Problem 257 - Intermediate Algebra

Question

In the following exercises, multiply. $$ (94.69)(-12.678) $$

Step-by-Step Solution

Answer

-1199.07402

Step 1: Understand the Problem

Identify that the exercise involves the multiplication of two numbers: 94.69 and -12.678.

Step 2: Multiply the Absolute Values

First, multiply the absolute values of the numbers: \( 94.69 \times 12.678 \). Align the numbers according to their decimal points and perform the multiplication as if they were whole numbers.

Step 3: Perform the Multiplication

Calculate \( 94.69 \times 12.678 \) exactly using long multiplication or a calculator. The multiplication yields approximately \( 1199.07402 \).

Step 4: Apply the Sign Rule

Since one of the factors is negative, the product will also be negative. Therefore, the result is \( -1199.07402 \).

Step 5: Round if Necessary

Unless specified, the result might need to be rounded. But in this case, no specific instruction for rounding is given, so keep the result as \( -1199.07402 \).

Key Concepts

Absolute Values

Absolute values are essential when dealing with multiplication, especially when the numbers have different signs. The absolute value of a number is its distance from zero on the number line, irrespective of its direction. This means both -5 and 5 have an absolute value of 5, represented as \( |5| = 5 \) and \( |-5| = 5 \).

When multiplying \( 94.69 \) and \( -12.678 \), first disregard the signs and focus on their absolute values: \( 94.69 \) and \( 12.678 \). This simplifies the problem and lets you handle the multiplication step without worrying about the signs.

Once the multiplication is done, you can apply the appropriate sign to the result, following the sign rule in multiplication.

Sign Rule in Multiplication

The sign rule is a fundamental part of multiplying numbers, especially when they include both positive and negative values. Here are the simple rules:

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Positive = Negative
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Negative = Positive

In our case, \( 94.69 \) is positive and \( -12.678 \) is negative. According to the sign rule, when you multiply a positive number by a negative number, the result is negative. Therefore, after calculating the product of the absolute values (which is \( 1199.07402 \)), you apply the negative sign to get the final answer: \( -1199.07402 \).

Long Multiplication

Long multiplication is a step-by-step method to multiply larger numbers manually. Here’s a simple way to approach it:

• Write the numbers down, aligning the decimal points vertically.
• Temporarily ignore the decimal points and multiply the numbers as if they were whole numbers.
• Once the multiplication is complete, count the total number of decimal places in both of the original numbers.
• Place the decimal in the product, counting from the right, according to the total number of decimal places.

For \( 94.69 \) and \( 12.678 \), ignoring the decimal points gives us \( 9469 \) and \( 12678 \). Multiply these as whole numbers, yielding \( 119907402 \). Finally, account for the 5 decimal places altogether (2 from \( 94.69 \) and 3 from \( 12.678 \)), and place the decimal point in the product: \( 1199.07402 \).

While it's faster to use a calculator, understanding long multiplication is crucial, especially when learning foundational math methods.