Problem 44 - 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. $$\frac{3}{8}+\frac{1}{8}$$

Step-by-Step Solution

Answer

\(\frac{3}{8} + \frac{1}{8} = \frac{1}{2}\) in lowest terms.

Step 1: Understanding the Problem

We are asked to add two fractions: \(\frac{3}{8}\) and \(\frac{1}{8}\). These fractions have the same denominator.

Step 2: Finding the Common Denominator

Since both fractions, \(\frac{3}{8}\) and \(\frac{1}{8}\), already have the same denominator, which is 8, we can directly proceed to addition. There is no need to find a different common denominator.

Step 3: Adding the Numerators

Add the numerators of the fractions while keeping the denominator the same. So, we calculate \(3 + 1 = 4\). The resulting fraction is \(\frac{4}{8}\).

Step 4: Reducing to Lowest Terms

The fraction \(\frac{4}{8}\) can be simplified. Find the greatest common divisor of 4 and 8, which is 4, and divide both the numerator and denominator by this number: \(\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\).

Key Concepts

Mixed Numbers and Their Role in Addition

Mixed numbers are numbers that consist of a whole number and a fraction together. For example, the number 2 1/2 is a mixed number where 2 is the whole number and 1/2 is the fraction.
They can be tricky when adding because you need to add both the whole numbers and the fractions separately.
In exercises that involve mixed numbers, you follow these steps:

• Add the whole numbers together.
• Then add the fractions using the same denominator if possible.
• If the fractions have different denominators, you first find the common denominator.

Understanding mixed numbers is essential for handling more complex addition problems involving fractions.
This makes the process of adding not just numbers but parts of numbers truly seamless.

Importance of a Common Denominator

Fractions can only be added when they have the same denominator, known as the common denominator. This is because fractions with the same denominator share equal parts. In our problem, \(\frac{3}{8} + \frac{1}{8}\), the common denominator is already given, which is 8.
A quick way to find a common denominator when one isn't provided is by finding the least common multiple (LCM) of the two denominators.
Once the common denominator is found:

• Align the fractions by converting them to have this common denominator.
• Add only the numerators and keep this common denominator.

This simple practice helps ensure accurate and easy summation of fractions, making the calculation more straightforward.

Simplifying Fractions The Right Way

Once you have added the fractions, you might end up with a fraction that can be simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.
For instance, if you add \(\frac{3}{8} + \frac{1}{8} = \frac{4}{8}\), you simplify this by finding the greatest common divisor (GCD) of 4 and 8, which is 4.
Then divide both the numerator and the denominator by their GCD:

• Divide 4 by 4 to get 1.
• Divide 8 by 4 to get 2.

Thus, \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\).
This simplification makes fractions easier to understand and use, especially when performing further mathematical operations.