Problem 4 - Algebra 2 and Trigonometry

Question

Write each ratio in simplest form. \(21 : 14\)

Step-by-Step Solution

Answer

The simplest form of the ratio \(21 : 14\) is \(3 : 2\).

Step 1: Identify the terms of the ratio

Identify the two numbers in the ratio. For the ratio \(21 : 14\), the terms are 21 and 14.

Step 2: Find the greatest common divisor (GCD)

Find the greatest common divisor of 21 and 14 to simplify the ratio. The factors of 21 are 1, 3, 7, and 21. The factors of 14 are 1, 2, 7, and 14. The greatest common divisor is 7.

Step 3: Divide each term by the GCD

Divide each term of the ratio by the GCD. Divide 21 by 7 to get 3, and divide 14 by 7 to get 2.

Step 4: Write the simplified ratio

Express the ratio as \(3 : 2\) since 21 divided by 7 is 3, and 14 divided by 7 is 2. This is the simplest form of the ratio.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a crucial concept in mathematics, especially useful for comparing and simplifying ratios. The GCD of two or more numbers is the largest positive integer that can exactly divide each of the numbers without leaving a remainder.
Determining the GCD can help reduce ratios to their simplest form, making them easier to understand and work with.
To find the GCD:

• List all the factors of each number. Factors are numbers that divide another number completely with no remainder.
• Identify the largest factor that both numbers share. This is the GCD.

For instance, to find the GCD of 21 and 14, you list their factors:

• The factors of 21 are 1, 3, 7, 21.
• The factors of 14 are 1, 2, 7, 14.

Both 21 and 14 share the factor 7, which is the largest common factor. Therefore, 7 is the GCD. By using the GCD, you can simplify ratios like 21:14 effectively.

Simplifying Ratios

Simplifying a ratio involves reducing it to its simplest form, which makes relationships between numbers clearer and more manageable. When a ratio is simplified, the numbers are as small as possible while still maintaining the same relationship.
Here's how to simplify a ratio:

• First, find the greatest common divisor (GCD) of the numbers involved in the ratio.
• Then, divide each term of the ratio by this GCD.

For example, with the ratio 21:14, you first determine the GCD, which is 7. Next, divide each number by 7:

• 21 divided by 7 equals 3.
• 14 divided by 7 equals 2.

Thus, the simplified ratio is 3:2.
Simplifying ratios helps provide a clearer understanding of the proportional relationship between two quantities.

Mathematical Factors

Mathematical factors are numbers you can multiply together to get another number. Identifying these factors is crucial when working with ratios because they can help simplify them by finding the greatest common divisor.
Here's how to find the factors:

• Start with the number 1 and proceed up to the number itself.
• Test each possible factor by dividing it into the original number to see if it results in a whole number (no remainder).

Consider the number 21:

• Divide 21 by 1, 3, 7, and 21, and note which divisions have no remainder. These are your factors.

Similarly, for 14:

• Using numbers 1, 2, 7, and 14, all divide cleanly into 14 with no remainder.

Knowing factors aids you in understanding the components and relationships in numbers, facilitating easier problem-solving in mathematics, particularly in simplifying ratios.