Problem 39 - Prealgebra

Question

Divide. Find the product of \(2 \frac{2}{3}\) and \(1 \frac{1}{2}\).

Step-by-Step Solution

Answer

The product of \(2 \frac{2}{3}\) and \(1 \frac{1}{2}\) is 4.

Step 1: Convert Mixed Numbers to Improper Fractions

Firstly, convert the mixed numbers into improper fractions. To convert \(2 \frac{2}{3}\), multiply the whole number (2) by the denominator (3), and add the numerator (2). This gives \(2\cdot3+2=8\), so \(2 \frac{2}{3} = \frac{8}{3}\). For \(1 \frac{1}{2}\), multiply the whole number (1) by the denominator (2), and add the numerator (1). This gives \(1\cdot2+1=3\), so \(1 \frac{1}{2} = \frac{3}{2}\).

Step 2: Multiply the Improper Fractions

Now multiply the improper fractions \(\frac{8}{3}\) and \(\frac{3}{2}\) by multiplying their numerators together and their denominators together. So the product is \(\frac{8 \cdot 3}{3 \cdot 2} = \frac{24}{6}\).

Step 3: Simplify the Resulting Fraction

The fraction \(\frac{24}{6}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Therefore, \(\frac{24\div 6}{6\div 6} = \frac{4}{1} = 4\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This typically means that the fraction is greater than or equal to one whole. Understanding improper fractions is important because they often appear when converting mixed numbers.

Mixed numbers combine a whole number with a fraction, making computation a bit tricky. So, converting them to improper fractions simplifies many operations.

• To convert a mixed number, multiply the whole number by the denominator of the fractional part.
• Add the numerator of the fractional part to the result.
• This sum becomes the new numerator, and the denominator stays the same.

For example, converting the mixed number \(2 \frac{2}{3}\) involves calculating \(2 \times 3 + 2 = 8\), so it becomes \(\frac{8}{3}\).

Multiplying Fractions

Multiplying fractions involves a straightforward method:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

This simple process contrasts other fraction operations like addition and subtraction, which need a common denominator. Once you multiply the fractions, you get a new fraction that may need further simplification.

Consider the example: multiplying \(\frac{8}{3}\) and \(\frac{3}{2}\). By multiplying the numerators (8 and 3) and the denominators (3 and 2) separately, you get \(\frac{24}{6}\). Notice how this product is an improper fraction, which is common when dealing with multiplication of fractions.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. The goal is to find the simplest form, where the numerator and denominator are as small as possible while still representing the same value. This is done by:

• Finding the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our example, the fraction \(\frac{24}{6}\) is simplified by identifying the GCD of 24 and 6, which is 6. Dividing both terms by 6, the fraction becomes \(\frac{4}{1}\), or simply \(4\). Simplification not only makes fractions clearer, but also more convenient for further calculations.