Problem 44 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{210}{462}$$

Step-by-Step Solution

Answer

The fraction \(\frac{210}{462}\) reduces to \(\frac{5}{11}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and the denominator. We list the factors of 210 and 462. Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. Factors of 462: 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462. The largest common factor is 42.

Step 2: Divide Both Numerator and Denominator by GCD

Now that we've identified the GCD as 42, we divide both the numerator and the denominator by this number to reduce the fraction. Divide the numerator: \[\frac{210}{42} = 5\]Divide the denominator:\[\frac{462}{42} = 11\]

Step 3: Write the Reduced Fraction

After dividing both the numerator and the denominator by the GCD of 42, the reduced fraction is obtained:\[\frac{210}{462} = \frac{5}{11}\]

Key Concepts

Greatest Common Divisor

Understanding the "Greatest Common Divisor," or GCD, is essential when it comes to reducing fractions. The GCD is the largest number that can evenly divide both the numerator and the denominator of a fraction. Think of it as the biggest "tool" you can use to cut down a fraction without leaving any leftovers.

To find the GCD, you start by listing all the factors of both the numerator and the denominator. For instance, in our example with the fraction \(\frac{210}{462}\), you would first list the factors of 210 and 462.

• Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.
• Factors of 462: 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462.

Once you've listed these factors, you look for the largest number that appears in both lists. In this example, that number is 42. This common factor that is as large as possible is called the greatest common divisor, and helps you simplify your fraction effectively.

Factors

When reducing fractions, it's important to understand what factors are and their role in the process. A factor is any number that can be multiplied by another number to reach a given product.

For example, if we're looking at the number 6, the factors of 6 include 1, 2, 3, and 6 because all these numbers can be divided into 6 without leaving a remainder.

In the context of fraction reduction, listing out the factors of both the numerator and the denominator is a key first step. It helps you determine the greatest common divisor. Consider the numbers 210 and 462. By identifying their factors, you can pinpoint which numbers are common to both, and from there, find the largest common one, which is crucial to reducing fractions.

Finding common factors allows us to see how much the numerator and the denominator can be simplified, making your fractions easier and more efficient to work with in mathematical calculations.

Lowest Terms

"Lowest terms" refers to the simplest form of a fraction, where the numerator and denominator are reduced as much as possible while still remaining whole numbers. This ensures that the fraction represents the simplest, clearest ratio possible.

After identifying the greatest common divisor, reducing to lowest terms is a straightforward process. Divide both the numerator and the denominator by the GCD. Using our example, once we've determined that the GCD is 42, we divide both 210 and 462 by 42.

• Numerator: \(210 \div 42 = 5\)
• Denominator: \(462 \div 42 = 11\)

This gives us the fraction \(\frac{5}{11}\), which is in lowest terms.

Putting a fraction in its lowest terms means it can't be simplified any further. This form is often easier to understand and use, whether you're comparing fractions, performing arithmetic operations, or applying them to real-world situations. It makes math less about complication and more about clarity.