Problem 12 - Prealgebra

Question

Multiply. $$9 \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The product of 9 and \(\frac{1}{3}\) is 3.

Step 1: Understand the Operation

The problem requires you to multiply a whole number by a fraction. Specifically, you need to multiply 9 by \(\frac{1}{3}\).

Step 2: Convert the Whole Number to a Fraction

To make the multiplication process easier, convert the whole number 9 into a fraction. This can be done by writing it as \(\frac{9}{1}\).

Step 3: Multiply the Fractions

Now multiply the fractions \(\frac{9}{1}\) and \(\frac{1}{3}\). To do this, multiply the numerators together and the denominators together: \(\frac{9 \times 1}{1 \times 3}\).

Step 4: Perform the Multiplications

Multiply the numerators: \(9 \times 1 = 9\). Multiply the denominators: \(1 \times 3 = 3\).

Step 5: Simplify the Resulting Fraction

After performing the multiplication, you have the fraction \(\frac{9}{3}\). Simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: \(\frac{9 \div 3}{3 \div 3} = \frac{3}{1}\).

Step 6: Convert Back to a Whole Number

Since \(\frac{3}{1}\) is equivalent to the whole number 3, the simplest form of the answer is 3.

Key Concepts

Converting Whole Numbers to Fractions

When multiplying a whole number by a fraction, it often helps to first convert the whole number into a fraction. This can simplify the multiplication process by treating both numbers in the same way. To transform a whole number into a fraction, place the number over 1. For instance, the whole number 9 becomes \( \frac{9}{1} \). This transformation works because any number divided by 1 remains unchanged. Converting in this way allows you to apply the same rules of multiplication for fractions effectively.

Next time you encounter a whole number being multiplied by a fraction, remember this simple conversion trick. It helps keep your work streamlined and tidy and is a useful tool across various math operations. By starting with identical numerical forms, you set your problem-solving up for success.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form. You achieve this by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this exercise, after multiplying to get \( \frac{9}{3} \), notice how both 9 and 3 share a GCD of 3.

To simplify, divide both the numerator and the denominator by this GCD: \( \frac{9 \div 3}{3 \div 3} = \frac{3}{1} \). This results in a simplified fraction, which can often be converted back into a whole number if the denominator is 1 (as is the case here). The process ensures that your answer is as clear and concise as possible, reflecting an optimal understanding of the problem.

Learning how to simplify fractions builds strong mathematical foundations and improves problem-solving efficiency in fraction-related exercises.

Mathematical Operations

Multiplication of fractions involves a straightforward method: multiply across. Specifically, you multiply the numerators together and multiply the denominators together. In the original problem, you begin with \( \frac{9}{1} \) and \( \frac{1}{3} \).

Multiply the numerators: \(9 \times 1 = 9\). Multiply the denominators: \(1 \times 3 = 3\). Thus, you arrive at \( \frac{9}{3} \), before proceeding to simplify it. This technique of multiplying fractions is consistent across all problems of this nature.

Understanding basic operations with fractions is crucial, as these operations are foundations for more complex mathematical concepts. By mastering multiplication, you open the door to effective learning in math, allowing for an easier transition to solving more advanced problems.