Problem 53 - Algebra A Combined Function

Question

Add or subtract as indicated. See Examples 15 through \(20,22,\) and \(23 .\) $$ 8 \frac{1}{8}-6 \frac{3}{8} $$

Step-by-Step Solution

Answer

The result is \(1 \frac{3}{4}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To subtract mixed numbers, first convert them to improper fractions. For the number \(8 \frac{1}{8}\): Multiply the whole number 8 by the denominator 8, then add the numerator 1. This gives \(8 \times 8 + 1 = 65\). Thus, \(8 \frac{1}{8} = \frac{65}{8}\). Similarly, for \(6 \frac{3}{8}\), multiply 6 by 8 and add 3, giving \(6 \times 8 + 3 = 51\). Hence, \(6 \frac{3}{8} = \frac{51}{8}\).

Step 2: Subtract the Improper Fractions

Now, subtract the improper fractions obtained from the previous step. Since both fractions have the same denominator, simply subtract the numerators: \(\frac{65}{8} - \frac{51}{8} = \frac{65 - 51}{8} = \frac{14}{8}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{14}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \(\frac{14}{8} = \frac{14 \div 2}{8 \div 2} = \frac{7}{4}\).

Step 4: Convert the Improper Fraction to a Mixed Number

Convert the improper fraction \(\frac{7}{4}\) back to a mixed number. Divide 7 by 4: 7 divided by 4 is 1 with a remainder of 3. Thus, the mixed number is \(1 \frac{3}{4}\).

Key Concepts

Understanding Improper Fractions

Improper fractions are those where the numerator is greater than or equal to the denominator. This can seem daunting at first, but they're quite useful, especially in calculations like addition or subtraction of fractions. For instance, in our exercise, we started with mixed numbers, which were then converted into improper fractions. This made it easier to perform our operations. Here's how it works:

• For a mixed number like \(8 \frac{1}{8}\), you multiply the whole number by the denominator (8) and add the numerator (1). Thus, \(8 \times 8 + 1 = 65\).
• The result, \(\frac{65}{8}\), is an improper fraction.

Improper fractions are crucial in easing calculations by avoiding mixed numbers during arithmetic operations.

Mastering Fraction Subtraction

Subtracting fractions is a fundamental skill that often trips up learners due to varying denominations. However, with improper fractions, the process simplifies significantly. Whenever you're subtracting fractions, ensure they have the same denominator. This is highlighted perfectly in our example:

• The fractions \(\frac{65}{8}\) and \(\frac{51}{8}\) share the common denominator of 8. This means we can easily subtract them.
• By subtracting the numerators directly, we get \(\frac{65 - 51}{8} = \frac{14}{8}\).

A common denominator allows straightforward subtraction directly between the numerators, emphasizing the utility of converting to improper fractions first.

Simplification of Fractions Explained

Simplifying fractions is the process of reducing them to their simplest form, making calculations and comparisons more manageable. After subtraction in our example, we got \(\frac{14}{8}\). This fraction isn't in its simplest form yet. To simplify, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 14 and 8 is 2.
• Divide both the numerator and the denominator by the GCD: \(\frac{14 \div 2}{8 \div 2} = \frac{7}{4}\).

Simplifying not only makes fractions easier to work with but also often necessary in achieving correct final answers.

Converting Fractions into Mixed Numbers

Sometimes, fractions are easier to understand when expressed as mixed numbers, particularly when tackling real-world problems. Converting improper fractions like \(\frac{7}{4}\) into mixed numbers involves a simple division process:

• Divide the numerator (7) by the denominator (4). The quotient forms the whole number, and the remainder becomes the new numerator.
• In our example, \(7 \div 4 = 1\) with a remainder of 3. This translates to \(1 \frac{3}{4}\) as the mixed number.
• The result, \(1 \frac{3}{4}\), is easier to interpret at a glance than \(\frac{7}{4}\).

Converting back to mixed numbers helps in clearly understanding and communicating fractional results.