Understanding Decimal Multiplication

The question asks us to find the product of 1.23 and 0.356, given that the product of their whole number counterparts, 123 and 356, is 43788.

This problem demonstrates a fundamental concept in decimal multiplication: the number of decimal places in the product is the sum of the number of decimal places in the numbers being multiplied.

Relating Whole Number and Decimal Products

We are given:

\(123 \times 356 = 43788\)

We need to find:

\(1.23 \times 0.356 = ?\)

Let's look at the relationship between the numbers:

• The number 1.23 is obtained by dividing 123 by 100 (moving the decimal point 2 places to the left). So, \(1.23 = \frac{123}{100}\).
• The number 0.356 is obtained by dividing 356 by 1000 (moving the decimal point 3 places to the left). So, \(0.356 = \frac{356}{1000}\).

Calculating the Decimal Product

Now, let's substitute these into the multiplication we need to perform:

\(1.23 \times 0.356 = \left(\frac{123}{100}\right) \times \left(\frac{356}{1000}\right)\)

\(1.23 \times 0.356 = \frac{123 \times 356}{100 \times 1000}\)

We already know that \(123 \times 356 = 43788\). So, we can substitute this value:

\(1.23 \times 0.356 = \frac{43788}{100000}\)

Dividing by 100000 means moving the decimal point 5 places to the left in the number 43788.

Starting with 43788, the decimal point is initially after the last digit (43788.0). Moving it 5 places to the left:

43788. → 4378.8 (1 place)

4378.8 → 437.88 (2 places)

437.88 → 43.788 (3 places)

43.788 → 4.3788 (4 places)

4.3788 → 0.43788 (5 places)

So, \(1.23 \times 0.356 = 0.43788\).

Counting Decimal Places Method

Alternatively, we can use the method of counting decimal places:

1. Multiply the numbers as if they were whole numbers: \(123 \times 356 = 43788\).
2. Count the total number of decimal places in the original numbers:
   • 1.23 has 2 decimal places.
   • 0.356 has 3 decimal places.
3. Add the number of decimal places: \(2 + 3 = 5\) decimal places.
4. Place the decimal point in the whole number product (43788) so that there are 5 decimal places from the right.

Starting with 43788, count 5 places from the right and place the decimal point:

43788 → 4378.8 (1)

4378.8 → 437.88 (2)

437.88 → 43.788 (3)

43.788 → 4.3788 (4)

4.3788 → 0.43788 (5)

The result is 0.43788.

Final Answer

Based on the calculation and the rule of decimal place counting, \(1.23 \times 0.356 = 0.43788\).

Revision Table: Decimal Multiplication Concept

Additional Information: Importance of Place Value

Understanding place value is crucial when working with decimals. Each digit's value depends on its position relative to the decimal point. For example, in 1.23, the '1' is in the ones place, the '2' is in the tenths place (\(0.2\)), and the '3' is in the hundredths place (\(0.03\)). Similarly, in 0.356, the '3' is in the tenths place (\(0.3\)), the '5' is in the hundredths place (\(0.05\)), and the '6' is in the thousandths place (\(0.006\)).

When you multiply \(1.23 \times 0.356\), you are essentially calculating \((1 + 0.2 + 0.03) \times (0.3 + 0.05 + 0.006)\). While distributing this would be complex, the rule of counting decimal places provides a shortcut based on the powers of 10 involved (\(10^{-1}\), \(10^{-2}\), \(10^{-3}\), etc.).

The product \(10^{-a} \times 10^{-b} = 10^{-(a+b)}\). This is why we add the number of decimal places. In 1.23, the smallest place value is hundredths (\(10^{-2}\)). In 0.356, the smallest place value is thousandths (\(10^{-3}\)). The smallest place value in the product will be hundred-thousandths (\(10^{-2} \times 10^{-3} = 10^{-5}\)), which corresponds to 5 decimal places.