Problem 41 - Prealgebra

Question

The problems below review some basic concepts of division with fractions and mixed numbers. Divide. $$\frac{1}{3} \div \frac{2}{3}$$

Step-by-Step Solution

Answer

The result of \( \frac{1}{3} \div \frac{2}{3} \) is \( \frac{1}{2} \).

Step 1: Understand the Problem

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

Step 2: Find the Reciprocal

The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). Switching the numerator and denominator of a fraction gives us its reciprocal.

Step 3: Multiply the Fractions

We multiply \( \frac{1}{3} \) by \( \frac{3}{2} \): \[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} \].

Step 4: Simplify the Fraction

Simplify \( \frac{3}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. \[ \frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \].

Key Concepts

Reciprocal

When it comes to dividing fractions, understanding the reciprocal is key. A reciprocal of a fraction is essentially what you flip the fraction into. In other words, you swap the numerator (the top part of the fraction) with the denominator (the bottom part). For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). This step is crucial because division of fractions isn't quite straightforward; instead, we turn it into a multiplication problem by multiplying by the reciprocal.

• The trick to finding a reciprocal is simple: just flip the fraction!
• This technique is only applicable when the fraction is not zero, as zero doesn’t have a reciprocal.

Make sure to be comfortable finding reciprocals because it is a fundamental skill necessary in fraction division.

Multiply Fractions

Once we have the reciprocal, the next step in dividing fractions is to multiply. Multiplying fractions is far simpler than it sounds.You take the numerators (top numbers) of both fractions and multiply them together. Then, do the same with the denominators (bottom numbers). For example, when multiplying \( \frac{1}{3} \) by its reciprocal of \( \frac{3}{2} \), you would do:\[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} \]

• You multiply straight across: top with top, bottom with bottom.
• Always simplify the result to avoid unnecessary complexity.
• This method works consistently no matter the fractions you are dealing with.

This step converts a division problem into an easily manageable multiplication problem.

Simplify Fractions

After multiplying, it's important to simplify the fraction to its simplest form. Simplifying fractions makes them easier to understand and compare.To simplify \( \frac{3}{6} \), divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD is 3, because 3 is the largest number that can divide both 3 and 6.\[ \frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \]Some helpful tips for simplifying fractions include:

• Identify the GCD for both numbers.
• Divide both the top and bottom by this number.
• Keep simplifying until no further reduction is possible."

Simplification keeps your final answer clean and is a significant part of fraction mathematics.