Problem 34 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-6 \frac{2}{3}\left(-1 \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The product is 10.

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For \(-6 \frac{2}{3}\):- Multiply the whole number 6 by the denominator 3. - Add the numerator 2 to get the new numerator.\[ 6 \times 3 + 2 = 18 + 2 = 20 \]Since the original number is negative, the improper fraction will be \(-\frac{20}{3}\).For \(-1 \frac{1}{2}\):- Multiply the whole number 1 by the denominator 2.- Add the numerator 1 to get the new numerator.\[ 1 \times 2 + 1 = 2 + 1 = 3 \]Since the original number is negative, the improper fraction will be \(-\frac{3}{2}\).

Step 2: Multiply the Improper Fractions

Next, multiply the improper fractions.\[ \left(-\frac{20}{3}\right) \times \left(-\frac{3}{2}\right) \] - Multiply the numerators: \(-20 \times -3 = 60\).- Multiply the denominators: \(3 \times 2 = 6\).So, the result of the multiplication is \[ \frac{60}{6} \].

Step 3: Simplify the Fraction

Finally, simplify the fraction \( \frac{60}{6} \).- Divide the numerator 60 by the denominator 6.\[ 60 \div 6 = 10 \]So, the simplified product is 10.

Key Concepts

Area Model

The area model is a visual representation used to simplify multiplication problems. Think of it like a rectangle divided into smaller sections, where each section represents a part of the multiplication. For multiplying two numbers, especially mixed numbers or larger numbers, the area model can break them down into more manageable parts.
To use an area model:

• Draw a rectangle and divide it into sections that represent the numbers you are multiplying.
• Calculate the product for each section.
• Add up all the results to find the final product.

In this context, the area model helps us understand how to multiply numbers by visually breaking down mixed numbers into whole and fractional parts, simplifying multiplication.

Improper Fractions

Improper fractions are key when dealing with mixed numbers in multiplication. An improper fraction is where the numerator (the top number) is larger than the denominator (the bottom number).
To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominators.
• Add the numerator to this result.
• Place this sum over the original denominator.

This conversion is essential because it aligns the numbers into a uniform fraction format, making multiplication straightforward and consistent.

Mixed Numbers

Mixed numbers combine whole numbers and fractions, like a small math sandwich! They appear frequently in problem-solving and require careful manipulation.
Working with mixed numbers involves converting them to improper fractions, to simplify calculations like multiplication. This pre-step of conversion allows computations to avoid the extra step of dealing with whole numbers separately.
When converting back for a final solution or when breaking down the numbers initially, remember that:

• The whole number represents complete units.
• The fraction adds more precision, detailing what portion of an unit remains.

This perspective is especially handy when simplifying back to mixed numbers after performing calculations.

Product

The product is the result of multiplication. It reveals the total when quantities are combined multiplicatively. In fraction multiplication, the process involves multiplying the numerators together and then the denominators together.
Here's a recap on how to find the product of two fractions:

• Multiply the numerators (top numbers).
• Multiply the denominators (bottom numbers).
• If necessary, simplify the resulting fraction.

For example, multiplying \(-\frac{20}{3}\) by \(-\frac{3}{2}\) yields a positive result, since two negatives make a positive. The calculated product is \(\frac{60}{6}\), which simplifies neatly to 10.

Negative Numbers

Negative numbers represent values less than zero and are denoted with a minus sign (\(-\)). These numbers pop up frequently in math problems, and understanding their rules is crucial.
When multiplying negative numbers:

• Two negative numbers give a positive product. The negatives cancel each other out.
• A positive and a negative number give a negative product.

In real-world terms, think of negative numbers as owing or subtracting weight from a balance. Thus, mastering their multiplication allows solving otherwise challenging problems smoothly.