Problem 38 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{2}{9}\( of \)\frac{6}{5}$$

Step-by-Step Solution

Answer

\(\frac{4}{15}\)

Step 1: Identify the Operation

The operation in the problem is multiplication. We need to find the product of the two fractions, \( \frac{2}{9} \) and \( \frac{6}{5} \).

Step 2: Multiply the Numerators

To find the product of two fractions, we first multiply the numerators of the fractions: \( 2 \times 6 = 12 \).

Step 3: Multiply the Denominators

Next, we multiply the denominators of the fractions: \( 9 \times 5 = 45 \).

Step 4: Form the Fraction

Combine the results from the previous steps to form the new fraction: \( \frac{12}{45} \).

Step 5: Simplify the Fraction

Now, simplify \( \frac{12}{45} \) by finding the greatest common divisor (GCD) of 12 and 45. The GCD is 3. Divide both the numerator and the denominator by 3: \( \frac{12 \div 3}{45 \div 3} = \frac{4}{15} \).

Key Concepts

Fraction Simplification

Fraction simplification is a crucial part of working with fractions efficiently. It involves reducing a fraction to its simplest form while maintaining the same value. When you have a fraction, like \( \frac{12}{45} \), simplification helps you express the fraction in the most straightforward terms possible, which often makes it easier to understand and work with.

This process requires finding a number that evenly divides both the numerator and the denominator. By dividing both by this number, you can reduce the fraction to its simplest form. It's like peeling away the unnecessary parts until you're left with the essence.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is an essential step in simplifying fractions. The GCD is the highest number that can exactly divide both the numerator and the denominator of a fraction. For example, to simplify \( \frac{12}{45} \), you first determine the GCD of 12 and 45.

To find the GCD, list out the factors of each number:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 45: 1, 3, 5, 9, 15, 45

The greatest common factor they both share is 3. Once you know this, you can divide the numerator and the denominator by this number, simplifying \( \frac{12}{45} \) to \( \frac{4}{15} \).

This step is vital because it ensures that the fraction you end up with is in its simplest form and easier to work with for any further mathematical operations.

Numerators and Denominators

Understanding the roles of numerators and denominators is fundamental in dealing with fractions. The numerator, the upper part of a fraction, represents how many parts we are considering. Meanwhile, the denominator, the bottom part, shows how many equal parts the whole is divided into.

When multiplying fractions, you perform the operation separately on the numerators and the denominators. For example, for the fractions \( \frac{2}{9} \) and \( \frac{6}{5} \), you calculate:

• Numerator: \( 2 \times 6 = 12 \)
• Denominator: \( 9 \times 5 = 45 \)

Thus, the resulting fraction is \( \frac{12}{45} \).

Finally, this fraction is simplified, as described earlier. Always remember that the numerator and denominator must be considered separately during operations before combining their results.

Mathematics Problem Solving

Solving mathematics problems often requires a clear understanding of the operations and the best strategies to tackle them. With fraction multiplication, the process involves distinct steps: identifying the operation, performing the calculations, and simplifying your results.

In our exercise, we began by recognizing that we needed to multiply two fractions. Next, we multiplied the numerators and denominators separately to obtain \( \frac{12}{45} \). The final step was to simplify the fraction by utilizing the greatest common divisor, ultimately arriving at \( \frac{4}{15} \).

This structured approach helps ensure accuracy and fosters a deeper comprehension of how different concepts work together in mathematics. For simplifying fractions, understanding how to find the GCD is especially beneficial, as it enhances problem-solving efficiency and improves the clarity of the final result.