Problem 33 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{5}{8}, \frac{15}{24}$$

Step-by-Step Solution

Answer

The fractions \(\frac{5}{8}\) and \(\frac{15}{24}\) are equivalent.

Step 1: Recognize Equivalent Fractions

Fractions are equivalent if they can be simplified to the same fraction or if they represent the same value when simplified. To check equivalence, we can cross-multiply or simplify either of the two given fractions.

Step 2: Simplify the First Fraction \(\frac{5}{8}\)

\(\frac{5}{8}\) is already in its simplest form as there is no number other than 1 that divides both 5 and 8. Hence, it cannot be simplified further.

Step 3: Simplify the Second Fraction \(\frac{15}{24}\)

Determine the greatest common divisor (GCD) of 15 and 24, which is 3. Divide both the numerator and the denominator by 3: \(\frac{15 \div 3}{24 \div 3} = \frac{5}{8}\). So, \(\frac{15}{24}\) simplifies to \(\frac{5}{8}\).

Step 4: Compare Simplified Fractions

Both \(\frac{5}{8}\) and \(\frac{15}{24}\) simplify to \(\frac{5}{8}\). Since they are identical after simplification, the fractions are equivalent.

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. For example, in the case of \(\frac{5}{8}\), the numerator is 5 and the denominator is 8. Since there are no numbers other than 1 that can divide both 5 and 8 evenly, this fraction is already simplified.

To simplify any fraction, follow these steps:

• Find the greatest number that divides both the numerator and the denominator. This is known as the greatest common divisor (GCD).
• Divide both the numerator and the denominator by the GCD.
• The result is the fraction in its simplest or most reduced form.

This process helps in comparing fractions for equivalence or simply making calculations easier. Remember, the value of the fraction doesn't change when you simplify it, only its appearance does.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that evenly divides two or more numbers. Identifying the GCD is a key step in simplifying fractions.

To find the GCD of two numbers, you can use several methods:

• Listing Factors: Write down all factors of each number and choose the largest number that appears in both lists.
• Prime Factorization: Break down each number into its prime factors and multiply the common primes.
• Euclidean Algorithm: Repeat division using the remainder until zero is reached. The last non-zero remainder is the GCD.

In the exercise, the GCD of 15 and 24 was found to be 3. Dividing both 15 and 24 by 3 leads to simplifying \(\frac{15}{24}\) to \(\frac{5}{8}\). This illustrates how finding the GCD helps in simplifying fractions to determine equivalence.

Fraction Equivalence

Fractions are equivalent if they represent the same portion or value, even if they look different. To check if two fractions are equivalent, they should simplify to the same smallest form or have equal cross products.

Here's how you can determine fraction equivalence:

• Simplification Method: Simplify each fraction to its simplest form. If both are reduced to the same fraction, they are equivalent.
• Cross Multiplication: Multiply the numerator of the first fraction with the denominator of the second, and vice versa. If the products are equal, the fractions are equivalent.

In our exercise, both \(\frac{5}{8}\) and \(\frac{15}{24}\) simplify to \(\frac{5}{8}\), showing they are equivalent. Understanding that different fractions can denote the same value is critical in mathematics.

Cross Multiplication

Cross multiplication is a quick method to determine if two fractions are equivalent without simplifying them first. It's a handy tool in many math problems involving fractions.

To use cross multiplication:

• Consider two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).
• Multiply the numerator of the first fraction by the denominator of the second, resulting in \(a \times d\).
• Multiply the numerator of the second fraction by the denominator of the first, resulting in \(b \times c\).
• If \(a \times d = b \times c\), the fractions are equivalent.

This technique was not necessary in our original exercise because simplification was straightforward, but cross multiplication offers a quick verification method when fractions are more complex. It's a reliable technique for students to use when assessing equivalence.