Problem 18 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-\frac{6}{7} \cdot \frac{2}{7}$$

Step-by-Step Solution

Answer

The product is \(\frac{-12}{49}\) in simplest form.

Step 1: Understand the Problem

We need to multiply two fractions: \(-\frac{6}{7}\) and \(\frac{2}{7}\). When multiplying fractions, the sign of the result will be the product of the signs of the numbers multiplied. In this case, one fraction is negative and the other is positive, so the product will be negative.

Step 2: Multiply the Numerators

To find the product, multiply the numerators (the top numbers) of the fractions together. \[-6 \times 2 = -12\]

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of the fractions together.\[7 \times 7 = 49\]

Step 4: Compose the Fraction

Combine the results of the previous two steps to create the resulting fraction by placing the product of the numerators over the product of the denominators.\[\frac{-12}{49}\]

Step 5: Simplify the Fraction

Check if the fraction \(\frac{-12}{49}\) can be simplified by finding the greatest common divisor of 12 and 49. Since 12 and 49 have no common factors other than 1, the fraction is already in simplest form.

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process if you break it down into simple steps. The process involves the numerators (the top numbers in a fraction) and the denominators (the bottom numbers in a fraction). When you multiply two fractions, you multiply the numerators together and then the denominators together. This gives you a new fraction that is your product. For example, when you multiply \(-\frac{6}{7}\) and \(\frac{2}{7}\), you multiply the numerators: \(-6 \times 2 = -12\), and the denominators: \(7 \times 7 = 49\). Here's a quick guide to multiply fractions efficiently:

• Multiply the numerators across to get a new numerator.
• Multiply the denominators across to get a new denominator.
• Combine them to form a new fraction.

Remember, practice with different numbers helps to solidify your understanding, so try various examples.

Negative Numbers

Working with negative numbers can initially seem tricky, but it's all about remembering the sign rules. When you're dealing with multiplication, keep these rules in mind:

• A positive number multiplied by a positive number is positive.
• A negative number multiplied by a negative number is positive.
• A positive number multiplied by a negative number is negative.
• A negative number multiplied by a positive number is negative.

In the context of multiplying fractions like \(-\frac{6}{7}\) and \(\frac{2}{7}\), the rule applied is the third one above: a negative times a positive is negative. Hence, the result is \(\frac{-12}{49}\). Understanding these rules will help you navigate through equations involving negative numbers more confidently.

Simplification of Fractions

Simplification of fractions is about making a fraction as simple as possible, meaning the numerator and the denominator have no common factors other than 1. When simplifying, your goal is to reduce the fraction so that you can't make it any smaller.To simplify a fraction, determine the greatest common divisor (GCD) of the numerator and the denominator. If they don't have common factors other than 1, then the fraction is already in its simplest form. For instance, with \(\frac{-12}{49}\), checking common factors reveals that 12 and 49 share no common divisors other than 1, so \(\frac{-12}{49}\) is in its simplest form. Remember:

• If the GCD is 1, the fraction is already simplified.
• Practice with finding GCDs can optimize the simplification process.

Understanding how to simplify fractions ensures they are as concise and accurate as possible.