Problem 94 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 4.6(-5.4)(-2) $$

Step-by-Step Solution

Answer

49.68

Step 1: Multiply the First Two Numbers

Start by multiplying the first two numbers, 4.6 and -5.4. Calculate: \[4.6 \times (-5.4) = -24.84\]

Step 2: Multiply the Result by the Third Number

Now, take the result from Step 1, which is -24.84, and multiply it by the third number, -2.Calculate:\[-24.84 \times (-2) = 49.68\]

Step 3: Understand the Sign

Notice that multiplying two negative numbers results in a positive number. Thus, the product of -24.84 and -2 is positive.

Key Concepts

Negative Numbers

Negative numbers represent values less than zero and are indicated by the minus sign (-). They play a significant role in mathematics, especially in multiplication and division.

When you multiply two negative numbers, the result is always positive. This might sound strange at first, but it's a core rule in mathematics: the product or quotient of an even number of negative numbers is positive. Consider a simple example to understand this idea better:

• If you multiply -3 by -3, you get 9 because two negatives multiply to make a positive.
• If there's an odd number of negative numbers (such as -3 times 3) the result remains negative.

Understanding this rule is crucial to solving problems that involve negative numbers, like our example, where multiplying -5.4 and -2 resulted in a positive number.

Positive Numbers

Positive numbers are greater than zero and lack any negative sign. They are what you naturally count with and are straightforward in arithmetic since they follow what might be called the 'standard' rules of multiplication.

When multiplying two positive numbers, the result is always positive, as you might expect. But here's an important detail: when you multiply a positive number by a negative number, the result will be negative. This was evident in the initial step of our exercise where 4.6, a positive number, was multiplied by -5.4, giving us -24.84.

In this case, it's the interaction with negative numbers that modifies their behavior, creating challenges that require attention to sign changes during calculations.

Decimal Multiplication

Decimal multiplication involves numbers that have digits following a decimal point, known as decimal points. These allow for greater precision and take a slightly different approach than whole numbers.

When multiplying decimal numbers, it helps to ignore the decimal point until the multiplication is complete. After finding the product, you then count the total number of decimal places in the numbers you are multiplying.

• For example, if one number has two decimal places, and the other has one decimal place, the product should have three decimal places.
• In our exercise, 4.6 has one decimal place and -5.4 has one decimal place. Hence, their product -24.84 correctly adopts two decimal places.

This attention to details with decimal places ensures your results are accurate and reflect the precision you're working with.