Problem 59 - Fundamentals of Mathematics

Question

Find \(\frac{3}{8}\) of \(\frac{32}{9}\).

Step-by-Step Solution

Answer

The solution is \(\frac{4}{3}\).

Step 1: Understand the Problem

We need to find the product of two fractions: \(\frac{3}{8}\) and \(\frac{32}{9}\). This means we will multiply these two fractions together.

Step 2: Multiply the Numerators

To find the product of the fractions, first multiply the numerators (the top numbers). Here it is \(3\) and \(32\). Thus, we calculate: \(3 \times 32 = 96\).

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers). Here it is \(8\) and \(9\). Thus, we compute: \(8 \times 9 = 72\).

Step 4: Form the New Fraction

Combine the results of the previous steps to create a new fraction: \(\frac{96}{72}\).

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{96}{72}\) by finding the greatest common divisor (GCD) of 96 and 72, which is 24. Divide both the numerator and the denominator by 24: \(\frac{96 \div 24}{72 \div 24} = \frac{4}{3}\).

Key Concepts

Understanding Numerators in Fraction Multiplication

Numerators are the top numbers in a fraction that tell us how many parts we are considering. When we multiply fractions, understanding numerators becomes crucial since they are the first elements to be multiplied. For instance, with fractions \( \frac{3}{8} \) and \( \frac{32}{9} \), the numerators are \(3\) and \(32\). To find the product of these fractions, you multiply the numerators together:

• Numerator of the first fraction: 3
• Numerator of the second fraction: 32

Then simply calculate \( 3 \times 32 \) which equals \( 96 \). It can be helpful to write the multiplication of numerators down beforehand, as a separate step, so there is no confusion during complex calculations. In real-world terms, multiplying numerators reflects combining parts from each whole into one number that represents all the parts collectively considered.

Denominators and Their Role in Multiplying Fractions

Denominators are the bottom numbers in fractions which signify the total parts that make up a whole. When you multiply fractions, calculating the product of the denominators follows the multiplication of numerators. Using our earlier fractions of \( \frac{3}{8} \) and \( \frac{32}{9} \), the denominators are \(8\) and \(9\). To multiply these, you will:

• Identify the denominator of the first fraction: 8
• Identify the denominator of the second fraction: 9

Compute the multiplication: \( 8 \times 9 \) which results in \( 72 \). This resulting number becomes the new denominator of the product fraction. Understanding denominators is essential as they help determine the size of each part in related fractions, ensuring they both are measured in consistent 'pieces' when combined. Proper handling of denominators ensures the integrity of the fractions remains intact during multiplication.

Simplifying Fractions to Achieve Simplest Terms

Simplifying fractions involves reducing them to their simplest form, which makes further calculations easier and makes the result easier to understand. After multiplying fractions, as in our example \( \frac{96}{72} \), you can simplify by finding the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD of 96 and 72 is 24. Simplifying the fraction involves dividing both the numerator and the denominator by this divisor:

• Divide 96 by 24 to get 4
• Divide 72 by 24 to get 3

Thus, \( \frac{96}{72} \) simplifies to \( \frac{4}{3} \). A fraction is considered simplified when the numerator and the denominator no longer share any common factors other than 1. Simplifying is not only important in math class, but it's also useful in everyday life to compare different quantities easily. It gives clarity and direction, ensuring that the results captured are most understandable and concise.