Question

Find each product without using a calculator \(1,000 \times 98.9\)

Short answer

98,900

Step-by-step solution

Express 98.9 as a Sum

Rewrite the number 98.9 as the sum of two numbers that are easier to work with. For example, 98.9 can be expressed as 100 - 1.1.

Use the Distributive Property

Apply the distributive property to simplify the multiplication: \(1,000 \times 98.9 = 1,000 \times (100 - 1.1)\).

Multiply Each Term Separately

Multiply 1,000 by each term inside the parentheses separately: \(1,000 \times 100 = 100,000\) and \(1,000 \times 1.1 = 1,100\).

Subtract the Results

Subtract the second result from the first result: \(100,000 - 1,100 = 98,900\).

Key concepts

Multiplication Techniques

Multiplication can sometimes seem tricky, especially with large numbers or decimal points. There are several techniques to make it easier. One useful method is breaking numbers into smaller parts. For instance, instead of directly multiplying 1,000 by 98.9, we can break down 98.9 into more manageable parts: 100 and -1.1. This simplification can make the multiplication steps more straightforward.
Another technique is using the distributive property, which allows us to multiply each part of the sum separately. This makes our calculations simpler and often reduces the potential for errors. Practicing these techniques can help you become more comfortable with complex multiplication problems.

Simplifying Expressions

Simplifying expressions is a crucial part of solving math problems efficiently. When you simplify an expression, you reduce it to its most basic form, making calculations easier. Using the distributive property is a key strategy in this process.
For example, in the exercise, we simplify the expression \(1,000 \times (100 - 1.1)\) instead of the original complex multiplication. By breaking down the problem, you turn a potentially difficult calculation into a series of simpler steps.
Remember to apply the properties of arithmetic, such as the associative and commutative properties, to combine like terms and simplify calculations further. With practice, you'll become more adept at quickly spotting ways to simplify problems.

Arithmetic Operations

Arithmetic operations are the foundation of all math problems. The four basic operations are addition, subtraction, multiplication, and division. Mastering these allows you to tackle more complex problems. In our exercise, we primarily focus on multiplication and subtraction.
Understanding how to multiply large numbers by breaking them into smaller parts, as shown, makes the calculations manageable. Similarly, subtraction is used in the final step of the solution where 1,100 is subtracted from 100,000.
Practicing these basic operations helps build confidence and proficiency, laying the groundwork for more advanced math topics. Always remember, breaking down complex problems into simpler steps is a powerful strategy in math.