Problem 15 - Basic College Mathematics

Question

Divide and simplify. \(\frac{1}{3} \div \frac{1}{6}\)

Step-by-Step Solution

Answer

2

Step 1 - Understand Division of Fractions

Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).

Step 2 - Identify the Reciprocal

Identify the reciprocal of \(\frac{1}{6}\). It is \(\frac{6}{1}\), because the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

Step 3 - Convert Division to Multiplication

Rewrite the division as multiplication by the reciprocal: \(\frac{1}{3} \div \frac{1}{6} = \frac{1}{3} \times \frac{6}{1}\).

Step 4 - Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{1}{3} \times \frac{6}{1} = \frac{1 \times 6}{3 \times 1}\).

Step 5 - Simplify the Result

Simplify the fraction: \(\frac{6}{3}\) simplifies to 2. Therefore, \(\frac{1}{3} \div \frac{1}{6} = 2\).

Key Concepts

Fractions

Let's start by understanding what fractions are. A fraction represents a part of a whole. It consists of two numbers: the numerator and the denominator. The numerator is the number on the top, and it tells you how many parts you have. The denominator is the number on the bottom, and it tells you how many equal parts the whole is divided into.
For example, in the fraction \(\frac{1}{3}\), 1 is the numerator and 3 is the denominator. This means you have 1 part out of 3 equal parts. Understanding fractions is essential, as they are the basic building blocks in many math concepts, especially when dealing with division and multiplication.

Reciprocal

A reciprocal is another key concept when working with fractions. The reciprocal of a fraction is created by swapping the numerator and the denominator. In other words, if you have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
Reciprocals are important when it comes to operations like division. For instance, if you need to divide by a fraction, you can multiply by its reciprocal instead. In our example, the reciprocal of \(\frac{1}{6}\) is \(\frac{6}{1}\), making the division problem easier to solve.

Multiplication of Fractions

Now, let's talk about multiplying fractions. To multiply two fractions, you simply multiply the numerators together and the denominators together. For example, to multiply \(\frac{1}{3}\) by \(\frac{6}{1}\), you do the following:
1. Multiply the numerators: 1 x 6 = 6
2. Multiply the denominators: 3 x 1 = 3
So, \(\frac{1}{3} \times \frac{6}{1} = \frac{6}{3}\). This method is straightforward and helps in simplifying the process of dealing with fractions in various mathematical operations.

Simplification

Simplification is the process of reducing fractions to their simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. For example, after multiplying \(\frac{1}{3}\) and \(\frac{6}{1}\), we get \(\frac{6}{3}\).
This fraction can be simplified because both 6 and 3 share a common factor, which is 3. Divide both the numerator and the denominator by 3: \(\frac{6}{3} = 2\).
Thus, simplification helps make fractions easier to work with and often makes the results cleaner and simpler to understand.