Question

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The difference of two numbers is 5 and the difference of their reciprocal is 1/10. Find the numbers.....
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Answer 1

Answer:

Let the two number be x and x+5.

We have, 1/x-1/x+5=1/10

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

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

5×10=x^2+5x

50=x^2+5x

x^2+5x-50=0

x^2+10x-5x-50=0

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

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

Therefore x-5=0 Or x+10=0

Hence, x=5,-10

Now as per question, given that the two numbers are natural nos.

Hence x=5

Second no.=5+5=10

I hope the helps you!!

Answer 2

Answer:

x = 10, y = 5

Step-by-step explanation:

First, let's start with what a reciprocal is.

A reciprocal is simply the reverse of the number. The numerator switches with the denominator. In mathematical terms, a reciprocal is 1 divided by the number.

We are now given two cases. The difference of the numbers and the difference of the reciprocals.

Let's take the first number as 'x' and the second number as 'y'

First case → x - y = 5

Reciprocal of 'x' is 1/x and the reciprocal of 'y' is 1/y

Second case → ( 1 / y ) - ( 1 / x ) = 1 / 10

Because, in the first equation, we get a positive output which is 5. This means that x > y.

In the second equation too we get positive solution that means the greater number goes first. But remember that we are taking is reciprocals. Hence, in the second equation 1/y > 1/x.

So, 1/y goes first.

Now, let's solve these equations.

we know that x - y = 5

Then, x = y + 5

(1 / y) - (1 / x) = 1/10

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

The LCM would be y² + 5y

This would become :

$\frac{(y+5)-y}{y^{2}+5y}$

$\frac{5}{y^{2}+5y}=\frac{1}{10}$

y² + 5y = 50

y² + 5y - 50 = 0

y - 5y + 10y - 50 = 0

y( y - 5 ) + 10( y - 5 ) = 0

( y + 10 )( y - 5 ) 0

y = -10 and y = 5

There will be two solutions.

Let's take y = -10

x - y = 5

x - (-10) = 5

x + 10 = 5

x = -5                      First solution : (-5,-10)

But if we substitute these values in the second equation, it wouldn't work. This is considered as an extraneous solution. THIS IS NOT THE SOLUTION.

Now, let's take y = 5

x - y = 5

x - 5 = 5

x = 10                          Second solution : (10,5)