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# Integer Division

## Understand the rules for dividing positive and negative numbers.

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Practice Integer Division
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Integer Division

### Let’s Think About It

Credit: Alan Cleaver
Source: https://www.flickr.com/photos/alancleaver/4222532649/
License: CC BY-NC 3.0

The company that Sam works for is doing a weight loss challenge in 2015. The three different departments in the company are competing against each other. Whichever department has the most negative average change in weight will win.

• Department A has 12 employees and its current overall change in weight is -36 pounds.
• Department B has 9 employees and its current overall change in weight is -45 pounds.
• Department C has 35 employees and its current overall change in weight is -70 pounds.

How could Sam determine which department is currently winning the weight loss challenge?

In this concept, you will learn how to divide integers.

### Guidance

Integers are the set of whole numbers and their opposites.

When you divide integers, your answer after dividing is called a quotient.

Here is an example.

15÷3=5\begin{align*}15 \div 3=5\end{align*}

In this example, 5 is the quotient.

Dividing integers is very similar to dividing whole numbers. The only difference is you will have to decide if your quotient is negative or positive. The rules for deciding if your quotient is positive/negative are very similar to the rules you used with integer multiplication.

To divide two integers:

1. First, divide the absolute value of the integers.
2. Next, determine the sign of your quotient according to the following rules:
• A positive number divided by a positive number equals a positive number.
• A positive number divided by a negative number equals a negative number.
• A negative number divided by a negative number equals a positive number.
• A negative number divided by a positive number equals a negative number.
• Any integer divided by zero is undefined.
• Zero divided by any integer except zero equals zero.
3. Your quotient will be your result from step 1 with the sign from step 2.

It might help to remember that if your original two integers have the same sign, then their quotient will be positive. If your original two integers have different signs, then their quotient will be negative. This is the same as it was for integer multiplication!

Let's look at an example.

Divide (33)÷(3)\begin{align*}(-33) \div (-3)\end{align*}

The first step is to divide the absolute value of the factors.

33÷3=11\begin{align*}33 \div 3=11\end{align*}

The next step is to decide on the sign of your answer. Both integers were negative and a negative divided by a negative equals a positive. This means your quotient is positive.

(33)÷(3)=11\begin{align*}(-33) \div (-3)=11\end{align*}

The answer is 11.

Here's another example.

Divide 205\begin{align*}\frac{-20}{5}\end{align*}.

Remember that the fraction bar is the same as division. This problem is the same as

20÷5\begin{align*}-20 \div 5\end{align*}

Again, the first step is to divide the absolute value of the factors.

20÷5=4\begin{align*}20 \div 5=4\end{align*}

The next step is to decide on the sign of your answer. The two factors had different signs. A negative divided by a positive is a negative. This means your quotient is negative.

205=20÷5=4\begin{align*}\frac{-20}{5}=-20 \div 5=-4\end{align*}

The answer is -4.

### Guided Practice

On 3 consecutive plays, a football team lost a total of 30 yards. The team lost the same number of yards on each play. Represent the change in the number of yards on each play as a negative integer.

First, represent the total number of yards lost as an integer.

Since the integer shows a loss of 30 yards, use a negative integer, -30.

To represent the loss for each of the 3 plays, divide the integer representing the total number of yards lost by 3.

30÷3=?\begin{align*}-30 \div 3 =?\end{align*}

First, divide the absolute value of the factors.

30÷3=10\begin{align*}30 \div 3=10\end{align*}

Next, decide on the sign for your quotient. A negative divided by a positive equals a negative, so your quotient is a negative.

30÷3=10\begin{align*}-30\div 3=-10\end{align*}

The answer is -10.

The integer -10 represents the change in the number of yards on each play.

### Examples

#### Example 1

Divide 12÷3\begin{align*}-12 \div -3\end{align*}.

First, divide the absolute value of the factors.

12÷3=4\begin{align*}12 \div 3=4\end{align*}.

Next, decide on the sign for your quotient. A negative divided by a negative equals a positive, so your quotient is a positive.

