Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{10}{16}$$

Step-by-Step Solution

Answer

The fraction \(\frac{10}{16}\) simplified to its lowest terms is \(\frac{5}{8}\).

Step 1: Identify the greatest common divisor

Identify the largest integer that can divide both 10 and 16 without a remainder. In this case, the greatest common divisor is 2.

Step 2: Divide numerator and denominator by the greatest common divisor

Now, divide both the numerator (10) and the denominator (16) by 2. This gives \(\frac{10}{2}\) for the numerator and \(\frac{16}{2}\) for the denominator.

Step 3: Simplify the fraction

Simplifying both \(\frac{10}{2}\) and \(\frac{16}{2}\) yields \(\frac{5}{8}\). This is the fraction \(\frac{10}{16}\) in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

Understanding the Greatest Common Divisor (GCD) is vital when simplifying fractions. The GCD of two numbers is the largest integer that divides both numbers without leaving a remainder.

For instance, let's look at the numbers 10 and 16. The divisors of 10 are 1, 2, 5, and 10, while the divisors of 16 are 1, 2, 4, 8, and 16. The common divisors are 1 and 2, hence the GCD is 2. This is the largest number that fits into both 10 and 16 equally.

Finding the GCD is the first crucial step in reducing fractions because it shows us by how much we can 'scale down' both the numerator and the denominator, ensuring the fraction stays equivalent but becomes simpler.

Numerator and Denominator

Every fraction consists of two parts: the numerator and the denominator. The numerator, situated at the top, tells us how many parts we have. The denominator, at the bottom, tells us into how many parts the whole is divided.

In the fraction \(\frac{10}{16}\), 10 is the numerator, meaning 10 parts, and 16 is the denominator, indicating that the whole is divided into 16 equal parts. When simplifying, the goal is to reduce both of these numbers equally to get the simplest form of the fraction that still represents the same proportion of the whole.

Reducing Fractions to Lowest Terms

Reducing fractions to their lowest terms means making the numerator and the denominator as small as possible without changing the value of the fraction. This is achieved by dividing both the numerator and the denominator by their GCD.

Let's revisit our example with \(\frac{10}{16}\). We already identified the GCD as 2. So, by dividing both 10 and 16 by 2, we get \(\frac{5}{8}\). Now, neither 5 nor 8 can be divided by the same number other than 1, so \(\frac{5}{8}\) is the fraction in its lowest terms. This simpler fraction is easier to work with and visually more understandable, showing us the true ratio between the numerator and the denominator.