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 determine which expression represents a rational number given that \( p \) and \( q \) are rational numbers and \( r \) is an irrational number, let's analyze each option.

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

B. \( p + pr \)
- The term \( pr \) (where \( p \) is rational and \( r \) is irrational) is irrational since the product of a nonzero rational number and an irrational number is irrational. Therefore, \( p + pr \) (rational + irrational) is also irrational.
- Thus, this expression is not rational.

C. \( p + q \)
- Both \( p \) and \( q \) are rational numbers. The sum of two rational numbers is rational.
- Thus, this expression is rational.

D. \( p + qr \)
- Here, \( qr \) (where \( q \) is rational and \( r \) is irrational) is irrational. Therefore, \( p + qr \) (rational + irrational) is also irrational.
- Thus, this expression is not rational.

The only expression that represents a rational number is:
**C. \( p + q \)**