Riya's approach to explaining the difference between rational and irrational numbers to Ankur could involve showing how these numbers can be expressed as fractions or using decimal expansions.

Explanation Method:

1. Rational Numbers: These numbers can be expressed as the ratio of two integers, i.e., in the form of a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$. Additionally, when written as decimals, rational numbers either terminate (end) or repeat in a regular pattern.

2. Irrational Numbers: These numbers cannot be written as a fraction of two integers. When written as decimals, they do not terminate and do not repeat in any regular pattern. Common examples of irrational numbers are $\pi$ and $\sqrt{2}$.

Step-by-Step Classification of the Numbers:

1. 1.125:

   • Decimal Form: 1.125 is a terminating decimal (it ends after the third decimal place).
   • Fraction Form: It can be written as $\frac{1125}{1000} = \frac{9}{8}$, which is a fraction of two integers.
   • Conclusion: Since it is a terminating decimal and can be expressed as a fraction, 1.125 is a rational number.

2. π (Pi):

   • Decimal Form: The decimal expansion of $\pi$ starts as 3.14159... and continues indefinitely without any repeating pattern.
   • Fraction Form: $\pi$ cannot be exactly written as a fraction of two integers.
   • Conclusion: Since $\pi$ has a non-terminating, non-repeating decimal expansion and cannot be written as a fraction, $\pi$ is an irrational number.

3. 1.67676767... (repeating):

   • Decimal Form: The decimal 1.67676767... repeats the sequence "67" indefinitely.
   • Fraction Form: A repeating decimal like this can be expressed as a fraction. In this case, it can be written as $\frac{167}{99}$.
   • Conclusion: Since it is a repeating decimal and can be written as a fraction, 1.67676767... is a rational number.

4. $\frac{22}{7}$:

   • Fraction Form: The number $\frac{22}{7}$ is already a fraction of two integers.
   • Decimal Form: When written as a decimal, $\frac{22}{7} = 3.142857142857...$ and repeats the sequence "142857" indefinitely.
   • Conclusion: Since it can be expressed as a fraction and has a repeating decimal, $\frac{22}{7}$ is a rational number.

Summary of Classification:

1. 1.125: Rational
2. $\pi$: Irrational
3. 1.67676767...: Rational
4. $\frac{22}{7}$: Rational

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Would you like further clarification on any of these points or more examples? Here are some related questions you might find interesting:

1. What are some other examples of irrational numbers besides $\pi$?
2. How can we convert repeating decimals into fractions?
3. How does the decimal expansion of a number help in determining its rationality?
4. What happens when a rational number is raised to a power?
5. Can an irrational number ever be represented as a fraction?

Tip: To quickly identify whether a decimal is rational, check if it terminates or repeats. If it does either, it is rational!