Problem 3 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 0.07 \\ \times 0.4 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product is 0.028.

Step 1: Remove Decimals

Convert the decimal numbers to whole numbers for easier multiplication. For 0.07, multiply by 100 to get 7. For 0.4, multiply by 10 to get 4.

Step 2: Multiply the Whole Numbers

Multiply the whole numbers obtained in the previous step. So, multiply 7 by 4. The product is 28.

Step 3: Adjust for Decimal Placement

Count the total number of decimal places in the original numbers. 0.07 has 2 decimal places, and 0.4 has 1 decimal place, so there are a total of 3 decimal places. Place the decimal point in the product (28) to reflect 3 decimal places, resulting in 0.028.

Key Concepts

Understanding Place Value in Decimals

Decimals are an extension of our base-10 number system. Each digit in a decimal number represents a fraction based on its place value. Place value for decimals starts immediately after the decimal point, going from tenths, to hundredths, to thousandths, and so on. This means in the decimal 0.07, the 7 is in the hundredths place, signifying that its value is 7 hundredths, or 0.07.

Understanding place value is crucial because it helps us convert decimals into whole numbers by scaling them up for easier multiplication. For example:

• In 0.07, moving the decimal two places to the right gives us the whole number 7.
• In 0.4, moving the decimal one place to the right gives the whole number 4.

Grasping this concept makes it easier to perform calculations that maintain accuracy by appropriately relocating the decimal point later.

The Multiplication Process with Decimals

When multiplying decimals, it's often helpful to first treat them as whole numbers, which simplifies the calculation. Here's how we can do that:

• Convert both decimal numbers to whole numbers as described above (moving the decimal places accordingly).
• Next, multiply these whole numbers. In our exercise, multiplying 7 by 4 gives 28.

This process mirrors how we handle multiplication with whole numbers we're familiar with, but by temporarily ignoring the decimal points, we make calculations straightforward and less prone to error. After obtaining the product, the next important step is adjusting the decimal point correctly, which ensures that our answer is accurate and meaningful.

Accurately Adjusting Decimals in the Product

Once the multiplication is done, it's time to adjust our result to account for the initial decimal places. This involves carefully placing the decimal point back into our product. Here's how you do it:

• Count all the decimal places present in the numbers you initially multiplied. In this scenario, 0.07 has 2 decimal places while 0.4 has 1 decimal place.
• Add these together to understand how many positions the decimal point needs to move in the final result. Adding 2 and 1, we find there are a total of 3 decimal places needed.
• Place the decimal point in the combined product, 28, resulting in 0.028.

This ensures that the answer is correctly scaled, maintaining its value in relation to the original numbers. Adjusting the decimal correctly is as important as the initial multiplication since it guarantees the calculation's integrity and precision.