Math, asked by khusheeyadav, 4 months ago

Q16. The denominator of a rational number is greater than its numerator by 5. If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3.Find the original number:​

Answers

Answered by Yuseong
12

Given:

• The denominator of a rational number is greater than its numerator by 5.

• If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3.

To calculate:

• The original number.

Calculation:

Let us assume the numerator as 'x'. So, the denominator becomes :

  • Denominator → x + 5
  • Numerator → x

 \sf { \longrightarrow Original \: number = \dfrac{x}{x+5} }

According to the question:

» If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3. So,

 \sf { \longrightarrow \dfrac{x + 8}{(x+5)-1} = \dfrac{5}{3} }

Here, an equation has been formed. So, at first we'll calculate the value of x and then we'll substitute the value of x in the original number expression in order to find the original number.

 \sf { \longrightarrow \dfrac{x + 8}{x+5-1} = \dfrac{5}{3} }

[Removing brackets.]

 \sf { \longrightarrow \dfrac{x + 8}{x+4}= \dfrac{5}{3}  }

[Performing addition.]

Now, by cross multiplication :

 \sf { \longrightarrow 3(x + 8) = 5(x+4)}

[Performing multiplication.]

 \sf { \longrightarrow 3(x) + 3(8) = 5(x) + 5(4)}

 \sf { \longrightarrow 3x + 24 = 5x + 20}

 \sf { \longrightarrow  24 - 20= 5x -3x}

 \sf { \longrightarrow  4= 2x}

 \sf { \longrightarrow  \dfrac{4}{2} = x}

 \longrightarrow  \boxed{\sf \orange{ 2 = x }}

Now, substituting the value of x in original fraction.

 \sf { \longrightarrow Original \: number = \dfrac{x}{x+5} }

 \sf { \longrightarrow Original \: number = \dfrac{2}{2+5} }

\longrightarrow \boxed{ \sf \red {  Original \: number = \dfrac{2}{7} }}

Therefore, original number is 2/7.

Verification:

As the question states that,

» If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3.

So, here :

  • LHS =  \sf{  \dfrac{x+8}{(x+5)-1} }

  • RHS =  \sf{ \dfrac{5}{3} }

Solving LHS :

 \sf { \longrightarrow LHS = \dfrac{2+8}{(2+5)-1} }

 \sf { \longrightarrow LHS = \dfrac{10}{7-1} }

 \sf { \longrightarrow LHS = \dfrac{10}{6} }

[Dividing denominator & numerator by 2.]

\longrightarrow \boxed{ \sf \red {  LHS = \dfrac{5}{3} }}

RHS :

\longrightarrow \boxed{ \sf \red {  RHS = \dfrac{5}{3} }}

Hence,

\longrightarrow \boxed{ \sf \red {  \dfrac{5}{3} = \dfrac{5}{3} }}

LHS= RHS

Hence, verified!

Answered by WaterPearl
26

Question

The denominator of a rational number is greater than its numerator by 5. If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3.Find the original number.

Given

  • The denominator of a rational number is greater than its numerator by 5.

  • If the numerator is increased by 8 and the denominator is decreased by 1, the new number becomes 5/3.

To Find

  • The original number.

Solution

Let the Numerator be x and denominator be ( x + 5)

According to the Question

When numerator is increased by 8 and the denominator by 1,So the fraction 5/3.

So,

\sf{ \frac{x + 8}{(x + 5) - 1} = \frac{5}{3}}

Now,

\sf{ \frac{x + 8}{(x + 5) - 1} = \frac{5}{3}}

\sf{ \frac{x + 8}{x + 4} = \frac{5}{3}}

→ 3 (x + 8) = 5 (x + 4)

→ 3x + 24 = 5x + 20

→ 5x - 3x = 24 - 20

→ 2x = 4

→ x = 4/2

→ x = 2

Now,substituting the value of x.

x + 5

→ 2 + 5

→ 7

Hence,The original number is 2/7.

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