Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{4}{6} \div \frac{1}{2}=\) ______

Short answer

\(1 \frac{1}{3}\)

Step-by-step solution

Understanding Division of Fractions

To divide fractions, we multiply by the reciprocal of the second fraction. Here, the problem is \( \frac{4}{6} \div \frac{1}{2} \). The reciprocal of \( \frac{1}{2} \) is \( \frac{2}{1} \). So, \( \frac{4}{6} \div \frac{1}{2} = \frac{4}{6} \times \frac{2}{1} \).

Multiply the Fractions

Multiply the numerators together and the denominators together. The expression \( \frac{4}{6} \times \frac{2}{1} \) becomes: \( \frac{4 \times 2}{6 \times 1} = \frac{8}{6} \).

Simplify the Fraction

To simplify \( \frac{8}{6} \), find the greatest common divisor (GCD) of 8 and 6, which is 2. Divide both the numerator and the denominator by 2: \( \frac{8 \div 2}{6 \div 2} = \frac{4}{3} \).

Convert to a Mixed Number

Since \( \frac{4}{3} \) is an improper fraction, convert it to a mixed number. Divide 4 by 3: 4 divided by 3 is 1 with a remainder of 1. So, \( \frac{4}{3} = 1 \frac{1}{3} \).

Key concepts

Reciprocal

When dividing fractions, a crucial step is understanding the term "reciprocal." The reciprocal of a fraction is simply obtained by swapping its numerator and denominator. So, for the fraction \( \frac{1}{2} \), its reciprocal is \( \frac{2}{1} \) or just \( 2 \). By multiplying by the reciprocal, we transform the division of fractions into a multiplication problem.
This makes it easier to solve. Remember:

• The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• If you have a whole number like 3, its reciprocal is \( \frac{1}{3} \).

Simplification

Simplification is an important process that involves reducing a fraction to its lowest possible terms. What does this mean? It means there are no common factors between the numerator and the denominator other than 1.
For example, in the step \( \frac{8}{6} \) becomes \( \frac{4}{3} \), we divide both numerator and denominator by their greatest common divisor (GCD), which is 2.
Here's how:

• Find the GCD of the numbers. For 8 and 6, it's 2.
• Divide both by their GCD: \( \frac{8 \div 2}{6 \div 2} \).
• Simplified version is \( \frac{4}{3} \).

Simplification helps in reducing fractions, making them easier to work with.

Improper Fractions

Improper fractions have a numerator larger than the denominator. In other words, the top number is greater than the bottom number.
In our exercise, \( \frac{4}{3} \) is an example of an improper fraction. While improper fractions are fine to use, they are often converted into mixed numbers for a more standardized representation.
Things to know:

• Improper fractions can be converted to mixed numbers easily.
• They are useful for expressing fractions greater than 1.

Recognizing and using improper fractions is a skill that boosts your number sense.

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction, helping in expressing quantities in a more digestible form.
To convert an improper fraction like \( \frac{4}{3} \) into a mixed number, you divide the numerator by the denominator. Here’s how it works.

• Whole number is the result of the division.
• Remainder becomes the new numerator, while the denominator stays the same.

So for \( \frac{4}{3} \), dividing 4 by 3 gives 1 as the whole number with a remainder of 1, making it \( 1 \frac{1}{3} \). Understanding mixed numbers can be especially helpful when working with measurements in everyday tasks.