Problem 271 - Prealgebra

Question

In the following exercises, multiply. $$(100)(36)$$

Step-by-Step Solution

Answer

3600

Step 1: Identify the Numbers

First, take note of the numbers we need to multiply. Here, the numbers are 100 and 36.

Step 2: Set Up the Multiplication

Write down the multiplication problem in standard form: \[ 100 \times 36 \]

Step 3: Multiply the Numbers

Perform the multiplication operation. When multiplying, you can break down the problem as: \[ 100 \times 36 = 100 \times (30 + 6) \] This can further be broken down as: \[ 100 \times 30 + 100 \times 6 \] Perform the individual multiplications: \[ 100 \times 30 = 3000 \] \[ 100 \times 6 = 600 \]

Step 4: Add the Products

Now, add the results from the previous step: \[ 3000 + 600 = 3600 \]

Step 5: Write the Final Answer

The result of multiplying 100 by 36 is 3600.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations. It involves finding the total of one number added to itself a certain number of times. Here, we multiply 100 by 36.
In general, multiplication can be denoted using the \( \times \) symbol or by writing one number adjacent to another with parentheses. So, \( 100 \times 36 \) or \( (100)(36) \) both mean the same thing.
Understanding multiplication is essential because it simplifies the addition of repeated numbers. Instead of adding 100 thirty-six times, you simply multiply to get the result much quicker. This is especially helpful for large numbers.

Distributive Property

The distributive property is a useful tool in multiplication. It states that \( a \times (b + c) = a \times b + a \times c \). This property helps to break down and simplify complex multiplication problems.
In our example, we used the distributive property to make multiplying 100 by 36 easier:
\[ 100 \times 36 = 100 \times (30 + 6) \]
Breaking it down further using the property, we get:
\[ 100 \times (30 + 6) = 100 \times 30 + 100 \times 6 \]
This approach converts one complex multiplication into two simpler ones:

• \( 100 \times 30 = 3000 \)
• \( 100 \times 6 = 600 \)

Finally, adding these products gives us the complete result, 3600. This makes understanding and performing multiplication easier, especially for larger numbers.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of mathematics.
In the given exercise, we perform a series of arithmetic operations to find the product of 100 and 36 using the distributive property.
The operations involved include:

• Identifying the numbers: Recognizing what numbers to multiply (100 and 36).
• Setting up the problem: Writing the multiplication in standard form \( 100 \times 36 \).
• Breaking down the multiplication: Using the distributive property to simplify.
• Performing the simpler multiplications:

• \( 100 \times 30 = 3000 \)
• \( 100 \times 6 = 600 \)

Adding the products: Summing up the simpler products to get the final result \[ 3000 + 600 = 3600 \].
These steps help ensure accuracy and make complex problems more manageable.