Problem 42 - Prealgebra

Question

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

Step-by-Step Solution

Answer

\( \frac{2}{3} \div \frac{1}{3} = 2 \).

Step 1: Understand the Division of Fractions

When dividing by a fraction, remember that dividing by a fraction is the same as multiplying by its reciprocal. For example, \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). This means you will flip the second fraction and then multiply.

Step 2: Find the Reciprocal

The reciprocal of \( \frac{1}{3} \) is \( \frac{3}{1} \). So, the division problem becomes a multiplication problem: \( \frac{2}{3} \times \frac{3}{1} \).

Step 3: Multiply the Fractions

Now, multiply the numerators and the denominators: \( \frac{2 \times 3}{3 \times 1} = \frac{6}{3} \).

Step 4: Simplify the Result

Simplify the fraction \( \frac{6}{3} \). Divide the numerator and the denominator by their greatest common divisor, which is 3 in this case. \( \frac{6}{3} = 2 \).

Key Concepts

Reciprocal of a Fraction

The concept of the reciprocal of a fraction is fundamental in the division and multiplication of fractions. If you want to find the reciprocal of a fraction, you simply swap (or flip) the numerator and the denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This process is sometimes referred to as finding the "multiplicative inverse," because a number multiplied by its reciprocal equals 1.

• Why use a reciprocal? When you divide by a fraction, you are actually multiplying by its reciprocal. This is what transforms a division problem into a multiplication problem.
• Example: For the fraction \( \frac{1}{3} \), its reciprocal is \( \frac{3}{1} \).

Knowing how to find the reciprocal quickly is essential when you deal with fraction division since it is the first step in converting a division problem into a simpler multiplication problem.

Multiplication of Fractions

Once you have converted your fraction division problem into a multiplication problem by using the reciprocal, the next step is to multiply the fractions. The multiplication of fractions involves a straightforward process: multiply the numerators together and then multiply the denominators together.

• Multiply the numerators: This gives you the new numerator of the resulting fraction.
• Multiply the denominators: This gives you the new denominator of the resulting fraction.

For instance, in the example \( \frac{2}{3} \times \frac{3}{1} \), you multiply 2 by 3 for the numerator, and 3 by 1 for the denominator, resulting in \( \frac{6}{3} \). This step combines both fractions into a single fraction.

Simplifying Fractions

After multiplying the fractions, the resulting fraction often needs to be simplified. Simplifying a fraction means reducing it to its simplest form, where the greatest common divisor (GCD) of the numerator and the denominator is 1.

• Find the GCD: Determine the highest number that divides both the numerator and denominator.
• Divide both terms: Divide both the numerator and the denominator by the GCD.

For the example \( \frac{6}{3} \), the GCD is 3. By dividing both 6 and 3 by 3, you simplify the fraction to \( 2 \). This process makes the fraction easier to understand and use, ensuring you work with the simplest possible terms.