Problem 28 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 1.23 \\ \times 100,000 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product is 123,000.

Step 1: Understand the Problem

We need to find the product of 1.23 and 100,000. Multiplying by 100,000 will effectively shift the decimal point in 1.23.

Step 2: Analyze Decimal Shifting

When you multiply a number by 100,000, you shift the decimal to the right by 5 places because 100,000 has 5 zeros.

Step 3: Shift the Decimal

Start with the number 1.23. Shift the decimal point in the number 1.23 five places to the right: 1.23 becomes 123000.

Step 4: Determine the Final Product

After shifting the decimal place, the number becomes 123000. Thus, the product of 1.23 and 100,000 is 123000.

Key Concepts

Decimal Point Shifting

Multiplying decimals sometimes involves shifting the decimal point instead of performing more complex calculations. Think of this as moving the decimal in a number to a new spot, depending on what you are multiplying by.

For example, when you multiply a number by 10,000, you move the decimal four places to the right because 10,000 has four zeros. Similarly, multiplying by 100,000 involves moving the decimal point five places to the right due to the five zeros in 100,000.

Let's take the number 1.23 as our starting point. If you multiply by 100,000, you'll shift the decimal point five places to the right. Start counting the moves from the decimal in 1.23. At each place move, you're effectively increasing the value. After all these shifts, 1.23 becomes 123,000.

This method saves time and helps you avoid mistakes when dealing with large numbers.

Basic Arithmetic Operations

Understanding basic arithmetic operations is crucial for solving mathematical problems involving decimals. These operations include addition, subtraction, multiplication, and division. Multiplication, in particular, is the focus when deciding how to handle decimals.

When multiplying two numbers, you combine their values by performing repetitive addition. With whole numbers, you multiply as usual. However, with decimal multiplication, you need to first shift the decimal point accordingly, then treat the numbers as whole numbers. After the multiplication, you finalize the result by ensuring the decimal point is correctly placed based on the numbers you interacted with.

• Identify each number's decimal places.
• Shift to deal with the decimals effectively.
• Calculate as if working with whole numbers.
• Position the decimal point in the final result.

Applying these steps ensures you handle decimals accurately during multiplication.

Place Value

The concept of place value is vital in mathematics, particularly when working with decimals. It refers to the value of each digit in a number based on its position. This feature means that the placement of a digit in a number determines its actual value.

For example, in the number 1.23, the '1' is in the ones place, '2' is in the tenths place, and '3' is in the hundredths place. As you multiply this number by 100,000, understanding that each digit's place value shifts significantly helps clarify the result.

By effectively applying place value knowledge, you can determine how each number and operation affects a decimal. This comprehension ensures that after moving the decimal point and possibly adjusting a number's scale, you still correctly interpret the numerical value of your result.

• Recognize the value of each digit.
• Shift place values while maintaining accuracy.
• Use place value to adjust decimal multiplication results.

Mastering place value will help you better understand and solve mathematical problems involving decimals.