The sum of a positive number and its reciprocal is twice the difference of the…

Concept: To translate a word problem about the sum and difference of a number and its reciprocal into an equation, let the number be x, write the sum as x + 1/x and the difference as x − 1/x, translate the given relationship between the two into a single equation, and solve for x — keeping only the root(s) that satisfy any stated constraint such as positivity.

Application:

1. Let the positive number be x, so its reciprocal is 1/x.

2. The sum of the number and its reciprocal is x + 1/x, and their difference is x − 1/x.

3. The problem states that the sum equals twice the difference: x + 1/x = 2(x − 1/x).

4. Expand the right-hand side: x + 1/x = 2x − 2/x.

5. Move the 1/x terms to the left and the x terms to the right: 1/x + 2/x = 2x − x.

6. Combine like terms on each side: 3/x = x.

7. Multiply both sides by x: 3 = x2.

8. Since the number is positive, take the positive square root: x = √3.

Cross-check: With x = √3, the reciprocal 1/x = √3⁄3. Sum = √3 + √3⁄3 = 4√3⁄3, and difference = √3 − √3⁄3 = 2√3⁄3, so twice the difference = 4√3⁄3 — equal to the sum, confirming the value.

Result: The number is √3.