Problem 67 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 16\left(\frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The simplified result is 24.

Step 1: Understand the Operation

The given operation involves multiplying the number 16 by the fraction \( \frac{3}{2} \). We want to find the product of these two values.

Step 2: Multiply the Numerator by 16

To multiply 16 by \( \frac{3}{2} \), first multiply 16 by the numerator, which is 3. This results in:\[ 16 \times 3 = 48 \]

Step 3: Divide by the Denominator

Next, take the result from Step 2 and divide it by the denominator of the fraction, which is 2:\[ \frac{48}{2} = 24 \]

Step 4: Simplify the Result

The division of 48 by 2 results in 24, which cannot be simplified further. Therefore, the simplified result of the operation is 24.

Key Concepts

Multiplication of Fractions

Multiplying fractions involves breaking down the problem into a simple two-step process. Unlike other operations, here, we focus primarily on the role each part of the fraction plays in the calculation. When you multiply a whole number by a fraction, you multiply the whole number by the fraction's numerator while keeping the denominator the same. This keeps the fraction's structure while scaling its value by the whole number.

• For our example with 16 and \( \frac{3}{2} \), you multiply the numerator 3 by 16.
• This gives us \( 16 \times 3 = 48 \).
• Next, you treat the denominator as a divider for this product.

Remember, multiplication with fractions is simply rearranging the whole number as a fraction with a denominator of 1 and proceeding with the same steps. This method ensures accuracy in calculations and helps in visualizing what happens to numbers in the equation.

Simplification

Simplification is a key part of working with fractions and algebraic expressions. After performing operations, always check if you can present the result in a more concise form. A number or a fraction is in its simplest form when it can't be further reduced while staying in the whole number or simpler fraction realms.

• For instance, after multiplying, we found our product as \( \frac{48}{2} \).
• You simplify by dividing the numerator by the denominator, leading to \( 24 \).

This process is crucial for making results more comprehensible and manageable. Simplified expressions offer a cleaner, clearer insight into the nature of the mathematical result without changing its inherent value. By displaying answers simply, we ease our understanding and maintain consistency.

Integer Multiplication

Integer multiplication is the foundation of performing arithmetic operations, especially when handling mixed operations involving fractions. It's important to understand how multiplication extends the value of numbers, especially between an integer and other forms of numerical representations such as fractions. In our exercise, multiplying 16 by 3 gives you an insight into how integer multiplication functions like adding a number to itself a particular number of times.

• Think of \( 16 \times 3 \) as adding 16 three times: \( 16 + 16 + 16 \).
• This results in 48, demonstrating how multiplication amplifies a number.

Effective integer multiplication, especially when multiplied by a number greater than one, increases the quantity based on the second number’s magnitude, proving vital in broader algebraic contexts.