QuestionIs $1.02 \overline{27}$ a rational number? Provide an explanation.

Studdy Solution

STEP 1

Assumptions1. A rational number is a number that can be expressed as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq0$ . . A number with a repeating decimal is a rational number if the repeating part can be expressed as a fraction.

STEP 2

We are given the number $1.02272727...$ , where $27$ is the repeating part. Let's denote this number as $x$ .
$x =1.02272727...$

STEP 3

To express $x$ as a fraction, we need to eliminate the repeating part. We can do this by multiplying $x$ by a power of10 that moves the decimal point to the right of the repeating part. Since the repeating part is two digits long, we multiply $x$ by $100$ .
$100x =102.272727...$

STEP 4

Now, subtract $x$ from $100x$ to eliminate the repeating part.
$100x - x =102.272727... -1.02272727...$

STEP 5

implify the left side of the equation.
$99x =102.272727... -1.02272727...$

STEP 6

implify the right side of the equation.
$99x =101.25$

STEP 7

Now, solve for $x$ by dividing both sides of the equation by $99$ .
$x = \frac{101.25}{99}$

STEP 8

implify the fraction on the right side of the equation.
$x = \frac{405}{396}$ Since $x$ can be expressed as a fraction where both the numerator and the denominator are integers, $x$ is a rational number.

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