Question

The denominator of a rational number is greater than its numerator by 7. If 3
is subtracted from the numerator and 2 is added to its denominator, the new
number becomes
Find the rational number.​

Answer 1

Answer:

let the equation be x / x + 7

if x - 3 / x + 7 + 2

it becomes y

x - 3 + 3 / x + 7 + 3 +2 = y

x / x + 12 = y

we cant find the no here since the result is not given

but we do know that denominator is not 0

so im going to make up a no by myself and try match it with equation

12 - 3 / 19 + 2

9/21

Answer 2

$\bold{\huge{\underline{\underline{\mathfrak{AnsWer:}}}}}$

$\bold{\boxed{\large{\sf{\pink{Rational\:Number\:=\:{\dfrac{x}{y}}\:=\:{\dfrac{6}{13}}}}}}}$

$\bold{\huge{\underline{\underline{\mathfrak{StEp\:by\:stEp\:explanation:}}}}}$

GIVEN :

The denominator of a rational number is greater than its numerator by 7

If 3 is subtracted from the numerator and 2 is added to the denominator the new number

$\bold{\dfrac{1}{5}}$

TO FIND :

The rational number

SOLUTION :

Let the numerator of the rational number be x.

Let the denominator of the rational number be y.

Rational Number =$\bold{\dfrac{x}{y}}$

$\bold{\underline{\underline{\sf{As\:per\:the\:first\:given\:condition:}}}}$

The denominator of a rational number is greater than its numerator by 7

Constituting it mathematically,

→$\bold{y=x+7}y=x+7$

→$\bold{x+7=y}x+7=y$

→ $\bold{x-y=-7}x−y=−7 --->(1)$

$\bold{\underline{\underline{\sf{As\:per\:the\:second\:given\:condition:}}}}$

If 3 is subtracted from the numerator and 2 is added to the denominator the new number $\bold{\dfrac{1}{5}}$

Numerator = x - 3

Denominator = y + 2

Rational number = $\bold{\dfrac{1}{5}}$

Constituting it mathematically,

→$\bold{\dfrac{x-3}{y+2}}$

=$\bold{\dfrac{1}{5}}$

Cross multiplying,

$→ \bold{5(x-3)=y+2}5(x−3)=y+2$

$→ \bold{5x-15=y+2}5x−15=y+2$

$→ \bold{5x-y=2+15}5x−y=2+15 ---> (2)$

Multiply equation 1 by 5,

→ x - y = - 7

→ 5x - 5y = - 35 ---> (3)

Solve equation 2 and equation 3 simultaneously by elimination method.

Subtract equation 3 from equation 2,

5x - 5y = - 35 ---> (3)

- (5x - y = 17) ------> (2)

--------------------------

- 4y = - 52

$\rightarrow→ \bold{y={\dfrac{-52}{-4}}}$

$\rightarrow→ \bold{y={\dfrac{52}{4}}}$

$\rightarrow→ \bold{y=13}$

Substitute y = 13 in equation 1,

→ x - y = - 7

→ x - 13 = - 7

→ x = - 7 + 13

→ x = 6

$\bold{\boxed{\large{\sf{\red{Numerator\:=\:x\:=\:6}}}}}$

$\bold{\boxed{\large{\sf{\red{Denominator\:=\:y\:=\:13}}}}}$

$\bold{\boxed{\large{\sf{\red{Rational\:Number\:=\:{\dfrac{x}{y}}\:=\:{\dfrac{6}{13}}}}}}}$