Problem 9 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-\frac{6}{10} \cdot \frac{1}{8}$$

Step-by-Step Solution

Answer

The product is \(-\frac{3}{40}\).

Step 1: Change Each Fraction to Simplest Form

First, identify that the fraction \(-\frac{6}{10}\) can be simplified. Divide the numerator and denominator by their greatest common divisor, which is 2. \(-\frac{6}{10} = -\frac{3}{5}\). The fraction \(\frac{1}{8}\) is already in its simplest form.

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions. You have \(-3\) from the fraction \(-\frac{3}{5}\) and \(1\) from \(\frac{1}{8}\), so the product of the numerators is: \(-3 \times 1 = -3\).

Step 3: Multiply the Denominators

Next, multiply the denominators of the two fractions. You have \(5\) from the fraction \(-\frac{3}{5}\) and \(8\) from \(\frac{1}{8}\), so the product of the denominators is: \(5 \times 8 = 40\).

Step 4: Write the Result as a Fraction

Combine the products of the numerators and the denominators to express the result as a fraction: \(-\frac{3}{40}\).

Step 5: Check if Simplification is Necessary

Finally, check if \(-\frac{3}{40}\) can be simplified further. Since 3 and 40 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Fraction Simplification

Simplifying a fraction means reducing it to its simplest form. To do this, find the greatest common divisor (GCD) of the numerator and the denominator. In the exercise, the fraction \(-\frac{6}{10}\) was simplified to \(-\frac{3}{5}\) by dividing the numerator and denominator by their GCD, which is 2.

Here's how you can simplify fractions:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

It's important to note that the fraction \(\frac{1}{8}\) could not be simplified further, as its numerator and denominator only share 1 as a common factor.

Numerator and Denominator

A fraction is composed of two parts: the numerator and the denominator. The numerator is the top part and represents how many parts of a whole are being considered. The denominator, on the bottom, tells you into how many equal parts the whole is divided.

In the fractions \(\-\frac{6}{10}\) and \(\frac{1}{8}\), we distinctively say:

• The numerator of \(\-\frac{6}{10}\), is -6.
• The denominator of \(\-\frac{6}{10}\), is 10.
• The numerator of \(\frac{1}{8}\), is 1.
• The denominator of \(\frac{1}{8}\), is 8.

Understanding the roles of the numerator and denominator is crucial when multiplying fractions. You multiply the numerators together and the denominators together to find the product.

Negative Number Operations

Working with negative numbers in fractions can seem tricky, but it's quite simple once you get the hang of it. A negative number can appear in the numerator, the denominator, or in front of the fraction.

For instance, \(-\frac{3}{5}\) contains a negative in the numerator, making the entire fraction negative. When multiplying two fractions where one or both are negative:

• If only one fraction is negative, the product will be negative.
• If both fractions are negative, the product will be positive.

In our exercise case, multiplying a negative fraction \(-\frac{3}{5}\) with a positive fraction \(\frac{1}{8}\) results in a negative product \(-\frac{3}{40}\). Understanding how negative numbers affect your calculations is essential in fraction multiplication.