Problem 9 - Elementary Algebra Within Reach

Question

Write the ratio as a fraction in simplest form. \(144: 16\)

Step-by-Step Solution

Answer

The simplified form of the given ratio 144:16 is 9.

Step 1: Convert Ratio to Fraction

To start, let's write the given ratio 144:16 as a fraction. So, we have the fraction \( \frac{144}{16} \) to start with.

Step 2: Identify the Greatest Common Divisor (GCD) of the Numerator and Denominator

The Greatest Common Divisor (GCD) of 144 and 16 is the largest number that divides evenly into both of them. The GCD can be found by using a method such as Euclid's Algorithm, or by listing the factors of each number and taking the largest one they have in common. In this case, the GCD of 144 and 16 is 16.

Step 3: Simplify the Fraction

Once the GCD is found, both the numerator and the denominator can be divided by it. In this case, divide 144 and 16 by 16. \[\frac{144}{16} = 9\ \]

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is the largest number that can divide a set of two or more numbers without leaving a remainder. When you simplify fractions, understanding the GCD is essential as it helps you to determine how much you can reduce the fraction by dividing both the numerator and the denominator by this number. For our example, with the numbers 144 and 16, the GCD would be 16. By dividing both numbers by 16, we can simplify the fraction \( \frac{144}{16} \) to 9.

• Divisibility: The GCD helps to identify the maximum number by which both numbers can be divided.
• Simplification: Using the GCD allows you to reduce a fraction to its simplest form.

The concept of the GCD is integral in simplifying any mathematical ratios or fractions.

Fractions

Fractions represent a part of a whole. They consist of a numerator and a denominator. The numerator is the top number that represents how many parts we have, while the denominator is the bottom number that indicates into how many equal parts the whole is divided. For instance, the fraction \( \frac{144}{16} \) means we have 144 parts out of a total 16 parts, which originally came from the ratio 144:16.

• Numerator: The top part of a fraction indicating the portion being considered.
• Denominator: The bottom part of a fraction indicating the total number of equal parts.

Simplifying fractions is about finding the smallest form of a fraction, keeping the same value, using the greatest common divisor of both the numerator and the denominator.

Euclid's Algorithm

Euclid's Algorithm is a simple yet powerful technique for finding the greatest common divisor of two numbers. It works based on the principle that the GCD of two numbers also divides their difference. This method is efficient, especially with large numbers, and involves a series of divide-and-remainder steps. Here's how you would apply it: 1. Take two numbers, say 144 and 16. 2. Divide 144 by 16, which gives a quotient of 9 and a remainder of 0. 3. Since the remainder is 0, the GCD is the divisor used, which is 16.

• Simplicity: Requires straightforward division steps to reach the result.
• Efficiency: Quickly finds the GCD without listing all factors.

By understanding and applying Euclid's Algorithm, simplifying ratios by finding their greatest common divisor becomes much more intuitive.