Question

The difference of two natural no. is 5 and the difference of their reciprocal is 5/14 find the no

Answer 1

Let the two natural number be x and y

Given x - y = 5

and

$\frac{1}{y} - \frac{1}{x} = \frac{5}{14}$

Since, y is less than x, so reciprocal of y would be greater than reciprocal of x.

Let's solve the second equation further

$\frac{1}{y} - \frac{1}{x} = \frac{5}{14} \\ \\ = > \frac{x}{xy} - \frac{y}{xy} = \frac{5}{14} \\ \\ = > \frac{x - y}{xy} = \frac{5}{14} \\ \\ = > \frac{ 5}{xy} = \frac{5}{14}$

since, (x - y) = 5

Further solving, we get

$5xy = 14 \times 5 \\ \\ = > xy = \frac{14 \times 5}{5} \\ \\ = > xy = 14$

Now, x - y = 5

So, square both the sides

(x - y)² = 5²

→ x² + y² - 2xy = 25

→ x² + y² -2(14) = 25

(since, xy = 14)

→ x² + y² - 28 = 25

→ x² + y² = 25 + 28

→ x² + y² = 53

Now, add 2xy on both sides

→ x² + y² + 2xy = 53 + 2xy

→ (x + y)² = 53 + 2(14)

→ (x + y)² = 53 + 28

→ (x + y)² = 81

→ x + y = √81 = 9

So, x - y = 5 and x + y = 9

Add the two equations

x - y + x + y = 5 + 9

→ 2x = 14

→ x = 14/2

→ x = 7

Hence, x - y = 5

→ 7 - y = 5

→ y = 2

So the numbers are 7 and 2