Problem 21 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\left(-\frac{1}{3}\right)\left(\frac{2}{5}\right)$$

Step-by-Step Solution

Answer

The result is \(-\frac{2}{15}\).

Step 1: Identify the Operation

The problem requires you to perform multiplication between two fractions: \(-\frac{1}{3}\) and \(\frac{2}{5}\).

Step 2: Multiply the Numerators

To multiply the fractions, first multiply the numerators. Multiply \(-1\) and \(2\) to get the new numerator: \(-1 \times 2 = -2\).

Step 3: Multiply the Denominators

Next, multiply the denominators of the fractions. Multiply \(3\) and \(5\) to get the new denominator: \(3 \times 5 = 15\).

Step 4: Combine the Results

Combine the results from the previous steps to form the new fraction: \(\{-2}{15}\).

Step 5: Simplify the Fraction

Check if the fraction \(-\frac{2}{15}\) can be simplified further. Since \(2\) and \(15\) have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process that involves two main parts: multiplying the numerators together and multiplying the denominators together. Imagine fractions as slices of pie. Each slice is a fraction of the whole pie.
When you multiply fractions, it's like you're finding a fraction of those slices, so the process is simple:

• Step 1: Multiply the Numerators: The numerators are the top numbers in the fractions. For the fractions \(-\frac{1}{3}\) and \(\frac{2}{5}\), you multiply \(-1\) and \(2\) to get \(-2\).
• Step 2: Multiply the Denominators: The denominators are the bottom numbers. Multiply \(3\) and \(5\) to get \(15\).

After these steps, you will have a new fraction \(-\frac{2}{15}\). At this point, you have multiplied the fractions successfully!

Simplifying Fractions

After finding the product of fractions, it is important to check if the resulting fraction can be made simpler. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
Think of simplifying as reducing clutter — like cleaning up a mess to make things as simple as they can be. Here’s how:

• Check for Common Factors: Look at both the numerator and denominator of \(-\frac{2}{15}\). The numbers \(2\) and \(15\) do not share any common factors other than 1.
• Simplify if Possible: If there were any common factors, you could divide both by the greatest common factor to make the fraction simpler. In this case, since \(-\frac{2}{15}\) cannot be simplified further, it's already in its simplest form.

Numerators and Denominators

Understanding numerators and denominators is crucial when dealing with fractions. They are the backbone of all fraction calculations and concepts.

• Numerators: The numerator is the top part of the fraction. It indicates how many parts of the whole you have. In \(-\frac{1}{3}\), \(-1\) is the numerator, representing a negative portion.
• Denominators: The denominator is the bottom part of the fraction. It shows into how many equal parts the whole is divided. In \(\frac{2}{5}\), \(5\) is the denominator, suggesting that the whole is split into five parts.

Together, numerators and denominators tell us everything about a fraction and how it compares to a whole. They may seem small, but they carry all the information needed for mathematical operations and understanding what a fraction represents.