Problem 27 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-\frac{7}{8} \cdot \frac{2}{5}$$

Step-by-Step Solution

Answer

The product is \\(-\frac{7}{20}\\).

Step 1: Understand the Problem

The exercise requires us to find the product of two fractions, \(-\frac{7}{8}\) and \(\frac{2}{5}\). Multiplying fractions involves multiplying the numerators together and the denominators together.

Step 2: Multiply the Numerators

To find the product of the fractions, multiply the numerators together: \(-7\times 2 = -14\).

Step 3: Multiply the Denominators

Now, multiply the denominators: \(8 \times 5 = 40\).

Step 4: Write the Fraction

Combine the results from Steps 2 and 3 to write the resulting fraction: \(-\frac{14}{40}\).

Step 5: Simplify the Fraction

Simplify the fraction \(-\frac{14}{40}\) by finding the greatest common factor (GCF) of 14 and 40, which is 2. Divide the numerator and the denominator by 2 to simplify: \(-\frac{7}{20}\).

Key Concepts

Area Model

The area model is a useful visual method to help understand the multiplication of fractions. Picture multiplying two fractions as finding a portion of a rectangle's area. First, draw a rectangle and then divide it into a grid according to the denominators of the fractions involved. For example, if you're multiplying \(-\frac{7}{8}\) by \(\frac{2}{5}\), you start by dividing the rectangle vertically into 8 equal parts and horizontally into 5 equal parts.

• Shade 7 of those 8 parts for the first fraction \(-\frac{7}{8}\).
• Next, shade 2 out of those 5 parts for the second fraction \(\frac{2}{5}\).
• The overlapping area represents the product of the fractions, showing \(\frac{14}{40}\). Counting overlapping squares can deepen your understanding.

The rectangle's shaded part symbolically represents \(-\frac{14}{40}\), which you can simplify further. This model is especially effective if you find visuals more intuitive than numbers alone.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form. It's like tidying up fractions by ensuring the numerator and denominator no longer share common factors. In our example, you start with \(-\frac{14}{40}\). The goal is to reduce the fraction, making it easier to interpret or compare.

• First, identify any common factors between the numerator and denominator. In this case, 14 and 40 can both be divided by 2.
• Perform the division for both: the numerator \(-14\) divided by 2 is \(-7\), and the denominator 40 divided by 2 is 20.

Now, you have the simplified fraction, \(-\frac{7}{20}\), which is more manageable. This makes it quicker to understand relationships between different values or can make further calculations easier. Remember, simplifying doesn't change the value, just how it's expressed!

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that can exactly divide both the numerator and the denominator of a fraction. It's a key tool for simplifying fractions. Determining the GCF helps make fractions more concise. For instance, with the fraction \(-\frac{14}{40}\), we need to find the GCF to simplify.

• First, list the factors of each number. For 14, the factors are 1, 2, 7, and 14. For 40, the factors are 1, 2, 4, 5, 8, 10, 20, and 40.
• Then, identify the largest common factor. In this case, it's 2.
• Use this GCF to divide both the numerator and the denominator, reducing the fraction from \(-\frac{14}{40}\) to \(-\frac{7}{20}\).

Using the GCF not only simplifies a fraction but also aids in quicker arithmetic calculations by reducing complexity. This not only makes numbers easier to work with, but it often becomes essential when working with large numbers or multiple fractions.