Problem 11 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 3.12 \\ \times 0.005 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product is 0.0156.

Step 1: Remove Decimals and Multiply

First, ignore the decimals in the numbers and multiply 312 by 5. This simplifies the numbers and makes the multiplication straightforward. Calculate \(312 \times 5 = 1560\).

Step 2: Count Decimal Places

Note the decimal places in the original numbers. For 3.12, there are two decimal places. For 0.005, there are three decimal places. In total, both numbers sum up to five decimal places.

Step 3: Apply the Decimal Places

Reintroduce the decimal point into the product (1560) by moving the decimal point five places to the left to account for the five decimal places. This results in 0.01560.

Step 4: Simplify the Final Answer

Remove any trailing zeros after the decimal point in 0.01560 to simplify the answer to 0.0156.

Key Concepts

Understanding Decimal Places

When working with decimal numbers, understanding 'decimal places' is crucial. Decimal places refer to the number of digits to the right of the decimal point. These digits represent fractions of a whole. For every step to the right, the number represents a smaller fraction:

• The first digit to the right of the decimal point is the tenths place.
• The second digit is the hundredths place.
• The third digit is the thousandths place, and so on.

In our example, 3.12 has two decimal places, as it extends to the hundredths. Similarly, 0.005 extends to the thousandths, with three decimal places.
Thus, adding both amounts, we find that there are a total of five decimal places in the multiplication problem.

Executing Product Calculation

Calculating the product of two decimals involves a few straightforward steps, especially if you focus on the integers first. Here's how you can simplify this operation:
Start by ignoring the decimal points in both numbers. This step is crucial, as it avoids complications in the initial multiplication.

• For example, multiply 312 by 5, treating them as whole numbers.
• The result, 1560, reflects the calculation without decimals interfering.

Once you have this integer product, the next step is reintroducing the decimal places. This reconciling of the original fractional nature of the numbers gives meaning to the calculation's result.

Integrating Mathematical Operations

Combining the mathematical operations of multiplication and the placement of decimals requires careful attention. After multiplying the whole numbers (like 312 and 5), it's time to bring back the decimals:

• Add the number of decimal places from both factors. For 3.12 and 0.005, this count totals to five decimal places.
• Move the decimal point five places to the left in your product (from 1560 to 0.01560).

Finally, crucial factors to remember include making your final answer as simplified as possible. Removing any unnecessary zeros at the end clarifies your final answer to 0.0156.
This refined result reflects not only the correct numerical value but also the neatness and practicality of well-executed mathematical operations.