Problem 12 - Algebra A Combined Function

Question

Multiply. 6(-2.5)

Step-by-Step Solution

Answer

The product is -15.

Step 1: Understand the Problem

You are asked to multiply two numbers: 6 and -2.5. The numbers have different signs. Remember that multiplying a positive number by a negative number results in a negative product.

Step 2: Ignore the Signs and Multiply the Absolute Values

First, ignore the negative sign of -2.5 and multiply the absolute values: 6 and 2.5. To do this, calculate the multiplication:\[ 6 \times 2.5 \]

Step 3: Perform the Multiplication Calculation

Now perform the multiplication. Break down 2.5 into easier parts, such as:\[ 6 \times 2 + 6 \times 0.5 \]First, calculate:\[ 6 \times 2 = 12 \]Then calculate:\[ 6 \times 0.5 = 3 \]Add the two results together:\[ 12 + 3 = 15 \]

Step 4: Apply the Sign

Since you originally multiplied a positive number (6) by a negative number (-2.5), the result of your multiplication is negative. Therefore, the answer is -15.

Key Concepts

Absolute Value

When multiplying numbers, understanding the absolute value can be very helpful. Absolute value is simply the distance a number is from zero on the number line, regardless of direction. It's always a positive value or zero. For example, the absolute value of 6 is 6, and the absolute value of -2.5 is 2.5.

• The absolute value of a negative number is its positive counterpart.
• It simplifies problems by temporarily removing any negative signs so we can work with positive numbers.

By using the absolute value, we can make calculations easier, as demonstrated in the exercise where initially only the absolute values (6 and 2.5) were multiplied to get the result of 15.

Multiplication Techniques

Different multiplication techniques can make calculations more manageable and less prone to error. One useful technique is breaking numbers into smaller, easier chunks. This helps us handle the math in steps.
For example, in the exercise:

• We broke down 2.5 into 2 and 0.5 to simplify the calculation when multiplying by 6.
• This way, we worked through \[ 6 \times 2 = 12 \] first and then \[ 6 \times 0.5 = 3 \].

Adding the results, \[ 12 + 3 = 15 \], helped us to piece together a manageable result before applying any negative signs back into the calculation later.

Negative Numbers

Negative numbers can make math look tricky, but they all follow the same basic rules. A negative number is a number less than zero and is usually indicated with a minus sign (-). When dealing with negative numbers, there's an important concept:

• They have opposite direction on the number line compared to positive numbers.
• Multiplying by a negative means reflecting the product on the number line.

In the exercise with 6 and -2.5, after finding the product of their absolute values, it is essential to apply the negative sign back, which means our final answer becomes -15.

Sign Rules in Multiplication

The sign rules in multiplication can help us predict the sign of our product efficiently without needing to find the result first. These rules state:

• A positive number multiplied by a positive number gives a positive product.
• A negative number multiplied by a negative number also gives a positive product.
• A positive multiplied by a negative, or a negative by a positive, results in a negative product.

In the exercise, we multiplied a positive (6) by a negative (-2.5), which according to the sign rules, gave us a negative result. Therefore, the product was -15, perfectly aligning with the rule outcomes.