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Chapter 5: Problem 35

Multiply. Simplify, if possible. $$ 25 \times 462 $$

Short Answer

Expert verified
11550

Step by step solution

01

Write the problem as it is

Start with the multiplication problem: \resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times 462\)}}
02

Break down the multiplication

Separate the numbers into more manageable parts. Use the distributive property: \resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times (400 + 60 + 2)\)}}
03

Apply the distributive property

Multiply 25 by each part separately and then sum the results: \resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times 400 + 25 \times 60 + 25 \times 2\)}}
04

Perform each multiplication

Calculate each multiplication:\resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times 400 = 10000\)}}\resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times 60 = 1500\)}}\resizebox{!}{40px}\text{\resizebox{!}{40px}{\(25 \times 2 = 50\)}}
05

Add the results together

Sum the individual products to get the final result:\resizebox{!}{40px}\text{\resizebox{!}{40px}{\(10000 + 1500 + 50 = 11550\)}}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is a handy tool in mathematics. It allows you to simplify complex multiplication by breaking it down into smaller, easier pieces. The property states that multiplying a number by a sum is the same as doing each multiplication separately. For example, in our problem, we transform \(25 \times 462\) into \(25 \times (400 + 60 + 2)\). This works because you are distributing the 25 across each part within the parentheses.
breaking down numbers
Breaking down numbers, also known as decomposition, is an essential technique to simplify calculations. Here, we break 462 into 400, 60, and 2. This simplification makes the multiplication more manageable. By dealing with smaller numbers, it's easier to perform quick mental calculations or ensure accuracy when using pen and paper. It's like cutting a big task into smaller, digestible pieces, making the overall task less daunting.
step-by-step calculation
Step-by-step calculation involves solving a problem in clear, sequential stages. This approach ensures that you understand and correctly perform each part of the process. We start with the initial problem: \(25 \times 462\). First, we express it as \(25 \times (400 + 60 + 2)\) using the distributive property. Next, we perform the multiplication individually: \(25 \times 400\), \(25 \times 60\), and \(25 \times 2\). Finally, we sum up the results: \(10000 + 1500 + 50 = 11550\). Breaking the problem into these small steps helps avoid mistakes and makes complex problems easier to handle.

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