Problem 14 - Fundamentals of Mathematics

Question

In the expression \(12 m\), how many \(m\) 's are indicated?

Step-by-Step Solution

Answer

The expression indicates 12 instances of the variable 'm'.

Step 1: Understand the Expression

The expression given is \(12m\) where \(12\) is a coefficient and \(m\) is a variable. The expression indicates that \(m\) is multiplied by \(12\).

Step 2: Identify the Coefficient

The coefficient \(12\) in the expression \(12m\) tells us how many times we are counting or multiplying the variable \(m\).

Step 3: Interpret the Expression

Since \(m\) is being multiplied by \(12\), it signifies that there are \(12\) occurrences or instances of \(m\) in the expression. This means \(12m\) is equivalent to adding \(m\) twelve times, i.e., \(m + m + m + \, ... \, + m \) (twelve times).

Key Concepts

Understanding Coefficients

In algebra, a coefficient is a number that is used to multiply a variable. It's like a multiplier that tells us how many times we count the variable. In the expression \(12m\), the number 12 is the coefficient. It tells us that the variable \(m\) is taken 12 times or that \(m\) is multiplied by 12. Coefficients help us understand the size or quantity of the variable in the context of the expression.

Coefficients can come in different forms:

• Whole numbers, like in our example \(12\).
• Fractions, such as \(\frac{1}{2}x\) which means half of \(x\).
• Negative numbers, for instance, \(-3y\) which can mean \(y\) subtracted three times.

Knowing the coefficient helps us interpret the expression in terms of real-world quantities, like distance or time.

What Are Variables?

Variables are symbols that represent unknown or replaceable numbers in algebraic expressions. They are like placeholders that can change or vary, depending on the context of the problem. In our expression, \(12m\), \(m\) is the variable.

Variables have a few common characteristics:

• They are usually represented by letters such as \(x, y, z, m\), etc.
• They can take different values; for example, \(m\) could be \(1, 5, or 10\).
• They help formulate general equations and rules that can solve a wide range of problems.

Understanding variables is crucial in forming and solving equations, as they allow us to work with unknowns and practical problems in mathematics.

Multiplication in Algebra

Multiplication in algebra works a bit like multiplication with numbers, but we often involve variables and coefficients. In the expression \(12m\), multiplication tells us that the variable \(m\) is repeated by the number (coefficient) 12.

Here are some core aspects of algebraic multiplication:

• It represents repeated addition. So, \(12m\) means \(m + m + m + \ldots\) for 12 times.
• It simplifies expressions; for example, instead of writing \(a + a + a\), you can write \(3a\).
• It is commutative, meaning \(ab = ba\); hence \(3x = x3\).

Grasping the concept of multiplication in algebra helps us work through complex expressions and simplifies our understanding of algebraic equations.