4.12: Integer Division
Cameron and his new diving partner Gina are going to be buddies on a 40 foot dive. Gina is a new diver and is still learning to make a descent. Cameron can make a free descent quite easily. This means that he doesn’t hold onto anything as he descends to the appropriate depth. Gina will hold onto the anchor line. Then they will meet on the bottom.
Cameron has decided to slow down his descent and to go down with Gina. He will do his free descent next to her descent holding onto the rope. He looks at his watch and sets the timer before they descend.
When they reach the bottom, Cameron looks at his watch. He sees that the descent took them 2 minutes. Not bad at all considering that Gina is a beginner. Cameron and Gina meet up with the group and check in with the Dive Master. Then they are off for a beautiful dive!!
How far did Cameron and Gina descend per minute?
To answer this question, you will need to understand how to divide integers. Pay attention and you will be able to answer these questions at the end of the Concept.
Guidance
Another important step in learning how to compute with integers is learning how to divide them. You can look for patterns in a sequence of quotients just as you looked for patterns in a sequence of products in an earlier lesson. These patterns will help you to understand the rules for dividing integers.
Let’s look at some integer patterns with division. We are looking at quotients. A quotient is the answer in a division problem.
Use a pattern to find the missing quotients below.
\begin{align*}6 \div 2 & = 3\\ 4 \div 2 & = 2\\ 2 \div 2 & = 1\\ 0 \div 2 & = 0\\ -2 \div 2 & = ?\\ -4 \div 2 & = ?\\\end{align*}
Look for a pattern among the quotients. Remember that a pattern has a rule that makes it repeat in a certain way. Look at the pattern below.
This pattern uses shapes and not numbers, but there is still a rule that has it happen the way that it does.
Now look at the number pattern.
You will see that you can subtract 1 from the previous quotient to find the next quotient. Remember, subtracting 1 is the same thing as adding its opposite, -1. Try adding -1 to the previous quotients to find the next quotients.
To find the quotient of \begin{align*}-2 \div 2\end{align*}, add \begin{align*}0+(-1)\end{align*}
\begin{align*}|0|=0\end{align*} and \begin{align*}|-1|=1\end{align*}, so subtract the lesser absolute value from the greater absolute value.
\begin{align*}1-0=1\end{align*}
The integer with the greater absolute value is -1, so give the answer a negative sign.
\begin{align*}0+(-1)=-1\end{align*}, so \begin{align*}-2 \div 2=-1\end{align*}
To find the quotient of \begin{align*}-4 \div 2\end{align*}, add \begin{align*}-1+(-1)\end{align*}
Both integers have the same sign, so add their absolute values.
\begin{align*}|-1|=1\end{align*}, so add
\begin{align*}1+1=2\end{align*}
Give that answer a negative sign.
\begin{align*}-1+(-1)=-2\end{align*}, so \begin{align*}-4 \div 2 =-2\end{align*}.
This shows the completed division facts.
\begin{align*}6 \div 2 & = 3\\ 4 \div 2 & = 2\\ 2 \div 2 & = 1\\ 0 \div 2 & = 0\\ -2 \div 2 & = -1\\ -4 \div 2 & = -2\\\end{align*}
Each quotient is still 1 less than the previous quotient.
What conclusions can we draw from this pattern?
You may notice the following.
- When a positive integer is divided by a positive integer, 2, the quotient is positive.
- When zero is divided by a positive integer, 2, the quotient is zero.
- When a negative integer is divided by a positive integer, 2, the quotient is negative.
These are the beginnings of our rules for dividing integers.
Let’s look at another pattern to complete these rules.
Look at the number facts below. Analyze the pattern of quotients shown.
\begin{align*}&9 \div (-3) = -3\\ &6 \div (-2) = -3\\ &3 \div (-1) = -3\\ &0 \div 0 = undefined\\ &-3 \div (-1) = 3\\ &-6 \div (-2) = 3\\ &-9 \div (-3) = 3\end{align*}
What do you notice about these facts?
You may notice the following rules.
- When a positive integer is divided by a negative integer, the quotient is negative.
- When zero is divided by zero, the quotient is undefined, not zero. (Note: Any number divided by zero is considered undefined.)
- When a negative integer is divided by a negative integer, the quotient is positive.
