Problem 3 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\frac{6}{5}

Step 1 - Simplify the Numerator

Combine \(\frac{1}{2}\) and \(\frac{1}{4}\) in the numerator. Find a common denominator: \(\frac{1}{2} = \frac{2}{4}\). So, \(\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}\).

Step 2 - Simplify the Denominator

Combine \(\frac{1}{2}\) and \(\frac{1}{8}\) in the denominator. Find a common denominator: \(\frac{1}{2} = \frac{4}{8}\). So, \(\frac{1}{2} + \frac{1}{8} = \frac{4}{8} + \frac{1}{8} = \frac{5}{8}\).

Step 3 - Write Fraction From Simplified Parts

Substitute the simplified numerator and denominator into the fraction: \(\frac{\frac{3}{4}}{\frac{5}{8}}\).

Step 4 - Perform the Division

Invert the denominator and multiply: \(\frac{3}{4} \times \frac{8}{5} = \frac{3 \times 8}{4 \times 5} = \frac{24}{20}\).

Step 5 - Simplify the Result

Simplify \(\frac{24}{20}\) by dividing both the numerator and the denominator by their greatest common divisor (4): \(\frac{24 \div 4}{20 \div 4} = \frac{6}{5}\).

Key Concepts

Common Denominator

When dealing with fractions, finding a common denominator is crucial for addition and subtraction. In the example, we added \(\frac{1}{2}\) and \(\frac{1}{4}\) by converting \(\frac{1}{2}\) into \(\frac{2}{4}\), so both fractions share the same denominator. This process ensures we can combine the fractions accurately.
To do this:

• Identify the least common multiple (LCM) of the denominators.
• Convert each fraction so both have the same denominator, using the LCM.
• Proceed with the arithmetic operation.

In our case, combining two fractions turns them into \(\frac{3}{4}\), which can then be processed further. Always ensure denominators match to simplify your calculations.

Fraction Multiplication

Fraction multiplication is straightforward compared to addition or subtraction. To multiply fractions, multiply the numerators together and then the denominators together. In step 4, we multiplied \(\frac{3}{4}\) by \(\frac{8}{5}\). Here's how:

• Multiply the numerators: \(3 \times 8 = 24\).
• Multiply the denominators: \(4 \times 5 = 20\).
• Put these products together to form \(\frac{24}{20}\).

This method simplifies the complex fraction by turning division into a more straightforward multiplication problem. Remember to align each process step-by-step to maintain accuracy.

Greatest Common Divisor

The greatest common divisor (GCD) helps simplify fractions by identifying the largest number that divides both the numerator and denominator. In the problem, after obtaining \(\frac{24}{20}\), we simplified it by determining the GCD of 24 and 20, which is 4.
Here's how to do it:

• List the factors of each number. For 24: 1, 2, 3, 4, 6, 8, 12, 24; for 20: 1, 2, 4, 5, 10, 20.
• Identify the largest common factor: 4.
• Divide both numerator and denominator by this GCD: \(\frac{24 \div 4}{20 \div 4} = \frac{6}{5}\).

This reduces the fraction to its simplest form, making it easier to interpret and use.

Fraction Simplification

Simplifying fractions is essential to arrive at the most concise and easily understood answer. After finding \(\frac{24}{20}\), we used the GCD to simplify it to \(\frac{6}{5}\). To simplify a fraction:

• Calculate the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

Simplifying fractions ensures you present the answer in its easiest form, aiding comprehension and further calculations. For example, \(\frac{24}{20}\) simplifies to \(\frac{6}{5}\), making it more understandable and reducing complexity.