Problem 2 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 0.8 \\ \times 0.3 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product is 0.24.

Step 1: Multiply the Numbers

First, ignore the decimal points and multiply 8 by 3. Calculate: \[8 \times 3 = 24\]

Step 2: Count Decimal Places

Determine the number of decimal places in each factor. Both 0.8 and 0.3 have 1 decimal place each, which totals to 2 decimal places in the product.

Step 3: Place the Decimal Point

Since the total number of decimal places should be 2, start from the right of 24 and count two places to place the decimal. You get 0.24.

Key Concepts

Multiplying Decimals

Multiplying decimals might seem challenging, but it's a straightforward process once you ignore the decimal points initially. Here's how you do it:

• First, treat the decimals as whole numbers, which makes the multiplication easier and reduces errors.
• Ignore the decimals temporarily and perform the multiplication as you usually would. In our exercise, multiply 8 by 3 to get 24.
• This simplification step helps you focus on the basic multiplication without worrying about where the decimal point will eventually go.

Once you have the product of these whole numbers, you need to find out where to place the decimal point in the final product. This is crucial for getting the correct answer. This step will be guided by the decimal places in the original numbers you multiplied.

Decimal Places

Decimal places play an essential role in accurately determining the position of the decimal point in the product when multiplying decimals. Each decimal number comprises a decimal point followed by digits, each of which is a decimal place.

• For example, 0.8 has one decimal place, and 0.3 also has one.
• To determine how many decimal places should be in the final answer, sum the decimal places of the numbers being multiplied.

In the exercise, we added the decimal places of 0.8 and 0.3, which equals two decimal places. Understanding and counting decimal places correctly ensures that the decimal point is placed in the exact position needed for the precise result.

Place Value

Place value is pivotal when dealing with decimals because it determines the value each digit holds in a number.

• Each digit to the right of a decimal point is worth a fraction of ten, such as tenths, hundredths, thousandths, and so on.
• For instance, in 0.8, the 8 is in the tenths place, making it 8/10 or 0.8.
• Understanding place value helps you interpret and assign value to digits in context, especially when shifting the decimal point after multiplication.

When you calculate the product—a count of two decimal places means counting toward the left two digits to correctly place the decimal point so that the product's value is accurately presented.

Step-by-Step Solution

Solving a multiplication problem involving decimals can be broken down into clear, achievable steps: - **Multiply as Whole Numbers**: Start by ignoring the decimal points. Multiply the numbers as if they were whole. For example, with 0.8 and 0.3, multiply 8 by 3 to get 24. - **Count Decimal Places**: Check each decimal number for the number of digits after the decimal. Add these together; 0.8 has one, and 0.3 has one, totaling two. - **Place the Decimal Point**: Use the total from the previous step to position the decimal point in the product. From the exercise, 24 becomes 0.24 as you count two places to the left. Following these steps ensures accuracy and provides a systematic approach to multiplying decimals. Mastery of this process can significantly improve your confidence and capability in handling more complex decimal multiplications.