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# Integer Multiplication

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

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Multiply and Divide Integers

The bald eagle has been estimated to dive at a speed of 75 miles per hour or 110 feet per second. Although the eagle will normally dive slower than this in order to have a more accurate attack, it is still possible they could dive at such speed. How far would the eagle be able to go at this speed in 2 seconds? If the eagle sees prey 750 feet below, how much farther must it dive to reach its prey?

In this concept, you will learn to multiply and divide integers.

### Multiplying and Dividing Integers

Integers are the values in the set of whole numbers and their opposites, {3,2,1,0,1,2,3}\begin{align*}\{\ldots -3, -2, -1, 0, 1, 2, 3 \ldots \}\end{align*}. When you multiply and divide integers, there are some rules:

1. When multiplying or dividing a positive integer by a positive integer, the product or quotient is positive.
2. 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.
3. When multiplying or dividing a negative integer by a negative integer, the product or quotient is positive.

Another way to remember multiplying and dividing integers is to remember if the signs are the same, the answer is positive. If the signs are different, the answer is negative.

Let’s look at an example where you can apply these rules.

12(5)\begin{align*}12(-5)\end{align*}

Notice that this is a multiplication problem. Parentheses are used around a single value to show multiplication.

First, multiply the two values.

12×5=60\begin{align*}12 \times 5 = 60\end{align*}

Next, add the sign. Remember rule 2, when you multiply a positive integer by a negative integer, the answer is negative.

Here is another example.

15050\begin{align*}\frac{-150}{-50}\end{align*}

First, divide the two values.

15050=3\begin{align*}\frac{150}{50} = 3\end{align*}

Next, add the sign. Remember rule 3, when you dividing a negative integer by a negative integer, the answer is positive.

### Examples

#### Example 1

Remember the bald eagle? You know the distance  (d)\begin{align*}(d)\end{align*}, speed (v)\begin{align*}(v)\end{align*}, and time (t)\begin{align*}(t)\end{align*}. When you use these three variables together, you use the formula: v=dt\begin{align*}v= \frac{d}{t}\end{align*}.

Now, look at the two questions in the original problem.

How far would the eagle be able to go at this speed in 2 seconds?

v=110ftsec,t=2 sec,d=?\begin{align*}v = 110 \frac{ft}{sec}, t = 2 \ sec, d =?\end{align*}

First, you need to find the distance so rearrange the formula to solve for the distance.

d=v(t)\begin{align*}d=v(t)\end{align*}

Next, put the values for speed and time into the formula to solve for the distance.

d=110(2)\begin{align*}d=110(2)\end{align*}

Then, solve for the distance. Remember rule 1, when you multiply a positive integer by a positive integer, the answer is positive.

d=220\begin{align*}d=220\end{align*}

The distance is 220 feet.

If the eagle sees prey 750 feet below, how much farther must it dive to reach its prey?

v=110ftsec,d=750 ft,t=?\begin{align*}v = 110 \frac{ft}{sec}, d = 750 \ ft, t =?\end{align*}

First, you need to find the time so rearrange the formula to solve for the time.

t=dv\begin{align*}t=\frac{d}{v}\end{align*}

Next, put the values for distance and speed into the formula to solve for the time.

t=750110\begin{align*}t = \frac{750}{110}\end{align*}

Then, solve for the time. Remember rule 1, when you divide a positive integer by a positive integer, the answer is positive.

t=6.82\begin{align*}t=6.82\end{align*}

The time is 6.82 seconds.

#### Example 2

Multiply.

(8)(5)(10)\begin{align*}(8)(-5)(10)\end{align*}

First, multiply the three values.

8×5×10=400\begin{align*}8 \times 5 \times 10 = 400\end{align*}

Next, add the sign. Remember rule 2, when you multiply a positive integer(s) by a negative integer, the answer is negative.

#### Example 3

Multiply.

9(7)\begin{align*}-9(7)\end{align*}

First, multiply the two values.

9×7=63\begin{align*}9 \times 7=63\end{align*}

Next, add the sign. Remember rule 2, when you multiply a positive integer by a negative integer, the answer is negative.

#### Example 4

Multiply.

3(12)\begin{align*}-3(-12)\end{align*}

First, multiply the two values.

3×12=36\begin{align*}3 \times 12=36\end{align*}

Next, add the sign. Remember rule 3, when you multiplying a negative integer by a negative integer, the answer is positive.

#### Example 5

Divide.

16913\begin{align*}\frac{-169}{13}\end{align*}

First, divide the two values.

16913=13\begin{align*}\frac{169}{13} = 13\end{align*}

Next, add the sign. Remember rule 2, when you divide a negative integer by a positive integer, the answer is negative.

### Review

Multiply the following integers.

1. 6(8)=\begin{align*}-6(-8) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

2. 5(10)=\begin{align*}5(-10) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

3. 3(4)=\begin{align*}3(-4) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

4. 3(4)=\begin{align*}-3(4) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

5. 8(9)=\begin{align*}8(-9) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

6. 9(12)=\begin{align*}-9(12) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

7. 8(11)=\begin{align*}8(-11) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

8. (5)(9)=\begin{align*}(-5)(-9) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

9. 7(8)=\begin{align*}-7(-8) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

10. (12)(12)=\begin{align*}(-12)(12) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Divide the following integers.

11. 12÷2=\begin{align*}-12 \div 2 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

12. 18÷6=\begin{align*}-18 \div -6 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

13. 24÷12=\begin{align*}-24 \div 12 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

14. 80÷4=\begin{align*}-80 \div -4 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

15. 60÷30=\begin{align*}-60 \div -30 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

16. 284=\begin{align*}\frac{28}{4}= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

17. 364=\begin{align*}\frac{-36}{4}= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

18. 459=\begin{align*}\frac{-45}{9}= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

19. 75÷25=\begin{align*}-75 \div 25= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

20. 68÷2=\begin{align*}-68 \div -2 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

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