The sum of two numbers is 11 and the sum of their 11 reciprocals is Find the numbers 28

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The sum of two numbers is $$11$$ and the sum of their reciprocals is $$\dfrac{{11}}{{28}}$$ Find the numbers.

Given sum of two numbers is 11

x+y=11 _____ (1)

sum of their reciprocals is $$\frac{11}{28}$$.

$$\frac{1}{x}+\frac{1}{y}=\frac{11}{48}$$

$$\frac{y+x}{xy}=\frac{11}{48}$$

$$28(x+y)= 11xy$$ _______ (2)

$$28(11)= 11.xy$$

$$xy=28$$

Now

$$(x-y)^{2}=x^{2}+y^{2}-2xy$$

$$(x-y)^{2}=(x+y)^{2}-4xy$$

$$(x-y)^{2}=(11)^{2}-4.28$$

$$(x-y)^{2}=121-112$$

$$(x-y)^{2}=9$$

$$x-y=\sqrt{9}$$

$$x-y=3$$ _______ (3)

On adding (1) & (3)

x+y=1

x-y=3

_________

2x=14

x=14/2

x=7

putting x=7 in equation _____ (1)

x+y=11

$$\pi +y=11$$

$$y=11-7$$

$$y=4$$

$$\therefore $$ The two numbers are $$(x,y)=(7,4)$$