Which number produces a rational number when added to 1/5?
The Correct Answer and Explanation is:
The number that produces a rational number when added to 15\frac{1}{5}51 is any rational number. For a specific example, consider the number 25\frac{2}{5}52.
Correct Answer: 25\frac{2}{5}52
Explanation
A rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero. This includes fractions such as 12\frac{1}{2}21, whole numbers like 3 (which is 31\frac{3}{1}13), and even negative numbers like −74-\frac{7}{4}−47.
The sum of two rational numbers is always a rational number. This is due to the properties of rational numbers under addition. When two rational numbers are added, their sum can be expressed as the quotient of two integers, which fits the definition of a rational number.
The number 15\frac{1}{5}51 is rational because it can be written as the ratio of two integers: 1 and 5. To produce a rational result after adding another number to 15\frac{1}{5}51, the added number must also be rational. For example, take 25\frac{2}{5}52, which is rational. Adding 15+25=35\frac{1}{5} + \frac{2}{5} = \frac{3}{5}51+52=53, which is also rational. This shows that the operation maintains rationality.
If the added number were irrational, such as 2\sqrt{2}2, then the result 15+2\frac{1}{5} + \sqrt{2}51+2 would be irrational, since adding a rational number to an irrational one always results in an irrational number.
Therefore, to ensure that the sum is rational, the added number must itself be rational. The choice 25\frac{2}{5}52 is just one example. Many other rational numbers—positive, negative, fractions, or whole numbers—could be added to 15\frac{1}{5}51 to yield a rational sum.
In conclusion, any rational number added to 15\frac{1}{5}51 will produce a rational number. One valid example is 25\frac{2}{5}52.
