Problem 24 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{30}{105}\)

Step-by-Step Solution

Answer

Question: Reduce the fraction \(\frac{30}{105}\) to its lowest terms. Answer: The fraction reduced to its lowest terms is \(\frac{2}{7}\).

Step 1: Find the GCD of the numerator and denominator

To find the GCD of 30 and 105, we have to find the largest number that can divide both 30 and 105. We can list down the factors of 30: 1, 2, 3, 5, 10, 15, 30. And for 105: 1, 3, 5, 7, 15, 21, 35, 105. From the list of factors, the highest common factor is 15, so the GCD of 30 and 105 is 15.

Step 2: Divide both numerator and denominator by the GCD

Now that we have the GCD, we'll divide the numerator and denominator by 15 to reduce the fraction to lowest terms. For the numerator: \(\frac{30}{15} = 2\). And for the denominator: \(\frac{105}{15} = 7\).

Step 3: Write the fraction in lowest terms

Now, we have the reduced numerator and denominator. So the fraction in lowest terms is \(\frac{2}{7}\).

Key Concepts

Greatest Common Divisor

Understanding how to find the greatest common divisor (GCD) is essential when reducing fractions. The GCD is the largest number that divides two or more numbers without leaving a remainder. In other words, it is the highest number both numbers are divisible by.

To find the GCD, you can list all the divisors of each number. For example, if we want to find the GCD of 30 and 105, first list the divisors of each:

• 30: 1, 2, 3, 5, 10, 15, 30
• 105: 1, 3, 5, 7, 15, 21, 35, 105

Looking at these lists, the largest common divisor is 15. Thus, the GCD of 30 and 105 is 15. Finding the GCD is crucial for simplifying fractions efficiently.

Numerator

In a fraction, the numerator is the top number and represents how many parts of the whole you have. For example, in the fraction \( \frac{30}{105} \), the numerator is 30. It indicates that there are 30 parts of a whole that is divided into 105 equal parts.

When reducing fractions, you use the GCD to simplify the numerator. For example, dividing 30 (the numerator) by the GCD of 15, we get \(\frac{30}{15} = 2\). This calculation reduces the numerator from 30 to 2, a smaller and simplified number.

Denominator

The denominator in a fraction, like \( \frac{30}{105} \), is the bottom number. It indicates the total number of equal parts the whole is divided into. In this example, the denominator is 105. It tells us that the whole is divided into 105 parts.

For reducing fractions, you also divide the denominator by the GCD. Using the earlier example with the denominator of 105, dividing by 15 (the GCD), we get \(\frac{105}{15} = 7\). This simplifies the denominator from 105 to 7, providing a clearer, reduced fraction. The denominator is essential in determining how much of the whole each part represents.

Factorization

Factorization involves breaking down a number into its component factors or divisors, essentially the numbers that multiply to form the original number. Factorization is a strategic tool in finding the GCD of two numbers, which in turn helps us simplify fractions.

To factorize a number, you systematically list its factors. For instance, 30 can be factorized into 1, 2, 3, 5, 10, 15, and 30. Similarly, 105 can be factorized into 1, 3, 5, 7, 15, 21, 35, and 105. Each of these factors results from multiplying two numbers together to achieve the original number.

Identifying common factors between numbers (like 15 in both 30 and 105) lets you pinpoint the GCD. Mastering factorization not only aids in reducing fractions but also strengthens overall number sense and arithmetic ability. It's a foundational skill that enhances mathematical problem-solving.