Question

The difference of two natural numbers is 5and the difference of their reciprocals is 1÷10 .find the number

Answer 1

Answer:

Required two natural numbers are 5 and 10.

Step-by-step explanation:

Let the required natural numbers are a and a - 5.

According to the question : -

⇒ Difference of their reciprocals is 1 / 10.

$\implies\dfrac{1}{a-5}-\dfrac{1}{a}=\dfrac{1}{10}$

$\implies \dfrac{a-(a-5)}{(a-5)a}=\dfrac{1}{10}$

$\implies \dfrac{a-a+5}{a^2-5a}=\dfrac{1}{10}$

$\implies \dfrac{5}{a^2-5a}=\dfrac{1}{10}$

$\implies 10 x 5 = a^2 - 5a$

$\implies 50 = a^2 - 5a$

$\implies a^2 - 5a - 50=0$

$\implies a^2 - ( 10 -5)a-50=0$

$\implies a^2 - 10a + 5a - 50=0$

$\implies a(a-10)+5(a-10)=0$

$\implies (a-10)(a+5)=0$

Therefore, by Zero Product Rule : -

a - 10 = 0 ⇒ a = 10

a + 5 = 0 ⇒ a = - 5

It is given that the required numbers are natural numbers, so taking +ve value of a,

a = 10

Hence,

Required numbers are : -

a = 10

a - 5 = 10 - 5 = 5

Answer 2

here is your answer OK ☺☺☺☺☺☺☺☺

let the number first = X

second number = 5x

different of the reciprocal 1/10

so 1/x - 1/x+5 = 1/10

x+5-x/x(x+5) = 1/10

5/x'2+5x= 1/10

X'2 + 5x=50 = X'2+5x-50=0
so.... x'2 + 10x -5x-50=0

x(x+10)-5 (x+10)=0
so (x-5) (x+10 ) =0


hence , by Zero Product Rule =>

X - 10 = 0 ⇒ X = 10

X + 5 = 0 ⇒ X = - 5


number are positive so ,

X = 10


so ,

Required numbers are =

X = 10

X - 5 = 10 - 5 = 5