Problem 4 - Prealgebra

Question

Find each sum. Write in simplest form. $$8 \frac{5}{12}+11 \frac{1}{4}$$

Step-by-Step Solution

Answer

The sum is \(19 \frac{2}{3}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(8 \frac{5}{12}\), multiply 8 by 12 and add 5 to get the numerator: \(8 \times 12 + 5 = 96 + 5 = 101\). So, \(8 \frac{5}{12} = \frac{101}{12}\). For \(11 \frac{1}{4}\), multiply 11 by 4 and add 1 to get the numerator: \(11 \times 4 + 1 = 44 + 1 = 45\). So, \(11 \frac{1}{4} = \frac{45}{4}\).

Step 2: Find a Common Denominator

Determine a common denominator for the fractions \(\frac{101}{12}\) and \(\frac{45}{4}\). The least common multiple of 12 and 4 is 12. Therefore, the common denominator is 12.

Step 3: Adjust the Fractions

Convert \(\frac{45}{4}\) to have a denominator of 12. Multiply both the numerator and denominator by 3: \(\frac{45}{4} \times \frac{3}{3} = \frac{135}{12}\). Now, both fractions have the same denominator: \(\frac{101}{12}\) and \(\frac{135}{12}\).

Step 4: Add the Fractions

Since the fractions now have the same denominator, add them by adding their numerators: \(\frac{101}{12} + \frac{135}{12} = \frac{101 + 135}{12} = \frac{236}{12}\).

Step 5: Simplify the Fraction

Simplify \(\frac{236}{12}\) by dividing both the numerator and denominator by their greatest common divisor, which is 4. \(\frac{236 \div 4}{12 \div 4} = \frac{59}{3}\).

Step 6: Convert to Mixed Number

Convert the improper fraction \(\frac{59}{3}\) back to a mixed number. Divide 59 by 3 to get 19 with a remainder of 2, so \(\frac{59}{3} = 19 \frac{2}{3}\).

Key Concepts

Improper Fractions

An improper fraction is when the numerator (the top number) is larger than the denominator (the bottom number). For instance, when you convert a mixed number like \(8 \frac{5}{12}\) into an improper fraction, you essentially stretch out the whole number part into the fraction form. Imagining this, think of the 8 whole parts multiplied by the 12 parts of a fraction, plus the remaining fraction, totaling a bigger numerator. In our example, this turns the given mixed number into \(\frac{101}{12}\).
Improper fractions might look a little different from the fractions you're used to. But don't worry, they just show parts of whole numbers in a different way. Whenever dealing with mixed numbers, it's useful to convert them to improper fractions to simplify calculations.

Mixed Numbers

Mixed numbers are a blend of whole numbers and fractions, like \(11 \frac{1}{4}\). They can be comfy to work with when directly counting or visualizing parts through everyday situations. However, they're not the best fit when adding fractions, in which case converting to improper fractions is more practical.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator of the fraction part. In \(11 \frac{1}{4}\), multiply 11 by 4 to get 44, then add 1 to get 45, and so, \(11 \frac{1}{4}\) becomes \(\frac{45}{4}\). It's like pulling apart the number into one complete fraction, making calculations simpler especially when adding or subtracting.

Common Denominator

Finding a common denominator is critical when adding or subtracting fractions since it aligns the fractions to the same basis. For example, with \(\frac{101}{12}\) and \(\frac{45}{4}\), the least common multiple (LCM) of 12 and 4 is 12, making it the simplest common denominator.
To adapt fractions to this denominator, \(\frac{45}{4}\) is transformed by multiplying both its numerator and denominator by 3 (as \(3 \times 4 = 12\)), which leads to \(\frac{135}{12}\). Now, the common framework sets the stage for seamless addition. It ensures both fractions speak the same "language" of parts, making the process straightforward.

Simplifying Fractions

After performing operations with fractions, like addition, you often end up with a fraction needing simplification. The purpose here is to make your answer as compact as possible. With our sum, \(\frac{236}{12}\), simplification involves identifying the greatest common divisor (GCD) for both the numerator and denominator.
For \(\frac{236}{12}\), dividing both by their GCD, which is 4, gives \(\frac{59}{3}\). This fraction is much neater and represents the same value as the unsimplified one. Often, the simplest form gives a clearer sense of proportion.
When you convert back to a mixed number, \(\frac{59}{3}\) turns to \(19 \frac{2}{3}\), offering an equally simplified and more easily understandable result. Remember, simplified fractions or the simplest form of a mixed number makes interpretation and further calculations simpler.