Question

change each repeating decimal to a ratio of two integers. $$ 0.217171717 \ldots $$

Short answer

The repeating decimal \(0.217171717\ldots\) is equal to \(\frac{217}{999}\).

Step-by-step solution

Understand the Repeating Part

Identify the repeating block in the decimal. In our case, the repeating section is "171" following the non-repeating part "2".

Set the Decimal as a Variable

Let \( x = 0.2171717\ldots \). This way, we can manipulate the decimal to eventually remove the repeating section.

Multiply to Shift the Decimal

Multiply \( x \) by 1000 to shift the decimal point right, covering the entire repeating block: \( 1000x = 217.171717\ldots \).

Subtract the Two Equations

Subtract the original equation \( x = 0.2171717\ldots \) from the new equation \( 1000x = 217.171717\ldots \):\( 1000x - x = 217.171717\ldots - 0.2171717\ldots \)Simplify this to:\( 999x = 217 \).

Solve for x

Divide both sides of the equation by 999 to solve for \( x \):\[ x = \frac{217}{999} \].

Simplify the Fraction if Possible

Check if \( \frac{217}{999} \) can be simplified by finding the greatest common divisor (GCD) of 217 and 999. Since the GCD is 1, the fraction is already in its simplest form.

Key concepts

Rational Numbers

Rational numbers are an essential part of mathematics. They are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. This concept forms a bridge between the familiar integers and the more complex real numbers.
If a number can be written as a fraction, such as \( \frac{1}{2} \) or \( \frac{-3}{4} \), it is a rational number. Often, when converting repeating decimals to fractions, we're essentially proving their rationality. For example, the decimal \(0.2171717\ldots\) is a repeating decimal representing a rational number.
Rational numbers are significant because they elegantly represent many real-world quantities. Whether in finance or measurement, understanding how to work with these numbers is key to solving various practical problems.

• They offer precise expression for fractions and repeating decimals.
• Easy to manipulate within addition, subtraction, multiplication, and division.

Fractions

Fractions are numbers expressed as the ratio of two integers. They look like \( \frac{a}{b} \), where \(a\) is the numerator and \(b\) is the denominator, and \(beq0\). Understanding fractions is vital for converting repeating decimals into rational expressions.
When dealing with repeating decimals like \(0.2171717\ldots\), we recognize that this is a number that can be expressed as a fraction. Through algebraic manipulation, we find that this decimal is equivalent to the fraction \( \frac{217}{999} \).
The process of converting involves a few key steps:

• Identify the repeating block in the decimal.
• Use multiplication to align the decimals and isolate the repeating part.
• Subtraction to simplify and eliminate the decimal form.

Understanding and simplifying fractions is crucial, as seen in our exercise, where the fraction \( \frac{217}{999} \) was already in its simplest form.

Algebraic Manipulation

Algebraic manipulation is a method used to rearrange equations to simplify or solve them. When it comes to converting repeating decimals into fractions, this skill is invaluable.
The key process is setting an equation for the repeating decimal, as done with \( x = 0.2171717\ldots \), and using algebraic operations like multiplication and subtraction to simplify the decimal. Here's how it works in our exercise:

• Assign the repeating decimal to a variable \(x\).
• Multiply by a power of 10 to shift the decimal point, aligning the repeating parts.
• Subtract the equations to eliminate repeating decimal parts.
• Solve for \(x\) to find the fractional form.

Algebraic manipulation provides a systematic approach to handle complex expressions. This strengthens problem-solving skills and enhances mathematical efficiency, which is crucial for students to master.