Question

the denominator of a rational number is greater than its numerator by 4 .if 3 is subtracted from the numerator the new number becomes 2/3, find the original number.​

Answer 1

$\bf{\huge{\underline{\boxed{\sf{\purple{Answer:}}}}}}$

$\bf{\underline{\underline{\sf{\blue{Given:}}}}}$

• The denominator of a rational number is greater than its numerator by 4
• if 3 is subtracted from the numerator the new number becomes $\frac{2}{3}$

$\bf{\underline{\underline{\sf{\blue{To\:find:}}}}}$

• The original number

$\bf{\underline{\underline{\sf{\blue{Solution:}}}}}$

Let the numerator be x.

Let the denominator be y.

Original number = $\bf\frac{x}{y}$

$\bf{\underline{\underline{\sf{\blue{As\:per\:first\:condition:}}}}}$

• The denominator of a rational number is greater than its numerator by 4

Representing it mathematically,

=> y = x + 4

=> x + 4 = y

=> x - y = - 4 -----> 1

$\bf{\underline{\underline{\sf{\blue{As\:per\:second\:condition:}}}}}$

• If 3 is subtracted from the numerator the new number becomes $\frac{2}{3}$

Numerator = x - 3

Representing the second condition mathematically,

=> $\frac{x-3}{y}$ = $\bf{2}{3}$

=> 3 ( x - 3) = 2y

=> 3x - 9 = 2y

=> 3x - 2y = 9 ----> 2

Multiply equation 1 by 2,

x - y = - 4 -----> 1

2 × x - 2 × y = 2 × - 4

2x - 2y = - 8 -----> 3

Solve equations 2 and 3 simultaneously by elimination method.

Subtract equation 3 from 2,

....+ 2x - 2y = - 8 -----> 3

- ( + 3x - 2y = 9 ) ------> 2

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

- x = - 17

x = 17

Substitute x = 17 in equation 2,

3x - 2y = 9 ----> 2

3 ( 17) - 2y = 9

51 - 2y = 9

- 2y = 9 - 51

- 2y = - 42

y = $\frac{-42}{-2}$

y = 21

Original number = $\bf\sf\frac{x}{y}$

Original number = $\bf\sf\frac{17}{21}$

$\bf{\huge{\underline{\boxed{\mathfrak{\green{Verificatiom:}}}}}}$

For first case :-

• The denominator of a rational number is greater than its numerator by 4

Numerator = x = 17

Denominator = y = 21

=> y = x + 4

=> 21 = 17 + 4

=> 21 = 21

LHS = RHS.

For second case :-

• if 3 is subtracted from the numerator the new number becomes $\frac{2}{3}$

Numerator = x - 3 = 17 - 3 = 14

Denominator = y = 21

=> $\bf\frac{x-3}{y}$ = $\bf\frac{2}{3}$

=> $\bf\frac{14}{21}$ = $\bf\frac{2}{3}$

Dividing LHS by 7,

=> $\bf\frac{2}{3}$ = $\bf\frac{2}{3}$

LHS = RHS.

Hence verified.