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The number rr is rational. Which statement about 10+r10 + r is true?\newlineChoices:\newline(A) 10+r10 + r is rational.\newline(B) 10+r10 + r is irrational.\newline(C) 10+r10 + r can be rational or irrational, depending on the value of rr.

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Q. The number rr is rational. Which statement about 10+r10 + r is true?\newlineChoices:\newline(A) 10+r10 + r is rational.\newline(B) 10+r10 + r is irrational.\newline(C) 10+r10 + r can be rational or irrational, depending on the value of rr.
  1. Identify Number Nature: Identify the nature of the number 1010.1010 is a whole number, which is also a rational number because it can be expressed as a fraction of two integers (101\frac{10}{1}).
  2. 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 pq\frac{p}{q} of two integers, with the denominator qq not equal to zero.
  3. 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=pqr = \frac{p}{q} and 10=10110 = \frac{10}{1}, then r+10=(pq)+(101)=p+10qq⋅1r + 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.
  4. Apply Property to Problem: Apply the property to the given problem.\newlineSince rr is given to be rational and 1010 is rational, their sum 10+r10 + r must also be rational.

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