4D Multiplying and Dividing Negative Quantities

Every month, Sophie pays her rent and her bank account balance decreases $457. If she pays her rent six months in advance, how much did her account decrease? Since a loss of $457 can be thought of as -$457, Sophie has -457 six times or $+6\times -457$, but how do we multiply and divide negative numbers? Let’s review.

If we think about the example above, Sophie loses $457 six times so she has to have lost a total of 6x457 = 2742. Hence her account balance decreased $2742. This gives us insight into the first rule for multiplication. The example shows that $+6\times -457=-2742$. When we multiply or divide two numbers with opposite signs the answer is always negative.

For example, look at $-8\times +14$. Since they have opposite signs, we multiply the numbers. The answer will be negative. Hence $-8\times +14=-112$.

To multiply or divide two numbers with the same sign, we multiply or divide and the answer will always be positive.

For example, look at $-9\times -7$. Since they have the same sign (both negative), we multiply the numbers (7x9=63). The answer will be positive. Hence $-9\times -7=+63$.

These rules apply to division as well. Look at $-18÷-3$. Since this problem involves division with the same sign the answer will be positive. Hence $-18÷-3=+6$. Another example is $+32÷-8$. Since this problem involves division with opposite signs the answer will be negative.

Hence $+32÷-8=-4$.

Do the following examples with your instructor:

Practice Problems Section 4D

For #1-30, perform the following operations.

1. $$-7\times +2$$

   Check answer

   ^-14

2. $$-5\times -13$$

   Check answer

   65

3. $$-12\times +8$$

   Check answer

   ^-96

4. $$(-16)(+3)$$

   Check answer

   ^-48

5. $$(+9)(-11)$$

   Check answer

   ^-99

6. $$(-21)(-2)$$

   Check answer

   42

7. $$-15÷+3$$

   Check answer

   ^-5

8. $$-35÷-5$$

   Check answer

   7

9. $$0÷-1$$

   Check answer

   0

10. $$-45\times 0$$

    Check answer

    0

11. $$-51÷+17$$

    Check answer

    ^-3

12. $$+85÷-17$$

    Check answer

    ^-5

13. $$-8\times +12$$

    Check answer

    ^-96

14. $$\frac{-13}{0}$$

    Check answer

    undefined

15. $$\frac{-36}{-4}$$

    Check answer

    9

16. $$-9\times +15$$

    Check answer

    ^-135

17. $$-12\times -20$$

    Check answer

    240

18. $$-14\times +6$$

    Check answer

    -84

19. $$(-25)(+7)$$

    Check answer

    ^-175

20. $$(+7)(-23)$$

    Check answer

    -161

21. $$(-42)(-10)$$

    Check answer

    420

22. $$-75÷+15$$

    Check answer

    ^-5

23. $$-144÷-9$$

    Check answer

    16

24. $$0÷-13$$

    Check answer

    0

25. $$-19\times 0$$

    Check answer

    0

26. $$-64÷+16$$

    Check answer

    ^-4

27. $$+1233÷-9$$

    Check answer

    ^-137

28. $$-76\times +1$$

    Check answer

    ^-76

29. $$\frac{-305}{0}$$

    Check answer

    undefined

30. $$\frac{-1696}{-8}$$

    Check answer

    212

As with all the rules in this chapter, they apply to decimals and fractions as well.

31. $$+3.4\times -1.8$$

    Check answer

    ^-6.12

32. $$(-\frac{9}{16})\left(\frac{8}{27}\right)$$

    Check answer

    $$-\frac{1}{6}$$

33. $$(-0.15)(-0.43)$$

    Check answer

    0.0645

34. $$-3.42÷+0.3$$

    Check answer

    ^-11.4

35. $$(-\frac{5}{14})÷\left(\frac{15}{7}\right)$$

    Check answer

    $$-\frac{1}{6}$$

36. $$-0.042÷-0.25$$

    Check answer

    0.168

There are many applications involving multiplication and division with negative numbers. Look at the following.

37. Juan pays $45 a month for his electric bill. His bank statement indicates that Juan has paid a total of $675 to the electric company. How many months has he paid this bill?

    Check answer

    15 months

38. The temperature Monday morning was $-3\text{°}F$. During the day the temperature dropped 4 degrees six times. What was the temperature Monday night?

    Check answer

    -27$\text{°F}$

39. Charlie is building a new house that will not be connected to any city water pipes, so he plans to dig a well that will reach $-100$ ft deep. If he can dig $-8$ ft each hour, how many hours will it take him to reach his goal of $-100$ ft?

    Check answer

    12.5 hours

40. A football team lost 6 yards on each of the first 3 rushing (running) plays. What was the team’s rushing total?

    Check answer

    ^-18

41. An investment banker was fired after losing $600,000 over his first year on the job. On average, how much did he net each month?

    Check answer

    ^-$50,000 per month

A common formula in statistics is $x=\mu +\left(z\right)\left(\sigma \right)$.

42. If $\mu =8.24$, $z=-1.35$ and $\sigma =3.61$, find x. (Remember to do multiplication before addition.)

    Check answer

    $$x=3.3665$$

43. If $\mu =62.9$, $z=-3.48$ and $\sigma =2.8$, find x. (Remember to do multiplication before addition.)

    Check answer

    $$x=53.156$$

44. If $\mu =-15$, $z=+2.4$ and $\sigma =+1.75$, find x. (Remember to do multiplication before addition.)

    Check answer

    $$x=-10.8$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.