Simplifying Decimal Multiplication

This question asks us to simplify the multiplication of two decimal numbers: \(0.004\) and \(0.5\). Multiplying decimals involves two main steps: multiplying the numbers as if they were whole numbers and then correctly placing the decimal point in the final answer.

Step-by-Step Solution for 0.004 × 0.5

Let's break down the multiplication:

1. Multiply the numbers ignoring the decimal points: Treat \(0.004\) as \(4\) and \(0.5\) as \(5\). \[4 \times 5 = 20\]
2. Count the total number of decimal places in the original numbers: The number \(0.004\) has three digits after the decimal point (0, 0, 4). The number \(0.5\) has one digit after the decimal point (5). Total number of decimal places = \(3 + 1 = 4\).
3. Place the decimal point in the result: The product of \(4\) and \(5\) is \(20\). We need to place the decimal point so that there are a total of 4 decimal places in the final answer. Starting from the right end of \(20\), we move the decimal point 4 places to the left.
   • Start with \(20\).
   • Move 1 place left: \(2.0\)
   • Move 2 places left: \(0.20\)
   • Move 3 places left: \(0.020\)
   • Move 4 places left: \(0.0020\)
   The resulting number is \(0.0020\), which is the same as \(0.002\).

So, the simplified result of \(0.004 \times 0.5\) is \(0.002\).

Understanding Decimal Multiplication

Multiplying decimals is essentially multiplying fractions involving powers of 10. For example:

So, \(0.004 \times 0.5 = \frac{4}{1000} \times \frac{5}{10} = \frac{4 \times 5}{1000 \times 10} = \frac{20}{10000}\)

To convert \(\frac{20}{10000}\) back to a decimal, we divide \(20\) by \(10000\). Dividing by \(10000\) means moving the decimal point 4 places to the left.

This confirms the result obtained by counting decimal places.

Revision Table: Key Points in Decimal Multiplication

• Multiply numbers as whole numbers first.
• Count total decimal places in the numbers being multiplied.
• Place decimal point in the product from the right, matching the total count.
• Trailing zeros after the last non-zero digit right of the decimal can be dropped (e.g., 0.0020 = 0.002).

Additional Information: Related Decimal Operations

Understanding other decimal operations is also crucial for quantitative aptitude problems:

• Decimal Addition and Subtraction: Align the decimal points vertically before adding or subtracting.
• Decimal Division: If the divisor is a decimal, move the decimal point in both the divisor and dividend to make the divisor a whole number. Then perform standard division.
• Multiplying by Powers of 10: Move the decimal point to the right by the number of zeros in the power of 10 (e.g., \(0.004 \times 100 = 0.4\)).
• Dividing by Powers of 10: Move the decimal point to the left by the number of zeros in the power of 10 (e.g., \(0.5 \div 10 = 0.05\)).