Problem 6 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$4 \frac{7}{10} \cdot 3 \frac{1}{10}$$

Step-by-Step Solution

Answer

The product is \(14 \frac{57}{100}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To multiply mixed numbers, first convert them to improper fractions. For the first number, \(4 \frac{7}{10}\), multiply the whole number 4 by the denominator 10, and add the numerator 7: \(4 \cdot 10 + 7 = 47\). So, \(4 \frac{7}{10}\) becomes \(\frac{47}{10}\). For the second number, \(3 \frac{1}{10}\), multiply the whole number 3 by the denominator 10, and add the numerator 1: \(3 \cdot 10 + 1 = 31\). Thus, \(3 \frac{1}{10}\) becomes \(\frac{31}{10}\).

Step 2: Multiply the Improper Fractions

Multiply the improper fractions \(\frac{47}{10}\) and \(\frac{31}{10}\). Multiply the numerators: \(47 \cdot 31 = 1457\). Multiply the denominators: \(10 \cdot 10 = 100\). The product is \(\frac{1457}{100}\).

Step 3: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{1457}{100}\) to a mixed number. Divide 1457 by 100, which is 14 with a remainder of 57. The whole number part is 14 and the fractional part is \(\frac{57}{100}\). Therefore, \(\frac{1457}{100} = 14 \frac{57}{100}\).

Key Concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means the fraction represents a number equal to or greater than one. Improper fractions are particularly useful in multiplication and division of fractions because they simplify the calculations. For example:

• Convert a mixed number like \(4 \frac{7}{10}\) to an improper fraction. First, multiply the whole number (4) by the denominator (10), then add the numerator (7). This gives you \(4 \times 10 + 7 = 47\).
• The improper fraction form is \(\frac{47}{10}\).

By converting mixed numbers to improper fractions, you streamline the multiplication process later on. This method ensures accuracy and helps maintain the proportion of the fractions.

Multiplication of Fractions

Multiplying fractions might initially seem tricky, but it follows a straightforward process. Here’s how it works:

• Firstly, make sure both numbers are in fraction form - whether proper or improper.
• For example, to multiply \(\frac{47}{10}\) and \(\frac{31}{10}\), just multiply the numerators together: \(47 \times 31 = 1457\).
• Then, multiply the denominators: \(10 \times 10 = 100\).

The result is the fraction \(\frac{1457}{100}\). Multiplying fractions can often result in an improper fraction, which you may need to convert into a mixed number or simplify for clarity. Multiplication of fractions is a scale transformation; it changes the size of the parts but not the relative values between them.

Proper Fractions

A proper fraction is when the numerator is less than the denominator, representing a number less than one. For instance, if after converting an improper fraction, you end up with a remainder smaller than the denominator, you can express this remainder as a proper fraction.

• Example: converting \(\frac{1457}{100}\) to a mixed number results in \(14 \frac{57}{100}\). Here, \(\frac{57}{100}\) is a proper fraction because 57 is less than 100.
• This ensures the mixed number accurately reflects both the whole number and the remaining fractional part.

Proper fractions offer a way to represent parts of a whole and are common in everyday contexts, such as splitting a cake into smaller portions. They are essential for conveying precise information about quantities less than one whole unit.