Problem 72 - Algebra 2

Question

Evaluate each expression. 2\(\left(\frac{1}{8}\right)\)

Step-by-Step Solution

Answer

The result of evaluating 2\(\left(\frac{1}{8}\right)\) is \(\frac{1}{4}\).

Step 1: Understand the Expression

The expression given is \(2\left(\frac{1}{8}\right)\), which means we need to multiply 2 by the fraction \(\frac{1}{8}\). This is an arithmetic operation involving fractions.

Step 2: Multiply the Whole Number and the Fraction

To multiply a whole number by a fraction, you multiply the whole number by the numerator of the fraction and keep the same denominator. Here, multiply 2 by the numerator of \(\frac{1}{8}\), which is 1: \(2 \times 1 = 2\).

Step 3: Write the Result as a Fraction

After multiplying, write the result over the same denominator. The new fraction is \(\frac{2}{8}\).

Step 4: Simplify the Fraction

Now simplify the fraction \(\frac{2}{8}\). Find the greatest common divisor (GCD) of 2 and 8, which is 2. Divide both the numerator and the denominator by 2: \(\frac{2}{8} = \frac{1}{4}\).

Key Concepts

Arithmetic Operations

In mathematics, arithmetic operations are a fundamental aspect of dealing with numbers, including operations like addition, subtraction, multiplication, and division. When working with fractions, these operations can initially seem tricky, but they follow simple rules that make them manageable.
Here, our task was to multiply a whole number by a fraction, specifically, multiplying 2 by \( \frac{1}{8} \). To perform this, you multiply the whole number (2) by the fraction's numerator (1), and the denominator (8) remains unchanged. This results in a fraction \( \frac{2}{8} \).
Understanding multiplication of fractions is key in arithmetic:

• Multiply the numerators together.
• Keep the denominator the same (when multiplying by a whole number).

Being clear on how arithmetic operations behave with fractions will aid in complex calculations later on.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where its numerator and denominator have no common divisors other than 1. After multiplying in our example, we ended up with the fraction \( \frac{2}{8} \).
To simplify \( \frac{2}{8} \):

• Identify common divisors of both the numerator (2) and the denominator (8).
• Use the greatest common divisor (GCD) to simplify.

For \( \frac{2}{8} \), both numbers share the divisor 2. Dividing the numerator and denominator by 2, we get \( \frac{1}{4} \), a simpler fraction. Simplification ensures that fractions are as easy to understand and use as possible.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial for simplifying fractions. The GCD between two numbers is the largest positive integer that divides both numbers without leaving a remainder.
In our example, we needed to simplify the fraction \( \frac{2}{8} \). This required finding the GCD of 2 and 8.
To find the GCD:

• List all divisors of each number. For 2, the divisors are 1 and 2. For 8, they are 1, 2, 4, and 8.
• The largest number common to both lists is the GCD. Here, it is 2.

Using the GCD, 2, we divided both the numerator and denominator of \( \frac{2}{8} \) to simplify it to \( \frac{1}{4} \). Knowing how to find the GCD helps make fractions less complex and is a valuable skill in arithmetic.