Problem 30 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified complex fraction is \(\frac{13}{10}\).

Step 1: Understand the structure

The given complex fraction is \(\frac{\frac{3}{4}+\frac{1}{3}}{\frac{2}{3}+\frac{1}{6}}\). It is a fraction where both the numerator and the denominator are themselves fractions.

Step 2: Simplify the numerator

Add \(\frac{3}{4}\) and \(\frac{1}{3}\) by finding a common denominator. The least common multiple of 4 and 3 is 12, so rewrite the fractions: \(\frac{3}{4} = \frac{9}{12}\) and \(\frac{1}{3} = \frac{4}{12}\). Thus, \(\frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12}\).

Step 3: Simplify the denominator

Add \(\frac{2}{3}\) and \(\frac{1}{6}\) by finding a common denominator. The least common multiple of 3 and 6 is 6, so rewrite the fractions: \(\frac{2}{3} = \frac{4}{6}\) and \(\frac{1}{6}\) stays the same. Thus, \(\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}\).

Step 4: Divide the simplified fractions

Now, divide the simplified numerator \(\frac{13}{12}\) by the simplified denominator \(\frac{5}{6}\). This is equivalent to multiplying \(\frac{13}{12}\) by the reciprocal of \(\frac{5}{6}\), which is \(\frac{6}{5}\).

Step 5: Perform the multiplication

Multiply the fractions: \(\frac{13}{12} \times \frac{6}{5} = \frac{13 \times 6}{12 \times 5} = \frac{78}{60}\).

Step 6: Simplify the final fraction

Simplify \(\frac{78}{60}\) by finding the greatest common divisor, which is 6. Divide the numerator and the denominator by 6: \(\frac{78 \div 6}{60 \div 6} = \frac{13}{10}\). The simplified form of the complex fraction is \(\frac{13}{10}\).

Key Concepts

Fractions Simplification

When it comes to breaking down complex fractions, **simplification** is your best friend. Let's start with the basics. Fractions can be intimidating, especially when they're fractions within fractions. But with the right approach, they become much simpler. For instance, look at the numerator and denominator of our complex fraction separately. Simplification is key here.

Ensure that the fractions you're dealing with are in their simplest form before proceeding to combine or divide. Simplifying involves finding a common denominator or factoring out common elements. This makes the next steps more manageable and prepares the fractions for easy addition or subtraction.

In our example, each part of the complex fraction (both the numerator and the denominator) needed to undergo simplification by finding a common base for addition. The ultimate goal is to make each fraction easier to manipulate.

Least Common Denominator

The **Least Common Denominator (LCD)** is crucial when performing operations with fractions. Why? Because it harmonizes the denominators, enabling you to add or subtract the fractions with ease. Think of the LCD as the smallest number that both denominators can divide into without leaving a remainder.

For instance, when working with the numerator \(\frac{3}{4} + \frac{1}{3}\), we identified the LCD to be 12, because 12 is the smallest number both 4 and 3 divide into perfectly. This allowed us to express these fractions with a unified denominator: \(\frac{9}{12} + \frac{4}{12}\). The same process applies to the denominator of our complex fraction, where the LCD between \(\frac{2}{3}\) and \(\frac{1}{6}\) is 6.

This critical step ensures no fraction is left unaccounted for in its actual value, providing a more straightforward path to simplification.

Dividing Fractions

Once we have simplified the complex fraction into two separate fractions, **Dividing Fractions** is the next step. Instead of directly dividing, we employ the widely known rule: "invert and multiply." This makes fractional division a lot easier.

Take the example from the exercise, where we have simplified our complex fraction to \(\frac{13}{12} \text{ and } \frac{5}{6}\). Instead of dividing \(\frac{13}{12}\) by \(\frac{5}{6}\), we multiply \(\frac{13}{12}\) by the reciprocal of \(\frac{5}{6}\). Thus, the problem transforms into: \(\frac{13}{12} \times \frac{6}{5}\).

This method eliminates any errors that may arise from direct division and is generally the more manageable and reliable technique for these operations.

Reciprocal in Fractions

The concept of a **reciprocal** is fundamental in dividing fractions. But what exactly is a reciprocal? It's simply the flip of a fraction.

So, if you have \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). In fraction division, we use reciprocity to transform a division problem into a multiplication one, which most people find simpler. By multiplying by the reciprocal, the numerator and denominator "switch roles," making the arithmetic straightforward.

In our exercise, the reciprocal of \(\frac{5}{6}\) is \(\frac{6}{5}\), and that's what we use to transform the division problem into multiplication. This operation is not just a trick; it's a valid mathematical transformation that ensures accuracy and ease in solving.