Problem 68 - Prealgebra

Question

Perform the indicated operations. $$960 \div 240$$

Step-by-Step Solution

Answer

960 divided by 240 equals 4.

Step 1: Understand the Operation

The operation given is division, represented by the division symbol \( \div \). We are required to divide 960 by 240.

Step 2: Set Up the Division

Write the division in fraction form: \( \frac{960}{240} \). This will help in simplifying the division by considering possible common factors.

Step 3: Simplify the Fraction

Identify common factors for 960 and 240. Both numbers are divisible by 240, which is the greatest common factor (GCF) for both.

Step 4: Divide Using the GCF

Divide both the numerator and the denominator by their GCF, which is 240: \[ \frac{960 \, \div \, 240}{240 \, \div \, 240} = \frac{4}{1} = 4. \]

Step 5: Conclude the Division

Thus, after simplifying, \( 960 \div 240 = 4 \).

Key Concepts

Understanding the Greatest Common Factor (GCF)

The greatest common factor, often abbreviated as GCF, is the largest number that can evenly divide two or more numbers without leaving a remainder. Knowing how to find the GCF is an essential skill when simplifying fractions because it helps you reduce numbers to their simplest form. For example, in the exercise where we divided 960 by 240, the GCF for both these numbers is 240. This means 240 is the largest number that divides both 960 and 240 evenly. Thus, by identifying the GCF, the process of division becomes straightforward, as it allows you to simplify the fraction neatly. To find the GCF, you can list out all factors of each number and choose the greatest one that appears in both lists. Alternatively, you can use the prime factorization method by breaking down each number into prime numbers, then multiplying the common prime factors. This ensures that you have the correct GCF and helps in simplifying more complex problems.

The Process of Fraction Simplification

Fraction simplification is a process of making a fraction as simple as possible. This is done by ensuring both the numerator and the denominator are divided by their greatest common factor. Taking the example of 960 divided by 240, the initial representation is the fraction \( \frac{960}{240} \). Here, 960 is the numerator and 240 is the denominator. Simplification begins by identifying the GCF, which is 240 in this case.

• Divide the numerator (960) by 240, resulting in 4.
• Divide the denominator (240) by itself, giving 1.

This simplification process turns the fraction into \( \frac{4}{1} \), which simplifies further to a whole number, 4. Simplifying fractions is not just limited to division problems; it applies to rational expressions in algebra, helping reduce expressions for easier handling.

Numerator and Denominator Explained

In any fraction, the number on the top is called the numerator, while the number on the bottom is referred to as the denominator. The numerator tells you how many parts you have, while the denominator indicates how many parts make up a whole. Understanding these components is crucial in performing division and simplification tasks. For the division problem \( \frac{960}{240} \), the numerator is 960, and the denominator is 240. Each plays a role in defining the fraction's value. When we simplify, both numbers are divided by the GCF, allowing us to reduce the fraction effectively.Knowing the purpose of both the numerator and the denominator helps in understanding their individual roles in division and other operations, laying a foundational understanding crucial in both basic and advanced mathematics. This understanding ensures accurate manipulations and transformations of numbers in various math problems.