Problem 29 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$5, \overline{6}$$

Step-by-Step Solution

Answer

The decimal 5, \overline{6} is equivalent to the mixed number 5 \frac{2}{3}.

Step 1: Recognize the Repeating Decimal

The number given is a decimal with a repeating part, specifically \(5, \overline{6}\). This means 5.666... where "6" is the repeating digit.

Step 2: Represent the Decimal as an Equation

Let \(x = 5.666...\). Since the repeating block (6) is one digit long, multiply both sides by 10: \(10x = 56.666...\).

Step 3: Subtract to Eliminate the Repeating Part

Subtract the original \(x = 5.666...\) from the new equation \(10x = 56.666...\): \[10x - x = 56.666... - 5.666...\]\[9x = 51\]

Step 4: Solve for x

Divide both sides by 9 to solve for \(x\): \[x = \frac{51}{9}\]Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3: \[x = \frac{51 \div 3}{9 \div 3} = \frac{17}{3}\]

Step 5: Convert to a Mixed Number

\(\frac{17}{3}\) is an improper fraction. To convert it to a mixed number, divide 17 by 3. 17 divided by 3 is 5 with a remainder of 2. Therefore, \[\frac{17}{3} = 5 \frac{2}{3}\]

Key Concepts

Mixed Numbers

What are mixed numbers? Simply put, a mixed number consists of a whole number and a fraction combined. When you see something like \(5 \frac{2}{3}\), it's a mixed number. It tells us we have 5 whole parts and an additional 2/3 of a part. This is especially useful when you're dealing with improper fractions, which might feel a bit unwieldy.How do you convert an improper fraction to a mixed number? Here's how:

• First, divide the numerator (the top number) by the denominator (the bottom number).
• The quotient (the result of the division) becomes the whole number part of the mixed number.
• The remainder of the division becomes the new numerator of the fractional part, still using the original denominator.

Using the exercise's example, \(\frac{17}{3}\) means 17 divided by 3. The quotient is 5 and the remainder is 2, thus the mixed number is \(5 \frac{2}{3}\). This makes mixed numbers not only more intuitive but also easier to visualize.

Improper Fractions

An improper fraction occurs when the numerator is larger than the denominator. Put another way, the fraction equates to a value greater than 1. For example, \(\frac{17}{3}\) from the exercise is an improper fraction.So why do we use improper fractions? They're particularly useful when you're performing arithmetic operations. When it's time to add, subtract, multiply, or divide fractions, improper fractions simplify the process because they eliminate the need to juggle whole numbers and parts.Converting from improper to mixed fractions is straightforward. You just divide the numerator by the denominator, as explained in the mixed numbers section. Remember:

• Improper fractions are not incorrect; they're just another form of expressing a number.
• Sometimes, it may be more convenient to work with improper fractions than with mixed numbers, especially in algebraic expressions.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) or Greatest Common Factor (GCF) is the largest number that divides exactly into two or more numbers. Finding the GCD is particularly useful when simplifying fractions.Why is the GCD important in fractions? Let's break down the steps:

• Once you have a fraction, check for common factors between the numerator and denominator.
• Identifying the greatest of these factors is your GCD.
• Then, divide both the numerator and the denominator by this GCD to simplify the fraction.

The exercise provided \(\frac{51}{9}\). Here, both 51 and 9 can be divided by 3, their GCD, reducing the fraction to \(\frac{17}{3}\). By simplifying, the fraction is easier to understand and use in further calculations.Using the GCD not only simplifies the fractions but also makes calculations more manageable without changing their value.