Problem 82 - Prealgebra

Question

Write your answers as whole numbers, proper fractions, or mixed numbers. Find each product. (Multiply) $$12 \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is 4.

Step 1: Understand the Problem

We need to multiply the whole number 12 by the fraction \( \frac{1}{3} \).

Step 2: Convert Whole Number to Fraction

Rewrite the whole number 12 as a fraction by placing it over 1. This gives us \( \frac{12}{1} \).

Step 3: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together. This means we multiply 12 (numerator) by 1 and 1 (denominator) by 3. So, \( \frac{12}{1} \times \frac{1}{3} = \frac{12 \times 1}{1 \times 3} = \frac{12}{3} \).

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{12}{3} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives \( \frac{12 \div 3}{3 \div 3} = \frac{4}{1} = 4 \).

Step 5: Present the Answer

The product of 12 and \( \frac{1}{3} \) is a whole number, 4.

Key Concepts

Fraction Multiplication

When you multiply fractions, the process involves a straightforward step of multiplying together both the numerators and the denominators of the fractions involved. If you have two fractions, such as \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is obtained by doing the following:

• Multiply the numerators: \( a \times c \)
• Multiply the denominators: \( b \times d \)

This results in a new fraction: \( \frac{a \times c}{b \times d} \).
Let's take the exercise you have: multiplying 12 by \( \frac{1}{3} \). First, you need to convert 12 into a fraction, which leads us to the next section.

Converting Whole Numbers to Fractions

It's essential to know how to convert whole numbers into fractions so you can use them in operations between fractions, like multiplication or addition.

• Every whole number can be expressed as a fraction by simply placing it over 1.
• This is because any number divided by 1 remains unchanged.

For example, in your exercise, we converted 12 to a fraction: \( \frac{12}{1} \).
Now, you have two fractions, \( \frac{12}{1} \) and \( \frac{1}{3} \), that you multiply using the method already described in the section above about fraction multiplication.

Simplifying Fractions

After multiplying fractions, you often need to simplify the resulting fraction. Simplifying makes fractions easier to understand and work with by presenting them in their smallest form.

• First, find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD to simplify the fraction.

In the exercise, you arrived at \( \frac{12}{3} \) after multiplying. The greatest common divisor of 12 and 3 is 3.
By dividing both the numerator and the denominator by 3, you simplify \( \frac{12}{3} \) to \( \frac{4}{1} \), which is simply 4, a whole number. Simplifying fractions is a key step to ensure that your final answer is clear and concise.