Problem 67 - Elementary Algebra Within Reach

Question

Express the statement as a ratio in simplest form. You study 4 hours per day and are in class 6 hours

Step-by-Step Solution

Answer

The simplest form of the ratio is '2:3'

Step 1: Identify the Quantities

Identify the quantities you are comparing. Here, the number of hours of study and day class are 4 and 6 respectively, to express this in ratio form, it will be written as '4:6' or '4/6'.

Step 2: Find the Greatest Common Divisor

Find the greatest common divisor of 4 and 6. The maximum number that divides both 4 and 6 is 2.

Step 3: Simplify the Ratio

Divide both numbers of the ratio by their greatest common divisor determined in the previous step. Therefore, the simplified ratio becomes (4/2):(6/2) or '2:3'.

Key Concepts

Greatest Common Divisor

When simplifying ratios, understanding the concept of the greatest common divisor (GCD) is crucial. The GCD is the largest number that evenly divides two or more numbers. For instance, when working with the quantities 4 and 6, as in the exercise provided, the GCD is important to finding a simpler, more understandable ratio. Finding the GCD involves:

• Listing the factors of each number.
• Identifying the largest number that appears in both lists.

For 4 and 6:

• Factors of 4 are: 1, 2, 4
• Factors of 6 are: 1, 2, 3, 6

The greatest factor they share is 2, which is their GCD. Knowing this helps reduce the ratio to its simplest form. By understanding the GCD, you can simplify not just numbers, but also expressions and fractions in a wider mathematical context.

Simplifying Ratios

Simplifying ratios makes them easier to interpret and compare. Ratios express a relationship between two numbers, much like fractions, and simplifying them involves reducing them to their smallest whole numbers. Using the GCD, you can simplify any given ratio.

• Start with the original ratio, e.g., 4:6.
• Divide both parts of the ratio by their GCD, which is 2 in our example.
• This results in 2:3, a simpler representation of the original ratio.

Simplified ratios are not just neat—understanding them helps deepen your comprehension of relationships in mathematical and real-world contexts.
Whether comparing study hours to class hours or any other figures, simplifying helps you focus on the core relationship. The process of simplifying also links strongly to work with fractions and mathematical expressions, often used interchangeably in solving problems.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that express a quantity or a relation. When working with ratios, you're dealing with a specific type of mathematical expression that compares two numbers.

• A ratio expresses a simple fraction form, like '4/6' in our exercise.
• To simplify an expression, we often employ operations using a GCD, similar to reducing fractions.
• Expressions can represent real-world scenarios, like time spent on activities or resources used.

Understanding ratios as expressions aids not only in solving equations but also in data interpretation, ensuring clarity.
By viewing ratio problems through the lens of mathematical expressions, you learn vital problem-solving skills applicable in various subjects, from business to science. Begin with mastering simple expressions before delving into complex scenarios, ensuring a solid foundation in mathematics.