Problem 76 - Algebra 2

Question

Evaluate each expression. \(\frac{1}{8}+\frac{5}{12}\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{13}{24}\).

Step 1: Find a Common Denominator

To add fractions, they need a common denominator. The denominators of the fractions are 8 and 12. The least common multiple of 8 and 12 is 24.

Step 2: Convert Fractions to Have the Common Denominator

Convert each fraction to an equivalent fraction with a denominator of 24. For \(\frac{1}{8}\): Multiply both numerator and denominator by 3 to get \(\frac{1}{8} = \frac{3}{24}\).For \(\frac{5}{12}\):Multiply both numerator and denominator by 2 to get \(\frac{5}{12} = \frac{10}{24}\).

Step 3: Add the Fractions

Now that both fractions have a common denominator, add their numerators: \(\frac{3}{24} + \frac{10}{24} = \frac{13}{24}\).

Step 4: Simplify the Result

Check if \(\frac{13}{24}\) can be simplified. Since 13 is a prime number and does not divide 24, the fraction is already in its simplest form.

Key Concepts

Least Common Multiple

The least common multiple, commonly abbreviated as LCM, is a crucial concept when dealing with fractions. It helps us find a common ground between two or more numbers. For example, to add the fractions \(\frac{1}{8}\) and \(\frac{5}{12}\), we need to determine the smallest number that both 8 and 12 can divide into wholly.

• 8: 8, 16, 24, 32, 40, ...
• 12: 12, 24, 36, 48, ...

The shared number here is 24, which makes it the least common multiple. This is essential for converting fractions so that they have the same denominator, allowing us to perform addition or subtraction.

Equivalent Fractions

Equivalent fractions are different fractions that name the same part of a whole. Understanding this concept is vital for addition or subtraction of fractions, as it enables us to convert them to have a common denominator without changing their value.For instance, in the example \(\frac{1}{8}\), multiplying both the numerator and denominator by 3 yields \(\frac{3}{24}\). Similarly, for \(\frac{5}{12}\), multiplying both the numerator and the denominator by 2 results in \(\frac{10}{24}\).

• \(\frac{1}{8} \times \frac{3}{3} = \frac{3}{24}\)
• \(\frac{5}{12} \times \frac{2}{2} = \frac{10}{24}\)

These are equivalent fractions, as they describe the same portion in terms of a common denominator of 24.

Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form, where the numerator and the denominator have no common factor other than 1. In basic terms, a simplified fraction cannot be reduced further. Once you have added or subtracted fractions, as we did with \(\frac{3}{24} + \frac{10}{24} = \frac{13}{24}\), check if the resulting fraction can be simplified.For \(\frac{13}{24}\), 13 is a prime number. Due to its prime nature and the fact it does not divide 24, \(\frac{13}{24}\) is already as simple as it gets.

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

This ensures your final answer is presented in its neatest form.

Common Denominator

The term "common denominator" is frequently encountered while working with fractions, especially during addition and subtraction. A common denominator is simply a shared multiple between the denominators of the fractions we are dealing with.Finding this number allows us to manipulate fractions in a way that is mathematically valid and visually understandable. In our example, after determining the least common multiple of 8 and 12, we adjust the fractions so they both have the common denominator, 24.

• Replace initial fractions: \(\frac{1}{8} = \frac{3}{24}\)
• Replace initial fractions: \(\frac{5}{12} = \frac{10}{24}\)

Once both fractions share this denominator, the operation becomes as straightforward as adding or subtracting whole numbers.