1C Formulas with Division

A common formula in Statistics is to calculate a z-score. The formula is given by $z=\frac{(x-\mu )}{\sigma }$. Many students have problems getting the correct z-score because they have trouble with division.

Two common applications of division is repeat subtraction and breaking a total into groups. For example Jimmy has $3000 in his account. How many times can he pay his rent of $400 before he runs out of money? Notice he is repeatedly subtracting the quantity $400. This is division. Remember the total should always come first in the division. So we can find the answer by dividing: $3000÷400$ = 7 remainder of 200. So he can pay his rent 7 times. After paying 7 times he will only have $200 in his account and will not be able to pay his rent.

Here is another example. Suppose a company has manufactured 13,600 paper bags. The bags are to be sent to five different supermarkets with each market getting an equal number of bags. How many bags were sent to each market? This is also division. Make sure to put the total first in the division. Dividing we get $13600÷5$ = 2,720. So each market will get 2,720 bags.

Let’s look at a few special division problems involving zero and one. First of all, any number divided by 1 is the same as the number itself. For example, let’s look at $53÷1$. This is asking how many times we can subtract 1 from 53. The answer is of course 53 times. Note, be careful of the order of division. $53÷1\ne 1÷53$. The 1 has to be the number you are dividing by (divisor). $1÷53$ can be written as a fraction or decimal. It is not 53!

Any non-zero number divided by itself is equal to 1. For example, let’s look at $\frac{37}{37}$. This is asking how many times you can subtract 37 from 37. The answer is of course 1 time.

You have to be careful with division problems involving zero. Zero divided by any non-zero number is equal to zero. For example, let’s look at $0÷19$. This is asking what number we can multiply by 19 and get an answer of 0. ($19\times ?=0$). The answer of course is zero.

Be careful though. A non-zero number divided by zero is undefined (does not exist). For example, let’s look at $13÷0$. This is asking what number we can multiply by 0 and get an answer of 13. ($0\times ?=13$). This is impossible! Remember any number times 0 is 0. It is impossible to get an answer of 13. That is why we say that a number divided by zero is undefined.

Do the following examples with your instructor. You can use the usual algorithm or the repeated subtraction approach. Leave your answer as quotient and remainder.

If you struggle with the standard algorithm of long division, then the repeat subtraction approach may work well for you. Here is how it works. Look at the bag problem. We were trying to figure out $13600÷5$. What we are asking is how many times can we subtract 5 from 13600? Take a guess. Say 1000 times. If we subtract 5x1000 we are left with 8600. Subtract 5 another 1000 times. We would be left with 3600. Keep guessing and subtracting. When you can’t subtract a 5 anymore you are done. Just keep track of how many times you subtracted 5. In the end, add up all the times you subtracted 5. The better your guesses get the fewer times you have to subtract.

$$\begin{array}{l}13600\\ \underset{\_}{-5000}\text{ (5 subtracted 1000 times)}\\ 8600\\ \underset{\_}{-5000}\text{ (5 subtracted 1000 times)}\\ 3600\\ \underset{\_}{-2000}\text{ (5 subtracted 400 times)}\\ 1600\\ \underset{\_}{-1000}\text{ (5 subtracted 200 times)}\\ 600\\ \underset{\_}{-500}\text{ (5 subtracted 100 times)}\\ 100\\ \underset{\_}{-100}\text{ (5 subtracted 20 times)}\\ 0\\ \text{ total number of times subtracted = 2720 with zero remainder}\end{array}$$

Do the following examples with your instructor. You can use the usual algorithm or the repeated subtraction approach. Leave your answer as quotient and remainder.

Practice Problems Section 1C

Divide the following using either repeated subtraction or the usual algorithm. You can leave your answer as quotient and remainder.

