Final answer:
To find the two numbers whose sum with their respective reciprocal is 82/9, we can set up a quadratic equation and solve for x using the quadratic formula. There will be two solutions representing the larger and smaller values.
Step-by-step explanation:
The question involves finding two numbers when given that the sum of a number and its reciprocal is 82/9. Let's denote the unknown number as x. Thus, we have the equation x + 1/x = 82/9. To solve for x, we first multiply both sides by x to get rid of the fraction, yielding x² + 1 = x(82/9). Rearranging the terms, we have x² - (82/9)x + 1 = 0. This is a quadratic equation, and we can solve it by applying the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a), where a = 1, b = -82/9, and c = 1. After computing, we find two solutions which represent the smaller and larger values asked for in the question.
Using numeric methods or further algebraic manipulation, we can identify these two numbers, remembering that the smaller value is less than 1 while the larger value is greater than 1, according to the properties of reciprocals given in the question information.