Problem 43 - Prealgebra

Question

Add and subtract as indicated. $$\frac{1}{2}+\frac{3}{8}$$

Step-by-Step Solution

Answer

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

Step 1: Find a Common Denominator

To add two fractions, they must have the same denominator. The denominators here are 2 and 8. The smallest common denominator is 8.

Step 2: Convert Fractions to Common Denominator

Convert \( \frac{1}{2} \) to a denominator of 8 by multiplying both the numerator and denominator by 4: \( \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \). Now both fractions are \( \frac{4}{8} \) and \( \frac{3}{8} \).

Step 3: Add the Fractions

Since both fractions have the same denominator, add the numerators: \( 4 + 3 = 7 \). The resulting fraction is \( \frac{7}{8} \).

Key Concepts

Common Denominator

When adding fractions, finding a common denominator is like creating a common platform for both numbers to speak the same language. Without sharing the same denominator, fractions are like apples and oranges; they're difficult to combine. This is because denominators tell you how many equal parts make up a whole. If one fraction is in halves and the other in eighths, you need to adjust them so they describe the same section of a whole.

To find a common denominator, identify a number that both original denominators can divide into evenly. Often, the easiest choice is the least common multiple (LCM) of the denominators. In our example, we look at fractions with denominators 2 and 8. The number 8 is the smallest that both can divide into without remainder. Having found this common denominator lets us move on to the next step: converting fractions.

Convert Fractions

Once we've found our common denominator, the next task is to convert each fraction so they share this new denominator. This involves finding an equivalent fraction with the denominator that was decided in the previous step.

Let's take the fraction \( \frac{1}{2} \). To convert it to have a denominator of 8, we need to multiply both the numerator and the denominator by the same number. In this case, that number is 4, since multiplying 2 by 4 gives us 8. Therefore, \( \frac{1}{2} \) becomes \( \frac{4}{8} \).

The fraction \( \frac{3}{8} \) already has the denominator 8, so no change is needed for that fraction. Now both fractions, \( \frac{4}{8} \) and \( \frac{3}{8} \), have a common denominator, paving the way for easy addition.

Adding Fractions Step by Step

With both fractions sharing the same denominator, we're now ready to add them together. This is the simplest part of the process. When adding fractions with a common denominator, you only need to add the numerators. The denominator remains unchanged.

For our fractions \( \frac{4}{8} \) and \( \frac{3}{8} \):

• Add the numerators: \( 4 + 3 = 7 \).
• Keep the denominator the same: 8.

Thus, the sum is \( \frac{7}{8} \).

Adding fractions becomes a straightforward task once you learn how to coordinate their denominators. This ability to find and use a common denominator is a foundational skill in working with fractions, and mastering it opens up many new possibilities in math. Remember, practice is key to becoming comfortable with these methods.