Question

change each repeating decimal to a ratio of two integers. $$ 2.56565656 \ldots $$

Short answer

The repeating decimal 2.565656... is \( \frac{254}{99} \) as a ratio of integers.

Step-by-step solution

Set Up an Equation

Let's represent the repeating decimal as a variable. Let \( x = 2.56565656... \). This means that the decimal part repeats every two digits, i.e., '56' is the repeating block.

Multiply to Remove Decimals

To eliminate the repeating part, multiply both sides by 100 to shift the decimal point two places to the right: \( 100x = 256.565656... \). Now, notice that the decimal part on both \( x \) and \( 100x \) are the same: '56'.

Set Up Subtraction Equation

Subtract \( x = 2.565656... \) from \( 100x = 256.565656... \): \( 100x - x = 256.565656... - 2.565656... \). This simplifies to \( 99x = 254 \).

Solve for x

Divide both sides of the equation by 99 to solve for \( x \):\[ x = \frac{254}{99} \].

Simplify the Fraction

Check if the fraction \( \frac{254}{99} \) can be simplified. Since 254 and 99 have no common factors other than 1, this fraction is already in its simplest form.

Key concepts

Converting Decimals to Fractions

Converting repeating decimals to fractions is a useful skill. It helps us express numbers as rational numbers, which are easier to work with in math. Let's take the repeating decimal given: 2.565656...
To transform a repeating decimal into a fraction, start by representing the decimal with a variable.

• Assign the repeating decimal to a variable, say, let \( x = 2.565656... \) to represent it.
• To visualize, '56' is the repeating part in this decimal.

Next, manipulate the variable to eliminate the repeating part. This clever trick is crucial:

• Multiply \( x \) by a power of 10 that moves the repeating block past the decimal point.
• For a two-digit repeat, multiplying by 100 is suitable: \( 100x = 256.565656... \)

Notice both sides have the same repeating '56', setting the stage for subtraction and further manipulation.

Rational Numbers

Rational numbers are numbers that can be written as a ratio of two integers, such as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). They include fractions and the entire set of integers themselves.
When we convert repeating decimals to fractions, we are producing rational numbers.

• This exercise takes a repeating decimal and expresses it as a rational number.
• Converting allows us to use this form for various algebraic operations.

Rational numbers form the bedrock of simple arithmetic calculations and are fundamental in algebra. Understanding this concept aids in better comprehending the world of numbers.
This concept ensures that what initially seemed infinite can actually be expressed in finite terms.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to solve for unknowns. It's like unpacking a problem to reveal its underlying components.
In this exercise, we employed algebraic manipulation to transform the initial equation into a simpler form.

• From \( x = 2.565656... \) and \( 100x = 256.565656... \), notice how both have the decimal part '56' repeating?
• Set up a subtraction: \( 100x - x = 256.565656... - 2.565656... \)
• This gives \( 99x = 254 \); a simple equation to solve for \( x \).

This trick of multiplication followed by subtraction efficiently eliminates the repeating part, turning the decimal into a clean fraction.
Mastering these algebraic techniques empowers you not just in math, but it’s applicable in solving real-world problems too.

Simplifying Fractions

Simplifying fractions is the final step in converting decimals to fractions. It involves finding the simplest form of a fraction, whereby the numerator and the denominator share no common factors apart from 1.
For our solution, we have \( \frac{254}{99} \).

• Check for any common factors between the numerator and the denominator.
• If none exist, the fraction is already in its simplest form.

In this particular case, our fraction cannot be simplified further because 254 and 99 share no common factors, meaning \( \frac{254}{99} \) remains as is.
Simplifying fractions makes them easier to work with and can help avoid errors in subsequent calculations. It's an appreciated practice that presents numbers in their most digestible form, leading to a cleaner and clearer understanding.