Problem 24 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 68 meters in 15 seconds

Step-by-Step Solution

Answer

The ratio 68 meters in 15 seconds expressed as a fraction in simplest form is \( \frac{68}{15} \).

Step 1: Understand the Ratio

The given problem asks you to express the ratio of 68 meters to 15 seconds as a fraction. A ratio is a comparison of two quantities, in this case, the distance in meters to the time in seconds.

Step 2: Write the Ratio as a Fraction

To express a ratio as a fraction, write the first quantity as the numerator and the second quantity as the denominator. Therefore, the ratio of 68 meters to 15 seconds can be written as the fraction \( \frac{68}{15} \).

Step 3: Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 68 and 15 is 1 since they have no common factors other than 1, which means the fraction is already in its simplest form.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. At their core, fractions express how many parts of something exist if it was divided into equal pieces. A fraction consists of two main parts: the numerator and the denominator.

• The numerator is the number on top, indicating how many parts we are considering.
• The denominator, at the bottom, shows how many equal parts the whole is divided into.

In our exercise, the fraction \( \frac{68}{15} \) is derived from a ratio, reflecting 68 meters per 15 seconds. This is key in translating real-world data into a manageable form to perform further mathematical operations. Fractions help us compare, measure, and understand relationships between quantities.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that involves expressing a fraction in a way that is easiest to understand. The goal is to reduce the fraction to its simplest form while maintaining the same value.

• Identify any common factors shared by the numerator and the denominator.
• Divide both parts of the fraction by their greatest common factor.

For instance, with the fraction \( \frac{68}{15} \), checking for common factors is crucial to know if the fraction can be simplified further. Simplification makes fractions easier to read and use, especially in subsequent calculations or comparisons, allowing clearer insights into the data.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that can evenly divide two numbers. Finding the GCD helps us simplify fractions because it shows us the greatest factor by which the numerator and denominator can both be divided.
Here's how you can find the GCD:

• List the factors of each number.
• Identify the largest factor that both numbers share.
• Divide the numerator and the denominator by this factor.

In the case of the fraction \( \frac{68}{15} \), since 68 and 15 share no common factors other than 1, their GCD is 1. This tells us that \( \frac{68}{15} \) is already in its simplest form. Understanding the GCD isn't just about simplifying fractions—it's a useful skill when comparing ratios and understanding portions in contexts like data analysis, cooking, and beyond.