Question

10 marisol divides to find the decimal form of \\(\frac{5}{12}\\). her work is shown. marisol says that \\(\frac{5}{12}\\) is not a rational number because the decimal form neither terminates nor repeats. is marisol correct? explain your reasoning. show your work.

Explanation:

Step1: Recall the definition of a rational number

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$. Also, the decimal form of a rational number either terminates or repeats.

Step2: Analyze the fraction $\frac{5}{12}$

First, check if $\frac{5}{12}$ is a rational number by the definition. Since $5$ and $12$ are integers and $12
eq0$, $\frac{5}{12}$ is in the form $\frac{p}{q}$ with integer $p = 5$ and $q=12$. So by definition, it should be a rational number.

Step3: Perform the division to find the decimal form

Let's divide $5$ by $12$:
\[

$$\begin{align*} 5\div12&=0.41666\cdots \end{align*}$$

\]
When we do the long division (as shown in the problem's work, after the remainder $8$ we would bring down a $0$ to get $80$, and $12\times6 = 72$, remainder $8$ again. So the remainder $8$ repeats, which means the digit $6$ will repeat indefinitely. So the decimal form is $0.41\overline{6}$, which is a repeating decimal.

Step4: Conclude about Marisol's statement

Marisol said the decimal neither terminates nor repeats, but we see that the decimal form of $\frac{5}{12}$ is $0.41\overline{6}$, which is a repeating decimal. And since $\frac{5}{12}$ can be written as $\frac{5}{12}$ (with integer numerator and non - zero integer denominator), it is a rational number. So Marisol is incorrect.

Answer:

Marisol is not correct. A rational number is defined as a number that can be written as $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. Since $5$ and $12$ are integers and $12
eq0$, $\frac{5}{12}$ is a rational number. When we divide $5$ by $12$, we get $0.41666\cdots$ (or $0.41\overline{6}$), which is a repeating decimal (the digit $6$ repeats). The decimal form of a rational number either terminates or repeats, and here the decimal repeats, so $\frac{5}{12}$ is a rational number.