Problem 48 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{1}{4} \div \frac{1}{2}$$

Step-by-Step Solution

Answer

\frac{1}{2}\

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, multiply by the reciprocal of that fraction. The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\). So, rewrite \(\frac{1}{4} \div \frac{1}{2}\) as \(\frac{1}{4} \times \frac{2}{1}\).

Step 2: Multiply the Fractions

Multiply the numerators together: \(1 \times 2 = 2\). Multiply the denominators together: \(4 \times 1 = 4\). Therefore, \(\frac{1}{4} \times \frac{2}{1} = \frac{2}{4}\).

Step 3: Simplify the Fraction

Simplify \(\frac{2}{4}\) by dividing both the numerator and denominator by their greatest common divisor, which is 2: \(\frac{2 \/ 2}{4 \ / 2} = \frac{1}{2}\).

Key Concepts

Dividing Fractions

When you see a division between two fractions, it might look tricky at first. But, there is an easy way to handle it! To divide by a fraction, you multiply by its reciprocal.
The reciprocal of a fraction is simply flipping the numerator (top number) and the denominator (bottom number).
For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\) (also known as just 2).
So, if we have \(\frac{1}{4} \div \frac{1}{2}\), we rewrite it as \(\frac{1}{4} \times \frac{2}{1}\).

• Flip the second fraction to find the reciprocal.
• Change the division sign to a multiplication sign.
• Now, multiply the fractions.

Multiplying Fractions

Multiplying fractions is simpler than it might initially seem. Follow these easy steps:

• Multiply the numerators (top numbers) of the fractions to get the new numerator.
• Multiply the denominators (bottom numbers) to get the new denominator.

Using our example, after converting the division to multiplication, we have \(\frac{1}{4} \times \frac{2}{1}\). Let's multiply:

• Numerator: 1 \times 2 = 2
• Denominator: 4 \times 1 = 4

So, \(\frac{1}{4} \times \frac{2}{1}\) becomes \(\frac{2}{4}\).

Simplifying Fractions

Simplifying fractions makes them easier to work with. Here’s how you do it:

• Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 2 and 4 is 2.
• Divide both the numerator and the denominator by their GCD.

Applying this to our problem:
- For \(\frac{2}{4}\), divide both 2 and 4 by 2 to get \(\frac{2\/2}{4\/2} = \frac{1}{2}\).
So, \(\frac{2}{4}\) simplifies to \(\frac{1}{2}\).
This ensures the fraction is in its simplest form, making it easier to understand and use.