Understanding the Product of Decimals

Finding the product of two decimal numbers, such as 0.24 and 0.008, involves multiplying them and correctly placing the decimal point in the result. This process combines standard multiplication with a rule for decimal point placement based on the original numbers.

Step-by-Step Decimal Multiplication: 0.24 × 0.008

Here’s how to calculate the product of 0.24 and 0.008 in detail:

1. Multiply the Numbers as Whole Numbers: First, ignore the decimal points and multiply the numbers as if they were whole numbers. We multiply 24 by 8.

   		2	4
   ×			8
   	1	9	2

   
   The product of $24 \times 8$ is $192$.
2. Count the Total Number of Decimal Places: Next, count how many digits are after the decimal point in each of the original numbers (the factors).
   • In the number 0.24, there are 2 digits after the decimal point (the 2 and the 4).
   • In the number 0.008, there are 3 digits after the decimal point (the first 0, the second 0, and the 8).
   Now, add the number of decimal places from each factor to find the total number of decimal places the final product should have.

   Total decimal places = (Decimal places in 0.24) + (Decimal places in 0.008)

   Total decimal places = $2 + 3 = 5$

3. Place the Decimal Point in the Product: The product we got from multiplying 24 and 8 was 192. We need to place the decimal point in 192 so that there are a total of 5 digits after the decimal point. Starting from the rightmost digit of 192, move the decimal point 5 places to the left.

   The number is 192. We need to move the decimal 5 places left:

   192. becomes 19.2 (1 place)

   19.2 becomes 1.92 (2 places)

   1.92 becomes 0.192 (3 places)

   To move 4 places, we add a zero: 0.192 becomes 0.0192

   To move 5 places, we add another zero: 0.0192 becomes 0.00192

   Alternatively, you can think of having 192 and needing 5 decimal places. You add leading zeros until you can place the decimal:

   _ _ 1 9 2

   . _ _ 1 9 2 (after placing decimal 5 places left)

   0 . 0 0 1 9 2 (filling with zeros)

Result of the Product 0.24 × 0.008

Following these steps, the product of 0.24 and 0.008 is 0.00192.

$\qquad 0.24 \times 0.008 = 0.00192$

Revision Table: Key Rules for Multiplying Decimals

Additional Information: Relating Decimals to Fractions

Understanding that decimals are just another way to write fractions can also help. For example:

• $0.24 = \frac{24}{100}$
• $0.008 = \frac{8}{1000}$

Multiplying these fractions gives:

$\qquad 0.24 \times 0.008 = \frac{24}{100} \times \frac{8}{1000}$

$\qquad = \frac{24 \times 8}{100 \times 1000}$

$\qquad = \frac{192}{100000}$

To convert the fraction $\frac{192}{100000}$ back to a decimal, we divide 192 by 100,000. Dividing by 100,000 (which has 5 zeros) means moving the decimal point in 192 (which is 192.0) 5 places to the left.

$192.0 \div 100,000 = 0.00192$

This fractional approach confirms the result obtained by counting decimal places and can provide a deeper understanding of why the decimal placement rule works for decimal multiplication.