Problem 84 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$3(550+2)$$

Step-by-Step Solution

Answer

The result of the expression is 1656.

Step 1: Distribute the Three

To solve the expression using the distributive property, first distribute the 3 across the terms inside the parentheses. This means we will multiply 3 by each of the terms inside the parentheses separately. So, the expression becomes: \[3 \times (550 + 2) = 3 \times 550 + 3 \times 2\]

Step 2: Multiply by 550

Now, multiply the first term inside the parentheses by 3:\[3 \times 550 = 1650\] This is because multiplying by 3 means adding the number to itself two more times: 550 + 550 + 550.

Step 3: Multiply by 2

Next, multiply the second term inside the parentheses by 3:\[3 \times 2 = 6\] This is straightforward multiplication of a whole number.

Step 4: Add the Products

Add the results from Step 2 and Step 3 to find the final value:\[1650 + 6 = 1656\] So, the solution to the expression \(3(550+2)\) is 1656.

Key Concepts

Distributive Property

The distributive property is a fundamental idea in mathematics that helps us simplify expressions and perform calculations more easily. It involves distributing or spreading out a factor across terms in parentheses. This is useful when you want to multiply a number by a sum or difference. In our example, the distributive property allows us to break down a complex multiplication problem into simpler parts.

Here's how it works:

• Take the number outside the parentheses (in this case, 3) and multiply it by each term inside the parentheses (550 and 2).
• This process creates two separate multiplication problems (i.e., 3 × 550 and 3 × 2) which are usually easier to solve independently.

After distributing, simply combine the results. This makes solving multiplication problems more straightforward and helps prevent errors. Just remember to multiply each term inside the parentheses separately before adding the products together.

Multiplying Whole Numbers

Multiplying whole numbers is one of the basic arithmetic operations that forms the foundation of more advanced math concepts. It's essentially repeated addition, where one number is added to itself a specified number of times.

In our example, when we multiply 3 by 550, we're really adding 550 three times (550 + 550 + 550), resulting in 1650. Similarly, multiplying 3 by 2 involves adding 2 to itself three times, which equals 6.

Here's a quick tip for multiplying large numbers:

• Break down large numbers into easier chunks, using place value. For example, you could break 550 into 500 and 50 and multiply these separately before adding the results.
• This helps you manage the calculation step-by-step, making it easier to follow and reducing the chance of mistakes.

Understanding how to multiply whole numbers is crucial because it lays the groundwork for tackling more complex problems, like those involving decimals and fractions.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are the building blocks of all mathematics and are used in various aspects of daily life and advanced studies.

When we look at a problem like 3(550 + 2), we are combining addition and multiplication:

• Addition helps us combine numbers together, like totaling the products from each part of the distributive property calculation (1650 from 3 × 550 and 6 from 3 × 2), giving us 1656.
• Multiplication, on the other hand, is efficient repeated addition, allowing us to quickly find sums like 550 three times.

Mastering these basic operations enables you to solve more complex problems with confidence, as they serve as the backbone for all math concepts. By making these processes second nature, you improve both your speed and accuracy in solving various types of math problems.