Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{1}{3} \div \frac{1}{2}=\) ______

Short answer

\(\frac{2}{3}\)

Step-by-step solution

Understand the Problem

We need to divide two fractions: \( \frac{1}{3} \) and \( \frac{1}{2} \). When dividing fractions, we multiply the first fraction by the reciprocal of the second.

Find the Reciprocal

The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For \( \frac{1}{2} \), the reciprocal is \( \frac{2}{1} \).

Multiply Fractions

To divide \( \frac{1}{3} \) by \( \frac{1}{2} \), we multiply \( \frac{1}{3} \) by \( \frac{2}{1} \): \[ \frac{1}{3} \times \frac{2}{1} \].

Calculate the Product

Multiply the numerators: \(1 \times 2 = 2\).Multiply the denominators: \(3 \times 1 = 3\).So, \( \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} \).

Simplify the Result

Check if the fraction \( \frac{2}{3} \) can be simplified. Since 2 and 3 have no common factors other than 1, the fraction is already in its simplest form.

Key concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. For instance, if you have 2\(\frac{1}{3}\), the 2 is the whole number, and \(\frac{1}{3}\) is the fractional part. Unlike simple fractions, mixed numbers represent quantities greater than one.
To work with mixed numbers in fraction division, they often need conversion to improper fractions. An improper fraction has a numerator larger than its denominator. Here's how to convert a mixed number:

• Multiply the whole number by the denominator of the fraction part.
• Add this result to the numerator of the fractional part.
• Place this sum over the original denominator.

For example, converting 2\(\frac{1}{3}\) results in the improper fraction \(\frac{7}{3}\). This conversion is crucial for performing operations like multiplication and division.

Reciprocals

Reciprocals play a key role in the division of fractions. To find a reciprocal, you simply swap the numerator and the denominator of a fraction.
For example, given \(\frac{1}{2}\), its reciprocal is \(\frac{2}{1}\). This reciprocal will help in turning the division operation into a multiplication one.
Without reciprocal, dividing fractions would be more complicated.

• The original division problem \(\frac{1}{3} \div \frac{1}{2}\) becomes a multiplication problem \(\frac{1}{3} \times \frac{2}{1}\) once we find the reciprocal of \(\frac{1}{2}\).
• This simplifies the arithmetic, as multiplying fractions is a more straightforward operation.

Remembering to take the reciprocal is crucial whenever you divide by a fraction.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it is one of the simpler operations in fraction arithmetic. In essence, you multiply the numerators together and the denominators together.
Applying this to \(\frac{1}{3} \times \frac{2}{1}\):

• Multiply the numerators: 1 and 2, resulting in 2.
• Multiply the denominators: 3 and 1, resulting in 3.
• This yields the fraction \(\frac{2}{3}\).

The process can be broken down into manageable steps.
Additionally, once you've multiplied, it's important to check if the resulting fraction can be simplified.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. A fraction is in its simplest form when the numerator and denominator do not have any common factors other than 1.
To simplify, you find the greatest common divisor (GCD) of both numbers and divide them by it.
However, in the example of \(\frac{2}{3}\), the only common factor is 1, meaning the fraction is already simplified.

• Look for common factors between the numerator and denominator.
• Divide both by their GCD if any exist.
• If no other factor exists beyond 1, you're done!

This step ensures the fraction is fully reduced, making it easier to interpret and work with in subsequent operations.