Problem 78 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{2}{3} \cdot 12 \cdot \frac{3}{4}$$

Step-by-Step Solution

Answer

The product is 6.

Step 1: Multiply the Fractions and Whole Number

First, multiply the fractions and the whole number together. The expression is \( \frac{2}{3} \cdot 12 \cdot \frac{3}{4} \). Start by simplifying \( 12 \cdot \frac{2}{3} \). Multiply 12 by the numerator of \( \frac{2}{3} \), which gives \( 12 \times 2 = 24 \). Then divide 24 by the denominator of \( \frac{2}{3} \), which is 3. So, \( \frac{24}{3} = 8 \).

Step 2: Multiply the Result with Second Fraction

The result from Step 1 is 8. Now multiply 8 by the second fraction \( \frac{3}{4} \). Multiply the whole number 8 by the numerator of \( \frac{3}{4} \), which results in \( 8 \times 3 = 24 \). Divide this product by the denominator of \( \frac{3}{4} \), which is 4. Therefore, \( \frac{24}{4} = 6 \).

Step 3: Write the Simplified Product

The final product after simplifying across all steps is 6.

Key Concepts

Simplifying Fractions

When we simplify fractions, we are essentially making them easier to work with by expressing them in their most reduced form. Fractions are simpler when their numerator and denominator are as small as possible, yet they still represent the same value. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
For instance, if you have the fraction \( \frac{24}{36} \), the GCD of 24 and 36 is 12. Therefore, you divide both the numerator and the denominator by 12, giving you the simplified fraction \( \frac{2}{3} \).
This process serves as a key step in various math operations involving fractions, ensuring that results are neat and easy to understand.

Multiplying Fractions and Whole Numbers

When you're faced with multiplying fractions and whole numbers, the process involves a few simple steps. Let's dive into these steps using our example: \( \frac{2}{3} \cdot 12 \).

• First, represent the whole number as a fraction. In this case, 12 can be written as \( \frac{12}{1} \).
• Next, multiply the numerators (top numbers) of both fractions together: \( 2 \times 12 = 24 \).
• Then, multiply the denominators (bottom numbers): \( 3 \times 1 = 3 \).
• The result is a new fraction: \( \frac{24}{3} \).

Now, simplify the resulting fraction. Dividing 24 by 3 gives you a simplified product of 8.
By treating whole numbers as a fraction over 1, multiplying fractions and whole numbers becomes straightforward.

Reducing Fractions

Reducing fractions, often called "simplifying," is the process of making fractions as concise as possible without changing their value. This is an essential skill in fraction calculations, ensuring that answers are always presented in their simplest form.
After multiplication, like in our example, you might find yourself with a fraction result such as \( \frac{24}{4} \).

• To reduce, look for the greatest common factor (GCF) that both the numerator and the denominator share. Here, both 24 and 4 can be divided evenly by 4.
• Divide the numerator and the denominator by this GCF: \( 24 \div 4 = 6 \) and \( 4 \div 4 = 1 \).
• The reduced fraction is \( \frac{6}{1} \), or simply 6.

Reducing is crucial after any operation involving fractions to ensure the result is clear and in its simplest form, providing an answer that is tidy and easy to interpret.