Problem 73 - Algebra 2

Question

Evaluate each expression. 3\(\left(\frac{1}{6}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{2} \).

Step 1: Understand the Problem

We need to evaluate the expression \( 3 \left( \frac{1}{6} \right) \). This is essentially asking us to multiply 3 by \( \frac{1}{6} \).

Step 2: Set up the Multiplication

The expression \( 3 \left( \frac{1}{6} \right) \) can be rewritten as \( 3 \times \frac{1}{6} \). This shows clearly that we are multiplying a whole number by a fraction.

Step 3: Perform Multiplication

To multiply a whole number by a fraction, you multiply the numerator of the fraction by the whole number and keep the denominator the same. Therefore, \( 3 \times \frac{1}{6} = \frac{3}{6} \).

Step 4: Simplify the Fraction

The fraction \( \frac{3}{6} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Thus, \( \frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \).

Step 5: State the Final Result

The simplified result of the expression \( 3 \left( \frac{1}{6} \right) \) is \( \frac{1}{2} \).

Key Concepts

Whole Number Multiplication

When we talk about whole number multiplication, especially with fractions, it might seem tricky at first. But it's pretty straightforward! Imagine you have three blocks, and you want to know how much one-sixth of them would be. Instead of actual blocks, we're using numbers. So, when we multiply 3 by \( \frac{1}{6} \), we're finding \( 1/6 \) of 3.

Here’s how you do it:

• Take the whole number, in our case, 3.
• Multiply it by the numerator of the fraction, which is also 3 in \( \frac{3}{6} \).
• Leave the denominator, 6, the same.

That gives us \( \frac{3}{6} \). Remember, the denominator (the bottom number of the fraction) remains unchanged because it just tells you that the whole is divided into 6 equal parts.
This method helps simplify the process and makes multiplying whole numbers by fractions a breeze.

Simplifying Fractions

Once you have your fraction, like \( \frac{3}{6} \), you might notice that the numbers can be reduced. This is called simplifying the fraction, and it makes your results look neat and straightforward.
Simplifying is key:

• Look for a number that divides evenly into both the numerator and the denominator.
• In \( \frac{3}{6} \), both 3 and 6 can be divided by 3.
• So dividing the top and bottom by 3 gives \( \frac{1}{2} \).

This means \( \frac{1}{2} \) is the simplest form of \( \frac{3}{6} \). It's always good practice to have your fractions in their simplest form, making them easier to understand and use in further calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is such a handy tool when simplifying fractions! It's the biggest number that can evenly divide both the numerator and the denominator.

Let’s find the GCD:

• Check if there’s a number that can divide into both the top and the bottom of a fraction without leaving a remainder.
• For \( \frac{3}{6} \), the numbers 1, 2, and 3 are all divisors. But 3 is the largest, making it the GCD.

Once you identify the GCD, you divide both the numerator and the denominator by this number.
This technique helps simplify fractions like \( \frac{3}{6} \) into \( \frac{1}{2} \), making your math work cleaner and more manageable!