Problem 13 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 4.003 \\ \times 6.07 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

24.30821

Step 1: Ignore Decimal Points

First, multiply the numbers as if they were whole numbers: 4003 and 607. This is like ignoring the decimal points temporarily for easier calculation.

Step 2: Multiply the Whole Numbers

Compute the product of 4003 and 607. Multiply them just like you would with any regular whole numbers.

Step 3: Calculate the Product

Execute the multiplication: \[4003 \times 607 = 2,430,821.\]

Step 4: Restore Decimal Places

Now, consider the decimal places from the original problem. 4.003 has three decimal places, and 6.07 has two. Therefore, the actual product will need five decimal places. Count five places from the right of the product to place the decimal point.

Step 5: Format the Final Answer

Place the decimal and adjust the number: \[2,430,821 \rightarrow 24.30821.\] The final answer to the original problem, considering decimal places, is 24.30821.

Key Concepts

Understanding Place Value

When multiplying decimals, understanding place value is crucial. Place value helps us know the value of a digit in a number based on its position. It’s like knowing how much each digit contributes to the number as a whole. For example, in the number 4.003, the '4' is in the units place, the first '0' is in the tenths place, the second '0' is in the hundredths place, and '3' is in the thousandths place. In multiplication, knowing the place value aids in aligning the numbers correctly, especially when converting a decimal multiplication problem into one involving whole numbers temporarily. This understanding makes it easier to perform operations such as multiplication and eventually helps in placing the decimal point correctly in the product.

Product Calculation in Decimal Multiplication

When calculating the product of two decimal numbers, an important first step is to treat them like whole numbers. This means for the exercise given, remember to temporarily ignore the decimal points and just multiply the numbers as if there were no decimals in them.

• Here, 4.003 becomes 4003 and 6.07 becomes 607.
• Multiply these whole numbers as usual: \[4003 \times 607 = 2,430,821.\]

This approach simplifies the process because you're already familiar with multiplying whole numbers. Once you have this product, the next steps involve adjusting for the decimal points you initially ignored.

Decimal Places Adjustment for Accuracy

After finding the product of the whole numbers, the next crucial step is to adjust for the decimal places. This adjustment ensures that the final answer has the correct level of precision as required by the original decimal numbers.

• For example, 4.003 has three decimal places, and 6.07 has two. When you multiply these, count the total number of decimal places from both numbers, which is five in this case.
• Therefore, the result of our whole number multiplication, which was 2,430,821, needs a decimal point adjusted for five places.
• Counting five digits from the right in 2,430,821 gives us the final answer: 24.30821.

This step ensures the result accurately reflects the original numbers' magnitudes, maintaining mathematical integrity.