Problem 23 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-\frac{1}{2}\left(-\frac{2}{7}\right)$$

Step-by-Step Solution

Answer

The product is \(\frac{1}{7}\).

Step 1: Understanding Multiplication of Fractions

When multiplying fractions, multiply the numerators together and the denominators together. With signs, remember that a negative times a negative is a positive.

Step 2: Identify the Numerators and Denominators

The first fraction is \(-\frac{1}{2}\) with a numerator of -1 and a denominator of 2. The second fraction is \(-\frac{2}{7}\) with a numerator of -2 and a denominator of 7.

Step 3: Multiply the Numerators

Multiply the numerators: \(-1 \times -2 = 2\).

Step 4: Multiply the Denominators

Multiply the denominators: \(2 \times 7 = 14\).

Step 5: Combine the Results

Combine the results of the numerators and denominators: \(\frac{2}{14}\).

Step 6: Simplify the Fraction

The fraction \(\frac{2}{14}\) can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 2: \(\frac{2 \div 2}{14 \div 2} = \frac{1}{7}\).

Key Concepts

Understanding the Area Model for Fractions

The area model is a visual representation used to understand multiplication, including with fractions. It helps in picturing how two quantities interact by breaking them down into more manageable parts. When we apply the area model to fractions, we imagine a rectangle.
One side of the rectangle represents one fraction, while the other side represents the second fraction.

• Divide the rectangle into sections based on the denominators of the fractions.
• Each section corresponds to a part of the whole, scaling according to the numerator.

By filling in part of the rectangle representing the fractions, you can visualize the product, helping to understand the multiplication process. This visual aid bolsters comprehension, especially when working with signed numbers like in our initial problem.

Basics of Fractions

Fractions represent parts of a whole. They consist of a numerator and a denominator.
The numerator tells us how many parts we have, while the denominator tells us how many parts make up a whole.

• The fraction \( \frac{a}{b} \) means "a" parts of a whole divided into "b" equal parts.
• Fractions can represent less than one whole, one whole, or more than one whole depending on the relationship between the numerator and the denominator.

When multiplying fractions, the process includes multiplying the numerators to find the new numerator and the denominators to find the new denominator.
It's important to note the rules for multiplying signed numbers, where two negative numbers result in a positive product.

Working with Negative Numbers in Multiplication

Negative numbers can often seem challenging, but understanding how they behave in multiplication is straightforward. The rule to remember is simple: when you multiply two negative numbers, you get a positive result.

• This is because the negatives "cancel out" each other.
• For example, \((-1) \times (-2) = 2\).

Thus, even with fractions, when you multiply a negative fraction by another negative fraction, your result will be positive. This holds true for our example, where \(-\frac{1}{2} \times -\frac{2}{7} = \frac{2}{14}\). Understanding this concept helps avoid common pitfalls in calculations.

Simplifying Fractions

Simplifying fractions makes them easier to interpret and use. This involves reducing a fraction to its simplest form, where the numerator and denominator are as small as possible.

• To simplify, find the greatest common divisor (GCD) shared by the numerator and denominator.
• Divide both the numerator and the denominator by this number.

For example, with \(\frac{2}{14}\), the GCD of 2 and 14 is 2. Dividing both by 2 yields \(\frac{1}{7}\). This final fraction is simpler and more intuitive to work with, maintaining the same value as the original fraction. Simplifying is always a valuable step in presenting your final answer.