Problem 258 - Intermediate Algebra

Question

In the following exercises, multiply. $$ (-8.5)(1.69) $$

Step-by-Step Solution

Answer

-14.365

Step 1 - Understand the Problem

The problem asks to multiply two numbers: (-8.5) and 1.69.

Step 2 - Multiply the Numbers

Multiply the absolute values of the numbers: \[ 8.5 \times 1.69 \] First, calculate the multiplication: \[ 8.5 \times 1.69 = 14.365 \]

Step 3 - Determine the Sign of the Product

Since we are multiplying a positive number (1.69) by a negative number (-8.5), the product will be negative: \[ (-8.5) \times 1.69 = -14.365 \]

Key Concepts

negative and positive numbers

Understanding the difference between negative and positive numbers is crucial in multiplication. Negative numbers are less than zero and represented with a minus sign (-), while positive numbers are greater than zero and have no sign or a plus sign (+). When multiplying numbers, it's important to consider their signs:

- When you multiply two positive numbers, the product is positive.
- When you multiply two negative numbers, the product is also positive (since two negatives cancel each other out).
- When you multiply a positive number by a negative number, the product is negative. In our example, we are multiplying -8.5 and 1.69: a negative and a positive number, so the result is negative.

absolute values

Absolute value is the distance of a number from zero on the number line, without considering direction. So, it's always a non-negative number. For example, the absolute value of -3 is 3, written as \(|-3| = 3\). The absolute value function strips any negative sign if it's present.

In the given exercise, to multiply -8.5 by 1.69, first, we find the absolute values of -8.5 and 1.69. The absolute value of -8.5 is 8.5, and the absolute value of 1.69 is 1.69. We ignore the negative sign during the multiplication process and just consider 8.5 and 1.69, then apply the negative sign at the end.

multiplication steps

To multiply two decimal numbers, follow these steps:

1. **Ignore the decimal points**: Treat the numbers as if they were whole numbers. For example, 8.5 and 1.69 become 85 and 169.
2. **Multiply** the whole numbers: Using the multiplication method you prefer, multiply 85 by 169. Let's break it down:

\[ 85 \times 169 \]
= \[ 85 \times (100 + 60 + 9) \]
= \[ (85 \times 100) + (85 \times 60) + (85 \times 9) \]
= \[ 8500 + 5100 + 765 \]
= \[ 14365 \]

3. **Count the total decimal places**: Count the number of digits to the right of the decimal points in the original numbers. In 8.5, there is 1 digit, and in 1.69, there are 2 digits, totaling to 3 decimal places.
4. **Place the decimal point**: In the product, place the decimal point so that there are 3 digits to its right. Therefore, 14365 becomes 14.365.

5. **Apply the sign**: Finally, since we are multiplying a negative number by a positive number, the product will be negative: -14.365.