Question

the sum of two number is 5 if sum of their reciprocal is 5 opon 6 then the number are

Answer 1

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Answer 2

Answer:

The required numbers are 3 & 2.

Step-by-step-explanation:

Let the first number be x.

And the second number be y.

From the first condition,

$\sf\:x\:+\:y\:=\:5\\\\\implies\red{\sf\:x\:=\:5\:-\:y}\sf\:\:\:-\:-\:(\:1\:)$

From the second condition,

$\sf\:\frac{1}{x}\:+\:\frac{1}{y}\:=\:\frac{5}{6}\\\\\implies\sf\:\dfrac{y\:+\:x}{xy}\:=\:\frac{5}{6}\\\\\implies\sf\:6\:(\:y\:+\:x\:)\:=\:5xy\\\\\implies\sf\:6x\:+\:6y\:=\:5xy\\\\\implies\sf\:6\:(\:5\:-\:y\:)\:+\:6y\:=\:5\:\times\:(\:5\:-\:y\:)\:\times\:y\:\:\:-\:-\:[\:From\:(\:1\:)\:]\\\\\implies\sf\:30\:-\:\cancel{6y}\:+\:\cancel{6y}\:=\:(\:25\:-\:5y\:)\:\times\:y\\\\\implies\sf\:30\:=\:25y\:-\:5y^{2}\\\\\implies\sf\:5y^{2}\:-\:25y\:+\:30\:=\:0\\\\\implies\sf\:y^{2}\:-\:5y\:+\:6\:=\:0\:\:\:-\:-\:-\:[\:Dividing\:both\:sides\:by\:5\:]\\\\\implies\sf\:y^{2}\:-\:3y\:-\:2y\:+\:6\:=\:0\\\\\implies\sf\:y\:(\:y\:-\:3\:)\:-\:2\:(\:y\:-\:3\:)\:=\:0\\\\\implies\sf\:(\:y\:-\:3\:)\:(\:y\:-\:2\:)\:=\:0\\\\\implies\sf\:y\:-\:3\:=\:0\:\:\:or\:\:\:y\:-\:2\:=\:0\\\\\implies\boxed{\red{\sf\:y\:=\:3\:}}\:\sf\:\:or\:\:\:\boxed{\red{\sf\:y\:=\:2}}$

Now, by substituting y = 3 in equation ( 1 ), we get,

$\sf\:x\:=\:5\:-\:y\:\:\:-\:-\:(\:1\:)\\\\\implies\sf\:x\:=\:5\:-\:3\\\\\implies\boxed{\red{\sf\:x\:=\:2}}$

Now, by substituting y = 2 in equation ( 1 ), we get,

$\sf\:x\:=\:5\:-\:y\:\:\:\:-\:-\:(\:1\:)\\\\\implies\sf\:x\:=\:5\:-\:2\\\\\implies\boxed{\red{\sf\:x\:=\:3}}$

The required numbers are 3 & 2.