Problem 101 - Prealgebra

Question

Add. $$\frac{5}{8}+\frac{3}{4}$$

Step-by-Step Solution

Answer

The sum \( \frac{5}{8} + \frac{3}{4} \) is \( 1 \frac{3}{8} \).

Step 1: Identify the Least Common Denominator (LCD)

To add fractions, they must have the same denominator. Here, the denominators are 8 and 4. The least common multiple of 8 and 4 is 8. Therefore, the least common denominator is 8.

Step 2: Adjust the Fractions to Have the LCD

The first fraction \( \frac{5}{8} \) already has a denominator of 8, so it remains the same. The second fraction \( \frac{3}{4} \) needs to be converted to have a denominator of 8. Multiply both the numerator and denominator by 2 to get \( \frac{6}{8} \).

Step 3: Add the Fractions

Now that both fractions have the same denominator, you can add them. \( \frac{5}{8} + \frac{6}{8} = \frac{5+6}{8} = \frac{11}{8} \).

Step 4: Simplify the Fraction (If Needed)

The fraction \( \frac{11}{8} \) is an improper fraction. It can be converted to a mixed number. Divide 11 by 8 to get 1 with a remainder of 3, so \( \frac{11}{8} = 1 \frac{3}{8} \).

Key Concepts

Understanding the Least Common Denominator

When adding fractions, it’s crucial to have the same denominator for both fractions so that you can combine them easily. The least common denominator (LCD) is the smallest number that can be a common multiple of the denominators of the fractions involved.

For example, in the exercise combining \( \frac{5}{8} \) and \( \frac{3}{4} \):

• Identify the denominators: 8 and 4.
• Find the least common multiple: In this case, the answer is 8 because it is the smallest number that both 4 and 8 divide into without a remainder.

Having the least common denominator allows you to adjust the numerators of both fractions proportionally, ensuring that you're adding equivalent parts.

Exploring Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). They might look a bit unusual at first but are completely valid and common in mathematics.

In our solution:

• The calculation yields \( \frac{11}{8} \) after adding \( \frac{5}{8} + \frac{6}{8} \).
• Here, 11 is greater than 8, making it an improper fraction.

Improper fractions are particularly helpful for computations, as they straightforwardly indicate a value greater than one whole. To work with them, you can also convert them into mixed numbers, which provide a different perspective.

Converting to Mixed Numbers

Mixed numbers provide a combination of whole numbers and fractions, offering an alternative way to express improper fractions. For many, they are easier to interpret and use in everyday situations.

To convert an improper fraction like \( \frac{11}{8} \) to a mixed number:

• Divide the numerator by the denominator: 11 divided by 8 equals 1 with a remainder of 3.
• The quotient (1) becomes the whole number part.
• Use the remainder (3) as the new numerator over the original denominator (8), forming \( \frac{3}{8} \).
• Combine them to get the mixed number: \( 1 \frac{3}{8} \).

This way, \( \frac{11}{8} \) is expressed as \( 1 \frac{3}{8} \), showing clearly that it’s a bit more than one whole.