Final answer:
To find the number that produces a rational number when added to 1/5, we can represent the number as x and set up the equation as x + 1/5 = a/b, where a and b are integers. We then find a common denominator, subtract 1/5 from both sides, simplify the right side, and solve for x by dividing both sides by 5.
Step-by-step explanation:
When a number produces a rational number when added to 1/5, it means that the result is a fraction where the numerator and denominator are both integers. To find this number, we can express it as a fraction with a common denominator of 5. Let's represent the number as x. So, the equation would be x + 1/5 = a/b, where a and b are integers. To get a common denominator, we multiply x by 5, resulting in 5x + 1/5 = a/b. Now, the equation has a common denominator of 5, and we can continue solving for x.
Next, we subtract 1/5 from both sides of the equation to isolate 5x, giving us 5x = a/b - 1/5. To simplify the right side of the equation, we find a common denominator by multiplying 1/5 by b/b, which gives us b/5b. So, the equation becomes 5x = (a - b)/5b. To solve for x, we divide both sides of the equation by 5, resulting in x = (a - b)/(5b * 5).