Problem 2 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \frac{1}{4} \cdot \frac{8}{9} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2}{9} \).

Step 1: Identify the Multiplication

First, we need to understand that we are multiplying two fractions: \( \frac{1}{4} \) and \( \frac{8}{9} \). This means we need to multiply the numerators together and the denominators together.

Step 2: Multiply Numerators

Take the numerators of each fraction, 1 and 8, and multiply them: \[ 1 \times 8 = 8 \]

Step 3: Multiply Denominators

Take the denominators of each fraction, 4 and 9, and multiply them: \[ 4 \times 9 = 36 \]

Step 4: Form New Fraction

Combine the results to form a new fraction, using the products from the numerator and denominator: \[ \frac{8}{36} \]

Step 5: Simplify the Fraction

To simplify \( \frac{8}{36} \), find the greatest common divisor (GCD) of 8 and 36, which is 4. Divide both the numerator and the denominator by their GCD:\[ \frac{8 \div 4}{36 \div 4} = \frac{2}{9} \]

Key Concepts

Understanding Numerators and Denominators

When dealing with fractions, knowing how to handle numerators and denominators is crucial. **The numerator** is the top part of a fraction. It tells you how many parts of a whole you have. For example, in \( \frac{1}{4} \), the number **1** is the numerator.

• Numerators show the counts of a divided whole.
• They can be any integer, positive or negative.

**The denominator**, on the other hand, is the bottom part of the fraction. It informs you about the number of equal parts the whole is divided into. Thus, in \( \frac{1}{4} \), **4** is the denominator.

• Denominators cannot be zero, since division by zero isn't defined.
• They give context to the quantity the numerator describes.

In multiplication of fractions, multiply the numerators together and the denominators together separately. This gives you a new fraction where both components are derived from the original fractions.

The Simplification of Fractions

Simplifying fractions is an important step to make them easier to understand or compare. Simplification reduces a fraction to its simplest form, where the numerator and the denominator have no common factors other than one. Consider the fraction \( \frac{8}{36} \):

• Both numerator (8) and denominator (36) can be divided evenly by 4.
• After division, it becomes \( \frac{2}{9} \).

**Why simplify fractions?** Simplification can help in easily recognizing or comparing values:

• **Easier comparison:** Simpler fractions can be more straightforward to compare with other fractions.
• **Lesser components:** Reducing numbers can simplify calculations in future operations involving the fraction.

To simplify, always look for the greatest common factors that both numbers share and divide accordingly.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that divides two numbers without a remainder. It's also sometimes called the greatest common factor. For example, to simplify \( \frac{8}{36} \), we found the GCD of 8 and 36, which is 4.

• **Finding the GCD:** List integers that divide both the numerator and the denominator evenly and pick the largest one.
• Other methods include prime factorization or using the Euclidean algorithm.

**Why use the GCD?**

• It's essential for simplifying fractions to their lowest terms.
• Makes multiplication or addition of fractions more efficient in future calculations.

Understanding and using the GCD helps in breaking down fractions into a form that is easy to work with, which is beneficial both in arithmetic calculations and real-world applications.