Problem 1 - Technical Mathematics with Calculus

Question

Multiply and reduce. Do some by calculator. $$\frac{1}{3} \times \frac{2}{5}$$

Step-by-Step Solution

Answer

The result of \frac{1}{3} \times \frac{2}{5} is \frac{2}{15}.

Step 1: Multiply Numerators

Start by multiplying the numerators (the top numbers) of the fractions. In this case, multiply 1 by 2.

Step 2: Multiply Denominators

Next, multiply the denominators (the bottom numbers) of the fractions. Here, you multiply 3 by 5.

Step 3: Simplify the Fraction

If necessary, simplify the result by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number. In this case, the fraction is already in its simplest form.

Key Concepts

Numerators and Denominators

When multiplying fractions, understanding the role of numerators and denominators is crucial. A fraction consists of two parts: the numerator, which represents the number of parts you have, and the denominator, which represents the total number of equal parts that make up a whole.

For example, \( \frac{1}{3} \) has a numerator of 1 and a denominator of 3, indicating it represents one part out of three equal parts of a whole. To multiply fractions, you multiply the numerators together to find the numerator of the product, and you multiply the denominators together to find the denominator of the product.

Thus, when you calculate \( \frac{1}{3} \times \frac{2}{5} \), you multiply the numerators 1 and 2 to get 2, and the denominators 3 and 5 to get 15, resulting in the fraction \( \frac{2}{15} \). This straightforward process is the fundamental operation behind multiplying fractions. It's important to ensure the integrity of each part: the numerator reflects the product of the parts, and the denominator signifies the new total number of parts.

Simplifying Fractions

Simplifying fractions is integral to expressing them in their most reduced form, making them easier to understand and work with. After multiplying the numerators and denominators of fractions, the next step is to simplify the resulting fraction if possible.

A fraction is simplified when the numerator and denominator are as small as possible, meaning they cannot be divided evenly by any number other than 1. This is done by finding the greatest common divisor (GCD) of both the numerator and the denominator and then dividing both by this GCD.

For instance, if our fractional multiplication yields \( \frac{4}{10} \), we can simplify it by noting that both 4 and 10 can be divided by 2. When we do this, we obtain the simplified fraction \( \frac{2}{5} \). However, in our example problem \( \frac{2}{15} \), no common divisor other than 1 exists, so the fraction is already in its simplest form. Understanding simplification is key to working with fractions across all areas of math.

Greatest Common Divisor

The GCD, or greatest common divisor, is the highest number that evenly divides two or more numbers with no remainder. It plays a vital role in simplifying fractions, because dividing the numerator and denominator of a fraction by their GCD reduces the fraction to its simplest form.

To find the GCD of two numbers, list the factors of each number, then identify the largest factor they have in common. For larger numbers, using the Euclidean algorithm—a process of division and remainder—is more efficient.

In the context of our example where the numerator is 2 and the denominator is 15, you would separately list the factors of 2 (1, 2) and 15 (1, 3, 5, 15). The only common factor is 1, which confirms that \( \frac{2}{15} \) cannot be simplified further and is already in simplest form. Recognizing the GCD and being able to determine it allows for the simplification process to be accurate and efficient.