Question

the sum of two numbers is 8 and the sum of their reciprocals is 8 by 15 find the number

Answer 1

Hey!!

Good Evening

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Let the two numbers be x and y

Then

=> x + y = 8 -----------(1)

=> x = 8 - y ----------------(2)

Also, given that sum of reciprocals

$= > \frac{1}{x} + \frac{1}{y} = \frac{8}{15}$

$= > \frac{x + y}{xy} = \frac{8}{15}$

Replacing (1) in above equation

$= > \frac{8}{xy} = \frac{8}{15}$

Cancelling 8 on Numerator sides and cross Multiplying

=> xy = 15

Using (2) in above equation

=> y(8 - y) = 15

=> y² - 8y + 15 = 0

By middle term splitting method

=> y² - 3y - 5y + 15 = 0

=> y(y - 3) - 5(y - 3) = 0

Thus

=> (y - 3)(y - 5) = 0

Thus y = 3 or y = 5

Using this in (2) we have

Thus x = 5 or x = 3

Thus the two numbers are 3 and 5

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Hope this helps ✌️

Good Night

Answer 2

$\textbf{Answer}$

Suppose the first number is x

Since sum of two numbers = 8

=> Other number = (8 - x)

We know that for any number x,

$\textbf{Reciprocal of x = 1/x}$

$\textbf{According to the question,}$

1/x + 1/(8 - x) = 8/15

=> (8 - x + x)/(x)(8 - x) = 8/15

=> 8/(8x - x^2) = 8/15

=> 8x - x^2 = 15

=> x^2 - 8x + 15 = 0

=> x^2 - 5x - 3x + 15 = 0

=> x(x - 5) - 3(x - 5) = 0

=> (x - 3) (x - 5) = 0

=> x = 3 or x = 5

=> 8 - x = 8 - 3 = 5 (if x = 3)
=> 8 - x = 8 - 5 = 3 (if x = 5)

$\textbf{Two such numbers are 3 and 5}$


$\textbf{Hope My Answer Helped}$
$\textbf{Thanks}$