Problem 43 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or a mixed number in simplest form. $$ \frac{2}{8}+\frac{1}{8} $$

Step-by-Step Solution

Answer

The sum is \( \frac{3}{8} \).

Step 1: Add the fractions with the same denominator

When two fractions have the same denominator, you can add the numerators directly and keep the denominator the same. So, \( \frac{2}{8}+\frac{1}{8} = \frac{2+1}{8} \).

Step 2: Calculate the sum

Calculating the sum in the numerator, \( 2+1 = 3 \). So the addition results in \( \frac{3}{8} \).

Step 3: Simplify the fraction

The fraction \( \frac{3}{8} \) is already in its simplest form, as 3 and 8 do not share any common factors other than 1. Therefore, no further simplification is needed.

Key Concepts

Understanding Fractions

Fractions are numbers that represent a part of a whole. They are made up of two main components: the numerator and the denominator. The numerator is the top number, and it tells us how many parts we have. The denominator is the bottom number, and it shows how many equal parts the whole is divided into.
For example, in the fraction \( \frac{2}{8} \), 2 is the numerator and 8 is the denominator. This fraction tells us that we have 2 parts out of a total of 8 parts.
Understanding fractions is essential when adding or subtracting them, as we must pay attention to these two components to ensure a correct calculation.

• Numerator: The number above the line, representing how many parts are being considered.
• Denominator: The number below the line, indicating the total number of equal parts.

By grasping these basics, one can more easily manipulate and work with fractions in various mathematical operations.

Finding the Simplest Form

When a fraction is in its simplest or lowest form, this means that the numerator and the denominator have no common factors other than 1. Simplifying a fraction makes it easier to understand and work with.
To simplify a fraction, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCF.

For instance, even though \( \frac{3}{8} \) is already in its simplest form, the simplification process ensures that both numbers can't be divided by any other number except 1. This is a vital step in ensuring clarity and precision in mathematical expressions.
Always aim to express results in the simplest form so that the solutions are clear and concise.

Working with Numerators and Denominators

To effectively add fractions, understanding how numerators and denominators interact is crucial. If fractions have the same denominator, you can directly add up the numerators. The denominator stays the same because it indicates how many total parts make up the whole.
Consider this: if you have \( \frac{2}{8} \) and add \( \frac{1}{8} \), only the numerators (2 and 1) are added to get 3, keeping the denominator as 8. That's because both fractions are parts of the same whole.
It's also important to remember:

• Denominators should ideally be the same when adding fractions to avoid complex calculations.
• If denominators are different, you'll need to find a common denominator before proceeding.

By mastering how numerators and denominators work together, you'll find it much easier to handle fraction problems efficiently and accurately.