Problem 2 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$2 \frac{1}{3} \cdot 6 \frac{3}{4}$$

Step-by-Step Solution

Answer

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

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers to improper fractions. For the first mixed number, \(2 \frac{1}{3}\), multiply the whole number (2) by the denominator (3) and add the numerator (1): \((2 \times 3) + 1 = 7\). So, \(2 \frac{1}{3}\) becomes \(\frac{7}{3}\). For the second mixed number, \(6 \frac{3}{4}\), multiply the whole number (6) by the denominator (4) and add the numerator (3): \((6 \times 4) + 3 = 27\). Thus, \(6 \frac{3}{4}\) becomes \(\frac{27}{4}\).

Step 2: Multiply the Fractions

Multiply the two improper fractions: \(\frac{7}{3}\) and \(\frac{27}{4}\). Multiply the numerators: \(7 \times 27 = 189\). Multiply the denominators: \(3 \times 4 = 12\). Thus, the product of the fractions is \(\frac{189}{12}\).

Step 3: Simplify the Improper Fraction

Now, we simplify \(\frac{189}{12}\). First, find the greatest common divisor (GCD) of 189 and 12, which is 3. Divide both the numerator and denominator by 3:\[\frac{189 \div 3}{12 \div 3} = \frac{63}{4}\]. The improper fraction after simplification is \(\frac{63}{4}\).

Step 4: Convert to Mixed Number

To express \(\frac{63}{4}\) as a mixed number, divide 63 by 4. The quotient is 15 and the remainder is 3. Therefore, \(\frac{63}{4}\) can be written as the mixed number \(15 \frac{3}{4}\).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process that doesn't involve much complication. When you multiply fractions, you simply multiply the numerators together, and you multiply the denominators together.
For example, to multiply two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), you follow these steps:

• Multiply the numerators: \(a \times c\).
• Multiply the denominators: \(b \times d\).

The result of multiplying fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) would be \(\frac{a \times c}{b \times d}\). This is how we got the intermediate result in the exercise.
While measuring the product, it's important to simplify the resulting fraction if possible just like we do with whole numbers, ensuring everything is as concise as possible.

Improper Fractions

In mathematics, an improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. This is different from a proper fraction, where the numerator is less than the denominator.
For instance, \(\frac{7}{3}\) and \(\frac{27}{4}\) are examples of improper fractions and were derived during the initial step in the given exercise.
Some features of improper fractions include:

• They can be converted into mixed numbers.
• They are useful in mathematical operations such as multiplying and dividing fractions because they simplify calculations.

To convert a mixed number to an improper fraction, multiply the whole number part by the fraction's denominator and add the fraction's numerator. Place this result as the new numerator, while the denominator remains the same, just like we did in the exercise.

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction, and they are useful for showing quantities greater than one in a simple form. For example, \(2 \frac{1}{3}\) and \(6 \frac{3}{4}\) are mixed numbers.
Converting between mixed numbers and improper fractions is a common skill needed in math, especially when performing operations like multiplication. Here’s how you convert them:

• To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
• To convert an improper fraction back into a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder is the new numerator of the fractional part.

Mixed numbers provide a practical way to view portions, especially when simplifying answers is necessary, like in the final step of the exercise where the improper fraction is turned back into a mixed number for clarity.