Problem 79 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{3}{8} \cdot 24 \cdot \frac{2}{3}$$

Step-by-Step Solution

Answer

The product is 6.

Step 1: Understand the Problem

The exercise asks us to multiply three numbers: \( \frac{3}{8} \), \( 24 \), and \( \frac{2}{3} \). After calculating the product, we must also simplify the result.

Step 2: Multiply the Fractions and Whole Number

To multiply these terms, consider \( 24 \) as \( \frac{24}{1} \) to treat it as a fraction. The expression becomes \( \frac{3}{8} \cdot \frac{24}{1} \cdot \frac{2}{3} \). Multiply the numerators together, and multiply the denominators together:\[\frac{3 \cdot 24 \cdot 2}{8 \cdot 1 \cdot 3}\] which simplifies to \( \frac{144}{24} \).

Step 3: Simplify the Fraction

To simplify \( \frac{144}{24} \), find the greatest common divisor (GCD) of 144 and 24, which is 24. Divide both the numerator and denominator by 24:\[\frac{144 \div 24}{24 \div 24} = \frac{6}{1}\] which simplifies to 6.

Step 4: Verify the Solution

Multiply through as a check:\[ \frac{3}{8} \times \frac{24}{1} = \frac{72}{8} = 9 \] and now, \( 9 \times \frac{2}{3} = \frac{18}{3} = 6 \). Verifying each step confirms that the simplified product is indeed correct.

Key Concepts

Reduce Fractions

Reducing fractions is an essential skill in mathematics that makes dealing with fractions much easier. When you reduce a fraction, you transform it to its simplest form, meaning that the numerator and the denominator have no common factors other than 1.
Let's say you have the fraction \( \frac{144}{24} \). To reduce it, you need to find the greatest common divisor (GCD) and then divide both the numerator and the denominator by this number. This process will give you the simplest version of the fraction.

• Start with the fraction \( \frac{144}{24} \).
• Find the GCD, which is 24.
• Divide both 144 and 24 by 24 to get \( \frac{6}{1} \).

Now, the fraction is fully reduced, and \( \frac{6}{1} \) is simply 6 in whole number form.

Greatest Common Divisor

The greatest common divisor, or GCD, is a key concept when working with fractions because it helps us reduce them to their simplest form. The GCD of two numbers is the largest positive integer that divides both numbers without a remainder. Here's how you can find the GCD:

• List the factors of each number. For example, for 144, the factors are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. For 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.
• Identify the common factors from both lists. They are 1, 2, 3, 4, 6, 8, 12, and 24.
• The largest of these is 24, so the GCD of 144 and 24 is 24.

Finding the GCD helps simplify calculations and ensures the fraction is in its simplest form, making it easier to work with in equations and operations.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, which means the numerator and denominator share no common factors other than 1.
It's like cleaning up your work to make it clearer and more straightforward. Simplifying fractions is crucial in math because it allows you to focus on the essential parts of a problem without unnecessary complexity.Consider the fraction \( \frac{144}{24} \). Simplifying this fraction makes it easier to work with:

• Identify the GCD, which is 24 in this case.
• Divide the numerator and the denominator by the GCD.
• This gives you \( \frac{6}{1} \), or simply 6.

By simplifying, you ensure that the fraction represents the same value but in an easier-to-understand format. This process is a fundamental part of fraction multiplication and helps in efficiently solving math problems.