Problem 43 - Prealgebra

Question

The problems below review some basic concepts of addition of fractions and mixed numbers. Add each of the following and reduce all answers to lowest terms. $$\frac{1}{3}+\frac{2}{3}$$

Step-by-Step Solution

Answer

The answer is 1.

Step 1: Identify Like Denominators

Observe that both fractions share the same denominator, which is 3. This indicates that the fractions can be directly added together.

Step 2: Add the Numerators

Since the denominators are the same, we add the numerators. Thus, \ \( 1 + 2 = 3 \).

Step 3: Combine the Fractions

Combine the numerators over the common denominator. This gives us \ \( \frac{3}{3} \).

Step 4: Simplify the Fraction

The fraction \(\frac{3}{3}\) simplifies to 1 because any fraction where the numerator is equal to the denominator is equal to 1.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction, such as \(2 \frac{1}{3}\). They are useful for representing quantities larger than one, where part of the quantity is a whole number and part is a fraction.

In math problems, mixed numbers are often present when dealing with addition or subtraction of fractions. Understanding how to handle them can be crucial for grasping these operations effectively.

• To work with mixed numbers, it is sometimes easier to convert them to improper fractions. An improper fraction has the numerator larger than the denominator, which simplifies arithmetic operations.

For example, converting \(2 \frac{1}{3}\) into an improper fraction involves multiplying the whole number by the denominator and adding the numerator:
\(2 \times 3 + 1 = 7\), so \(2 \frac{1}{3}\) becomes \(\frac{7}{3}\).

Combining these concepts allows us to add, subtract, or even multiply and divide mixed numbers by managing them as improper fractions first, then possibly converting back into mixed numbers for final presentation.

Like Denominators

When adding fractions, like denominators mean that the denominators – the bottom numbers of the fractions – are the same. This is crucial because it allows direct addition of the fractions' numerators without modifying the fractions themselves. For example, with \(\frac{1}{3} + \frac{2}{3}\), both fractions have 3 as the denominator.

Here’s why like denominators are important:

• Fractions with like denominators maintain their relationship to whole units, making addition straightforward without the need for conversion.
• Having a common denominator standardizes the unit of measure, ensuring consistent addition results without recalculating values relative to others.

When fractions have different denominators, it becomes necessary to find a common denominator which involves more steps, like finding the least common multiple, before the fractions can be combined.

Overall, having like denominators simplifies the process and reduces the chance for errors, making basic arithmetic with fractions more intuitive.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form such that the numerator and denominator have no common factors other than 1. The fraction \(\frac{3}{3}\), for example, simplifies to 1. Simplification is an essential step for neatness and clarity in final answers and is vital for understanding equivalent fractions.

Here’s how you can easily simplify fractions:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide the numerator and the denominator by their GCD to reduce the fraction to its simplest form.

In addition and subtraction of fractions like our example \(\frac{3}{3}\), simplification often leads to easier further operations or more understandable results. It’s beneficial not only for presenting the cleanest form of a calculation but also for making sense of the fractional relationships themselves.

Remember, a fraction representing a whole can be identified by having the same numerator and denominator, simplifying directly to the whole number 1.