Question

Find each quantity. \(\frac{2}{6}\) of 120

Short answer

The quantity is 40.

Step-by-step solution

- Simplify the Fraction

Simplify the fraction \(\frac{2}{6}\) by finding the greatest common divisor of 2 and 6, which is 2. Divide both the numerator and the denominator by 2 to get \(\frac{1}{3}\).

- Set up the Multiplication

Once the fraction is simplified, set up the multiplication equation using the simplified fraction and the quantity 120. This will be \(\frac{1}{3} \times 120\).

- Perform the Multiplication

Multiply the simplified fraction by 120. This means multiplying the numerator 1 by 120 and then dividing by the denominator 3. The calculation is \(\frac{1 \times 120}{3} \) which simplifies to \(\frac{120}{3} \).

- Simplify the Result

Divide the numerator by the denominator to find the final answer. So, \(\frac{120}{3} = 40\).

Key concepts

Simplifying Fractions

To understand multiplication of fractions better, it's crucial to know how to simplify them. Simplifying a fraction means reducing it to its lowest possible terms. For example, in the problem \(\frac{2}{6}\), we simplify by finding a common factor for both numerator and denominator. The highest common factor for 2 and 6 is 2. When you divide both by their greatest common divisor (GCD), you get \(\frac{1}{3}\). This step makes calculations easier.

• Identify the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and the denominator by that number.

This technique helps streamline the subsequent steps in your calculations, making complex problems simpler.

Multiplication of Fractions

Once you have a simplified fraction, the next step is to handle multiplication. The key idea is to multiply the numerators and then the denominators. In your specific exercise, after you simplify \(\frac{2}{6}\) to \(\frac{1}{3}\), you then multiply it by 120. The formula is:
\[ \frac{1}{3} \times 120 = \frac{1 \times 120}{3} \]

• Write the simplified fraction.
• Set up your multiplication.
• Multiply the numerator with the whole number.
• Divide the result by the denominator.

This approach ensures you get accurate results. In this case, \(\frac{120}{3}\) simplifies to 40.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is foundational for simplifying fractions. It is the largest number that can divide both the numerator and the denominator evenly. For instance, with \(\frac{2}{6}\), the GCD is 2. Knowing how to find the GCD can make fraction problems much simpler:

• List the factors of each number (e.g., factors of 2 are 1, 2; factors of 6 are 1, 2, 3, 6).
• Find the largest factor common to both lists.
• Divide both numerator and denominator by the GCD.

This helps in reducing fractions to their simplest form, which in turn simplifies further calculations.

Basic Arithmetic

Basic arithmetic operations like multiplication and division are crucial when dealing with fractions. If you're multiplying a fraction and a whole number, it's helpful to break things down:

• Multiplication: Multiply the numerator of the fraction by the whole number. For example, \(\frac{1}{3} \times 120\) becomes \(\frac{1 \times 120}{3}\).
• Division: After multiplying, divide the product by the denominator. In this case, \(\frac{120}{3} = 40\).

Mastering these basic operations ensures that you can handle more complex problems with confidence.