Numbers Finder by Difference and Reciprocal Difference

Input the difference and reciprocal difference of two numbers to calculate them quickly and easily.

Calculate Two Numbers Using Their Difference and Reciprocal Difference

Result

How to Calculate Two Numbers Using Their Difference and Reciprocal Difference

Let \( x \) and \( y \) be the two numbers. Assume their difference \( D \) and reciprocal difference \( R \) are known. The initial formulas are: Difference of the numbers: \( x - y = D \) Reciprocal difference: \(\frac{1}{y} - \frac{1}{x} = R\)

Expand the Reciprocal Difference Formula: \( \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy} \) This simplifies to: \( R = \frac{D}{xy} \)
Solve for \( xy \): \( xy = \frac{D}{R} \)
Calculate \( x + y \) Using the Sum and Product Formula: \( (x + y)^2 = (x - y)^2 + 4xy \) Substituting the known values: \( x + y = \sqrt{D^2 + 4 \cdot \frac{D}{R}} \)
Solve for \( x \) and \( y \) Using the Sum and Difference Formula: \( x = \frac{(x + y) + (x - y)}{2} = \frac{(x + y + D)}{2} \) \( y = \frac{(x + y) - (x - y)}{2} = \frac{(x + y - D)}{2} \)

Examples

Example 1: Find two numbers whose difference is 4 and reciprocal difference is \( \frac{4}{21} \).

Solution:

1. Calculate \( xy \):

\( xy = \frac{D}{R} = \frac{4}{\frac{4}{21}} = 21 \)

2. Calculate \( x + y \):

\( x + y = \sqrt{D^2 + 4 \cdot xy} = \sqrt{4^2 + 4 \cdot 21} = \sqrt{16 + 84} = \sqrt{100} = 10 \)

3. Solve for \( x \) and \( y \):

\( x = \frac{10 + 4}{2} = 7 \)

\( y = \frac{10 - 4}{2} = 3 \)

Result: The two numbers are 7 and 3.

Example 2: Find two numbers whose difference is 5 and reciprocal difference is \( \frac{1}{10} \).

Solution:

1. Calculate \( xy \):

\( xy = \frac{D}{R} = \frac{5}{\frac{1}{10}} = 50 \)

2. Calculate \( x + y \):

\( x + y = \sqrt{D^2 + 4 \cdot xy} = \sqrt{5^2 + 4 \cdot 50} = \sqrt{25 + 200} = \sqrt{225} = 15 \)

3. Solve for \( x \) and \( y \):

\( x = \frac{15 + 5}{2} = 10 \)

\( y = \frac{15 - 5}{2} = 5 \)

Result: The two numbers are 10 and 5.