4C Subtracting Negative Quantities

Suppose we want to find the difference between 7 degrees below zero and 16 degrees below zero. Remember to find the difference between quantities we need to subtract, but how do we subtract negative numbers? That is the topic of this section.

The key to subtracting signed numbers is the idea of an opposite. Opposites have the same absolute value. Nonzero opposites have opposite signs; zero is its own opposite. For example the opposite of -4 is +4. The opposite of +7.241 is -7.241.

Look at the following key example. Suppose Julie has $24 and then spends $5, how much will she have left? We know from previous experience that this is a take away and can be found by the subtraction 24 – 5 = $19. Notice that losing $5 can also be thought of as adding -5. In fact if we added -5, we get the following. $+24+-5=+19$. Did you notice that we get the same answer? This shows that $$24-5=+24-+5=+24+-5$$. Notice subtracting a quantity is the same as adding the opposite of the quantity. Subtracting +5 is the same as adding the opposite of +5. Since the opposite of +5 is -5, we see that subtracting +5 is the same as adding -5. This principle of adding the opposite can be applied to any subtraction problem. We simply add the opposite.

For example, suppose we want to subtract the following. $+7--4$. Again we can change the subtraction into adding the opposite. Remember that we want to take the opposite of the number being subtracted. This is the second number. Do not take the opposite of the first number. The opposite of -4 is +4. So subtracting -4 is the same as adding +4. Hence

$$+7--4=+7++4=+11$$. Notice the +7 stayed the same.

Let’s look at another example. $-9--10$. Again the opposite of -10 is +10 so we get $-9--10=-9++10$. Now we need to remember our addition rules for negative numbers. To add two numbers with opposite signs, we subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. If the absolute values are equal, the sum is zero. Hence 10 – 9 = 1. Since the 10 was positive, our answer will also be positive (+1).

What happens when we add opposites? Look at $-13++13$. To add two numbers with opposite signs, we compare their absolute values. Equal absolute values cancel to zero. So 13-13 = 0. 0 is the only number that is neither positive nor negative. So $-13++13=0$. Hence when we add two opposites, the answer is always zero.

Subtract the following examples with your instructor.

Practice Problems Section 4C

Subtract the following.

1. $$+3-+9$$

   Check answer

   ^-6

2. $$-12-+6$$

   Check answer

   ^-18

3. $$+8-+7$$

   Check answer

   1

4. $$+3--10$$

   Check answer

   13

5. $$-9--3$$

   Check answer

   ^-6

6. $$+8--12$$

   Check answer

   20

7. $$-14--5$$

   Check answer

   ^-9

8. $$-17-+6$$

   Check answer

   ^-23

9. $$-8--8$$

   Check answer

   0

10. $$+21-+30$$

    Check answer

    ^-9

11. $$-16-+11$$

    Check answer

    ^-27

12. $$+7-+7$$

    Check answer

    0

13. $$-14-+6$$

    Check answer

    ^-20

14. $$-17--12$$

    Check answer

    ^-5

15. $$+18-+11$$

    Check answer

    7

16. $$+13--16$$

    Check answer

    29

17. $$-17--21$$

    Check answer

    4

18. $$-28--19$$

    Check answer

    ^-9

19. $$-24-+16$$

    Check answer

    ^-40

20. $$-52-+36$$

    Check answer

    ^-88

21. $$-26--26$$

    Check answer

    0

22. $$+19-+34$$

    Check answer

    ^-15

23. $$-23-+14$$

    Check answer

    ^-37

24. $$+19-+19$$

    Check answer

    0

Remember that the idea of adding the opposite also applies to fractions and decimals. Try the following.

25. $$+0.074-+0.183$$

    Check answer

    ^-0.109

26. $$-\frac{2}{3}-(+\frac{1}{6})$$

    Check answer

    $$-\frac{5}{6}$$

27. $$+5.7-+2.61$$

    Check answer

    3.09

28. $$+\frac{1}{8}-(-\frac{1}{5})$$

    Check answer

    $$\frac{13}{40}$$

29. $$-9.17--6.8$$

    Check answer

    ^-2.37

30. $$\frac{3}{7}-2⁤\frac{5}{7}$$

    Check answer

    ^- $2⁤\frac{2}{7}$

There are many applications of subtraction.

31. One scuba diver is 24 feet below sea level. A second scuba diver is 37 feet below sea level. How much deeper is the second scuba diver than the first?

    Check answer

    The second diver is 13 feet deeper than the first.

32. Death Valley has a height of 86 meters below sea level. Mount Everest has a height of 8,848 meters above sea level. How much taller is Mount Everest than Death Valley?

    Check answer

    Mt. Everest is 8,934 meters taller than Death Valley.

33. Mark and Ryan both gamble on horse races. One Sunday, Mark lost $235 and Ryan lost $197. How much more did Mark lose than Ryan?

    Check answer

    Mark lost $38 more than Ryan.

34. Aria owes the electric company $73 for June and owes $112 for July. What is the difference between the two bills?

    Check answer

    Aria’s electric bill is $39 more for July than that for June.

35. The daytime temperature on the moon’s surface can reach 253 degrees Fahrenheit, while the nighttime temperature can dip to 243 degrees Fahrenheit below zero. What is the difference between these two temperatures?

    Check answer

    496 degrees

36. On a winter day in Dallas, TX, the projected high temperature during the day is 12 degrees Celsius, which is 21 degrees higher than the projected low temperature that night. What is the projected low temperature?

    Check answer

    ^- 9$\text{°C}$

37. A small business that sells hats in a mall kiosk has a credit balance of $-\mathrm{11,239}$ dollars. If in the next 2 months they lose $2,460, what will their new balance be?

    Check answer

    ^-$13,699

38. A basketball player’s “plus/minus” is a number that is calculated by adding all of the positive points her team scores with all of the negative points her team gives up while she is in the game. While the starting center was in the game, her team scored 45 points and her plus/minus value was $-8$. Subtract 45 from $-8$ to find the negative contribution from points conceded. Use its absolute value to determine how many points her team gave up while she was in the game.

    Check answer

    The negative contribution is ^-53 points, so her team gave up 53 points.

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