Problem 44 - Algebra for College Students

Question

For Problems 41-64, simplify each complex fraction. $$ \frac{\frac{5}{9}+\frac{7}{36}}{\frac{3}{18}-\frac{5}{12}} $$

Step-by-Step Solution

Answer

The complex fraction simplifies to -3.

Step 1: Identify the Complex Fraction

The given complex fraction is \( \frac{\frac{5}{9}+\frac{7}{36}}{\frac{3}{18}-\frac{5}{12}} \). This is a fraction where both the numerator and the denominator are fractions themselves.

Step 2: Simplify the Numerator

The numerator is \( \frac{5}{9} + \frac{7}{36} \). To add these fractions, find a common denominator. The least common denominator of 9 and 36 is 36. Rewrite \( \frac{5}{9} \) as \( \frac{20}{36} \):\[ \frac{20}{36} + \frac{7}{36} = \frac{27}{36} \].Simplify \( \frac{27}{36} \) by dividing both the numerator and denominator by their greatest common factor, which is 9:\[ \frac{27}{36} = \frac{3}{4} \].

Step 3: Simplify the Denominator

The denominator is \( \frac{3}{18} - \frac{5}{12} \). Find the least common denominator, which is 36. Rewrite the fractions:\( \frac{3}{18} = \frac{6}{36} \) and \( \frac{5}{12} = \frac{15}{36} \).Subtract: \( \frac{6}{36} - \frac{15}{36} = \frac{-9}{36} \).Simplify \( \frac{-9}{36} \) by dividing both the numerator and denominator by 9:\[ \frac{-9}{36} = \frac{-1}{4} \].

Step 4: Simplify the Entire Fraction

Now simplify \( \frac{\frac{3}{4}}{\frac{-1}{4}} \). To divide by a fraction, multiply by its reciprocal:\[ \frac{3}{4} \times \frac{-4}{1} = \frac{-12}{4} \].Simplify \( \frac{-12}{4} \) by dividing both the numerator and denominator by 4:\[ \frac{-12}{4} = -3 \].

Key Concepts

Least Common Denominator

When working with fractions, the first step in simplifying a complex fraction is finding the least common denominator (LCD) for the fractions in the numerator and denominator. The least common denominator is the smallest multiple that is common to both denominators.

Finding the LCD allows you to rewrite each fraction with a common denominator, facilitating addition or subtraction. For instance, if we take the fractions \( \frac{5}{9} \) and \( \frac{7}{36} \), the denominators 9 and 36 must be converted to the same base. Here, the smallest shared multiple is 36.

Thus, \( \frac{5}{9} \) is rewritten as \( \frac{20}{36} \) since \( 9 \times 4 = 36 \), while \( \frac{7}{36} \) remains unchanged. This results in both fractions having a common denominator, making it easy to add them.

Adding Fractions

Adding fractions involves combining fractions that have like denominators. This is only possible after determining a common denominator. Once the fractions have the same denominator, you can add the numerators directly while keeping the denominator the same.

In our example, after finding the least common denominator of 36, we rewrite:

• \( \frac{5}{9} \) as \( \frac{20}{36} \)
• \( \frac{7}{36} \) as \( \frac{7}{36} \)

Now, these two fractions can be easily added: \( \frac{20}{36} + \frac{7}{36} = \frac{27}{36} \). Simplifying \( \frac{27}{36} \), we divide the numerator and the denominator by their greatest common factor (GCF), which is 9, resulting in \( \frac{3}{4} \). Simplifying helps bring the fraction to its simplest form.

Reciprocal of a Fraction

The reciprocal of a fraction is created by swapping its numerator and denominator. This concept becomes particularly useful in division because dividing by a fraction is equivalent to multiplying by its reciprocal.

When simplifying complex fractions, especially in the final step where one fraction is divided by another, the reciprocal is key. For our exercise, we simplify \( \frac{\frac{3}{4}}{\frac{-1}{4}} \) by multiplying \( \frac{3}{4} \) by the reciprocal of \( \frac{-1}{4} \), which is \( \frac{-4}{1} \): \[ \frac{3}{4} \times \frac{-4}{1} = \frac{-12}{4}. \]
So, the operation simplifies the expression by switching the division into a multiplication, making calculations straightforward and manageable.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form by dividing the top and bottom by their greatest common factor (GCF). This makes the fraction more concise and easier to understand.

In our exercise, after computing the result of \( \frac{\frac{3}{4}}{\frac{-1}{4}} \), we ended up with \( \frac{-12}{4} \).
To simplify it, we divide both the numerator and denominator by the GCF, which is 4:

• \( -12 \div 4 = -3 \)
• \( 4 \div 4 = 1 \)

Bringing us to the final simplified form of \( -3 \). Simplifying ensures mathematical expressions remain clear and concise, facilitating easier computation and understanding.