Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$130 \div(-10)$$

Step-by-Step Solution

Answer

-13

Step 1: Identify the Numbers to be Divided

The numbers to be divided are 130 and -10. In this division operation, 130 is the dividend and -10 is the divisor.

Step 2: Perform the Division

Divide 130 by -10. Because a positive number divided by a negative one is negative, the result will be negative as well.

Step 3: Write Down the Result

The result of the division \(130 \div -10\) is -13. Write it down as the final answer.

Key Concepts

Dividend

The concept of a "dividend" is fundamental in understanding division. In a division operation, the dividend is the number that you want to divide. It's the amount or the larger number that gets divided into smaller parts. For example, in the expression \(130 \div (-10)\), the number 130 is the dividend.

To remember this easily, you might think of the dividend as the "treasure chest". It's the value you're breaking down to distribute or divide. It's always the first number in a division equation.

• The dividend helps determine how much is being shared or divided.
• It can be a positive or negative number, affecting the outcome of the division.
• Knowing the dividend helps identify what is being divided in any given problem.

Next time you encounter a division task, start by identifying the dividend! It's your starting point in the division process.

Divisor

The "divisor" is equally as important, playing a key role opposite that of the dividend. In the same division expression \(130 \div (-10)\), here's where -10 comes into play. It's the divisor, the number by which you divide the dividend.

Think of the divisor as the tool you use to split the treasure chest into smaller pieces. It's always the number that follows the division symbol (\(\div\)) in a division operation.

• The divisor determines how many pieces or groups you're creating from the dividend.
• It can change the sign of the result if it's a negative number.
• If the divisor is zero, the division operation is undefined, as dividing by zero is mathematically incorrect.

Understanding the divisor allows you to determine how the dividend will be distributed or divided!

Negative Numbers

Negative numbers can sometimes trip us up when first learning about division, but they aren't too tricky once you understand a few basic rules. In the exercise \(130 \div (-10)\), both numbers have different signs, which impacts the outcome of the division.

To tackle operations involving negative numbers, keep these tips in mind:

• When you divide a positive number by a negative one, the outcome is a negative number, as seen with \(130 \div (-10) = -13\).
• If both numbers were negative, for example, \((-10) \div (-5)\), the result would be positive, because two negatives cancel each other out.
• The sign rules are the same whether adding, subtracting, multiplying, or dividing. It’s always about how the signs interact: same signs equal positive, different signs equal negative.

By following these guidelines, handling negative numbers in division becomes much simpler!

Division Operation

The "division operation" is a fundamental mathematical process for breaking up large numbers into smaller, evenly divided parts. The process involves sharing or distributing the dividend among the number of groups specified by the divisor. In our example \(130 \div (-10)\), the operation involves distributing the 130 into groups of -10 each.

Here's a simple breakdown of how division works:

• Identify the dividend and the divisor.
• Use the division symbol \(\div\) to perform the operation.
• Follow sign rules: positive \(\div\) negative results in a negative divisor.
• Calculate: Find how many times the divisor fits into the dividend, with considerations for sign.
• Express the quotient: Write the final result, such as our answer, -13.

Mastering division helps solve problems related to sharing, dividing resources, and understanding ratios and proportions in mathematics.