12÷3=4\begin{align*}-12 \div -3=4\end{align*}

The answer is 4.

#### Example 2

Divide 183\begin{align*}\frac{18}{-3}\end{align*}.

This is the same as 18÷3\begin{align*}18 \div -3\end{align*}.

First, divide the absolute value of the factors.

18÷3=6\begin{align*}18 \div 3=6\end{align*}.

Next, decide on the sign for your quotient. A positive divided by a negative equals a negative, so your quotient is a negative.

18÷3=6\begin{align*}18 \div -3=-6\end{align*}

The answer is -6.

#### Example 3

Divide 24÷8\begin{align*}-24 \div 8\end{align*}.

First, divide the absolute value of the factors.

24÷8=3\begin{align*}24\div 8=3\end{align*}

Next, decide on the sign for your quotient. A negative divided by a positive equals a negative, so your quotient is a negative.

24÷8=3\begin{align*}-24 \div 8=-3\end{align*}

The answer is -3.

Credit: -Paul H-
Source: https://www.flickr.com/photos/-paulh/4517197776/
License: CC BY-NC 3.0

Remember Sam and his weight loss challenge at work? He wants to find out which department is currently winning the competition. In order for a department to be in the lead, it needs to have the most negative average change in weight.

Department A with 12 employees has a current overall change in weight of -36 pounds. To find the average change in weight for the department, you need to divide the change in weight by the number of employees.

36÷12=?\begin{align*}-36 \div12=?\end{align*}

You know that 36÷12=3\begin{align*}36 \div 12=3\end{align*}. You also know that a negative divided by a positive equals a negative. This means the answer is negative.

36÷12=3\begin{align*}-36 \div 12=-3\end{align*}

Department A has an average change in weight of -3 pounds.

Department B with 9 employees has a current overall change in weight of -45 pounds. Divide to find its current average change in weight:

45÷9=?\begin{align*}-45 \div 9=?\end{align*}

You know that 45÷9=5\begin{align*}45 \div 9=5\end{align*} and a negative divided by a positive equals a negative.

45÷9=5\begin{align*}-45 \div 9=-5\end{align*}.

Department B has an average change in weight of -5 pounds.

Department C with 35 employees has a current overall change in weight of -70 pounds. Divide to find its current average change in weight:

70÷35=?\begin{align*}-70 \div 35=?\end{align*}

You know that 70÷35=2\begin{align*}70 \div 35=2\end{align*} and a negative divided by a positive equals a negative.

70÷35=2\begin{align*}-70 \div 35=-2\end{align*}.

Department C has an average change in weight of -2 pounds.

Currently, Department B has the most negative average change in weight with -5 pounds.

The answer is that Department B is currently winning the weight loss challenge.

### Video Review

The video below reviews dividing integers.

### Explore More

Find each quotient.

1. 18÷6\begin{align*}-18 \div 6\end{align*}
2. 18÷6\begin{align*}-18 \div -6\end{align*}
3. 48÷8\begin{align*}48 \div 8\end{align*}
4. 64÷(8)\begin{align*}64 \div (-8)\end{align*}
5. 28÷(4)\begin{align*}-28 \div (-4)\end{align*}
6. 35÷7\begin{align*}-35 \div 7\end{align*}
7. 80÷(4)\begin{align*}-80 \div (-4)\end{align*}
8. 50÷10\begin{align*}-50 \div 10\end{align*}
9. 18÷2\begin{align*}-18 \div -2\end{align*}
10. 42÷6\begin{align*}42 \div -6\end{align*}
11. 72÷9\begin{align*}-72 \div 9\end{align*}
12. 48÷12\begin{align*}-48 \div -12\end{align*}
13. 16÷4\begin{align*}-16 \div 4\end{align*}
14. 22÷11\begin{align*}-22 \div -11\end{align*}
15. 72÷12\begin{align*}72 \div -12\end{align*}

### Vocabulary Language: English

Quotient

Quotient

The quotient is the result after two amounts have been divided.
Undefined

Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.