Problem 5 - Elementary Algebra

Question

Express each ratio in reduced form. 96 feet 3 seconds

Step-by-Step Solution

Answer

The ratio in reduced form is 32 feet per second.

Step 1: Initial Ratio Setup

First, express the given ratio as a fraction. The problem asks us to express the ratio of 96 feet to 3 seconds. So, write this as a fraction: \( \frac{96 \text{ feet}}{3 \text{ seconds}} \).

Step 2: Simplify the Fraction

Next, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 96 and 3 is 3. Divide both the numerator and the denominator by 3: \( \frac{96 \div 3}{3 \div 3} = \frac{32}{1} \).

Step 3: Express in Simplified Form

Since \( \frac{32}{1} \) simplifies to just 32, the ratio of 96 feet to 3 seconds in reduced form is 32 feet per second.

Key Concepts

Greatest Common Divisor (GCD)

To simplify ratios and fractions, finding the Greatest Common Divisor (GCD) is a crucial step. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For the problem at hand, we have 96 feet and 3 seconds.

• List the factors: 96 has factors like 1, 2, 3, 4, 6, 8, ..., 96.
• 3 is simpler, with factors being 1 and 3.

The common factors between 96 and 3 are only 1 and 3. Hence, the GCD is 3.
Using the GCD helps in reducing the fraction by dividing both terms by this number, making calculations straightforward.

Fraction Simplification

Once you've identified the GCD, the next step is to simplify or reduce the fraction. A simplified fraction is easy to understand and further work with in calculations. For example, consider the fraction form of the original ratio:

• Start with the given: \( \frac{96 \text{ feet}}{3 \text{ seconds}} \).
• Divide the numerator and denominator by their GCD, which is 3.
• You have: \( \frac{96 \div 3}{3 \div 3} = \frac{32}{1} \).

This results in the simplified fraction: 32. Simplification makes it apparent that the final result is easier to grasp and use.

Unit Conversion

Conversions between different units involve expressing a quantity in different measurements without changing its value. In simpler terms, it rearranges how we look at dimensions. Using unit conversions can often help in understanding ratios.
In our case, the original ratio is 96 feet to 3 seconds, ending up as 32 feet per second. By simplifying the fraction \( \frac{96 \text{ feet}}{3 \text{ seconds}} \) to \( 32 \text{ feet per second} \), we see a clear, single unit rate.

• This conversion gives us a new perspective on how quickly something is moving.
• Unit conversion tells us the speed as a single number, like 32, where every second, 32 feet are covered.

Understanding unit conversion facilitates analyzing problems with ease, making complex comparisons more digestible.