Now we can use these rules to divide integers. Just like with the rules for multiplying, becoming great at dividing integers will require that you memorize these rules.
Next, let’s apply these rules to dividing integers.
Find the quotient \begin{align*}(-33) \div (-3)\end{align*}
To find this quotient, we need to divide two negative integers.
Divide the integers without paying attention to their signs. The quotient will be positive.
\begin{align*}(-33) \div (-3) = 33 \div 3 = 11\end{align*}
The quotient is 11.
Find the quotient \begin{align*}(-20) \div 5\end{align*}.
To find this quotient, we need to divide two integers with different signs.
Divide the integers without paying attention to their signs. Give the quotient a negative sign.
\begin{align*}20 \div 5 =4\end{align*}, so \begin{align*}(-20) \div 5 = -4\end{align*}.
The quotient is -4.
These problems used a division sign, but remember we can also show division using a fraction bar where the numerator is divided by the denominator.
Now, it’s time for you to practice applying these rules while figuring out quotients.
Example A
\begin{align*}-12 \div -3\end{align*}
Solution: \begin{align*}4\end{align*}
Example B
\begin{align*}\frac{18}{-3}\end{align*}
Solution: \begin{align*}-6\end{align*}
Example C
\begin{align*}-24 \div 8\end{align*}
Solution: \begin{align*}-3\end{align*}
Here is the original problem once again.
Cameron and his new diving partner Gina are going to be buddies on a 40 foot dive. Gina is a new diver and is still learning to make a descent. Cameron can make a free descent quite easily. This means that he doesn’t hold onto anything as he descends to the appropriate depth. Gina will hold onto the anchor line. Then they will meet on the bottom.
Cameron has decided to slow down his descent and to go down with Gina. He will do his free descent next to her descent holding onto the rope. He looks at his watch and sets the timer before they descend.
When they reach the bottom, Cameron looks at his watch. He sees that the descent took them 2 minutes. Not bad at all considering that Gina is a beginner. Cameron and Gina meet up with the group and check in with the Dive Master. Then they are off for a beautiful dive!!
How far did Cameron and Gina descend per minute?
First, we need to write integers to represent the depth and the time.
-40 feet is the depth
2 minutes is the time
We dive the depth by the time to find out the number of feet per minute.
\begin{align*}-40 \div 2 = -20\end{align*}
Cameron and Gina traveled -20 feet per minute.
Vocabulary
Here are the vocabulary words in this Concept.
- Quotient
- the answer in a division problem.
- Undefined
- when an integer is divided by 0, the answer of that is undefined.
Guided Practice
Here is one for you to try on your own.
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 number of yards lost on each play as a negative integer.
Answer
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.
We write this equation and then fill in the given values
Total yards lost \begin{align*}\div\end{align*} number of plays = yds lost on each play
\begin{align*}-30 \div 3 =?\end{align*}
To find this quotient, we need to divide two integers with different signs.
Divide the integers without paying attention to their signs. Give the quotient a negative sign.
\begin{align*}30 \div 3 = 10\end{align*}, so \begin{align*}(-30) \div 3 = -10\end{align*}.
The integer -10 represents the number of yards lost on each play.
Video Review
Here is a video for review.
- This is a James Sousa video on dividing integers.
Practice
Directions: Find each quotient.
1.\begin{align*}-18 \div 6\end{align*}
2. \begin{align*}-18 \div -6\end{align*}
3. \begin{align*}48 \div 8\end{align*}
4. \begin{align*}64 \div (-8)\end{align*}
5. \begin{align*}-28 \div (-4)\end{align*}
6. \begin{align*}-35 \div 7\end{align*}
7. \begin{align*}-80 \div (-4)\end{align*}
8. \begin{align*}-50 \div 10\end{align*}
9. \begin{align*}-18 \div -2\end{align*}
10. \begin{align*}42 \div -6\end{align*}
11. \begin{align*}-72 \div 9\end{align*}
12. \begin{align*}-48 \div -12\end{align*}
13. \begin{align*}-16 \div 4\end{align*}
14. \begin{align*}-22 \div -11\end{align*}
15. \begin{align*}72 \div -12\end{align*}
Image Attributions
Here you'll learn to divide integers.