Problem 47 - Algebra A Combined Function

Question

Fill in each blank. $$ 12=4 \cdot $$ _________

Step-by-Step Solution

Answer

The blank should be filled by 3.

Step 1: Understand the Equation

We need to identify a number that, when multiplied by 4, equals 12. The equation given is: \[12 = 4 \cdot ?\]

Step 2: Solve for the Blank

To find the number that should fill in the blank, divide 12 by 4, as the equation represents multiplication. We need to find: \[? = \frac{12}{4}\]

Step 3: Calculate the Result

Divide 12 by 4:\[ ? = 3\]Thus, the number that satisfies the equation is 3.

Key Concepts

Multiplication

In basic algebra, multiplication is one of the core arithmetic operations where we combine equal groups into one total. It involves two factors, which, when multiplied together, yield a product. For example, in the equation \( 12 = 4 \cdot ? \), 4 is a known factor, while we need to find the unknown factor. This operation allows us to calculate or determine amounts by increasing numbers by a set factor.

Recognizing that multiplication is simply repeated addition can help. For instance, \( 4 \times 3 \) is the same as adding 4 three times: \(4 + 4 + 4 = 12\). In this case, filling in the blank means finding the factor that, when multiplied by 4, equals 12.

Division

Division in arithmetic is essentially the opposite of multiplication. It involves splitting a number (dividend) into equal parts, using another number (divisor). It tells us how many times one number is contained within another.

For the equation \( 12 = 4 \cdot ? \), when we need to solve for the unknown, we can rearrange it to \( ? = \frac{12}{4} \). Performing this division allows us to find that the number is 3. It's about figuring out how many times 4 fits into 12.

• This process involves dividing 12 into 4 equal groups, resulting in a quotient of 3.
• Division is often useful in algebra to isolate variables or solve equations.

Equation Solving

Equation solving is the process of finding the value of the unknown variable that makes the equation true. It often requires understanding relationships among numbers and applying various arithmetic operations to isolate the variable in question.

In our example, the equation \( 12 = 4 \cdot ? \) requires solving for the unknown. By using division, we solve: \( ? = \frac{12}{4} \). This gives us 3.
The goal is to manipulate one side of the equation to exemplify the solution, ensuring both sides remain equal.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics, including addition, subtraction, multiplication, and division. These operations allow us to solve a broad range of problems by changing and comparing numbers.

In solving \( 12 = 4 \cdot ? \), both multiplication and division are applied.

• Multiplication helps in understanding how numbers can be added repeatedly to reach a certain value.
• Division helps determine one factor when the product and the other factor are known.

Mastery of these operations is essential as they form the basis for most algebraic manipulations and solutions.