Problem 95 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{1}{2}\right)^{2} \cdot \frac{8}{9}$$

Step-by-Step Solution

Answer

The product is \( \frac{2}{9} \).

Step 1: Multiply Fractions

First, we need to multiply the fractions. To do this, we multiply the numerators and then the denominators. Compute \( \left( \frac{1}{2} \right)^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). Now multiply this result with \( \frac{8}{9} \): \( \frac{1}{4} \times \frac{8}{9} = \frac{1 \times 8}{4 \times 9} = \frac{8}{36} \).

Step 2: Simplify the Fraction

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

Key Concepts

Numerator and Denominator

Let's dive into the basics of fractions, focusing on the two essential parts: the numerator and the denominator. A fraction represents a part of a whole and is written as one number over another. The number on the top is called the numerator, while the number on the bottom is called the denominator.

The numerator indicates how many parts of the whole are being considered. For instance, in the fraction \( \frac{3}{4} \), the numerator "3" tells us we're considering 3 parts of something that was divided into equal fractions.

Conversely, the denominator shows the total number of equal parts into which the whole is divided. In \( \frac{3}{4} \), the denominator "4" means the whole is divided into 4 parts.

• If the numerator is smaller than the denominator, the fraction is a proper fraction (e.g., \( \frac{1}{2} \)).
• If the numerator is equal to or larger than the denominator, it is an improper fraction (e.g., \( \frac{5}{3} \)).

Understanding these parts helps you perform operations like multiplication and division with ease.

Simplifying Fractions

Simplifying fractions makes them easier to understand by reducing them to their simplest form, meaning the numerator and denominator have no common factors other than 1. To simplify a fraction, divide both its numerator and denominator by their greatest common divisor (GCD).

Let's take our earlier multiplication result \( \frac{8}{36} \). To simplify this fraction:

• First, identify any common factors shared by 8 and 36.
• Both numbers can be divided by the GCD, which is 4 for this fraction.
• Divide the numerator and the denominator by 4: \( \frac{8 \div 4}{36 \div 4} = \frac{2}{9} \).

Now \( \frac{2}{9} \) is in its simplest form. Simplifying helps us better understand and work with fractions, ensuring clarity in calculations and results.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is a key step in simplifying fractions. The GCD is the largest number that can exactly divide both the numerator and the denominator without leaving a remainder.

Here's how you can find the GCD:

• List all divisors of each number.
• Identify the largest number common to both lists.
• This number is your GCD.

Let's apply this to 8 and 36.

• The divisors of 8 are 1, 2, 4, and 8.
• The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
• The greatest number common to both lists is 4.

Using the GCD allows us to reduce fractions efficiently, aiding in clearer mathematics and more relatable results.