Problem 4 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(5 \frac{2}{7}+3 \frac{3}{7}\)

Step-by-Step Solution

Answer

The sum is \(8 \frac{5}{7}\).

Step 1: Identify Whole Numbers and Fractions

The given mixed numbers are \(5 \frac{2}{7}\) and \(3 \frac{3}{7}\). Separate them into whole numbers and fractions:- Whole numbers: 5 and 3- Fractions: \(\frac{2}{7}\) and \(\frac{3}{7}\).

Step 2: Add Whole Numbers

Add the whole numbers from the mixed numbers:\(5 + 3 = 8\).

Step 3: Add Fractions

Now, add the fractions \(\frac{2}{7}\) and \(\frac{3}{7}\):\[\frac{2}{7} + \frac{3}{7} = \frac{5}{7}\].

Step 4: Combine Results into a Mixed Number

Combine the sums of the whole numbers and the fractions to form a new mixed number:\(8 + \frac{5}{7} = 8 \frac{5}{7}\).

Key Concepts

Understanding Mixed Numbers

In mathematics, mixed numbers are a combination of whole numbers and fractions. They are used to express quantities that are not whole but also not just a simple fraction. For example, \(5 \frac{2}{7}\) is a mixed number where "5" is a whole number and \(\frac{2}{7}\) is a fraction.

Recognizing the components of mixed numbers is important when performing operations like addition or subtraction. Start by identifying the whole number part and the fractional part of each mixed number. This process makes calculations easier by allowing you to handle the whole numbers and fractions separately.

• Example: In \(3 \frac{3}{7}\), "3" is the whole number, and \(\frac{3}{7}\) represents the fraction.
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, add the numerator, and place the result over the initial denominator.

Grasping Fractions

Fractions represent parts of a whole and are comprised of two parts: a numerator and a denominator. In the fraction \(\frac{2}{7}\), "2" is the numerator indicating the number of parts taken, and "7" is the denominator indicating the total number of equal parts.

When adding or subtracting fractions, the denominators must be the same. Fractions with the same denominator, like \(\frac{2}{7}\) and \(\frac{3}{7}\), are called like fractions. This means you can easily add the numerators while keeping the denominator the same:

• Adding like fractions: \(\frac{2}{7} + \frac{3}{7} = \frac{5}{7}\).
• For unlike fractions, find a common denominator before performing the operation.

Whole Numbers in Mixed Numbers

Whole numbers are integers that do not have a fractional or decimal component. They are the simplest part of mixed numbers to handle when performing arithmetic operations like addition and subtraction.

In the problem \(5 \frac{2}{7} + 3 \frac{3}{7}\), the whole numbers "5" and "3" can be added directly just like you would with any regular integers:

• Addition: \(5 + 3 = 8\)

Once the whole numbers are added, they form part of the new mixed number when combined with the sum of the fractions. Understanding and correctly managing the whole numbers separately provides clarity and simplifies the process of dealing with mixed numbers.