Problem 12 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$\frac{2}{3} \cdot 4 \frac{9}{10}$$

Step-by-Step Solution

Answer

The product is \(3 \frac{4}{15}\).

Step 1: Convert the Mixed Number

First, we need to convert the mixed number \(4 \frac{9}{10}\) into an improper fraction. To do this, multiply the whole number by the denominator of the fraction part and add the numerator: \(4 \times 10 + 9 = 49\). This gives us the improper fraction \(\frac{49}{10}\).

Step 2: Multiply the Fractions

Now that we have two fractions, \(\frac{2}{3}\) and \(\frac{49}{10}\), multiply the numerators together and the denominators together: \(2 \times 49 = 98\) and \(3 \times 10 = 30\). This gives us \(\frac{98}{30}\).

Step 3: Simplify the Fraction

Next, simplify the fraction \(\frac{98}{30}\). Find the greatest common divisor (GCD) of 98 and 30, which is 2. Divide both the numerator and denominator by 2: \(\frac{98}{2} = 49\) and \(\frac{30}{2} = 15\). This simplifies the fraction to \(\frac{49}{15}\).

Step 4: Convert to Proper Fraction or Mixed Number

To convert \(\frac{49}{15}\) to a mixed number, divide 49 by 15. This gives 3 whole parts, with a remainder of 4. Thus, the mixed number is \(3 \frac{4}{15}\).

Key Concepts

Understanding Mixed Numbers

A mixed number is a special way of expressing numbers that are more than a whole number. It combines a whole number and a proper fraction, like 4 \(\frac{9}{10}\). Mixed numbers make it easier to describe quantities that aren't whole, providing a clear distinction between the whole number portion and the fractional remainder. For instance, in cooking, you might come across a measurement like 1 \(\frac{1}{2}\) cups of sugar, which is a mixed number.

• **Whole Number:** This is the number you can count in entirety without any fraction, much like any simple count of items.
• **Fractional Part:** This represents the remaining portion that isn't enough to be considered a full whole number.

When dealing with calculations, mixed numbers may need to be converted to improper fractions. This simplifies the arithmetic operations, such as multiplication and division. We'll see more about improper fractions next.

Improper Fractions Explained

Improper fractions occur when the numerator (top number) is larger than the denominator (bottom number), like \(\frac{49}{10}\). They might look unwieldy, but they offer a straightforward form for computation. They are particularly significant in mathematics since they allow for simpler arithmetic operations.

• **Benefits:** Converting mixed numbers to improper fractions can simplify multiplication and division processes. This is because fractions can be easily multiplied directly without needing additional conversion steps.
• **Conversion:** To convert a mixed number like 4 \(\frac{9}{10}\) to an improper fraction, multiply the whole number by the denominator and add the numerator: \(4 \times 10 + 9 = 49\), resulting in \(\frac{49}{10}\).

After these conversions, operations like multiplication become more manageable, allowing us to focus on the mathematical process rather than fixing forms. It's important to then simplify the result, which is where the concept of the greatest common divisor comes into play.

The Role of Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is a key concept in simplifying fractions. It's the largest positive integer that divides two numbers without leaving a remainder. When multiplying fractions, finding the GCD is essential in reducing the resulting fraction to its simplest form.

• **Purpose:** A simplified fraction is easier to understand and work with. The GCD allows us to reduce fractions to this simplest form by eliminating any unnecessary complexity from larger numbers.
• **Calculation:** For a fraction like \(\frac{98}{30}\), finding the GCD involves identifying the largest number that can evenly divide both 98 and 30. In this case, the GCD is 2, which allows us to divide both the numerator and the denominator efficiently: \(\frac{98}{2} = 49\) and \(\frac{30}{2} = 15\), resulting in \(\frac{49}{15}\).

Understanding the GCD helps in refining the results of fraction operations and ensures that numbers are presented in a user-friendly format, enhancing clarity in calculations.