Problem 44 - Prealgebra

Question

Find the following sums. (Add.) $$\begin{array}{r}1 \frac{5}{6} \\\2 \frac{3}{4} \\\\+5 \frac{1}{2} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The sum is \(10 \frac{1}{12}\).

Step 1: Identify Whole and Fractional Parts

Separate the whole numbers and the fractional parts for each mixed number. We have \(1\), \(2\), and \(5\) as the whole numbers, and \(\frac{5}{6}\), \(\frac{3}{4}\), and \(\frac{1}{2}\) as the fractional parts.

Step 2: Add Whole Numbers

Add the whole numbers first: \(1 + 2 + 5 = 8\).

Step 3: Find Common Denominator for Fractions

Convert the fractions to have a common denominator. The denominators are \(6\), \(4\), and \(2\), and the least common denominator is \(12\). Convert each fraction:\[\frac{5}{6} = \frac{10}{12}, \quad \frac{3}{4} = \frac{9}{12}, \quad \text{and} \quad \frac{1}{2} = \frac{6}{12}.\]

Step 4: Add the Fractional Parts

Add the converted fractions together:\[\frac{10}{12} + \frac{9}{12} + \frac{6}{12} = \frac{25}{12}.\]Since \(\frac{25}{12}\) is an improper fraction, convert it to a mixed number. \(\frac{25}{12} = 2 \frac{1}{12}\).

Step 5: Combine Whole Numbers and Improper Fraction Conversion

Add the whole number result from Step 2 and the whole number from the improper fraction conversion: \(8 + 2 = 10\). Include the fractional part: \(10 \frac{1}{12}\).

Key Concepts

Understanding Common Denominator

When adding fractions, it's important they share the same denominator, known as having a common denominator. This ensures each fraction represents parts of the same whole. In our exercise, we had fractions with denominators 6, 4, and 2. To add these seamlessly, we needed a shared base, or least common denominator, which in this case was 12. By converting fractions to have a common denominator, we can easily add them: \(\frac{5}{6} = \frac{10}{12}\), \(\frac{3}{4} = \frac{9}{12}\), and \(\frac{1}{2} = \frac{6}{12}\). Having a common denominator simplifies the addition process of fractions.

What Makes a Fraction Improper

Improper fractions have numerators that are larger than their denominators. This means the fraction actually represents more than one whole. In the exercise, we ended up with the fraction \(\frac{25}{12}\) after adding the converted fractions. This is an improper fraction because 25, the numerator, is greater than 12, the denominator.

• Improper fractions are important because they show that adding fractions can result in whole numbers.
• They can also be converted into mixed numbers, making them easier to interpret as both wholes and parts.

In the solution, \(\frac{25}{12}\) was converted into \(2 \frac{1}{12}\), combining the whole number with the leftover part.

Explaining Mixed Numbers

Mixed numbers combine whole numbers and fractions, providing a convenient way to express amounts greater than one. In our task, we initially had mixed numbers like \(1 \frac{5}{6}\), \(2 \frac{3}{4}\), and \(5 \frac{1}{2}\).

• When adding mixed numbers, we separately add their whole parts and fractional parts.
• Mixed numbers can be especially useful for everyday measurements and understanding the size of numbers.

Throughout the solution, we converted between mixed numbers and improper fractions to facilitate easy addition, reflecting both exactness and interpretability.

Adding Fractions with Ease

Adding fractions involves a few key steps: ensuring you have a common denominator, performing the addition, and simplifying when possible. In this exercise, we took fractions from the mixed numbers, converted them to a common denominator, and combined them.

• This allowed us to avoid complications that arise when fractions have different denominators.
• Once added, we ended up with an improper fraction, which we simplified to a mixed number and added to the whole numbers.

The essence of adding fractions is about understanding how to work within their rule set, resulting in accurate sums expressed in different, understandable formats.