Chapter 2: Problem 54
When dividing two numbers that have the same sign, the result is ___. When dividing two numbers that have different signs, the result is ___. divide. $$ \frac{-91}{-7} $$
Short Answer
Expert verified
Answer: The result of the division $$\frac{-91}{-7}$$ is 13.
Step by step solution
01
Determine the sign of the result
The numbers -91 and -7 both have the same sign (negative). According to the rules mentioned in the exercise, when dividing two numbers with the same sign, the result is positive. So, the result of the given division will be positive.
02
Perform the division without considering the signs
Now, let's perform the division without considering the signs of the numbers, as if they were both positive. To do this, we'll just focus on the numbers 91 and 7:
$$
\frac{91}{7} = 13
$$
03
Apply the positive sign to the result
We've determined that the result of the division should be positive, based on the signs of the original numbers. We already know that $$\frac{91}{7}=13$$. Therefore, the result of the given division $$\frac{-91}{-7}$$ is also positive 13:
$$
\frac{-91}{-7} = 13
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arithmetic Operations
Arithmetic operations form the foundation of basic mathematics. These include addition, subtraction, multiplication, and division, which are essential tools for solving problems in mathematics and everyday life.
Understanding how to work with these operations allows students to navigate through more complex mathematical concepts smoothly. When dealing with division, which is one of the four main arithmetic operations, it's crucial to know that its purpose is to find out how many times a number (the divisor) is contained within another number (the dividend).
For instance, if you divide 10 by 2, you're essentially asking 'how many times does 2 go into 10?', and the answer is 5 times. This simple concept applies as well to more advanced topics, such as dividing integers, which may have positive or negative signs.
Understanding how to work with these operations allows students to navigate through more complex mathematical concepts smoothly. When dealing with division, which is one of the four main arithmetic operations, it's crucial to know that its purpose is to find out how many times a number (the divisor) is contained within another number (the dividend).
For instance, if you divide 10 by 2, you're essentially asking 'how many times does 2 go into 10?', and the answer is 5 times. This simple concept applies as well to more advanced topics, such as dividing integers, which may have positive or negative signs.
Positive and Negative Numbers
Positive and negative numbers are an integral part of the number system in prealgebra mathematics. A positive number is greater than zero and is often represented without a sign or with a plus sign (...).
Negative numbers are less than zero and are always represented with a minus sign (...). The concept of negative numbers is central to understanding how to work with different integer values, especially when performing basic arithmetic operations like division.
Negative numbers are less than zero and are always represented with a minus sign (...). The concept of negative numbers is central to understanding how to work with different integer values, especially when performing basic arithmetic operations like division.
Navigating through the Number Line
Imagine a horizontal line where the middle point is zero. Numbers to the right of zero are positive, and numbers to the left are negative. This mental image can help in visualizing operations involving both positive and negative numbers.Prealgebra Mathematics
Prealgebra mathematics is the stage where students learn about the properties and operations of basic arithmetic, setting the stage for algebra. It includes concepts like integers, fractions, decimals, and percents. This is where rules for the order of operations come in and where students learn how to manage different signs in numbers.
For example, in the exercise provided, we are involved in dividing integers, a vital skill in prealgebra that leads to understanding more complex algebraic expressions later on. Prealgebra ensures that the foundational skills are there for students to succeed in higher mathematics. Reinforcing these skills often involves repetitive practice to instill a deep understanding and fluency.
For example, in the exercise provided, we are involved in dividing integers, a vital skill in prealgebra that leads to understanding more complex algebraic expressions later on. Prealgebra ensures that the foundational skills are there for students to succeed in higher mathematics. Reinforcing these skills often involves repetitive practice to instill a deep understanding and fluency.
Dividing Negative Numbers
When dividing negative numbers, there's a simple rule to remember: dividing two numbers with the same sign results in a positive answer, whereas two numbers with different signs result in a negative answer. This is essential knowledge when you're working with integer division in prealgebra.
In the exercise we discussed, \( \frac{-91}{-7} \) both numbers are negative. Hence, according to our rule, the result should be positive. This might seem counterintuitive at first, but it aligns perfectly with the concept that multiplying or dividing two negatives results in a positive.
In the exercise we discussed, \( \frac{-91}{-7} \) both numbers are negative. Hence, according to our rule, the result should be positive. This might seem counterintuitive at first, but it aligns perfectly with the concept that multiplying or dividing two negatives results in a positive.