The number r is rational. Which statement about 10 + r is true? Choices: (A)10 + r is rational. (B)10 + r is irrational. (C)10 + r can be rational or irrational, depending on the value of r.

Identify Number Nature: Identify the nature of the number 10. 10 is a whole number, which is also a rational number because it can be expressed as a fraction of two integers (10/1).Define Rational Number: Understand the definition of a rational number. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.Analyze Sum of Rationals: Analyze the sum of two rational numbers. The sum of two rational numbers is always rational. This is because if r = p/q and 10 = 10/1, then r + 10 = (p/q) + (10/1) = (p + 10q)/(q*1), which is the quotient of two integers, hence rational.Apply Property to Problem: Apply the property to the given problem. Since r is given to be rational and 10 is rational, their sum 10 + r must also be rational.

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Math Problems

Algebra 1

Properties of operations on rational and irrational numbers

Full solution

Q. The number

r

is rational. Which statement about

10 + r

is true?

\newline

Choices:

\newline

(A)

10 + r

is rational.

\newline

(B)

10 + r

is irrational.

\newline

(C)

10 + r

can be rational or irrational, depending on the value of

r

Identify Number Nature: Identify the nature of the number $10$ . $10$ is a whole number, which is also a rational number because it can be expressed as a fraction of two integers ( $\frac{10}{1}$ ).
Define Rational Number: Understand the definition of a rational number. A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, with the denominator $q$ not equal to zero.
Analyze Sum of Rationals: Analyze the sum of two rational numbers. The sum of two rational numbers is always rational. This is because if $r = \frac{p}{q}$ and $10 = \frac{10}{1}$ , then $r + 10 = \left(\frac{p}{q}\right) + \left(\frac{10}{1}\right) = \frac{p + 10q}{q \cdot 1}$ , which is the quotient of two integers, hence rational.
Apply Property to Problem: Apply the property to the given problem. $\newline$ Since $r$ is given to be rational and $10$ is rational, their sum $10 + r$ must also be rational.