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Chapter 1: Problem 18

Fill in the blanks. The quotient of a ____________ number and a positive number is negative.

Short Answer

Expert verified
Negative

Step by step solution

01

Understanding Numeric and Algebraic Signs

When considering the quotient of two numbers, if the dividend (the number being divided) and the divisor (the number by which we divide) have the same signs, then the quotient is positive. Conversely, if the signs of dividend and divisor are different, then the quotient is negative.
02

Identifying the Missing Number Type

Since the quotient of the described division operation is negative and the divisor is a positive number, it is only logical that the dividend must be negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotients
A quotient is simply the result one gets when one number is divided by another. When you divide, you have two key elements involved: the dividend and the divisor. The dividend is the number you want to divide up. This is the number that you start with. The divisor is the number you are using to divide the dividend. Understanding the relationship between the signs of these numbers is crucial for determining the sign of the quotient:
  • If both numbers are positive or both are negative, the quotient will be positive.
  • However, if one number is positive and the other is negative, the resulting quotient will be negative.
So, when you see a division problem, look at the signs of the dividend and divisor. The rules are simple once you know them, and they help you quickly determine whether the answer to a division problem should be positive or negative.
Positive Numbers
Positive numbers are numbers that are greater than zero. These numbers are located to the right of zero on a number line. They are straightforward and often represent quantities like age, distance, or time in everyday life. In mathematical operations:
  • When you add two positive numbers, the result is always positive.
  • When you multiply two positive numbers, the result is also positive.
  • And as for division, dividing one positive number by another positive number yields a positive quotient.
It's important to remember these basic properties to easily manage positive numbers in any mathematical operation. The same applies when they are combined with negative numbers, especially when considering their impact on the sign of a quotient.
Negative Numbers
Negative numbers are numbers that are less than zero and they are found to the left of zero on a number line. These numbers usually have a minus sign in front of them. Negative numbers often represent losses or deficits, such as owing money or decreasing temperatures. Here are some key points about negative numbers:
  • When you add a negative number to a positive one, the sign of the result depends on the magnitude of the numbers involved.
  • Multiplying two negative numbers yields a positive product, while multiplying a positive number by a negative one gives a negative product.
  • As for division, a negative number divided by a positive number will result in a negative quotient.
Understanding these sign rules with negative numbers will help you navigate through mathematical operations more confidently, especially those dealing with division, where they define the sign of the quotient.

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Most popular questions from this chapter

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) The quotient obtained when the product of \(x\) and \(y\) is divided by \(z\)

Write each sentence as a mathematical expression. \(x\) is greater than or equal to 5

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) The quotient obtained when 10 greater than \(x\) is divided by \(z\) Consider the algebraic expression \(29 x y z+23 x y+19 x\)

If the values of both \(x\) and \(y\) were doubled, what would happen to the value of \(5 x y^{2} ?\)

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