Problem 39 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{\frac{1}{3}+\frac{3}{4}}{2-\frac{1}{6}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{13}{22} \).

Step 1: Simplify the Numerator

The numerator of the complex fraction is \( \frac{1}{3} + \frac{3}{4} \). To add these two fractions, find a common denominator, which is 12. Convert \( \frac{1}{3} \) to \( \frac{4}{12} \) and \( \frac{3}{4} \) to \( \frac{9}{12} \). Adding these gives \( \frac{4}{12} + \frac{9}{12} = \frac{13}{12} \).

Step 2: Simplify the Denominator

The denominator of the complex fraction is \( 2 - \frac{1}{6} \). Convert 2 into a fraction with a denominator of 6, which is \( \frac{12}{6} \). Subtract \( \frac{12}{6} - \frac{1}{6} = \frac{11}{6} \).

Step 3: Divide the Numerator by the Denominator

The complex fraction \( \frac{\frac{13}{12}}{\frac{11}{6}} \) can be simplified by multiplying the numerator by the reciprocal of the denominator: \( \frac{13}{12} \times \frac{6}{11} \). Multiply across: \( \frac{13 \times 6}{12 \times 11} = \frac{78}{132} \).

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{78}{132} \) by finding the greatest common factor (GCF) of 78 and 132, which is 6. Divide the numerator and the denominator by 6, yielding \( \frac{13}{22} \). This is the simplified form of the complex fraction.

Key Concepts

Fractions Simplification

Simplifying fractions means reducing a fraction to its simplest form so that the numerator and the denominator have no common factors other than 1. It is a crucial skill to master in mathematics as it makes calculations easier and helps in comparing fractions effectively.
To simplify a complex fraction like \( \frac{\left(\frac{1}{3} + \frac{3}{4}\right)}{\left(2 - \frac{1}{6}\right)} \), you need to first tackle what's inside the fraction. Start by simplifying the numerator and the denominator separately.

• For the numerator \( \frac{1}{3} + \frac{3}{4} \), find a common denominator. Here, that would be 12, turning \( \frac{1}{3} \) into \( \frac{4}{12} \) and \( \frac{3}{4} \) into \( \frac{9}{12} \). Adding these gives \( \frac{13}{12} \).
• For the denominator \( 2 - \frac{1}{6} \), rewrite 2 as a fraction \( \frac{12}{6} \), then subtract to get \( \frac{11}{6} \).

Now, you have a much simpler state: \( \frac{\frac{13}{12}}{\frac{11}{6}} \), which allows you to apply the concept of reciprocal multiplication next.

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. This concept is essential for simplifying fractions because dividing both the numerator and the denominator of a fraction by their GCF reduces the fraction to its simplest form.
In our example, you achieve the final simplification by finding the GCF of 78 and 132. Once you have the product of the numerator and the reciprocal of the denominator as \( \frac{78}{132} \), the next step is to simplify. The numbers 78 and 132 share common factors, but their greatest is 6.

• 78 divided by 6 gives 13.
• 132 divided by 6 gives 22.

Thus, the fraction \( \frac{78}{132} \) simplifies to \( \frac{13}{22} \). Understanding how to find and use the GCF quickly makes simplifying any fraction a much easier task.

Reciprocal Multiplication

Reciprocal multiplication is a method used to simplify division of fractions. When you divide by a fraction, you actually multiply by its reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \). This technique is particularly useful in dealing with complex fractions.
Observe the complex fraction \( \frac{\frac{13}{12}}{\frac{11}{6}} \). Dividing by \( \frac{11}{6} \) is the same as multiplying by its reciprocal, \( \frac{6}{11} \).

• So, instead of performing a cumbersome division, multiply across: \( \frac{13}{12} \times \frac{6}{11} \).
• This multiplication yields \( \frac{13 \times 6}{12 \times 11} \) or \( \frac{78}{132} \).

Using reciprocal multiplication not only simplifies the complexity of calculations but also gives a clear pathway to possible fraction simplification, bringing you closer to finding the final simplest form.