Problem 21 - Introductory Algebra for College Students

Question

perform the indicated multiplication. $$-0.2(-0.6)$$

Step-by-Step Solution

Answer

The result of the multiplication \(-0.2(-0.6)\) is \(0.12\).

Step 1: Understand the rule of signs in multiplication

In multiplication, a positive number times a positive number gives a positive result, a positive number times a negative number gives a negative result, a negative number times a positive number gives a negative result, and a negative number times a negative number gives a positive result. Since we're multiplying two negative numbers, our result will be positive.

Step 2: Perform the multiplication without considering the signs

Perform the multiplication of 0.2 and 0.6 without considering the negative signs. To do this, ignore the decimal places first. You would get \(2 \times 6 = 12\). Now, count the total number of decimal places in the numbers you multiplied. There are one digit after the decimal in 0.2 and one digit after the decimal in 0.6 so in total, that equals to 2. Place the decimal in your answer (12) so that there are two digits after it, resulting to 0.12.

Step 3: Apply the rule of signs to the result

Since the multiplication of two negative numbers yields a positive result, apply this by removing the negative sign from the answer. Therefore, -0.2 times -0.6 equals 0.12.

Key Concepts

Rules of Signs

When multiplying numbers, it's important to follow the rules of signs to determine the sign of the result. Here’s a quick recap:

• A positive number multiplied by a positive number gives a positive result.
• A positive number multiplied by a negative number yields a negative result.
• A negative number multiplied by a positive number also gives a negative result.
• A negative number multiplied by another negative number results in a positive number.

In our example \(-0.2 \times -0.6\), both numbers are negative. According to the rules, multiplying two negative numbers will ultimately give us a positive product. This foundational principle is crucial when handling algebraic multiplication. It helps in accurately predicting whether the final answer will be positive or negative.

Decimal Multiplication

Decimal multiplication can seem complex at first, but it becomes simple when broken down into steps. Initially, focus solely on multiplying the numbers as if they were whole numbers, irrespective of the decimal points. For the example \(0.2 \times 0.6\), ignore the decimals first and multiply \(2 \times 6 = 12\). Next, count the number of digits to the right of the decimal in both original numbers. In \(0.2\) and \(0.6\), each has one decimal place. Add them up, giving a total of two decimal places in the product. Finally, position the decimal point in your multiplication answer, \(12\), to have two digits after it: \(0.12\). This method provides a clear path to tackle decimal multiplication and helps to avoid common errors.

Negative Numbers

Negative numbers are numbers less than zero, represented with a "\(-\)" sign. Understanding how they behave in different mathematical operations is vital. In multiplication, as mentioned earlier, two negative numbers combine to form a positive number. This can be confusing at first due to their distinct behavior compared to positive numbers.To visualize, think of negative numbers as arrows pointing downwards, but when two such arrows are multiplied, they essentially reverse direction twice, resulting in an upward or positive direction.It is crucial to become proficient in handling negative numbers, especially in algebra, as they often appear in expressions and equations. Practice will help solidify your understanding, and soon you'll find it as intuitive as dealing with positive numbers. Remember, whenever in doubt, refer back to the rules of operations with signs to guide you.