Problem 21 - Beginning and Intermediate Algebra

Question

Multiply or divide as indicated. Write the answer in lowest terms. $$\frac{2}{3} \cdot \frac{3}{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Multiply the Numerators

Multiply the numerators directly:\[2 \times 3 = 6\]

Step 2: Multiply the Denominators

Multiply the denominators directly:\[3 \times 4 = 12\]

Step 3: Form the New Fraction

Combine the products from Step 1 and Step 2 to form the new fraction:\[\frac{6}{12}\]

Step 4: Simplify the Fraction

To simplify \(\frac{6}{12}\), divide both the numerator and the denominator by their greatest common factor, which is 6:\[\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}\]

Key Concepts

Numerator Multiplication

When multiplying fractions, the numerators (the top numbers) of the fractions are multiplied together. The numerator of a fraction indicates parts of the whole represented by the fraction. To find the new numerator after multiplication, simply multiply the numerators of the involved fractions directly, without involving any other numbers.
For instance, in the exercise \(\frac{2}{3} \cdot \frac{3}{4}\), the numerators are 2 and 3. Multiply them together to get the new numerator: \[2 \times 3 = 6\]
Keep this technique in mind as it helps simplify the process of multiplying fractions by treating it as a simple multiplication problem to handle the top numbers.

Denominator Multiplication

In the same way that we multiply the numerators, we also perform multiplication on the denominators of the fractions. The denominator is the bottom part of the fraction and represents how many equal parts the whole is divided into. When multiplying fractions, multiply the denominators together directly.
For our example \(\frac{2}{3} \cdot \frac{3}{4}\), the denominators are 3 and 4. Simply multiply them to find the new denominator: \[3 \times 4 = 12\]
This forms the base of our new fraction. Denominator multiplication is always direct; no need to alter or change the numbers initially.

Simplifying Fractions

Once you form a fraction from the results of the numerator and denominator multiplications, you may need to simplify it. Simplifying a fraction means reducing it to its simplest form where the greatest common factor (GCF) of the numerator and denominator is 1. It involves dividing both the numerator and the denominator by their GCF.
For the fraction \(\frac{6}{12}\), figure out the GCF of 6 and 12, which is 6. Divide both by this number to simplify the fraction: \[\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}\]
Using common factors to reduce fractions is essential in achieving the most comprehensible result, making numerical comparisons and further calculations easier.