Which of the following will result in a rational answer?
multiplying p by a fraction
adding the square root of a non perfect square to a whole
adding the square root of a perfect square to p
multiplying a fraction by a repeating decimal
The Correct Answer and Explanation is:
The correct answer is:
Multiplying p by a fraction
Explanation (300+ words):
To determine which of the options results in a rational number, we must first understand what a rational number is.
A rational number is any number that can be expressed as a ratio of two integers (a fraction), such as ½, -3, 0.75 (which is 3/4), or 5 (which is 5/1). Rational numbers include:
- All integers
- Fractions
- Terminating decimals
- Repeating decimals
Let’s evaluate each option:
1. Multiplying p by a fraction
Assume p is a rational number (e.g., an integer or a fraction).
When you multiply a rational number by another rational number (a fraction is rational by definition), the result will also be rational.
Example:
If p = 6 and the fraction is 2/3, then:
6 × 2/3 = 12/3 = 4 → which is a rational number.
So, this will result in a rational number. ✅
2. Adding the square root of a non-perfect square to a whole number
The square root of a non-perfect square (like √2, √5, √10) is irrational, meaning it cannot be written as a ratio of two integers.
Adding an irrational number to a whole number (which is rational) results in an irrational number.
Example: 3 + √2 is irrational. ❌
3. Adding the square root of a perfect square to p
The square root of a perfect square (like √4 = 2, √9 = 3) is rational.
If p is rational, and we add another rational number to it, the result will be rational.
Example: p = ½, √9 = 3 → ½ + 3 = 3.5 → Rational ✅
This also results in a rational number.
However, between this and the first option, the question asked which will result in a rational number — and both this and the first one are correct if p is rational.
4. Multiplying a fraction by a repeating decimal
A repeating decimal is rational, and a fraction is rational.
Multiplying two rational numbers yields another rational number.
Example: ½ × 0.333… = 0.1666… → This is also rational. ✅
Final Determination:
While more than one option can lead to rational answers, the most universally reliable and simplest is:
✅ Multiplying p by a fraction — always gives a rational result if p is rational.