Problem 8 - Fundamentals of Mathematics

Question

Find \(\frac{2}{3}\) of \(\frac{3}{4}\).

Step-by-Step Solution

Answer

The answer is \(\frac{1}{2}\).

Step 1: Understand the Problem

We need to find the product of two fractions: \(\frac{2}{3}\) and \(\frac{3}{4}\). The operation here is multiplication.

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions: \(2\) and \(3\). This gives us \(2 \times 3 = 6\).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions: \(3\) and \(4\). This gives us \(3 \times 4 = 12\).

Step 4: Form the New Fraction

Combine the results from the previous steps to form the new fraction: \(\frac{6}{12}\).

Step 5: Simplify the Fraction

Simplify \(\frac{6}{12}\) by finding the greatest common divisor (GCD) of 6 and 12, which is 6. Divide both the numerator and the denominator by 6 to get \(\frac{1}{2}\).

Key Concepts

Fraction Simplification

Fraction simplification is an essential math skill that helps in reducing a fraction to its simplest form. When you simplify a fraction, you don't change its value. You're merely expressing it in the simplest way possible. For the fraction \( \frac{6}{12} \), the simplification process involves identifying and dividing both the numerator (top part) and the denominator (bottom part) by their greatest common factor.

Simplification is important because:

• It makes fractions easier to understand and compare.
• It helps with performing additional operations like addition or subtraction.
• It provides a more straightforward view of the fraction's value.

Considering \( \frac{6}{12} \), we find it's equal to \( \frac{1}{2} \) after simplification, which is usually more intuitive to work with.

Numerator and Denominator

Understanding the roles of the numerator and denominator is crucial when dealing with fractions. The **numerator** is the top number in a fraction, representing how many parts we have. Meanwhile, the **denominator** is the bottom number, showing into how many equal parts the whole is divided.

For instance:

• In the fraction \( \frac{2}{3} \), the numerator is 2, and the denominator is 3.
• The fraction \( \frac{3}{4} \) has a numerator of 3 and a denominator of 4.

When multiplying fractions, you multiply the numerators together and the denominators together. This operation gives us a new fraction, sometimes requiring further simplification, like in our example where \( \frac{6}{12} \) becomes \( \frac{1}{2} \). Recognizing which is the numerator and which is the denominator helps keep these operations and simplifications consistent and accurate.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a key concept when simplifying fractions. It is the largest number that divides the numerator and the denominator without leaving a remainder. The GCD helps us reduce fractions to their simplest form so that they are as easy as possible to understand.

Let's consider the fraction \( \frac{6}{12} \):

• First, identify the factors of 6 (1, 2, 3, 6) and 12 (1, 2, 3, 4, 6, 12).
• The common factors are 1, 2, 3, and 6.
• The greatest common factor here is 6.

Once you find the GCD, you divide both the numerator and the denominator by this number, transforming \( \frac{6}{12} \) into \( \frac{1}{2} \). Understanding and using the GCD removes any unnecessary complexity from the fraction, making math problems more approachable.