Find a positive number which when increased by 16 is equal to 80 times the…

Concept:

A word problem that relates a number to its own reciprocal translates into an equation of the form x + a = b/x. Clearing the fraction by multiplying both sides by x converts it into a quadratic equation, which is then solved by rearranging into standard form and factoring (or using the quadratic formula). Since a quadratic has two roots, any extra condition stated in the problem (such as the number being positive) is used to pick the valid root.

Application:

1. Let the number be x. The problem statement translates to the equation x + 16 = 80 × (1/x).

2. Multiply both sides by x (x is not 0): x2 + 16x = 80.

3. Bring all terms to one side: x2 + 16x − 80 = 0.

4. Factor the quadratic: x2 + 16x − 80 = (x − 4)(x + 20) = 0.

5. So x = 4 or x = −20. The question asks for a positive number, therefore x = 4.

Check: 4 + 16 = 20 and 80 × (1/4) = 20, so both sides match and 4 satisfies the equation. The other root, −20, is negative and does not meet the problem's requirement.

Answer: 4