Problem 22 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(2 \frac{5}{12}+\left(2 \frac{7}{12}\right)\)

Step-by-Step Solution

Answer

The sum is 5.

Step 1: Convert Mixed Numbers to Improper Fractions

Start by converting the mixed numbers into improper fractions. For the first number: Multiply the whole number (2) by the denominator (12) and add the numerator (5). So, \(2 \frac{5}{12} = \frac{(2 \times 12) + 5}{12} = \frac{29}{12}\). Similarly, for the second number: \(2 \frac{7}{12} = \frac{(2 \times 12) + 7}{12} = \frac{31}{12}\). Thus, the expression becomes \(\frac{29}{12} + \frac{31}{12}\).

Step 2: Add the Improper Fractions

To add fractions, they must have the same denominator. Since both fractions do, simply add the numerators and keep the denominator. Calculate \(\frac{29}{12} + \frac{31}{12} = \frac{29 + 31}{12} = \frac{60}{12}\).

Step 3: Simplify the Result

Simplify the fraction by dividing the numerator by the denominator. Thus, \(\frac{60}{12} = 5\). Therefore, the final answer in simplest form is 5.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that contain both a whole number and a fraction. They are used to represent quantities greater than one in an easy-to-understand format. For example, in the mixed number \(2\frac{5}{12}\), the "2" is the whole number part, and \(\frac{5}{12}\) is the fractional part.
It's often easier to visualize, but not always as convenient for calculations. To convert a mixed number into an improper fraction, simply multiply the whole number by the fraction's denominator and add the fraction's numerator. This result becomes the new numerator, while the denominator remains the same.

• Example: Convert \(2\frac{7}{12}\) to an improper fraction:
• Multiply the whole number (2) by the denominator (12): \(2 \times 12 = 24\)
• Add the numerator (7): \(24 + 7 = 31\)
• The improper fraction is \(\frac{31}{12}\)

Improper Fractions

Improper fractions have numerators equal to or larger than their denominators. These fractions often represent values greater than one. For example, \(\frac{29}{12}\) is an improper fraction because 29 is greater than 12. While they might seem less intuitive than mixed numbers, they are often more manageable for arithmetic operations.
When adding or multiplying fractions, improper fractions can make calculations simpler. The improper form consolidates the whole and fraction parts into one single fraction, often making operations like addition more straightforward because you don’t need to handle whole numbers separately.

• Conversion tip: Every mixed number can be expressed as an improper fraction by following the conversion steps.
• Remember: The denominator doesn't change when converting between mixed numbers and improper fractions.

Simplification

Simplification involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Simplified fractions make results clearer and easier to understand. In our original problem, the fraction \(\frac{60}{12}\) simplifies directly to 5 through division.
To simplify:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Your fraction will now be in its simplest form.

In our exercise, \(60\) divided by \(12\) equals 5, a whole number with no fractional part, making the simplified answer straightforward and exact.

Addition of Fractions

Adding fractions requires that they share the same denominator, making them easy to combine. If fractions do not originally have the same denominator, they must first be adjusted to a common denominator before adding. In our exercise, both fractions \(\frac{29}{12}\) and \(\frac{31}{12}\) already share a denominator, allowing us to focus solely on the numerators.
Steps to add fractions:

• Ensure the denominators are the same. If not, find a common denominator.
• Add the numerators together.
• Keep the denominator the same.
• Convert the sum to its simplest form if necessary.

In our example, adding the numerators gives \(29 + 31 = 60\), resulting in \(\frac{60}{12}\), which simplifies to 5, providing our final answer in its simplest form.