Problem 76 - Prealgebra

Question

Perform the indicated operations. $$3.5 \times 1,000$$

Step-by-Step Solution

Answer

The product of multiplying 3.5 by 1,000 is 3,500.

Step 1: Understand the Multiplication

In this exercise, you need to multiply the numbers 3.5 and 1,000 to find the product. This means you will find how much 3.5 groups of 1,000 is in total.

Step 2: Apply Basic Multiplication Rules

Multiply 3.5 by 1,000. When multiplying a decimal by a power of 10, you can move the decimal point to the right by the number of zeros in the power of 10. Since 1,000 has 3 zeros, move the decimal in 3.5 three places to the right.

Step 3: Perform the Calculation

Start with 3.5. Move the decimal point three places to the right. After moving, since there's only one digit after the decimal point and we need to move three, add 2 zeros at the end of 35 to accommodate the move. This gives us 3,500.

Key Concepts

Understanding Decimal Point Movement

When multiplying decimal numbers, especially by powers of ten, understanding how the decimal point moves is crucial. In the exercise, we multiplied 3.5 by 1,000. Here, the decimal point movement plays an essential role in simplifying the process. To move the decimal point, look at the number of zeros in the factor–since 1,000 has three zeros, move the decimal point in 3.5 three places to the right.

• Start with 3.5, which is written as 3.5.
• Moving it one place changes it to 35 (3.5 becomes 35.0).
• Continue moving it two more places to get 3,500.

Always remember to fill any empty decimal places with zeros if necessary. This ensures that the number retains its magnitude as required by its multiplication.

Grasping Powers of Ten

Powers of ten are fundamental in performing efficient multiplication and division with decimals. Each power represents how many times to multiply 10 by itself, like 10, 100, 1,000, and so on. In our example, multiplying by 1,000 means we use 10 raised to the power of 3. Here’s how it works:

• Power of zero (10⁰) is 1.
• Power of one (10¹) is 10.
• Power of two (10²) is 100.
• Power of three (10³) is 1,000.

The exponents indicate how many zeros are added when multiplied by 10. Therefore, 1,000 signifies that 3 zeros will be added when adjusting the decimal point, which transforms 3.5 into 3,500 effortlessly.

Mastering Basic Multiplication Rules with Decimals

Basic multiplication rules apply in decimal multiplication similarly to whole numbers, but with a focus on handling the decimal point. Follow these steps to ensure accuracy:

• Ignore the decimal point initially and multiply the numbers as if they were whole numbers.
• After finding the product, correctly place the decimal point. In our exercise, shift it based on the powers of ten rule.

For example, with 3.5 x 1,000:

• Consider 35 and 1,000 as the numbers.
• 35 x 1,000 equals 35,000.
• Adjust for decimals: Moving the point three places right leads to the product 3,500.

By sticking to these steps, you'll effectively handle decimal multiplications without error.