Problem 19 - Prealgebra

Question

Multiply. $$48 \cdot \frac{1}{12} \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The result is \(\frac{4}{3}\).

Step 1: Setup the Multiplication

We are given the expression to multiply: \(48 \cdot \frac{1}{12} \cdot \frac{1}{3}\). Our goal is to perform the multiplication step by step.

Step 2: Multiply 48 by 1/12

First, multiply 48 by \(\frac{1}{12}\) to simplify the expression: \(48 \cdot \frac{1}{12} = \frac{48}{12}\). Performing this division gives us \(\frac{48}{12} = 4\). Thus, the expression simplifies to 4.

Step 3: Multiply the Result by 1/3

Next, take the result from Step 2, which is 4, and multiply it by \(\frac{1}{3}\): \(4 \cdot \frac{1}{3} = \frac{4}{3}\). This fraction represents the final result of the multiplication.

Key Concepts

Understanding Fractions

Fractions are a way to express parts of a whole. They consist of a numerator, the top number, which indicates how many parts we have, and a denominator, the bottom number, which tells us into how many parts the whole is divided. For example, in the fraction \(\frac{1}{12}\), the numerator is 1 and the denominator is 12. This means we have 1 part out of a total of 12 parts.

Fractions can make complex mathematical operations intuitive. They often arise in situations where division is necessary to express an amount smaller than the whole. Understanding fractions is key when performing operations like addition, subtraction, and, as in our exercise, multiplication. Being well-versed in fractions allows us to navigate these mathematical landscapes with ease.

Multiplication Steps with Fractions

When multiplying numbers by fractions, the process involves a straightforward step-by-step approach:

• Identify and Prepare the Numbers: Begin with the numbers provided in the problem. In our exercise, we have 48, \(\frac{1}{12}\), and \(\frac{1}{3}\).
• Multiply the Whole Number by the Fraction: When multiplying a whole number by a fraction, treat the whole number as a fraction by giving it a denominator of 1. For example, \(48\) becomes \(\frac{48}{1}\). First, multiply 48 by \(\frac{1}{12}\), resulting in \(\frac{48}{12}\) or 4 after simplifying.
• Continue with Remaining Fractions: Take the result and multiply it by the next fraction. For instance, multiply the result 4 by \(\frac{1}{3}\) to obtain \(\frac{4}{3}\).

Multiplying fractions is essentially about multiplying the numerators together and the denominators together. Simplify when possible at each step to keep the numbers manageable.

Simplifying Expressions

Simplifying expressions is about reducing a complex mathematical sentence into its simplest form. This makes it easier to understand and use. In the context of our exercise, after each multiplication step, the results are simplified to make calculations easier.

• Factor and Reduce: In expressions like \(\frac{48}{12}\), reducing involves dividing both the numerator and the denominator by their greatest common factor, which in this case gives us 4 after simplifying.
• Check for Simplification Opportunities: Always look for fractions that can be reduced to smaller terms. In our final result, \(\frac{4}{3}\) is already in its simplest form since 4 and 3 have no common factors other than 1.

Simplifying expressions ensures that the process of multiplication stays clean and clear, ultimately leading to the simplest possible expression.