Problem 81 - Prealgebra

Question

Write your answers as whole numbers, proper fractions, or mixed numbers. Find each product. (Multiply) $$8 \cdot \frac{3}{4}$$

Step-by-Step Solution

Answer

The product of \(8\) and \(\frac{3}{4}\) is 6.

Step 1: Write the Whole Number as a Fraction

Convert the whole number 8 into a fraction by placing it over 1. This makes it easier to multiply with another fraction. So, write 8 as \( \frac{8}{1} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: 8 from \( \frac{8}{1} \) and 3 from \( \frac{3}{4} \). This gives \( 8 \times 3 = 24 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: 1 from \( \frac{8}{1} \) and 4 from \( \frac{3}{4} \). This gives \( 1 \times 4 = 4 \).

Step 4: Form the Resulting Fraction

Combine the results from Step 2 and Step 3 to form the new fraction: \( \frac{24}{4} \).

Step 5: Simplify the Fraction

Simplify \( \frac{24}{4} \) by dividing both the numerator and the denominator by their greatest common factor, which is 4. Simplification results in \( \frac{24 \div 4}{4 \div 4} = \frac{6}{1} \), which is equal to 6.

Key Concepts

Whole Numbers

Whole numbers are the numbers we use for counting, and include zero and all the positive integers like 1, 2, 3, and so on. When multiplying whole numbers, you are essentially adding that number together multiple times. For instance, if you multiply 4 by 3, it is the same as adding 4 to itself three times: \( 4 + 4 + 4 = 12 \).

When you wish to multiply a whole number by a fraction, you can convert the whole number to a fraction by placing it over 1. By doing so, you prepare it for multiplication with another fraction. For example, the whole number 8 becomes \( \frac{8}{1} \). This helps keep the multiplication process straightforward, as you can then follow the regular rules for multiplying fractions.

Proper Fractions

Proper fractions are fractions where the numerator (the top number) is smaller than the denominator (the bottom number). This type of fraction represents a part of a whole, but less than one whole unit. Examples include \( \frac{1}{2} \), \( \frac{3}{4} \), and \( \frac{5}{8} \).

When you multiply proper fractions, you multiply the numerators together and then the denominators together. Consider multiplying \( \frac{3}{4} \) by another fraction or a whole number (converted to a fraction, of course). It involves multiplying the numerators (3 and 8 if 8 is converted to \( \frac{8}{1} \)) and then the denominators (4 and 1). This process results in a new fraction that may need simplification. Keeping track of steps helps prevent any errors in calculation.

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction. For instance, \( 2 \frac{1}{2} \) is a mixed number. They're useful in everyday measurements and cooking, where quantities rarely come in whole numbers.

To multiply mixed numbers, it's often easier to first convert them into improper fractions. An improper fraction has a numerator larger than its denominator. To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction part, then add the numerator. This sum becomes the new numerator, while the denominator stays the same. For \( 2 \frac{1}{2} \), you'd convert it by calculating \( (2 \times 2) + 1 = 5 \), giving you \( \frac{5}{2} \).

Once converted, multiply the fractions by multiplying numerators together and denominators together. Simplifying the resulting improper fraction can then yield a mixed number or a whole number, depending on the result.