Problem 46 - Prealgebra

Question

Add and subtract as indicated. $$\frac{5}{8}+\frac{3}{4}$$

Step-by-Step Solution

Answer

The sum is \(1\frac{3}{8}\).

Step 1: Find a Common Denominator

Identify the denominators of the fractions, which are 8 and 4. The least common multiple of 8 and 4 is 8, so this will be our common denominator.

Step 2: Convert Fractions to Have the Same Denominator

We already have \( \frac{5}{8} \) with the denominator of 8. Now convert \( \frac{3}{4} \) to an equivalent fraction with a denominator of 8 by multiplying the numerator and the denominator by 2: \( \frac{3}{4} \times \frac{2}{2} = \frac{6}{8} \).

Step 3: Add the Fractions

Now that the fractions \( \frac{5}{8} \) and \( \frac{6}{8} \) have the same denominator, we can add them by adding their numerators: \( \frac{5+6}{8} = \frac{11}{8} \).

Step 4: Simplify the Result

\( \frac{11}{8} \) is an improper fraction. Convert it to a mixed number by dividing 11 by 8. 11 divided by 8 is 1 with a remainder of 3, so \( \frac{11}{8} = 1\frac{3}{8} \).

Key Concepts

Common Denominator

When dealing with fractions, finding a common denominator is a crucial step. Consider the fractions in the problem, such as \( \frac{5}{8} \) and \( \frac{3}{4} \). Their denominators, 8 and 4, need to match for us to perform operations like addition or subtraction. The denominator reflects how many equal parts make up one whole.

• To find a common denominator, you determine the least common multiple (LCM) of the individual denominators.
• In this case, the denominators 8 and 4 have a least common multiple of 8.
• This means both fractions can be expressed with the same denominator of 8.

Using a common denominator simplifies the process of adding and subtracting fractions by aligning the fractions to have matching 'part sizes', making calculations straightforward.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. Imagine cutting a pizza: whether it's in eight slices or four, the pizza's size remains the same if you eat half.
To manipulate a fraction without changing its value, we scale both the numerator and denominator equally. For example, converting \( \frac{3}{4} \) to a fraction with a denominator of 8 involves multiplying both the numerator and denominator by 2. This yields:

• \( \frac{3}{4} \times \frac{2}{2} = \frac{6}{8} \)
• The fractions \( \frac{3}{4} \) and \( \frac{6}{8} \) are equivalent.

Recognizing equivalent fractions helps us handle operations like addition, because it allows us to express different fractions under a single, common denominator.

Improper Fractions

An improper fraction occurs when the numerator is greater than or equal to the denominator. This signifies that the fraction represents a value greater than one. For instance, after adding \( \frac{5}{8} \) and \( \frac{6}{8} \), the result \( \frac{11}{8} \) is an improper fraction. This indicates that our sum exceeds the whole. To convert it to a mixed number, perform the following:

• Divide the numerator by the denominator: 11 divided by 8 gives 1 with a remainder of 3.
• This can be rewritten as a mixed number: \( 1\frac{3}{8} \).
• The whole number 1 represents the complete wholes, and \( \frac{3}{8} \) is the remaining part.

Understanding improper fractions and their conversion to mixed numbers helps interpret results that indicate more than a complete unit.