Question

the denominator of a rational number is less than its ne numerator than 5 if the five is added to the numerator the new number become 11 by 6 find the original rational number​ ​

Answer 1

$\huge\mathtt{\fbox{\red{Answer✍︎}}}$

$\large\underline\red{GIVEN,}$

$\sf\dashrightarrow \blue{THE\:GIVEN\:FRACTION\:IS\:A\:RATIONAL\:NUMBER.}$

$\sf\dashrightarrow {\blue{\mathbb{\text{ denominator is less than its numerator by 5.}}}}$

$\sf\therefore \blue{let\:the\:numerator\:be\:x}$

$\sf\dashrightarrow \blue{denominator= x-5}$

$\sf\dashrightarrow \blue{\dfrac{x}{x-5}}$

$\sf\dashrightarrow \bold\pink{if \:5 \:is \:added\: to \:the \:numerator, \:numerator \:becomes\: ; \dfrac{11}{6}}$

$\sf\dashrightarrow \red{numerator= x+5}$

THE EQUATION FORM IS,

$\rm{\boxed{\sf{\green{ \circ\:\: \dfrac{x+5}{x-5} = \dfrac{11}{6}\:\: \circ}}}}$

$\large\underline\purple{TO\:FIND,}$

$\sf\dashrightarrow \red{\:THE\:ORIGINAL\:RATIONAL\:NUMBER}$

$\sf\implies \green{\dfrac{x + 5}{x - 5} = \dfrac{11}{6}}$

$\sf\implies \green{6 \times (x+5)= 11 \times (x-5)}$

$\sf\implies \green{6x+30= 11x-55}$

$\sf\implies \green{30+55=11x-6x}$

$\sf\implies \green{85= 5x}$

$\sf\implies \green{x= \dfrac{85}{5}}$

$\sf\implies \green{x= \cancel \dfrac{85}{5}}$

$\sf\implies \orange{x = 17}$

$\rm{\boxed{\sf{ \circ\:\: x= 17 \:\: \circ}}}$

THE ORIGINAL NUMBERS ARE,

$\sf\implies \red{numerator=x =17}$

$\sf\implies \red{d enominator= x-5}$

$\sf\implies \red{denominator=17-5}$

$\sf\implies \pink{denominator=12}$

$\large\underline\orange{FRACTION,}$

$\sf\dashrightarrow \purple{\dfrac{x}{x-5}= \dfrac{17}{12}}$

$\sf\dashrightarrow \purple{\dfrac{NUMERATOR}{DENOMINATOR}= \dfrac{17}{12}}$

$\rm\underline\blue{NUMERATOR\:IS\:17\:AND\: DENOMINATOR\:IS\:12}$

$\rm{\boxed{\sf{ \circ\:\: \dfrac{NUMERATOR}{DENOMINATOR}= \dfrac{17}{12} \:\: \circ}}}$

Answer 2

Answer:

$GIVEN,</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \blue{THE\:GIVEN\:FRACTION\:IS\:A\:RATIONAL\:NUMBER.}⇢THEGIVENFRACTIONISARATIONALNUMBER.</p><p></p><p>\sf\dashrightarrow {\blue{\mathbb{\text{ denominator is less than its numerator by 5.}}}}⇢ denominator is less than its numerator by 5.</p><p></p><p>\sf\therefore \blue{let\:the\:numerator\:be\:x}∴letthenumeratorbex</p><p></p><p>\sf\dashrightarrow \blue{denominator= x-5}⇢denominator=x−5</p><p></p><p>\sf\dashrightarrow \blue{\dfrac{x}{x-5}}⇢ </p><p>x−5</p><p>x</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \bold\pink{if \:5 \:is \:added\: to \:the \:numerator, \:numerator \:becomes\: ; \dfrac{11}{6}}⇢if5isaddedtothenumerator,numeratorbecomes; </p><p>6</p><p>11</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \red{numerator= x+5}⇢numerator=x+5</p><p></p><p>THE EQUATION FORM IS,</p><p></p><p>\rm{\boxed{\sf{\green{ \circ\:\: \dfrac{x+5}{x-5} = \dfrac{11}{6}\:\: \circ}}}} </p><p>∘ </p><p>x−5</p><p>x+5</p><p> </p><p> = </p><p>6</p><p>11</p><p> </p><p> ∘</p><p> </p><p> </p><p></p><p>\large\underline\purple{TO\:FIND,} </p><p>TOFIND,</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \red{\:THE\:ORIGINAL\:RATIONAL\:NUMBER}⇢THEORIGINALRATIONALNUMBER</p><p></p><p>\sf\implies \green{\dfrac{x + 5}{x - 5} = \dfrac{11}{6}}⟹ </p><p>x−5</p><p>x+5</p><p> </p><p> = </p><p>6</p><p>11</p><p> </p><p> </p><p></p><p>\sf\implies \green{6 \times (x+5)= 11 \times (x-5)}⟹6×(x+5)=11×(x−5)</p><p></p><p>\sf\implies \green{6x+30= 11x-55}⟹6x+30=11x−55</p><p></p><p>\sf\implies \green{30+55=11x-6x}⟹30+55=11x−6x</p><p></p><p>\sf\implies \green{85= 5x}⟹85=5x</p><p></p><p>\sf\implies \green{x= \dfrac{85}{5}}⟹x= </p><p>5</p><p>85</p><p> </p><p> </p><p></p><p>\sf\implies \green{x= \cancel \dfrac{85}{5}}⟹x= </p><p>5</p><p>85</p><p> </p><p> </p><p> </p><p> </p><p></p><p>\sf\implies \orange{x = 17}⟹x=17</p><p></p><p>\rm{\boxed{\sf{ \circ\:\: x= 17 \:\: \circ}}} </p><p>∘x=17∘</p><p> </p><p> </p><p></p><p>THE ORIGINAL NUMBERS ARE,</p><p></p><p>\sf\implies \red{numerator=x =17}⟹numerator=x=17</p><p></p><p>\sf\implies \red{d enominator= x-5}⟹denominator=x−5</p><p></p><p>\sf\implies \red{denominator=17-5}⟹denominator=17−5</p><p></p><p>\sf\implies \pink{denominator=12}⟹denominator=12</p><p></p><p>\large\underline\orange{FRACTION,} </p><p>FRACTION,</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \purple{\dfrac{x}{x-5}= \dfrac{17}{12}}⇢ </p><p>x−5</p><p>x</p><p> </p><p> = </p><p>12</p><p>17</p><p> </p><p> </p><p></p><p>\sf\dashrightarrow \purple{\dfrac{NUMERATOR}{DENOMINATOR}= \dfrac{17}{12}}⇢ </p><p>DENOMINATOR</p><p>NUMERATOR</p><p> </p><p> = </p><p>12</p><p>17</p><p> </p><p> </p><p></p><p>\rm\underline\blue{NUMERATOR\:IS\:17\:AND\: DENOMINATOR\:IS\:12} </p><p>NUMERATORIS17ANDDENOMINATORIS12</p><p> </p><p> </p><p></p><p>\rm{\boxed{\sf{ \circ\:\: \dfrac{NUMERATOR}{DENOMINATOR}= \dfrac{17}{12} \:\: \circ}}} </p><p>∘ </p><p>DENOMINATOR</p><p>NUMERATOR</p><p> </p><p> = </p><p>12</p><p>17</p><p> </p><p> ∘</p><p>$