Problem 60 - Algebra 1: Concepts and Skills

Question

Find the product. \(0.7 \times 0.8\)

Step-by-Step Solution

Answer

The product of 0.7 and 0.8 is 0.56.

Step 1: Setup the Multiplication

Start by setting up the multiplication. Ensure that 0.7 is setup vertically and 0.8 horizontally: \n0.7\nx 0.8

Step 2: Ignore the Decimal Points and Multiply as Whole Numbers

For now, ignore the decimal points and multiply 7 and 8 like you would multiply whole numbers. Doing so gives you 56. Therefore, based on this calculation, you'll write down 56 below the line.

Step 3: Place the Decimal Point

Finally, count how many numbers there are after the decimal point in the original problem. In this case, the number is 2. Hence, place the decimal point before the last two digits of the answer 56 to get 0.56.

Key Concepts

Understanding the Decimal Point in Multiplication

Decimals are a way of representing fractions without using a fraction bar. In decimal numbers, the position of the decimal point is crucial because it determines the value of the digits in the number.
When multiplying decimals, it is important to first set up the multiplication without worrying about the decimal point position. This simplifies the process, making it more like multiplying whole numbers. Once the multiplication is complete, the position of the decimal point is determined based on the total number of decimal places in both numbers being multiplied.
To clarify:

• Each digit to the right of the decimal point represents a fraction of ten, like tenths, hundredths, etc.
• To find the correct position for the decimal in the final product, count the total number of decimal places from both numbers involved in the multiplication.
• Place the decimal point in the result so that it has the same number of decimal places.

Simplifying Multiplication by Treating Decimals as Whole Numbers

Multiplying decimal numbers can initially seem complicated, but simplifying the process by treating them as whole numbers makes it easier. The key is to temporarily ignore the decimal points.
Here's how it works:

• Remove the decimal points from both numbers and treat them as whole numbers.
• Perform the multiplication exactly as you would with whole numbers.
• Write down the result of this multiplication, remembering it for the next step.

This simplification allows you to focus on the core multiplication before dealing with decimal placements. By breaking the problem down, it becomes less intimidating and much more manageable.

The Role of Whole Numbers in Decimal Multiplication

Whole numbers form the foundation for performing operations like multiplication. Even when working with decimals, the core multiplication process relies on understanding how to multiply whole numbers.
In the case of decimal multiplication, once you've set aside the decimal point, what remains is a whole number multiplication problem.
Consider the example from the original exercise:

• Both 0.7 and 0.8 are treated as 7 and 8, respectively.
• Multiply these whole numbers which gives you 56.

After determining this product, the decimal point placement is decided based on previous calculations regarding decimal places. The familiarity with whole number multiplication greatly aids in performing accurate decimal multiplication. Thus, reinforcing a strong grasp of whole number operations is invaluable for understanding this concept.