Question

The difference of two numbers is 4. If the difference of their reciprocals is 4/21, find the two numbers.

Answer 1

Answer:

the numbers are 3 and 7

refer to the attachment for the solution.

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Answer 2

Answer:

The required numbers are - 3 and - 7.

$\sf\:\:\:\:\:\:\:OR$

The required numbers are 7 and 3.

Step-by-step-explanation:

Let the greater number be x.

And the smaller number be y.

From the first condition,

x - y = 4

$\implies\:$ x = y + 4 - - ( 1 )

From the second condition,

1 / x - 1 / y = 4 / 21

$\implies\:$ y - x / xy = 4 / 21

$\implies\:$ 21 ( y - x ) = 4xy

$\implies\:$ 21y - 21x = 4xy - - ( 2 )

Now, by substituting the value of x from the equation ( 1 ) in equation ( 2 ), we get,

21y - 21x = 4xy - - ( 2 )

$\implies\:$ 21y - 21 ( y + 4 ) = 4 × ( y + 4 ) × y

$\implies\:$ 21y - 21y - 84 = ( 4y + 16 ) × y

$\implies\:$ - 84 = 4y² + 16y

$\implies\:$ 4y² + 16y + 84 = 0

$\implies\:$ y² + 4y + 21 = 0 - - [ Dividing both sides by 4 ]

$\implies\:$ y² + 7y - 3y + 21 = 0

$\implies\:$ y ( y + 7 ) - 3 ( y + 7 ) = 0

$\implies\:$ ( y + 7 ) ( y - 3 ) = 0

$\implies\:$ y + 7 = 0 $\:\:\:\sf\:or\:\:\:$ y - 3 = 0

$\implies\boxed{\red{\sf\:y\:=\:-\:7}}\:\:\:\sf\:or\:\:\:\boxed{\red{\sf\:y\:=\:3}}$

Now, by substituting y = - 7 in equation ( 1 ), we get,

x = y + 4

$\implies\:$ x = - 7 + 4

$\implies\boxed{\red{\sf\:x\:=\:-\:3}}$

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

x = y + 4

$\implies\:$ x = 3 + 4

$\implies\boxed{\red{\sf\:x\:=\:7}}$

The required numbers are - 3 and - 7.

$\sf\:\:\:\:\:\:\:OR$

The required numbers are 7 and 3.