Problem 26 - Fundamentals of Mathematics

Question

Determine the missing factor(s). \(36=9\) ______.

Step-by-Step Solution

Answer

The missing factor is 4.

Step 1: Understand the Given Equation

In the equation given, we have 36 equals 9 times an unknown factor. We need to determine what number, when multiplied by 9, will result in 36. Mathematically, this can be expressed as \(36 = 9 \times x\), where \(x\) is the missing factor.

Step 2: Set Up the Equation

We have the equation \(36 = 9 \times x\). This implies that 9 times a certain number equals 36.

Step 3: Solve for the Missing Factor

To find the missing factor \(x\), we divide both sides of the equation by 9. This operation isolates \(x\) on one side of the equation: \(x = \frac{36}{9}\).

Step 4: Perform the Division

Now, calculate \(\frac{36}{9}\). Divide 36 by 9, which simplifies to 4.

Step 5: Conclusion

The missing factor is 4, because 9 multiplied by 4 equals 36.

Key Concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. In our problem, we have the equation \(36 = 9 \times x\). This equation tells us that the product of 9 and an unknown variable \(x\) is equal to 36. To better understand equations, remember:

• An equation shows a relationship between numbers or variables.
• In our example, 36 is the total product of multiplying 9 and some unknown number \(x\).
• The equals sign \(=\) indicates that both sides of the equation have the same value.

Understanding how to set up and interpret equations is crucial for solving problems in algebra.

Multiplication

Multiplication is one of the basic arithmetic operations, and it involves combining equal groups. In this exercise, we are using multiplication to find a product. Here's how it works:

• In the equation \(36 = 9 \times x\), 9 is a factor, and \(x\) is the unknown factor we need to find.
• To "multiply" means to add a number to itself a certain number of times. For example, 9 multiplied by 4 means 9 added together 4 times.
• Multiplication is commutative, meaning \(a \times b = b \times a\). It is also associative, which allows us to group numbers differently, like \((a \times b) \times c = a \times (b \times c)\).

This fundamental operation helps us understand how quantities relate to each other.

Division

Division is another fundamental operation in mathematics, which is essentially the inverse of multiplication. When solving for a missing factor like in our problem, division helps us isolate the unknown variable:

• To solve \(36 = 9 \times x\), we divide both sides by 9 to find \(x\). This results in \(x = \frac{36}{9}\).
• Division helps us separate a total into equal parts.\(36 \div 9\) asks, "how many times does 9 fit into 36?" The answer is 4.
• We use division here to balance the equation by doing the same operation on both sides.

Understanding division ensures you can solve equations and re-discover factors in a relationship.

Solving for Variables

Solving for variables in an equation involves finding the value of the variable that makes the equation true. Let’s break it down:

• In our problem, the variable \(x\) holds the place of the missing number in the equation \(36 = 9 \times x\).
• To determine \(x\), we perform operations that isolate \(x\) on one side of the equation.
• In our example, dividing both sides of the equation by 9 simplifies it to \(x = 4\).
• This operation of isolating \(x\) reveals that the missing number is 4 because when 9 is multiplied by 4, the product equals 36.

Knowing how to solve for variables enables you to find unknowns in various mathematical contexts, which is a fundamental skill in algebra.