Problem 69 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{3 a}{5 b}}{2}$$

Step-by-Step Solution

Answer

The simplified result is \[\frac{3 a}{10 b}\].

Step 1: Understand the Problem

The given expression is \[\frac{\frac{3 a}{5 b}}{2}\]. Here, we need to simplify the fraction within a fraction.

Step 2: Rewrite the Expression

First, rewrite the complex fraction as a division problem: \[\frac{3 a}{5 b} \div 2\].

Step 3: Simplify the Division

To divide by 2, multiply by the reciprocal of 2. So, we have: \[\frac{3 a}{5 b} \times \frac{1}{2}\].

Step 4: Multiply the Fractions

Now multiply the fractions: \[\frac{3 a \times 1}{5 b \times 2} = \frac{3 a}{10 b}\].

Key Concepts

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both are also fractions. They might look a bit intimidating at first because they contain layers of fractions inside a larger fraction. However, simplifying them can be straightforward if broken down into manageable steps.

To simplify a complex fraction, we typically rewrite it as a division problem. This aligns with our basic understanding of fractions, where fraction division is simply multiplying by the reciprocal.
For example, consider the complex fraction \(\frac{\frac{3a}{5b}}{2}\). This can initially look tricky but following systematic steps can make it easier.

Division of Fractions

Division of fractions may seem challenging, but it follows a simple rule: instead of dividing, multiply by the reciprocal. The reciprocal of a fraction is inverted, meaning the numerator and the denominator are switched.

For instance, in the problem \(\frac{\frac{3a}{5b}}{2}\), we first rewrite it as \(\frac{3a}{5b} \div 2\). Next, we change the division into multiplication by the reciprocal of 2, which becomes \(\frac{3a}{5b} \times \frac{1}{2}\). This step simplifies the fraction, making it easier to handle.
Simplifying division problems in this way helps break them down into more straightforward multiplication tasks.

Multiplication of Fractions

Multiplication of fractions is a straightforward process. When you multiply two fractions, you multiply their numerators together and their denominators together.

In our example, we are multiplying \(\frac{3a}{5b} \times \frac{1}{2}\). You multiply the numerators: 3a and 1, which gives 3a. Then multiply the denominators: 5b and 2, which gives 10b. Thus, \(\frac{3a}{5b} \times \frac{1}{2} = \frac{3a}{10b}\).
This step-by-step approach clarifies the overall process and ensures accuracy in simplification.

Reciprocals

The reciprocal of a number is essentially its 'flipped' version. For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). Reciprocals are crucial in fraction division because dividing by a number is the same as multiplying by its reciprocal.

In our exercise, we divided by 2 by multiplying by \(\frac{1}{2}\). This transformation simplifies the operation and helps solve the problem efficiently.
Understanding reciprocals helps in simplifying complex fractions and performing division of fractions seamlessly.