Question

the difference of two natural number is 5 and difference of their recipocal is 1/10 then find the two numbers.
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Answer 1

Answer:

Let the two numbers be x and x - 5 .

According to the question.

⇒ x(x-5) = 50

⇒ x2 - 5x - 50 = 0

⇒ x2 - 10x + 5x - 50 = 0

⇒ x (x - 10) + 5 (x - 10) = 0

⇒ (x+5) (x-10) = 0

⇒ (x+5) (x-10) = 0

⇒ x = -5 or 10

⇒ x = 10 (x = -5 , rejected)

∴ Two numbers are 10 and (10-5) = 5.

Answer 2

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The difference of two natural number is 5 and difference of their reciprocal is 1/10.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The two number.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

Let the two natural number be r and m respectively.

$\bigstar\underline{\boldsymbol{According\:to\:the\:question\::}}}}}$

$\implies\rm{r-m=5................(1)}$

&

$\implies\rm{\dfrac{1}{m} -\dfrac{1}{r} =\dfrac{1}{10} }\\\\\\\implies\rm{\dfrac{r-m}{rm} =\dfrac{1}{10} }\\\\\\\implies\rm{10(r-m)=rm}\\\\\\\implies\rm{10(5)=rm\:\:\:[from(1)]}\\\\\\\implies\rm{50=rm}\\\\\\\implies\sf{r=\dfrac{50}{m} ........................(2)}$

Putting the value of r in equation (1),we get;

$\implies\rm{\dfrac{50}{m}-m=5}\\\\\implies\rm{50-m^{2} =5m}\\\\\implies\rm{m^{2} -5m-50=0}\\\\\implies\rm{m^{2} -10m+5m-50=0}\\\\\implies\rm{m(m-10)+5(m-10)=0}\\\\\implies\rm{(m-10)(m+5)=0}\\\\\implies\rm{m-10=0\:\:\:Or\:\:\:m+5=0}\\\\\implies\rm{\orange{m=10\:\:\:Or\:\:\:m\neq -5}}$

Putting the value of m = 10 in equation (2),we get;

$\implies\rm{r=\cancel{\dfrac{50}{10} }}\\\\\implies\rm{\orange{r=5}}$

Thus;

The two number is r = 5 & m = 10 .