Problem 268 - Prealgebra

Question

In the following exercises, multiply. $$53 \times 98$$

Step-by-Step Solution

Answer

53 × 98 = 5194

Step 1: Write the problem

The given exercise is to multiply 53 and 98. It can be written as: 53 × 98

Step 2: Break down the multiplication

To make the multiplication easier, break down 98 into (100 - 2) since multiplying by 100 and then subtracting from it is easier.So, rewrite the equation as: 53 × (100 - 2)

Step 3: Distribute the multiplication

Distribute the multiplication over the subtraction. This means multiply 53 by 100 and then subtract 53 multiplied by 2.So, it becomes:53 × 100 - 53 × 2

Step 4: Perform individual multiplications

Now, calculate the individual multiplications:53 × 100 = 530053 × 2 = 106

Step 5: Subtract results

Subtract the second result from the first one:5300 - 106 = 5194

Key Concepts

Distributive Property

The distributive property is a powerful tool in multiplication. It allows us to break down complex problems into simpler parts. The property states that for any numbers a, b, and c, \(a \times (b + c) = a \times b + a \times c\). Similarly, it works for subtraction: \(a \times (b - c) = a \times b - a \times c\).
In our exercise, we used the distributive property to make multiplying 53 by 98 simpler.
Instead of directly multiplying 53 and 98, we split 98 into more manageable parts (100 - 2). This made the whole process easier by allowing us to work with simpler numbers first.
When you're faced with tough multiplication problems, remember the distributive property. It can transform a difficult task into a series of easier steps.

Breaking Down Numbers

Breaking down numbers is a technique that makes complex calculations more manageable.
In the original exercise, we broke down 98 into 100 - 2. This made it easier to manage because multiplying by 100 and subtracting from it is simpler than directly tackling 98.
Here's how it works:

• Identify a nearby rounded number (like 100 in our example).
• Express the original number as a simple equation involving that round number (98 becomes 100 - 2).

This method simplifies the calculation into smaller, easier steps.
Other numbers can be broken down similarly. For instance, 47 can be split into 50 - 3, or 125 can be thought of as 100 + 25.
The overall idea is to use numbers that are easier to multiply or work with, making the entire problem simpler.

Step-by-Step Problem Solving

Step-by-step problem solving is crucial for understanding and solving math problems.
Let's recap the steps we took for our multiplication problem, where we calculated 53 × 98.

• Step 1: Write down the initial problem: 53 × 98.
• Step 2: Break down the more challenging number (98) into parts: 100 - 2.
• Step 3: Use the distributive property to split the multiplication into two simpler parts: 53 × (100 - 2) becomes 53 × 100 - 53 × 2.
• Step 4: Perform the simpler multiplications independently: 53 × 100 = 5300 and 53 × 2 = 106.
• Step 5: Subtract the results of the simple multiplications: 5300 - 106 = 5194.

By breaking down the problem and solving it step by step, we ensure that each part is understandable and manageable.
This systematic approach is not just useful for multiplication; it can help with a variety of math problems.
Always break the problem into smaller steps, understand each one, and then put it all together.