2.12: Multiply and Divide Integers
After the first few weeks of sales, the students were very excited because the school store was a thriving success. Unfortunately, students often were asking for calculators and there weren’t any for purchase.
“We really need to buy some calculators,” Mallory said at the Wednesday afternoon meeting. “I have kids asking for them all the time.”
“I agree,” said Trevor. “But we have the big dance coming up and all of our funds are going to towards that, so we don’t really have the money for calculators.”
“But we will sell them, so we can make the money back within a few months,” Kelly said.
“Well, how about a loan?” Mr. Janus asked. “I could loan you $250.00 and you could pay it back over the next four months.”
“You can do that?” Kelly asked.
“Yes, I have some money in the student council account that you may borrow from. How’s that?”
“That is terrific. Now with $250.00 as a loan, we will have four months to pay it back. How much is that per month?” Trevor said taking out a notebook.
We can think of the loan as a negative number. The students did not have the money for the calculators, so they were loaned the money for the purchase. This problem is asking how much the students will need to pay back each month to repay the loan. To work with this problem, you will need to divide integers. Pay close attention, you will see this problem again.
Guidance
Here you will learn how to multiply and divide integers. Because we are working with integers once again, remember the definition of an integer.
Integers are the values in the set of whole numbers and their opposites, {... 3, 2, 1, 0, 1, 2, 3...}.
When you multiply and divide integers, there are some rules that need to be committed to memory.
Here are the rules that need to be remembered.
 When multiplying or dividing a positive integer by a positive integer, the product or quotient is positive.
 When multiplying or dividing a positive integer by a negative integer, or a negative integer by a positive integer, the product or quotient is negative.
 When multiplying or dividing a negative integer by a negative integer, the product or quotient is positive.
You can remember these rules in a quicker way: if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Take a few minutes to write these rules down in your notebook.
Now let's apply these rules.
\begin{align*}12(5)\end{align*}
Notice that this is a multiplication problem. We use the parentheses around a single value to show multiplication. Now we can multiply the two values and then add the sign.
\begin{align*}12 \times 5 = 60\end{align*}
A negative value times a positive value is a negative value.
Our answer is 60.
Here is another one.
\begin{align*}\frac{150}{50}\end{align*}
Notice that this is a division problem. We use the fraction bar to show division. We do the division itself first.
\begin{align*}150 \div 50 = 3\end{align*}
A negative divided by a negative is a positive.
The answer is 3.
Example A
\begin{align*}9(7)\end{align*}
Solution:\begin{align*}63\end{align*}
Example B
\begin{align*}3(12)\end{align*}
Solution:\begin{align*}36\end{align*}
Example C
\begin{align*}169 \div 13\end{align*}
Solution:\begin{align*}13\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
First, let’s write an equation and then solve for the quotient.
To write an equation, we can use \begin{align*}x\end{align*}
We know that the loan amount is $250.00. That is a negative number.
250.00
We can divide that amount by the four months that the students have to repay the loan.
\begin{align*}x=\frac{250}{4}\end{align*}
Next we complete the division.
\begin{align*}x=\$62.50\end{align*}
If the students reimburse Mr. Janus $62.50 for each month for the next four months, then they will successfully pay back the loan.
Vocabulary
 Integer
 the set of whole numbers and their opposites {... 3, 2, 1, 0, 1, 2 ...}
Guided Practice
Here is one for you to try on your own.
\begin{align*}(8)(5)(10)\end{align*}
Solution
Here we have three values. Notice that the parentheses around single values means that we are going to multiply.
Multiply the values first to find the product then figure out the sign.
\begin{align*}8 \times 5 \times 10 = 400\end{align*}
Now let’s work through the signs. A negative times a positive is negative and then times another positive is still negative.
The answer is \begin{align*}400\end{align*}
Video Review
Multiplying and Dividing Integers
Practice
Directions: Multiply the following integers.

\begin{align*}6(8) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−6(−8)=−−−−− 
\begin{align*}5(10) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5(−10)=−−−−− 
\begin{align*}3(4) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3(−4)=−−−−− 
\begin{align*}3(4) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−3(4)=−−−−− 
\begin{align*}8(9) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8(−9)=−−−−− 
\begin{align*}9(12) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−9(12)=−−−−− 
\begin{align*}8(11) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8(−11)=−−−−− 
\begin{align*}(5)(9) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
(−5)(−9)=−−−−− 
\begin{align*}7(8) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−7(−8)=−−−−− 
\begin{align*}(12)(12) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
(−12)(12)=−−−−−
Directions: Divide the following integers.

\begin{align*}12 \div 2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−12÷2=−−−−− 
\begin{align*} 18 \div 6 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−18÷−6=−−−−− 
\begin{align*}24 \div 12 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−24÷12=−−−−− 
\begin{align*}80 \div 4 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−80÷−4=−−−−− 
\begin{align*}60 \div 30 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−60÷−30=−−−−− 
\begin{align*}\frac{28}{4}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
284=−−−−− 
\begin{align*}\frac{36}{4}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−364=−−−−− 
\begin{align*}\frac{45}{9}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−45−9=−−−−− 
\begin{align*}75 \div 25 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−75÷25=−−−−− 
\begin{align*}68 \div 2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
−68÷−2=−−−−−
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Here you'll learn to multiply and divide integers.