Problem 37 - Elementary Algebra

Question

For exercises 35-38, evaluate. $$ 2 \div \frac{1}{2} $$

Step-by-Step Solution

Answer

4

Step 1: Understand the division of fractions

To divide by a fraction, multiply by its reciprocal. The division of any number by a fraction is equivalent to multiplying the number by the reciprocal of that fraction.

Step 2: Identify the reciprocal of the fraction

The reciprocal of \(\frac{1}{2}\) is \(2\). Since the reciprocal of any fraction \(\frac{a}{b}\) is \(\frac{b}{a}\), the reciprocal of \(\frac{1}{2}\) is \(2\).

Step 3: Multiply the numbers

Multiply \(2\) by \(2\). That is, \(2 \times 2 = 4\).

Key Concepts

Reciprocal

When we talk about reciprocals, we mean flipping a fraction. For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
Not only fractions have reciprocals: every number does. The reciprocal of a whole number like 4 is \(\frac{1}{4}\). The reciprocal of 1 remains 1 because \(\frac{1}{1} = 1\).
In the exercise above, we need to find the reciprocal of \(\frac{1}{2}\). Flipping \(\frac{1}{2}\) gives us 2 because \(\frac{2}{1} = 2\).
Reciprocals are useful because they help us transform division problems into multiplication problems, a concept that will be clear in the next section.

Multiplication of Fractions

Once we have the reciprocal, we turn the division problem into a multiplication problem. For example, dividing by \(\frac{1}{2}\) becomes multiplying by 2.
This is due to a fundamental rule in arithmetic: dividing by a number is the same as multiplying by its reciprocal.
Let's apply this rule to our exercise. We start with 2 divided by \(\frac{1}{2}\). Instead of dividing, we multiply by the reciprocal of \(\frac{1}{2}\), which is 2.
So, our new problem is now \(2 \times 2\), giving us the result of 4. Thus, switching division to multiplication using reciprocals makes the problem much simpler to solve.

Basic Arithmetic

Understanding basic arithmetic is crucial when working with fractions. Arithmetic includes addition, subtraction, multiplication, and division of numbers. In our example, we specifically deal with multiplication.
Here are some key points:

• Multiplication of whole numbers: When you multiply two whole numbers, you add the first number to itself as many times as the second number indicates. For instance, \(2 \times 3\) means adding 2 three times, which is 6.
• Multiplication of fractions: To multiply fractions, multiply the numerators together and the denominators together. For example, \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12}\), which simplifies to \(\frac{1}{2}\).
• Multiplying by a whole number: To multiply a fraction by a whole number, simply multiply the numerator by the whole number. For instance, \(\frac{2}{5} \times 3 = \frac{6}{5}\).

In the problem \(2 \times 2 = 4\), we see basic multiplication applied. Knowing these basic arithmetic rules ensures that you can solve fraction division problems accurately.