Problem 76 - Beginning and Intermediate Algebra

Question

Divide. $$ \frac{-45}{-9} $$

Step-by-Step Solution

Answer

The result is 5.

Step 1: Understand the Problem

We are asked to divide -45 by -9. The division of two numbers involves finding how many times the divisor can fit into the dividend.

Step 2: Consider Division with Negative Numbers

When dividing two negative numbers, the negatives cancel each other out. This means the result of dividing -45 by -9 will be positive.

Step 3: Perform the Division

Divide the absolute values of the numbers: 45 divided by 9. Think about how many times 9 fits into 45. Since 9 times 5 is 45, 9 fits into 45 exactly 5 times.

Step 4: Write the Final Result With Correct Sign

Since both the dividend and divisor were negative, and dividing two negative numbers results in a positive quotient, the final answer is 5.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are opposite to positive numbers. In mathematics, negative numbers are used to represent a deficit or a loss.
For example, if you owe someone $10, you can think of it as having -$10. To indicate a negative number, we place a minus sign (-) in front of it.

• Negative numbers are crucial in various real-life situations, like measuring temperatures below freezing or when dealing with financial losses.
• They come into play often in math especially during operations such as addition, subtraction, and division.
• Understanding negative numbers helps in advancing problem-solving skills in complex mathematical problems.

Absolute Values

The absolute value of a number refers to its distance from zero on the number line, regardless of its direction. It is always a non-negative number.
To find the absolute value, we simply ignore the minus sign if it is there.

• The absolute value of -45 is 45, and the absolute value of 45 is also 45.
• In mathematical notation, we write the absolute value of a number as two vertical bars, like this: \(|-45| = 45\).
• Absolute values are important because they convey pure magnitude without considering the direction.

Integer Division

Integer division involves dividing two integers and examining how many times the divisor fits into the dividend evenly.
For example, dividing 45 by 9 involves seeing how many 9s can fit into 45, which is 5 times.

• Unlike regular division that may result in a fraction or decimal, integer division yields a whole number result.
• It is useful in situations where only complete units are considered, such as distributing items evenly among people.
• It simplifies computations in mathematical algorithms and is extensively used in programming and computer science.

Positive Quotient

When we divide a negative number by another negative number, the result is a positive quotient. This happens because the two negative signs cancel each other out.
In the division of \(-45\) by \(-9\), both numbers are negative, so the quotient is positive.

• This rule is consistent with the properties of multiplication and division in arithmetic operations, where a negative times a negative equals a positive.
• Understanding why the quotient is positive is important because it ensures accuracy in solving problems involving division.
• Therefore, whenever you see two negative numbers in a division problem, you can anticipate a positive result.