Problem 20 - Algebra 2 and Trigonometry

Question

In \(13-22,\) write each decimal as a common fraction. $$ 0.5 \overline{7} $$

Step-by-Step Solution

Answer

The decimal \(0.5\overline{7}\) is \(\frac{26}{45}\) as a fraction.

Step 1: Express the Decimal with Variables

Let \( x = 0.57777\ldots \). This represents the decimal as repeating.

Step 2: Set Up the Equation for Repetition

Multiply both sides by 10 to shift the decimal point to move the repetition: \[ 10x = 5.7777\ldots \] and by 100 to make another equation: \[ 100x = 57.7777\ldots \]

Step 3: Subtract the Equations

Subtract the first equation from the second to eliminate the repeating part: \[ 100x - 10x = 57.7777\ldots - 5.7777\ldots \]This simplifies to \[ 90x = 52 \]

Step 4: Solve for x

Divide both sides by 90 to solve for \( x \):\[ x = \frac{52}{90} \]This simplifies to \[ x = \frac{26}{45} \] after dividing the numerator and the denominator by 2.

Key Concepts

Repeating Decimals

When you have a decimal number where one or more numbers repeat indefinitely, it's known as a repeating decimal. For example, in the decimal \(0.5\overline{7}\), the digit 7 repeats endlessly. Repeating decimals can be tricky, but there's a systematic way to convert them into fractions.

To start, you associate a variable, like \(x\), with your decimal number. This is useful for setting up algebraic equations to help you find an equivalent fraction. With \(0.57777\ldots\) as an example, let \(x = 0.57777\ldots\). By thinking of the repeating part as going to infinity, you prepare yourself for the algebraic manipulation needed to remove the endlessly repeated portion.

This repetition pattern allows for subtraction after aligning decimal points, which can help eliminate the repeated portion, simplifying the conversion to a fraction.

Fraction Simplification

Once you express a repeating decimal as a fraction, it's essential to simplify it. Simplifying helps you reach the simplest, easy-to-understand form of your fraction. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing them both by this number.

In the previous example, after expressing \(0.5\overline{7}\) as \(\frac{52}{90}\), you simplify it by finding the GCD of 52 and 90, which is 2. Divide both the numerator and the denominator by 2 to get \(\frac{26}{45}\). This process ensures your fraction is in its most reduced form.

Simplification is crucial because it makes fractions easier to interpret, and comparisons between fractions become straightforward when they're in simpler form.

• Find GCD of numerator and denominator
• Divide both by the GCD
• Check if the resulting fraction is fully reduced

Algebraic Expressions

Algebraic expressions play a key role when dealing with repeating decimals. They involve using variables to represent numbers, which can simplify complex mathematical concepts.

In converting repeating decimals to fractions, you set your repeating decimal equal to a variable, such as \(x\). This helps in forming equations that allow you to manipulate and isolate parts of the number that repeat.

For example, if you have \(x = 0.57777\ldots\), you multiply by powers of 10 to shift the decimals and form another equation, such as \(10x = 5.7777\ldots\) and \(100x = 57.7777\ldots\). Subtract these equations to remove the repeating part by producing a simpler expression \(90x = 52\). Solving this equation gives the fraction.

This manipulation highlights how algebra makes it easier to visualize and solve problems that would otherwise be more complex.