Problem 97 - Elementary Algebra

Question

Evaluate: \(8 \div \frac{1}{2}\)

Step-by-Step Solution

Answer

16

Step 1: Express the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1} \) or simply 2. Thus, \( 8 \div \frac{1}{2} \) can be written as \( 8 \times 2 \).

Step 2: Perform the Multiplication

Now, multiply 8 by 2. \( 8 \times 2 = 16 \).

Step 3: Write the Final Answer

Therefore, \( 8 \div \frac{1}{2} = 16 \).

Key Concepts

reciprocal

In mathematics, the concept of the reciprocal is fundamental when working with fractions. To find the reciprocal of a fraction, you simply swap its numerator and denominator.

For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\)\ , or simply 2. Essentially, you are flipping the fraction upside down. The reciprocal of a number is also known as its multiplicative inverse because multiplying a number by its reciprocal always gives 1.

Consider the fraction \(\frac{3}{4}\):

• The reciprocal is \(\frac{4}{3}\).

When dealing with whole numbers such as 5, the reciprocal is \(\frac{1}{5}\). This principle is crucial for understanding how to reverse operations, like division with fractions.

multiplication

Multiplication is one of the four basic arithmetic operations. When we talk about multiplying fractions, the process is straightforward: Simply multiply the numerators together and then the denominators together.

Let's break it down:

• Take the fractions \(\frac{a}{b} \times \frac{c}{d}\).
• Multiply the numerators: \(a \times c\).
• Multiply the denominators: \(b \times d\).

For example, if you have \(\frac{2}{3} \times \frac{4}{5}\), you would calculate:

• Numerator: 2 \times 4 = 8
• Denominator: 3 \times 5 = 15

So, \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\).

Remember, the key idea is multiplying the top numbers (numerators) together and the bottom numbers (denominators) together. This simplicity is why multiplication often replaces division when handling fractions.

fraction division

Fraction division might seem complex at first, but it becomes straightforward when you understand the steps involved. To divide by a fraction, you multiply by its reciprocal.

Here’s a step-by-step breakdown:

• Step 1: Find the reciprocal of the divisor (the fraction you are dividing by).
• Step 2: Change the division operation to multiplication.
• Step 3: Multiply as you would with regular fractions.

Consider the exercise: \(8 \div \frac{1}{2}\).

Step 1: Find the reciprocal of \(\frac{1}{2}\), which is 2.

Step 2: Change the division sign to multiplication: \(8 \times 2\).

Step 3: Multiply: 8 \times 2 = 16.

By converting the division into a multiplication problem, we simplify the process.
This method is useful for all fractions. Just remember: Division by a fraction is the same as multiplication by its reciprocal.