Question

Express the following ratios as fractions. Reduce to lowest terms. 4: 6 ______

Short answer

\( \frac{2}{3} \)

Step-by-step solution

Identify the Numbers in the Ratio

The given ratio is 4:6. We need to take these two numbers and express them as a fraction.

Write the Ratio as a Fraction

Express the numbers from the ratio as a fraction: \( \frac{4}{6} \).

Find the Greatest Common Divisor (GCD)

To reduce the fraction, we need to find the greatest common divisor of 4 and 6. The factors of 4 are 1, 2, and 4. The factors of 6 are 1, 2, 3, and 6. The greatest common factor is 2.

Divide the Numerator and Denominator by the GCD

Divide both the numerator (4) and the denominator (6) by their greatest common divisor (2): \( \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \).

Express the Simplified Fraction

The ratio 4:6 expressed as a fraction in lowest terms is \( \frac{2}{3} \).

Key concepts

Understanding Ratios

Ratios are a way to compare two or more quantities. They show how many times one value contains or is contained within the other. A ratio can be expressed in different forms: as two numbers separated by a colon (4:6), as a fraction \( \frac{4}{6} \), or in a decimal form.

Ratios are useful in many real-life scenarios, such as cooking where you might need to maintain a specific proportion of ingredients.

• Example: If you have a recipe that requires a 4:6 ratio of sugar to flour, it means for every 4 parts sugar, you use 6 parts flour.

Ratios help in understanding the relative size of one number to another, making it easier to compare sizes or quantities. You can also scale them up or down, as the ratio remains consistent, exhibiting a consistent relationship between the compared quantities.

The Role of Greatest Common Divisor (GCD)

Finding the greatest common divisor (GCD) is crucial for simplifying fractions. The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder.

For instance, consider the numbers 4 and 6. The factors of 4 are 1, 2, and 4, while the factors of 6 are 1, 2, 3, and 6. The largest number common to both sets is 2, which is the GCD. The primary use of the GCD in context to fractions is to simplify them by reducing the fraction to its lowest terms.

• Example: To simplify \( \frac{4}{6} \), divide both the numerator and denominator by their GCD, which is 2, resulting in \( \frac{2}{3} \).

By dividing both parts of the fraction by the GCD, you maintain the value of the fraction while reducing its size. This process reveals the simplest form of the given ratio or fraction.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their lowest terms. In mathematics, this means you make the numerator and denominator as small as possible, without changing the fraction's value. This is done by dividing both the numerator and the denominator by their GCD.

For example, the fraction \( \frac{4}{6} \) can be simplified by dividing both 4 and 6 by their GCD of 2, resulting in \( \frac{2}{3} \). This fraction is now in its simplest form. Simplifying fractions is important because it makes calculations easier and helps in clearly understanding the proportion or ratio that the fraction represents.

• Simplification can help in performing operations like addition and subtraction, as they require the fractions involved to have a common denominator.

Simplified fractions are more intuitive and less cumbersome, hence aiding in better interpretation and usage in problems.

Importance of Mathematics Education

Mathematics education lays the groundwork for logical thinking and problem-solving skills. Understanding key concepts such as fractions, ratios, and the greatest common divisor equips students with tools for everyday decision making and professional applications.

Math is not just about numbers; it's about patterns, structures, and logical rigor. Concepts like simplifying fractions help develop clarity of thought and precision. Mathematics education encourages curiosity and critical thinking, essential traits for innovation and discovery.

• Applying math in real-world scenarios teaches students the relevance of what they learn, making education more engaging and meaningful.

As students progress, they build upon these foundational concepts, paving the way for more complex topics. Such a structured approach ensures a deep comprehension of mathematical ideas, empowering students for future challenges.