Problem 225 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}} $$

Step-by-Step Solution

Answer

1

Step 1: Find a Common Denominator for Addition

To add the fractions \(\frac{5}{8}\) and \(\frac{1}{6}\), first find a common denominator. The least common denominator (LCD) of 8 and 6 is 24.

Step 2: Convert the Fractions

Convert \(\frac{5}{8}\) and \(\frac{1}{6}\) to have the denominator of 24. This gives \(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\) and \(\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}\).

Step 3: Add the Fractions

Now add the fractions: \(\frac{15}{24} + \frac{4}{24} = \frac{15 + 4}{24} = \frac{19}{24}\).

Step 4: Simplify the Expression

The expression now looks like \(\frac{\frac{19}{24}}{\frac{19}{24}}\).

Step 5: Divide the Fractions

Dividing \(\frac{19}{24}\) by \(\frac{19}{24}\) results in 1 because any number divided by itself equals 1.

Key Concepts

Least Common Denominator (LCD)

When working with fractions, especially in addition, it is essential to find a common denominator. The least common denominator (LCD) is the smallest number that both denominators share as a multiple. Here, with denominators 8 and 6, the LCD is 24. Finding the LCD simplifies the process by ensuring both fractions have the same denominator, making addition straightforward.

Fraction Addition

Adding fractions requires matching their denominators. After finding the LCD, convert each fraction to an equivalent fraction with the LCD. For example, \(\frac{5}{8}\) becomes \(\frac{15}{24}\) and \(\frac{1}{6}\) becomes \(\frac{4}{24}\). With the same denominator, you can directly add them: \(\frac{15}{24} + \frac{4}{24} = \frac{19}{24}\).

Division of Fractions

Dividing fractions may initially seem complex, but it becomes simple by rearranging the division into multiplication. Dividing by a fraction is the same as multiplying by its reciprocal (flipping the numerator and denominator). For instance, \(\frac{\frac{19}{24}}{\frac{19}{24}}\) turns into \(\frac{19}{24} \times \frac{24}{19}\), simplifying to 1.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. To convert to equivalent fractions, multiply or divide both the numerator and the denominator by the same number. For instance, to convert \(\frac{5}{8}\) to a denominator of 24, you multiply both by 3, resulting in \(\frac{15}{24}\). This preserves the value while changing the appearance.