Question

The denominator of a rational number is greater than its numerator by 7. if numerator is increased by 17 and denominator decreased by 6, the new number becomes 2. find the original number​

Answer 1

GIVEN :-

• The denominator of a rational number is greater than its numerator by 7.
• Numerator is increased by 17 and denominator decreased by 6, the new number becomes 2.

TO FIND :-

• The original fraction.

SOLUTION :-

Let the numerator of a fraction be "x" and the denominator be "x + 7".

⇒ Original fraction = Numerator/Denominator

⇒ Original fraction = x/(x + 7).

Now , According to the Question,

⇒ Numerator = x + 17

similarly,

⇒ Denominator = x + 7 - 6

⇒ Denominator = x + 1

Now,

⇒ New fraction = Numerator/Denominator

⇒ New fraction = (x + 17)/(x + 1).

Now According to the Question,

⇒ New fraction = 2

⇒ (x + 17)/(x + 1) = 2/1

⇒ x + 17 = 2(x + 1)

⇒ x + 17 = 2x + 2

⇒ x - 2x = 2 - 17

⇒ -x = -15

⇒ x = 15

Now substitute the value of x in original fraction,

⇒ Original fraction = x/(x + 7).

⇒ Original fraction = 15/(15 + 7)

⇒ Original fraction = 15/22

Hence The required original fraction is 15/22.

Answer 2

Answer:

$\huge \bf \: given$

The denominator of a rational number is greater than its numerator by 7. if numerator is increased by 17 and denominator decreased by 6, the new number becomes 2. find the original number.

$\huge \bf \: lets \: solve$

Let us assume the numerator as y

Numerator - y

Denominator - 7+y

$\sf \implies \: fraction \: = \frac{y}{y + 7}$

Now,

When numerator is increased bye 17 and denominator decreased by 6. the number become 2

So,

Numerator = y + 17

Denominator = y + 7 - 1

Denominator = y + 6

Now,

New fraction = 2

$\sf \frac{y + 17}{y + 1} = \frac{2}{1}$

$\sf \: y + 17 = 2(y + 1)$

$\sf \: y + 17 = 2y + 2$

$\sf \: y - 2y = 17 - 2$

$\sf - y = - 15$

[Substituting value of y in original fraction]

$\sf \implies \: \frac{15}{(15 + 7)}$

$\sf \implies \: \frac{15}{22}$

Verification

Verification of Case 1

$\mapsto \sf case \: 1 \: = denominator \: greater \: by \\ 7$

$\dfrac{15}{22}$

Case 1 is right

Verification of case 2

$\frac{15 + 17}{22 - 6} = 2$

$\frac{32}{16} = 2$

$2 = 2$

LHS = RHS

$\huge \bf fraction \: = \frac{15}{22}$