Problem 6 - Algebra for College Students

Question

Simplify each complex fraction. Use either method. $$\frac{\frac{1}{2}-\frac{1}{4}}{\frac{1}{6}-\frac{1}{8}}$$

Step-by-Step Solution

Answer

The simplified value is 6.

Step 1: Simplify the Numerator

First, simplify the expression in the numerator: \ \ \( \frac{1}{2} - \frac{1}{4} \). Find a common denominator, which is 4. Rewrite each fraction with this common denominator: \ \ \( \frac{1 \times 2}{2 \times 2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} \). Now, subtract the fractions: \ \ \( \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \). So, the simplified numerator is \( \frac{1}{4} \).

Step 2: Simplify the Denominator

Next, simplify the expression in the denominator: \ \ \( \frac{1}{6} - \frac{1}{8} \). Find a common denominator, which is 24. Rewrite each fraction with this common denominator: \ \ \( \frac{1 \times 4}{6 \times 4} - \frac{1 \times 3}{8 \times 3} = \frac{4}{24} - \frac{3}{24} \). Now, subtract the fractions: \ \ \( \frac{4}{24} - \frac{3}{24} = \frac{1}{24} \). So, the simplified denominator is \( \frac{1}{24} \).

Step 3: Divide the Numerator by the Denominator

Now, write the simplified fraction from the previous steps: \ \ \( \frac{\frac{1}{4}}{\frac{1}{24}} \). To divide fractions, multiply the numerator by the reciprocal of the denominator: \ \ \( \frac{1}{4} \times \frac{24}{1} = \frac{1 \times 24}{4 \times 1} = \frac{24}{4} = 6 \). Thus, the simplified value of the complex fraction is 6.

Key Concepts

common denominator

A common denominator makes adding and subtracting fractions much easier. A common denominator is a shared multiple of the denominators of two or more fractions. When fractions have the same denominator, it's simpler to combine them. For example, with the fractions \(\frac{1}{2}\) and \(\frac{1}{4}\), we find a common denominator: 4. Rewrite each fraction as \(\frac{1 \times 2}{2 \times 2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4}\). This makes it easy to subtract them, resulting in \(\frac{1}{4}\).Finding a common denominator can be straightforward.
If the denominators are small numbers, you can list the multiples to find the least common multiple (LCM).
Sometimes you might need to multiply the denominators of both fractions to find the common denominator.

subtracting fractions

Subtracting fractions follows a straightforward process once you have a common denominator. Consider \(\frac{1}{6}\) and \(\frac{1}{8}\) as an example. First, find a common denominator. Here, it is 24. Rewrite each fraction with the common denominator: \(\frac{1 \times 4}{6 \times 4} - \frac{1 \times 3}{8 \times 3} = \frac{4}{24} - \frac{3}{24}\). Now, subtract the numerators: \(\frac{4}{24} - \frac{3}{24} = \frac{1}{24}\).It's crucial to simplify fractions whenever possible.
If the numerator and denominator have a common factor, divide both by this number to reduce the fraction to its simplest form.

reciprocal

The reciprocal of a number or a fraction is what you get when you flip the numerator and the denominator. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). To simplify a complex fraction like \(\frac{\frac{1}{4}}{\frac{1}{24}}\), you multiply by the reciprocal of the denominator. So, \(\frac{1}{4} \times \frac{24}{1} = \frac{1 \times 24}{4 \times 1} = \frac{24}{4} = 6\).Reciprocals are essential in arithmetic.
Especially, when you need to divide by a fraction.
Always remember to flip the fraction to find its reciprocal before multiplying.