1. $$17÷1$$

   Check answer

   17

2. $$21÷0$$

3. $$0÷4$$

   Check answer

   0

4. $$\frac{0}{6}$$

5. $$\frac{13}{0}$$

   Check answer

   Undefined

6. $$\frac{245}{1}$$

7. $$\frac{\mathrm{2,635}}{1}$$

   Check answer

   2,635

8. $$375÷0$$

9. $$\frac{64}{64}$$

   Check answer

   1

10. $$17÷17$$

11. $$0÷16$$

    Check answer

    0

12. $$33÷0$$

13. $$\frac{20}{0}$$

    Check answer

    Undefined

14. $$\frac{0}{12}$$

15. $$\frac{382}{1}$$

    Check answer

    382

16. $$\frac{85}{85}$$

17. $$0÷26$$

    Check answer

    0

18. $$\frac{527}{1}$$

19. $$35÷3$$

    Check answer

    11 R 2

20. $$99÷12$$

21. $$76÷10$$

    Check answer

    7 R 6

22. $$17÷6$$

23. $$73÷9$$

    Check answer

    8 R 1

24. $$92÷11$$

25. $$50÷13$$

    Check answer

    3 R 11

26. $$133÷7$$

27. $$748÷4$$

    Check answer

    187

28. $$96÷12$$

29. $$420÷6$$

    Check answer

    70

30. $$300÷18$$

31. $$83÷5$$

    Check answer

    16 R 3

32. $$724÷8$$

33. $$\mathrm{4,000}÷15$$

    Check answer

    266 R 10

34. $$564÷9$$

35. $$\mathrm{5,000}÷13$$

    Check answer

    384 R 8

36. $$7461÷20$$

37. $$368÷12$$

    Check answer

    30 R 8

38. $$\mathrm{6,000}÷17$$

39. $$\mathrm{4,813}÷25$$

    Check answer

    192 R 13

40. $$\frac{3450}{82}$$

41. $$\frac{1256}{28}$$

    Check answer

    44 R 24

42. $$\frac{\mathrm{40,000}}{300}$$

43. $$\frac{2481}{42}$$

    Check answer

    59 R 3

44. $$\frac{2546}{19}$$

45. $$\frac{\mathrm{20,007}}{130}$$

    Check answer

    153 R 117

46. $h=\frac{2A}{b}$ is a formula in geometry that can be used to calculate the height (h) of a triangle if we know the area (A) and the base (b). A sail boat has a sail that has an area of 120 square feet. If the base is 6 feet, what is the height of the sail?

47. $h=\frac{2A}{b}$ is a formula in geometry that can be used to calculate the height (h) of a triangle if we know the area (A) and the base (b). Another boat sail has an area of 170 square feet and a base of 20 feet. What is the height of the sail?

    Check answer

    17 ft

48. Jerome needs to save $10,080 to buy a car. If he can save $420 per month, how many months will it take him to reach his goal?

49. Mia has $5160 in her checking account. Her monthly rent is $890. If she does not deposit any money into the account, how many months can she pay her rent before the money runs out?

    Check answer

    5 mo R $710

50. Jed is saving up to buy a guitar. The guitar he wants is $2470. He makes $330 per week at his work. If he saves his entire weekly paychecks, how many weeks will he have to work to afford the guitar?

51. Trianna’s loves to buy candy at the corner store before school each morning. Her mom gave her $19 for candy. The candy she loves the best costs $2. How many days can she buy her favorite candy before she runs out of money. Will she have any money left over?

    Check answer

    9 days R $1

52. Let’s look at the statistics formula for z-score we looked at in the beginning of this section $z=\frac{(x-\mu )}{\sigma }$. Because the subtraction is in a parenthesis, we need to do this first. Find the z-score if $x=23,\mu =17,\sigma =6$. A z-score larger than two is considered unusual. Is this an unusual z-score?

53. Let’s look at the statistics formula for z-score we looked at in the beginning of this section $z=\frac{(x-\mu )}{\sigma }$. Because the subtraction is in a parenthesis, we need to do this first. Find the z-score if $x=55,\mu =27,\sigma =4$. A z-score larger than two is considered unusual. Is this an unusual z-score?

    Check answer

    7 (unusual)

54. Let’s look at the statistics formula for z-score we looked at in the beginning of this section $z=\frac{(x-\mu )}{\sigma }$. Because the subtraction is in a parenthesis, we need to do this first. Find the z-score if $x=237,\mu =141,\sigma =32$. A z-score larger than two is considered unusual. Is this an unusual z-score?

55. Let’s look at the statistics formula for z-score we looked at in the beginning of this section $z=\frac{(x-\mu )}{\sigma }$. Because the subtraction is in a parenthesis, we need to do this first. Find the z-score if $x=143,\mu =101,\sigma =14$. A z-score larger than two is considered unusual. Is this an unusual z-score?

    Check answer

    3 (unusual)

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