Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{2}{5} \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The simplified multiplication is \(\frac{2}{15}\).

Step 1: Multiply the Numerators and Denominators

Multiply the numerators: \(2 * 1 = 2\). Multiply the denominators: \(5 * 3 = 15\). So, \(\frac{2}{5} * \frac{1}{3} = \frac{2}{15}\).

Step 2: Reducing the Fraction to its Lowest Terms

Check if the new fraction can be simplified. Since the numerator 2 and the denominator 15 have no common divisors except for 1, the fraction \(\frac{2}{15}\) is already in its simplest form.

Key Concepts

Reducing Fractions to Lowest Terms

When it comes to fractions, reducing to the lowest terms means making the fraction as simple as possible. This is done by identifying the greatest common divisor (GCD) that both the numerator (top number) and the denominator (bottom number) share, then dividing both by this number.

For example, if you have the fraction \(\frac{6}{9}\), both 6 and 9 can be divided by 3. When you do this, 6 divided by 3 equals 2, and 9 divided by 3 equals 3, so the fraction reduces to \(\frac{2}{3}\).

This process does not change the value of the fraction; rather, it simplifies it, making it easier to understand or further manipulate in mathematical operations. In the exercise \(\frac{2}{5} \times \frac{1}{3}\), since 2 and 15 have no common factors other than 1, the fraction \(\frac{2}{15}\) cannot be reduced any further and is already in its simplest form.

Numerators and Denominators

The numerator and denominator are fundamental parts of a fraction. The numerator, located above the fraction bar, indicates how many parts of the whole you are dealing with, while the denominator, below the fraction bar, indicates the total number of equal parts the whole is divided into.

In multiplying fractions, you multiply numerator with numerator and denominator with denominator. If we take the exercise \(\frac{2}{5} \times \frac{1}{3}\), we multiply 2 (the numerator of the first fraction) by 1 (the numerator of the second fraction), which equals 2, and 5 (the denominator of the first fraction) by 3 (the denominator of the second fraction), which equals 15. So, the resultant fraction is \(\frac{2}{15}\).

Understanding the role of numerators and denominators is crucial because it helps you grasp why certain mathematical rules for fractions work the way they do and allows you to simplify fractions correctly by reducing them to their lowest terms.

Simplifying Fractions

Simplifying fractions, often called 'reducing' fractions, is the process of converting a fraction into its simplest form where the numerator and the denominator are as small as possible. The goal is to strip the fraction down to its essential value without changing the quantity it represents.

This is typically done by dividing both the numerator and the denominator by their greatest common divisor. For example, with the fraction \(\frac{8}{20}\), the greatest common divisor is 4. Divide both parts of the fraction by 4, and you get \(\frac{2}{5}\), which is the simplified form.

Sometimes, you'll find fractions that are already in the simplest form, like \(\frac{3}{7}\) or \(\frac{5}{11}\). These cannot be simplified any further because the numerator and the denominator have no common divisors other than 1. In our original exercise, the fraction \(\frac{2}{15}\) is already simplified as 2 and 15 do not share any common factors apart from 1.