Problem 13 - Elementary Algebra

Question

For the following problems, answer the question of how many. $$ (a+b) \text { 's in } 12(a+b) ? $$

Step-by-Step Solution

Answer

Answer: The expression (a+b) appears 12 times in the expression 12(a+b).

Step 1: Identify the expression

The given expression is 12(a+b), and we are asked to determine how many times (a+b) appears in it.

Step 2: Use the properties of multiplication

Multiplication is defined as repeated addition. So, when we see 12(a+b), it means that we are adding (a+b) to itself 12 times, or: $$ (a+b) + (a+b) + \cdots + (a+b) \\ $$ There are 12 copies of (a+b) in the given expression.

Step 3: Count the number of times (a+b) appears

Since we have established that the expression 12(a+b) is equivalent to adding (a+b) 12 times, we can conclude that (a+b) appears in the expression 12 times.

Key Concepts

Algebraic Expressions

In mathematics, an algebraic expression is a combination of numbers, variables, and operators (such as addition, subtraction, multiplication, and division). An example of an algebraic expression is \( 12(a+b) \), which is a sum of multiplications and additions involving variables \( a \) and \( b \).
Understanding algebraic expressions is crucial because they serve as the foundation for many mathematical concepts. When dealing with these expressions, it is essential to recognize the different components:

• **Terms**: The parts of the expression that are added or subtracted. In \( a+b \), \( a \) and \( b \) are terms.
• **Operators**: Symbols that indicate operations like addition (+) or multiplication (×).

Algebraic expressions allow us to write mathematical problems generally and solve for specific values, providing a versatile tool for analysis in both math and applied sciences.

Repeated Addition

Repeated addition is one of the simplest ways to understand multiplication, especially when dealing with whole numbers. Instead of adding a number to itself multiple times, we multiply it.
For instance, the expression \( 12(a+b) \) can be understood as adding \( (a+b) \) twelve times. Imagine writing it out like this:

• \( (a+b) + (a+b) + \ldots + (a+b) \)

This pattern continues until you have a total of twelve \((a+b)\) terms. Repeated addition helps us to see how multiplication condenses what would be a lengthy addition operation into something more manageable. This concept makes it easier to handle complex problems in arithmetic and algebra without having to manually perform large numbers of additions.

Coefficients

In an algebraic expression, a coefficient is the numerical factor that multiplies a variable or a set of variables. Consider \( 12(a+b) \), where 12 is the coefficient. It tells us how many times the variable term \((a+b)\) is included in the expression.
Coefficients are not always whole numbers; they can be fractions or decimals as well. Their role is crucial in understanding how relationships between variables are quantified in expressions and equations. Here are some key points about coefficients:

• If an expression has a term such as \( 5x \), the coefficient of \( x \) is 5.
• When no number is explicitly written before a term, the coefficient is implicit and usually taken as 1 (e.g., \( x \) implies \( 1x \)).
• Coefficients can also be negative, indicating a negative relationship between the variables, such as in \(-3y\).

Understanding coefficients unlocks deeper insights into how algebraic expressions represent relationships and changes in variables.