Problem 26 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 20 tiles to 24 tiles

Step-by-Step Solution

Answer

The ratio 20 to 24 as a fraction in simplest form is \( \frac{5}{6} \).

Step 1: Identify the Ratio

Start by identifying the given ratio, which is 20 tiles to 24 tiles.

Step 2: Write as a Fraction

Express the ratio 20 to 24 as a fraction: \( \frac{20}{24} \).

Step 3: Find the Greatest Common Divisor (GCD)

Find the greatest common divisor of 20 and 24. Both numbers are divisible by 4, which is their GCD.

Step 4: Divide by the GCD

Simplify the fraction by dividing both the numerator and the denominator by their GCD. \( \frac{20 \div 4}{24 \div 4} = \frac{5}{6} \).

Step 5: Verify the Simplified Fraction

Ensure the fraction \( \frac{5}{6} \) is in its simplest form by checking that 5 and 6 have no common divisors other than 1.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is essential for simplifying fractions. It is the largest number that can divide two or more numbers without leaving a remainder. For instance, consider the numbers 20 and 24. By finding the greatest common divisor of these numbers, we can simplify any fraction formed by them. To determine the GCD:

• List the divisors of each number. For 20, the divisors are 1, 2, 4, 5, 10, and 20. For 24, they are 1, 2, 3, 4, 6, 8, 12, and 24.
• Identify the common divisors. Both 20 and 24 are divisible by 1, 2, and 4.
• Of these, 4 is the largest common divisor, so the GCD is 4.

Using the GCD helps in reducing fractions to their simplest form, as in the example of transforming \( \frac{20}{24} \) to \( \frac{5}{6} \). This process is crucial in mathematics where precision and simplicity are valued.

Exploring Ratios in Mathematics

Ratios are a fundamental concept in mathematics that compares two quantities. Understanding ratios helps in solving problems involving proportional relationships, which pop up frequently in real life. A ratio can be expressed in several forms, such as "20 to 24" or "20:24." When dealing with ratios, you often want to observe the relationship in its simplest form. This is achieved by converting the ratio into a fraction and simplifying it. For the given example of 20 tiles to 24 tiles as a ratio:

• Express the ratio as a fraction: \( \frac{20}{24} \).
• Use the greatest common divisor to simplify. Divide both terms by the GCD, 4.
• This results in the simplified fraction \( \frac{5}{6} \).

Simplifying ratios to their simplest form is crucial. It expresses the most direct relationship between the quantities, making interpretation and further calculations easier.

Mastering Fractions and Their Simplifications

Fractions are numerical expressions representing the division of a whole into parts. They are composed of a numerator and a denominator. Simplifying fractions is essential for clear and effective mathematical communication. Simplification involves converting the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Let's revisit how this is applied:

• Take the original fraction derived from a ratio, such as \( \frac{20}{24} \).
• Identify the greatest common divisor, which in this case is 4.
• Divide the numerator and the denominator by 4, resulting in \( \frac{5}{6} \).

This process ensures accuracy and efficiency, especially in complex computations. By understanding fractions thoroughly, you build a strong foundation for solving varied mathematical problems.

Significance in Mathematics Education

Mathematics education equips learners with critical thinking and problem-solving skills. Concepts like simplifying fractions and understanding ratios are introduced early to build a strong mathematical foundation. These skills are applicable across many disciplines:

• In science, for balancing chemical equations or calculating probabilities.
• In life skills, such as dividing resources, cooking, or financial budgeting.
• In technology and engineering for measurements and model scaling.

Formal education aims to make these concepts intuitive. Teachers often emphasize understanding over memorization, demonstrating practical applications. Emphasizing real-life examples and interactive activities help students grasp these abstract concepts, ensuring they are well-prepared for future academic and real-world challenges.