Problem 23 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{3}{8}+\frac{5}{8} $$

Step-by-Step Solution

Answer

The simplified result is 1.

Step 1: Identify Common Denominator

In the expression \( \frac{3}{8} + \frac{5}{8} \), notice both fractions share a common denominator, which is 8. This makes it easier to add the fractions directly.

Step 2: Add the Numerators

Since the denominators are the same, we can directly add the numerators. So, \( 3 + 5 = 8 \).

Step 3: Write the Sum with the Common Denominator

Combine the result from the addition of the numerators over the common denominator. This gives us \( \frac{8}{8} \).

Step 4: Simplify the Fraction

Notice that the fraction \( \frac{8}{8} \) is equal to 1, since the numerator and denominator are the same.

Key Concepts

Common Denominator

When adding fractions, having a common denominator is crucial. It refers to the denominator that two or more fractions share. In our example with \( \frac{3}{8} + \frac{5}{8} \), both fractions already have the denominator 8 in common. This makes the process of addition much simpler, because you don't have to do additional work to make the denominators the same.

Here's why a common denominator matters:

• Allows direct addition or subtraction of numerators.
• Makes calculations easier and reduces errors.
• Saves time compared to finding a least common denominator.

Understanding the importance of a common denominator helps in quickly assessing whether fractions can be added or require adjustments. It's like having a common ground where both fractions can meet!

Numerators

The numerator is the top part of a fraction. It's the number that tells us how many parts of the whole we are considering. In the problem \( \frac{3}{8} + \frac{5}{8} \), the numerators are 3 and 5. After establishing a common denominator, fractions can be combined by simply adding or subtracting these numerators.

Here’s how adding numerators works in this context:

• Add the two numerators directly: \( 3 + 5 \).
• Keep the denominator the same, since it’s common (8 in this case).
• Write the sum as a new fraction: \( \frac{8}{8} \).

This simple addition of numerators over the unchanged common denominator helps create a new fraction, which might sometimes still need simplification.Understanding numerators properly is key to managing fractions easily and efficiently.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form. This means making the fraction as "small" as possible while still having the same value. Once you've combined the numerators, as we did with \( \frac{3}{8} + \frac{5}{8} \), you end up with \( \frac{8}{8} \).

Steps of simplification include:

• Check if the numerator and denominator are the same. If they are, like in our example, the fraction equals 1.
• If not the same, divide the numerator and the denominator by their greatest common factor (GCF).
• Rewrite the fraction using the results from the division.

Simplifying a fraction not only makes it neater but also easier to understand in terms of proportionality. In this case, \( \frac{8}{8} = 1 \), which clearly indicates one whole. Knowing how and when to simplify fractions is a key step in solving any fraction-related problem effectively.