Question

The difference of two numbers is 4 and the difference of their reciprocal is 4/21. Find the numbers

Answer 1

Answer

Given,

• Difference of two numbers is '4'

• Difference of their reciprocal is 4/21.

Two find,

The two numbers

Solution,

Let the numbers be 'x' & 'y'

Let the larger number be 'x'

According to the question :

x - y = 4

x = 4 + y -----------(1)

Reciprocal of 'x' & 'y' will be:

$\implies \rm \frac { 1 } { x } and \frac { 1 } { y }$

According to condition 2

Reciprocal of larger number will be smaller than the smaller number hence

$\implies \rm \frac { 1 } { y} - \frac { 1 } { </p><p>x} = \frac { 4 } { 21 }$

$\implies \rm \frac { x-y } { xy } = \frac { 4 } { 21 }$

Cross multiplying ,

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

Substituting the value of x we got from (1)

We get,

$\implies \rm 4(4+y)y = -21y + 21(4+y)$

$\implies \rm 16 + 4y^2 = -21y + 84 + 21y$

Dividing the equation by '4' We get,

$\implies \rm y^2 + 4y - 21 = 0$

Here We got a quadratic equation.

Here We got a quadratic equation.Let us solve it by splitting the middle term method

$\implies \rm y^2 + 7y - 3y - 21= 0$

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

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

• y = -7
• y = 3

Here if y = -7 then by equation (1)

We get x = -3

Here if y = 3 then by equation (1)

We get x = 7

Hence the numbers would be

• -7 , -3
• 3 , 7