Problem 5 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 0.03 \\ \times 0.09 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product of 0.03 and 0.09 is 0.0027.

Step 1: Understanding the Problem

We are asked to find the product of two numbers: 0.03 and 0.09. This involves multiplying these two decimal numbers.

Step 2: Set Up the Multiplication

Ignore the decimal points in 0.03 and 0.09 for a moment. Treat them as whole numbers, so we need to multiply 3 by 9.

Step 3: Multiply Whole Numbers

Calculate the product of 3 and 9. We have:\[3 \times 9 = 27\]

Step 4: Account for Decimal Places

Determine the total number of decimal places in the original numbers. The number 0.03 has 2 decimal places, and 0.09 has 2 decimal places, making a total of 4 decimal places.

Step 5: Adjust the Product with Decimal Places

Place the decimal point in the product (27) so that it has 4 decimal places. That leads to 0.0027 as the product of 0.03 and 0.09.

Key Concepts

Understanding Place Value

Place value is crucial in understanding how decimal numbers work. Each digit in a decimal number has a specific 'place' based on a power of ten. For instance, in the number 0.03:

• The digit '0' is in the tenths place.
• The digit '3' is in the hundredths place, which represents \(3 \times \frac{1}{100}\).

In multiplication, these place values impact the final result. By knowing the position of each digit, you can determine how significant it is in terms of tens, hundreds, and so on. This understanding helps in maintaining accuracy when aligning numbers during multiplication.

Decimal Places Matter

Decimal places indicate how many digits are to the right of the decimal point. For example, 0.03 has two decimal places. When multiplying two decimal numbers, you sum the number of decimal places. Here's why:

• 0.03 has two decimal places.
• 0.09 has two as well.
• These add up to a total of four decimal places.

This total is essential for positioning the decimal point in the final product. A mistake in counting decimal places can lead to a huge error in your answer.

Mastering Multiplication Steps

Multiplying decimal numbers might seem tricky, but following clear steps makes it easier. First, disregard the decimals and consider the numbers as whole. So, for 0.03 and 0.09:

• Multiply 3 by 9, giving you 27.

Next, apply the total decimal places calculated earlier. Since there are four decimal places, convert 27 into 0.0027. The steps: 1. Set aside decimal points. 2. Multiply the numbers as whole numbers. 3. Reintroduce the decimal, ensuring the proper place based on the total decimal places counted at the beginning. This ensures accuracy and avoids confusion when dealing with decimals.