Problem 30 - Prealgebra

Question

Find each product. Use an area model if necessary. $$6 \frac{2}{3} \cdot \frac{1}{2}$$

Step-by-Step Solution

Answer

The product is \(3 \frac{1}{3}\).

Step 1: Convert Mixed Number to Improper Fraction

The first number is a mixed number, so we need to convert it to an improper fraction. The mixed number is \( 6 \frac{2}{3} \). To convert, multiply the whole number by the denominator and add the numerator: \(6 \times 3 + 2 = 18 + 2 = 20\). Thus, \(6 \frac{2}{3} = \frac{20}{3}\).

Step 2: Multiply Fractions

Now, we have two fractions: \(\frac{20}{3}\) and \(\frac{1}{2}\). To find the product, multiply the numerators and the denominators: \(\frac{20 \times 1}{3 \times 2} = \frac{20}{6}\).

Step 3: Simplify the Fraction

The fraction \(\frac{20}{6}\) can be simplified. Find the greatest common divisor (GCD) of 20 and 6, which is 2. Divide both the numerator and the denominator by 2: \(\frac{20 \div 2}{6 \div 2} = \frac{10}{3}\).

Step 4: Convert Improper Fraction to Mixed Number

\(\frac{10}{3}\) is an improper fraction, so we convert it to a mixed number. Divide 10 by 3, which goes 3 times with a remainder of 1. Therefore, \(\frac{10}{3} = 3 \frac{1}{3}\).

Key Concepts

Area Model

The area model is a visual way to understand multiplication, especially when dealing with fractions. It's like drawing a rectangle to represent the entire fraction problem. The length and width of the rectangle are our two fractions that we want to multiply. Imagine one fraction is the length and the other is the width. When you multiply these, the resulting area is your product.

For example, when multiplying fractions, you can draw a rectangle and partition it based on each fraction. This makes it easier to see how the numbers interact. By splitting the rectangle into smaller sections or areas, you show the parts that overlap, representing the multiplication. While it’s not as common to use with all types of problems, it’s especially helpful for introducing the concept to those just learning how to multiply fractions.

Improper Fractions

Fractions can look pretty straightforward, but improper fractions might seem confusing at first. An improper fraction occurs when the numerator (the top number) is larger than the denominator (the bottom number). This just means you have more than one whole.

• For instance, in our exercise, we initially converted the mixed number \(6 \frac{2}{3}\) into an improper fraction, \(\frac{20}{3}\).
• Here, 20 is larger than 3, indicating it’s more than one whole unit.

To convert, multiply the whole number by the denominator, then add the numerator. This gives you a single numerator that fits "inside" the denominator multiple times, showing the whole parts plus any extra.

Mixed Numbers

Mixed numbers combine a whole number with a fraction, like \(6 \frac{2}{3}\). They're useful for expressing quantities that are more than whole but less than another whole number.

• In our problem, we quickly converted the mixed number to an improper fraction so we could multiply more easily.

When you're done multiplying, it’s often helpful to convert your answer back to a mixed number, because it’s easier to understand for most people. You accomplish this by dividing the numerator by the denominator. The whole number from this division becomes your whole part, whereas the remainder becomes the new numerator of the fraction.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with, reducing them to their simplest form. To do this, find the greatest common divisor (GCD) of the numerator and the denominator. Divide both by this number to simplify.

For the fraction \(\frac{20}{6}\):

• Find the GCD of 20 and 6. It is 2.
• Divide both the top and bottom by 2 to get \(\frac{10}{3}\).

Simplifying lets you express the fraction in the tiniest possible parts, making calculations simpler and the answers clearer to interpret.