Problem 28 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{1}{6}, \frac{7}{42}$$

Step-by-Step Solution

Answer

The fractions \(\frac{1}{6}\) and \(\frac{7}{42}\) are equivalent.

Step 1: Understand the Concept of Equivalent Fractions

Equivalent fractions are fractions that represent the same value even if they look different. Two fractions are equivalent if they have the same simplest form or if their cross-products are equal.

Step 2: Find the Simplest Form of Each Fraction

Begin by simplifying each fraction to its simplest form. For \(\frac{1}{6}\), it is already in its simplest form because 1 and 6 have no common factors other than 1. Now, simplify \(\frac{7}{42}\). Since both 7 and 42 can be divided by 7, divide them by 7. Thus, \(\frac{7}{42} \rightarrow \frac{1}{6}\).

Step 3: Compare the Simplest Forms

Now compare the simplest forms of each fraction. Both \(\frac{1}{6}\) and the simplified form of \(\frac{7}{42}\) is \(\frac{1}{6}\), indicating that the fractions are indeed equivalent.

Key Concepts

Fraction Simplification

When simplifying a fraction, our goal is to make it as simple as possible while maintaining its overall value. For example, if you have a fraction like \(\frac{7}{42}\), you might notice that both the numerator (7) and the denominator (42) can be divided by the same number. This number is called a 'common factor'.
To simplify, you can divide the numerator and the denominator by any common factors until the fraction can no longer be simplified.Here's how you simplify \(\frac{7}{42}\):

• Identify the greatest common factor (GCF) of 7 and 42, which is 7.
• Divide both the numerator and the denominator by 7:
• \[\frac{7}{42} = \frac{7 \div 7}{42 \div 7} = \frac{1}{6}\]

This process has reduced the fraction to its simplest form, \(\frac{1}{6}\). Now, both fractions, \(\frac{1}{6}\) and the simplified form of \(\frac{7}{42}\), are the same, showing they are equivalent.

Common Factors

Common factors are numbers that can evenly divide both the numerator and the denominator of a fraction. From our example, the common factor is a number that can go into both 7 and 42 without leaving any remainder.
To find them, list out the factors of each number:

• Factors of 7: 1, 7
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

In this list, the number 7 is common, making it a common factor. Recognizing common factors helps simplify fractions to the most basic form by dividing the numerator and denominator by the greatest common factor (GCF). This step is crucial because it often reveals the simplest form of a fraction. Knowing how to find common factors quickly can simplify math problems significantly.

Cross-Products

Cross-products provide another way to check if two fractions are equivalent. This method doesn't require simplifying the fractions. Instead, you multiply the numerator of the first fraction by the denominator of the second, and vice versa, then compare the products.
Let's use the fractions \(\frac{1}{6}\) and \(\frac{7}{42}\):

• First, calculate the cross-products:
• \[1 \times 42 = 42\]
• \[6 \times 7 = 42\]

Since both cross-products are equal (42), the fractions are equivalent.
Cross-products are especially useful when fractions are large or difficult to simplify quickly. This method provides a quick check, ensuring you can determine equivalency without needing to simplify each fraction.