Question

Suppose that p and q are rational numbers and r is an irrational number. Which expression represents a rational number?

A. p+r

B. p+pr

C. p+q

D. p+qr

Answer 1

To evaluate which expression represents a rational number, we need to analyze each option given that \( p \) and \( q \) are rational numbers, and \( r \) is an irrational number.

A. \( p + r \)
- This is the sum of a rational number \( p \) and an irrational number \( r \). The sum of a rational number and an irrational number is always irrational. So, this expression is **not** rational.

B. \( p + pr \)
- Here, \( pr \) is the product of a rational number \( p \) and an irrational number \( r \). The product of a non-zero rational number and an irrational number is irrational. Therefore, \( pr \) is irrational, and adding it to \( p \) (a rational number) gives us \( p + pr \), which is also irrational. So, this expression is **not** rational.

C. \( p + q \)
- This is the sum of two rational numbers \( p \) and \( q \). The sum of two rational numbers is always rational. Therefore, this expression is rational.

D. \( p + qr \)
- Here, \( qr \) is the product of a rational number \( q \) and an irrational number \( r \). As mentioned earlier, this product is irrational. Thus, adding it to \( p \) (a rational number) results in \( p + qr \), which is also irrational. So, this expression is **not** rational.

Based on the analysis, the expression that represents a rational number is:

**C. \( p + q \)**