Problem 286 - Prealgebra

Question

In the following exercises, multiply. $$485(602)$$

Step-by-Step Solution

Answer

291970

Step 1 - Write the Multiplication Problem

Given the multiplication problem, write it down: $$485 \times 602$$

Step 2 - Distribute the First Number

Distribute the number 485 to each digit of 602. This can be visualized as: $$485 \times (600 + 2)$$

Step 3 - Multiply Each Term Separately

Now, multiply each part of the distributed equation separately. First, multiply 485 by 600: $$485 \times 600 = 291000$$ Then, multiply 485 by 2: $$485 \times 2 = 970$$

Step 4 - Add the Products Together

Add the two products obtained in the previous step: $$291000 + 970$$ Perform the addition to get the final result: $$291970$$

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that helps break down multiplication problems into simpler parts. When you have an expression like the one in the exercise, 485 and 602, it can be challenging to multiply these large numbers directly. The distributive property allows you to rewrite the problem in a way that makes the multiplication more manageable. For example, you can express 602 as a sum of 600 and 2. This way, the multiplication problem can be transformed into 485 times (600 + 2). Now, instead of multiplying 485 by 602 in one step, you perform two simpler multiplications: 485 by 600 and 485 by 2. The distributive property states that:

• a(b + c) = ab + ac

This means you distribute a to both b and c, then add the products. This method is particularly helpful for students learning multiplication as it makes the process more intuitive and less prone to errors. You apply the distributive property to decompose tough problems into easy-to-manage chunks!

Multiplicative Operations

Multiplicative operations in math involve the process of repeated addition. That means when you multiply two numbers, you are essentially adding one number to itself a certain number of times. In the given exercise, the task is to multiply 485 by 602. Breaking down this big task using simpler multiplicative operations, you can first understand what multiplication by a whole number entails. The basic steps are:

• Break down the larger number using the distributive property.
• Multiply each term separately.

To multiply 485 by 600, you can think of it as multiplying 485 by 6 and adding two zeros at the end, because 600 is 6 times 100. Similarly, multiplying 485 by 2 is straightforward. By breaking it down—first dealing with 600, and then with 2—you can see how much easier the task becomes. This drill makes the multiplication of large numbers more accessible and builds a strong foundation for tackling even more complex problems.

Step-by-Step Solutions

Using a step-by-step approach is key for understanding and solving complex multiplication problems. Here, we'll break down the multiplication of 485 by 602 in detailed steps:
Step 1: Write down the multiplication problem:

• 485 × 602

Step 2: Distribute the first number, 485, to each digit of 602:

• Rewrite 602 as (600 + 2).
• 485 × (600 + 2)

Step 3: Multiply each term separately:

• First, multiply 485 by 600. Visualize this as 485 × 6 = 2910, then add two zeros: 2910 × 100 = 291000.
• Next, multiply 485 by 2: 485 × 2 = 970.

Step 4: Add the products together:

• 291000 + 970 = 291970

This step-by-step solution ensures that you don't miss out on any crucial detail and helps you keep track of your progress. By following each step carefully, you arrive at the correct answer: 291970. Practice this method, and you'll become more confident in handling even the most challenging multiplication tasks!