a rational number is greater than its numerator by 8. if the numerator is increased by 17 and the denominator is decreased by 1 , the number obtained is 3/2 . find the rational number .​

Answer 1

Given:

• Denominator of a rational number is greater than its numerator by 8.
• Numerator is increased by 17 and the denominator is decreased by 1.
• Number obtained is 3/2.

To Find:

• Find the rational number

Solution:

As per given in the question, denominator of a rational number is greater than its numerator by 8.After letting the value we also have to suppose Numerator is increased by 17 and the denominator is decreased by 1 and after supposing it also then we obtain 3/2 (which is given in the question).So let's find

Let the numerator of the rational number be x.

So as per given,the denominator will be x +8.

The rational number will be $\dfrac{x}{x+8}$

According to the question,

$\: \: \sf \: \frac{x + 17}{x + 8 - 1} = \frac{3}{2} \\ \\ \: \: \sf \: \frac{x + 17}{x + 7} = \frac{3}{2} \\ \\ \: \: \sf \: 2(x + 17) = 3(x + 7) \\ \\ \: \: \sf \: 2x + 34 = 3x + 21 \\ \\ \: \: \sf \: 34 - 21 = 3x - 2x \\ \\ \: \: \sf \: 13 = x \\ \\ \: \sf \dagger \therefore \: x = 13$

$\therefore$

Therefore,the value of x is 13.

The rational number will be

$\: \: \sf \: \frac{x}{x + 8} \\ \\ \: \: \sf \: \frac{13}{13 + 8} \\ \\ \: \sf \: = \frac{13}{21}$

Hence, The rational number is 13/21.

Answer 2

★ The rational number is 13/21.

Step-by-step explanation:

Analysis -

‎ ‎ ‎ ‎ ‎ ‎In the question, it is conditioned that the denominator of a rational number is greater than its numerator by 8. If its numerator is increased by 17 and denominator is decreased by 1, we will obtain the rational number 3/2. Now, we are asked to find the rational number.

Solution -

Let us consider the value of numerator of the rational number be 'k'.

So, according to the question, its denominator will be equal to 'k + 8'.

The expression of the rational number is:

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \boxed{\sf \pmb{\dfrac{k}{k + 8}}}}$

It is also given that, if we add 17 to its numerator and subtract 1 from its denominator, we will get 3/2 as the result. This can be formed as an equation:

$\: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \boxed{\sf \pmb{\dfrac{k + 17}{k + 8 - 1} = \dfrac{3}{2}}}}$

By solving this equation, we will get the value of 'k'.

$\dashrightarrow{ \sf{\dfrac{k + 17}{k + 8 - 1} = \dfrac{3}{2}}} \\ \\ \textsf{[By cross - multiplying]}\\ \\ \dashrightarrow{ \sf{2(k + 17) = 3(k + 8 - 1)}} \\ \\ \dashrightarrow{ \sf{2k + 34 = 3k + 24 - 3}} \\ \\ \dashrightarrow{ \sf{2k + 34 = 3k + 21}} \\ \\ \dashrightarrow{ \sf{2k - 3k = 21 - 34}} \\ \\ \dashrightarrow{ \sf{ - 1k = - 13}} \\ \\ \dashrightarrow{ \sf{k = \dfrac{ \cancel{-} 13}{ \cancel{ -} 1} }} \\ \\ \dashrightarrow{ \underline{ \underline{ \sf \pmb{k =13 }}}}$

The value of 'k' is 13. Now, on substituting the value of 'k' in the expression formed and simplifying it will give us the required answer!

$\twoheadrightarrow{ \sf{ \dfrac{k}{k + 8} = \dfrac{13}{13 + 8} = \boxed{ \frak{\pmb{\red{\dfrac{13}{21} }}}}}}$

Therefore, the rational number is 13/21.

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