Problem 36 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{1}{4}\( of \)\frac{8}{9}$$

Step-by-Step Solution

Answer

\( \frac{2}{9} \)

Step 1: Understand the Problem

We are asked to find the value of \( \frac{1}{4} \) of \( \frac{8}{9} \). This means we need to calculate what one-fourth of eight-ninths is.

Step 2: Set Up the Multiplication

To find one fraction of another, we multiply the two fractions together. Therefore, we need to multiply \( \frac{1}{4} \) by \( \frac{8}{9} \). The multiplication of fractions involves multiplying the numerators and the denominators separately.

Step 3: Multiply the Numerators

Multiply the numerators of the fractions: \( 1 \times 8 = 8 \). This gives us the numerator for the new fraction.

Step 4: Multiply the Denominators

Multiply the denominators of the fractions: \( 4 \times 9 = 36 \). This gives us the denominator for the new fraction.

Step 5: Write the Fraction

Combine the results from Steps 3 and 4 to write the fraction: \( \frac{8}{36} \).

Step 6: 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 4: \( \frac{8 \div 4}{36 \div 4} = \frac{2}{9} \).

Step 7: Conclusion

The simplified fraction \( \frac{2}{9} \) is the value of \( \frac{1}{4} \) of \( \frac{8}{9} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a way to make fractions easier to understand and work with. When a fraction is simplified, it expresses the same value as a smaller, more reduced form. This is sometimes called reducing a fraction to its lowest terms.

To simplify a fraction, you must divide both the numerator (top part) and the denominator (bottom part) by the greatest common divisor (GCD). This is the highest number that can evenly divide both the numerator and the denominator.

Here's a quick step-by-step on how to simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Rewrite the fraction using these new values.

Doing this keeps the fraction equivalent to the original, but often in a more convenient form for calculating or comparing. It's essential in ensuring numbers are manageable and results are optimal.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It's also known as the greatest common factor (GCF). The GCD is the largest number that can evenly divide two or more numbers without leaving a remainder. Identifying it is an important skill in math, especially when working with fractions.

To find the GCD, you can use these methods:

• **Listing Factors:** Write down all the factors of each number and identify the largest one they share.
• **Prime Factorization:** Break down each number into its prime factors, then multiply the common prime factors.
• **Euclidean Algorithm:** Use subtraction or division to repeatedly reduce the numbers until you find their GCD.

By learning how to find the greatest common divisor, you can effectively simplify fractions and understand their basic properties. This concept helps especially in fraction operations like addition, subtraction, and, as we've seen, multiplication.

Numerator and Denominator Multiplication

When multiplying fractions, you multiply the numerators and denominators separately. This process forms the foundation of multiplying fractions correctly. Let's break it down further.

When you have two fractions, say \( \frac{a}{b} \times \frac{c}{d} \), follow these steps:

• Multiply the numerators: Take the top numbers (\(a\) and \(c\)) and multiply them. This becomes the numerator of your result.
• Multiply the denominators: Take the bottom numbers (\(b\) and \(d\)) and multiply them. This becomes the denominator of your result.
• Combine: Write your new fraction as \( \frac{ac}{bd} \).

It’s crucial to remember not to cross-multiply when multiplying fractions directly. This method helps you correctly compute products of fractions, leading to results that can often be simplified. By attending to both the numerators and denominators separately and understanding how multiplication works in this context, you'll be equipped to tackle any fraction multiplication challenge.