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Integer Division

Understand the rules for dividing positive and negative numbers.

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Integer Division
License: CC BY-NC 3.0

Cat wants to do the 24 hour comics challenge, but she is a little overwhelmed at the idea of drawing 24 pages from scratch in one day. She has a story idea so she tries to break it into manageable pieces. Of the 24 pages, she knows that she wants one full page for the title page, and she wants one full page splash at the climax. She also knows that she wants the first story page to be three panels of establishing shots. When she does a very rough story-board, she figures out that she needs 147 distinct panels to tell her story. So how many panels will she need to average per page?

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

Dividing Integers

An integer is the set of whole numbers and their opposites.

A quotient is the answer in a division problem. Division is the number of times one number goes into another number.

The divisor is the number divided into the other number.

Here are the sign rules for division:

Positive \begin{align*}\div \end{align*} positive = positive
Negative \begin{align*}\div \end{align*} positive, or vice versa=negative
Negative \begin{align*}\div \end{align*} negative = positive

Another way to think of it is if there are an odd number of negatives in the problem, the answer is negative. If there are an even number, the answer is positive. These rules are the same as for multiplication.

Here is an example.

10 \begin{align*}\div\end{align*} 2 = 

In this example, ten is being divided into increments of two. So first, break 10 into increments of 2.

2 + 2 = 4

2 + 2 + 2 = 6

2 + 2 + 2 + 2 = 8

2 + 2 + 2 + 2 + 2 = 10

Next, count how many increments of the divisor added up to the other number.

In this case, there were five 2s.

The answer is 5.

Here is another example.

\begin{align*}30\div -5\end{align*}= ____

First, break 30 into increments of 5, ignoring the sign.

5 + 5 + 5 + 5 + 5 + 5 = 30

Next, count the number of increments.

In this case, there are six 5s. If you know the multiplication table, you can simplify the process without converting to addition every time. Then, you just need to worry about the signs.

Then, take into account the sign.

The original divisor was negative. In order to divide a negative number into a positive number, the answer must also be negative.

The answer is -6.

Examples

Example 1

Earlier, you were given a problem about Cat and her comic book.

She wants to figure out how many panels she needs to draw per page. She knows she needs 147 panels to tell her story. Her comic book is 24 pages. But of those, the first is a title page, the second is a page of establishing shots, and her climax page only has one panel. So she only really has 21 pages to work with.

In order to decide how many panels to draw on each page, she sets up a division problem.

\begin{align*}147\div 21\end{align*}

Next, she divides the number of panels my the number of pages.

She finds that she needs to plan to have an average of 7 panels per page to tell her story.

The answer is 7.

Example 2

Carry out the division problem.

\begin{align*}18 \div -2 \div 3\end{align*}

First, carry out the division of the first two integers ignoring the third integer and the signs

\begin{align*}18\div 2\end{align*} = 9

Next, carry out the division with that answer and the 3. (Remember: the number on the right always goes into the number on the left.)

 \begin{align*}9\div 3=3\end{align*}

Then, count the number of negatives.

In this case, there is one negative in the original problem. One is odd, so the final answer is negative.

The answer is -3.

Example 3

Carry out the division problem.

-16 \begin{align*}\div\end{align*} -2 = ____

First, divide the second number into the first, ignoring signs.

 \begin{align*}16\div 2=8\end{align*}

Then, count the number of negatives in the original problem.

In this case, there are two. Two is even, so the final answer is positive.

The answer is 8.

Example 4

Carry out the division problem.

-24 \begin{align*}\div\end{align*} -12 =____

First, divide the second number into the first, ignoring signs.

 \begin{align*}24\div 12=2\end{align*}

Next, count the number of negatives in the original problem.

In this case, there are two. Two is even, so the final answer is positive.

The answer is 2.

Example 5

Carry out the division problem.

-64 \begin{align*}\div\end{align*} 2 = ____

First, divide the second number into the first, ignoring signs.

 \begin{align*}64\div 2=32\end{align*}

Next, count the number of negatives in the original problem.

In this case, there is one. One is odd, so the final answer is negative.

The answer is -32.

Review

Find the quotient of each integer pair.

  1. -18 \begin{align*}\div\end{align*} 9 = ____
  2. -22 \begin{align*}\div\end{align*} -11 = ____
  3. -32 \begin{align*}\div\end{align*} 8 = ____
  4. 32 \begin{align*}\div\end{align*} 8 = ____
  5. -21 \begin{align*}\div\end{align*} 7 = ____
  6. -72 \begin{align*}\div\end{align*} 12 = ____
  7. -80 \begin{align*}\div\end{align*} -10 = ____
  8. 56 \begin{align*}\div\end{align*} -7 = ____
  9. 63 \begin{align*}\div\end{align*} -9 = ____
  10. -121 \begin{align*}\div\end{align*} -11 = ____
  11. 144 \begin{align*}\div\end{align*} -12 = ____
  12. 200 \begin{align*}\div\end{align*} -4 = ____
  13. -50 \begin{align*}\div\end{align*} -2 = ____
  14. 28 \begin{align*}\div\end{align*} -2 = ____
  15. 66 \begin{align*}\div\end{align*} 3 = ____
  16. 150 \begin{align*}\div\end{align*} -3 = ____
  17. 180 \begin{align*}\div\end{align*} -90 = ____
  18. 70 \begin{align*}\div\end{align*} -35 = ____
  19. -44 \begin{align*}\div\end{align*} -22 = ____
  20. 75 \begin{align*}\div\end{align*} 3 = ____

Evaluate each numerical expression.

  1. \begin{align*}\frac{-9+ -3}{6}\end{align*}
  2. \begin{align*}\frac{-9(-6)}{2}\end{align*}
  3. \begin{align*}\frac{(15)(3)}{-5}\end{align*}
  4. \begin{align*}\frac{-18(4)}{9}\end{align*}
  5. \begin{align*}\frac{-3-12}{-5}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.13. 

Resources

 

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Vocabulary

Fraction Bar

A fraction bar is a line used to divide the numerator and the denominator of a fraction. The fraction bar means division.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

Quotient

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

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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