Show that the following number is rational by writing it as a ratio of two integers.

0.71717171…

The number \( 0.71717171\ldots \) is a repeating decimal, and we can express it as a fraction, proving that it is rational. Here's how we do it step by step:

Let \( x = 0.71717171\ldots \)

### Step 1: Multiply both sides by 100
Since the decimal repeats every two digits, we multiply by \( 100 \) to shift the decimal point two places to the right:

\[
100x = 71.71717171\ldots
\]

### Step 2: Subtract the original equation from this new equation
Now subtract \( x = 0.71717171\ldots \) from \( 100x = 71.71717171\ldots \):

\[
100x - x = 71.71717171\ldots - 0.71717171\ldots
\]

This simplifies to:

\[
99x = 71
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \) by dividing both sides by 99:

\[
x = \frac{71}{99}
\]

Thus, \( 0.71717171\ldots \) is equal to \( \frac{71}{99} \), which is a ratio of two integers, so it is a rational number.

Would you like further details or clarifications?

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Here are 5 related questions for further exploration:
1. How can we convert other repeating decimals into fractions?
2. What is the difference between rational and irrational numbers?
3. Can a repeating decimal ever be an irrational number?
4. How would you convert \( 0.25252525\ldots \) into a fraction?
5. What properties do rational numbers have in relation to repeating decimals?

**Tip:** A decimal that repeats after a fixed number of digits is always a rational number.

Show that the following number is rational by writing it as a ratio of two integers: 0.71717171…

Learn how to prove that the repeating decimal 0.71717171… is a rational number by converting it into the fraction 71/99. This step-by-step guide covers key algebraic techniques and is suitable for students in Grades 8-10.

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