Problem 7 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(9 \frac{1}{6}+2 \frac{5}{6}\)

Step-by-Step Solution

Answer

The sum of \(9 \frac{1}{6}\) and \(2 \frac{5}{6}\) is 12.

Step 1: Convert Mixed Numbers to Improper Fractions

The first step is to convert the mixed numbers into improper fractions. For the mixed number \(9 \frac{1}{6}\), multiply the whole number 9 by the denominator 6 and add the numerator 1: \((9 \times 6) + 1 = 55\), so it becomes \(\frac{55}{6}\). For \(2 \frac{5}{6}\), multiply the whole number 2 by the denominator 6 and add the numerator 5: \((2 \times 6) + 5 = 17\), so it becomes \(\frac{17}{6}\).

Step 2: Add the Improper Fractions

Both fractions have the same denominator, 6, so you can add the numerators directly: \(\frac{55}{6} + \frac{17}{6} = \frac{55 + 17}{6} = \frac{72}{6}\).

Step 3: Simplify the Improper Fraction to a Mixed Number

To simplify \(\frac{72}{6}\), divide 72 by 6: \(72 \div 6 = 12\). Therefore, \(\frac{72}{6}\) simplifies to the whole number 12.

Key Concepts

Improper Fractions

Improper fractions may sound daunting at first, but they're actually quite straightforward.
Let's break it down. An improper fraction is a type of fraction where the numerator, which is the top number, is larger than or equal to the denominator, the bottom number.
This is different from a proper fraction, where the numerator is smaller than the denominator.When dealing with mixed numbers, often the first step is to convert them into improper fractions. For example, if you have a mixed number like \(9 \frac{1}{6}\), you transform it into an improper fraction by

• multiplying the whole number by the denominator,
• adding the numerator,
• finally writing the result over the original denominator.

This operation is crucial because it allows us to perform operations, like addition, much more easily. Just remember that despite having a larger numerator, improper fractions still represent a quantity more straightforward to work with in calculations.

Addition of Fractions

Adding fractions, whether they're proper or improper, follows specific rules to ensure the result is correct. When the fractions share the same denominator, the process is especially simple. You simply add the numerators directly and retain the common denominator.For instance, with fractions like \(\frac{55}{6}\) and \(\frac{17}{6}\), you can add them by:

• Adding the numerators \(55 + 17\) to get 72,
• keeping the denominator the same, resulting in \(\frac{72}{6}\).

This method works seamlessly when the denominators match, and you can quickly find the sum. If the denominators were different, a few additional steps, like finding a common denominator, would be necessary. But for the simplification seen above, knowing they share the same bottom number eases the approach.

Simplifying Fractions

Simplifying fractions can be thought of as converting them into their simplest, cleanest form.
When a fraction can no longer be divided by a common factor other than 1, it's considered simplified. In the context of our example, we began with the improper fraction \(\frac{72}{6}\). To simplify this, you determine how many times the denominator fits into the numerator.
Here,

• divide 72 by 6,
• which equals 12.

Thus, \(\frac{72}{6}\) simplifies directly to the whole number 12.If a fraction doesn't simplify to a whole number, you aim to reduce the numerator and denominator by their greatest common divisor (GCD). Mastery of simplifying fractions not only helps in clean mathematical expression but also checks results for accuracy.