Question

which numbers are rational? the dots, …, indicate that the pattern continues.\\(\boldsymbol{\text{i. } 3.3333333}\\\\ \boldsymbol{\text{ii. } 3.303030333\dots}\\\\ \boldsymbol{\text{iii. } 3.456}\\\\ \text{which numbers are rational?}\\\\ \text{a. i only}\\\\ \text{b. i and iii}\\\\ \text{c. ii and iii}\\\\ \text{d. iii only}\\\\ \text{e. ii only}\\\\ \text{f. none of these}

Explanation:

Step1: Recall Rational Number Definition

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$. This includes terminating decimals, repeating decimals, and integers (since integers can be written as $\frac{n}{1}$).

Step2: Analyze Number I (3.3333333)

Number I is a terminating decimal (or a repeating decimal with all 3s, but here it's written as a finite decimal). It can be written as $\frac{33333333}{10000000}$, so it's rational.

Step3: Analyze Number II (3.303303333...)

Number II has a non - repeating, non - terminating decimal expansion (the pattern of 303 and 333... doesn't repeat in a fixed cycle). So it's an irrational number.

Step4: Analyze Number III (3.456)

Number III is a terminating decimal. It can be written as $\frac{3456}{1000}=\frac{432}{125}$, so it's rational.

Step5: Determine Rational Numbers

From the above analysis, numbers I and III are rational.

Answer:

C. I and III