Problem 4 - Algebra for College Students

Question

Simplify each complex fraction. Use either method. $$\frac{\frac{1}{3}+\frac{1}{4}}{\frac{1}{5}+\frac{1}{6}}$$

Step-by-Step Solution

Answer

\(\frac{35}{22}\)

Step 1: Combine the numerators

First, add the fractions in the numerator of the complex fraction. The fractions to add are \(\frac{1}{3}\) and \(\frac{1}{4}\). Find a common denominator, which is 12. \[\frac{1}{3} = \frac{4}{12} \text{ and } \frac{1}{4} = \frac{3}{12}\]So, \[\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\]

Step 2: Combine the denominators

Next, add the fractions in the denominator of the complex fraction. The fractions to add are \(\frac{1}{5}\) and \(\frac{1}{6}\). Find a common denominator, which is 30. \[\frac{1}{5} = \frac{6}{30} \text{ and } \frac{1}{6} = \frac{5}{30}\]So, \[\frac{1}{5} + \frac{1}{6} = \frac{6}{30} + \frac{5}{30} = \frac{11}{30}\]

Step 3: Divide the results

Now, divide the result of step 1 by the result of step 2. This means finding the fraction \[\frac{\frac{7}{12}}{\frac{11}{30}}\] To divide by a fraction, multiply by its reciprocal: \[\frac{7}{12} \times \frac{30}{11} = \frac{7 \times 30}{12 \times 11} = \frac{210}{132}\]

Step 4: Simplify the final fraction

Simplify \(\frac{210}{132}\). The greatest common divisor (GCD) of 210 and 132 is 6. \[\frac{210 \text{ ÷ } 6}{132 \text{ ÷ } 6} = \frac{35}{22}\]Thus, the simplified form is \(\frac{35}{22}\).

Key Concepts

common denominator

When dealing with fractions, finding the common denominator is crucial, especially when adding or subtracting them. A common denominator is a shared multiple of the denominators of two or more fractions. For example, to add \(\frac{1}{3}\) and \(\frac{1}{4}\), we look for the smallest number that both 3 and 4 can divide into evenly. This number is 12, as it is the least common multiple (LCM) of both denominators.

• Convert each fraction to an equivalent fraction with the common denominator. For \(\frac{1}{3}\), multiply the numerator and denominator by 4 to get \(\frac{4}{12}\).
• Do the same for \(\frac{1}{4}\), but multiply both the numerator and denominator by 3 to get \(\frac{3}{12}\).

Now, both fractions have the same denominator, allowing you to easily add them: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\). This setup makes complex problems manageable by breaking them down into simpler steps.

reciprocal

The reciprocal of a number or fraction is simply flipping the numerator and the denominator. In other words, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This concept is particularly useful when dividing fractions.

In the context of the exercise, we had to find \(\frac{\frac{7}{12}}{\frac{11}{30}}\). To divide by \(\frac{11}{30}\), multiply by its reciprocal instead: \(\frac{30}{11}\). This is a core technique for simplifying complex fractions.

• Start by multiplying the numerators: \(\frac{7}{12} \times \frac{30}{11}\) becomes \(\frac{7 \times 30}{12 \times 11}\)
• This gives you \(\frac{210}{132}\)

The reciprocal allows for a clearer, more straightforward multiplication process rather than dealing directly with division.

greatest common divisor

Simplifying a fraction involves dividing both the numerator and the denominator by the greatest common divisor (GCD), which is the largest number that divides both of them evenly.

For \(\frac{210}{132}\), the GCD is 6. This can be found by listing the factors of both numbers and identifying the largest common factor.

• Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
• Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132

The largest common number is 6. Therefore, divide both the numerator and the denominator by 6 to simplify: \(\frac{210 ÷ 6}{132 ÷ 6} = \frac{35}{22}\).

Simplifying fractions using the GCD makes your final answer neat and easier to understand.