2C Formulas with Dividing Decimals

Look at the following z-score formula again from statistics. $z=\frac{(x-\mu )}{\sigma }$. Suppose we want to calculate the z-score if $$x=17.6\text{ pounds},\mu =13.8\text{ pounds},\text{ and }\sigma =2.5\text{ pounds}$$. Not only do we need to be able to subtract decimals, but we also need to be able to divide decimals. So let’s review dividing decimals some.

Much like multiplying decimals, the key to dividing decimals is that we need to make sure to put the decimal in the correct place. We need to make sure that the number we are dividing by (divisor) is a whole number. If the divisor is a whole number then the decimal in the answer will be directly above where it is in the number we are dividing. That is pretty confusing, but I hope an example will help clear it up.

Look at $1.46÷2.5$. One thing to note is that the first number (dividend) always goes on the inside of the long division. The number you are dividing by (divisor) goes on the outside.

$\begin{array}{cc}2.5& 1.46\end{array}$ Since the divisor is not a whole number, we need to move the decimal one place to the right. We will have to move the decimal inside 1 place to the right as well. Hence the division becomes:

$\begin{array}{cc}25& 14.6\end{array}$ Notice the dividend does not have to be a whole number. Only the divisor does. Now divide as if they are whole numbers. You may need to add zeros to the end of the dividend if it does not come out even.

$$\begin{array}{cc}& 0.584\\ 25& 14.600\\ & -125\\ & 210\\ & -200\\ & 100\\ & -100\\ & 0\end{array}$$

Notice we keep adding zeros and dividing until we get a zero remainder or we see a repeating pattern. In this case we got a zero remainder after the thousandths place. Notice also the decimal point is directly above where it is in the dividend. This again is only the case when the divisor is a whole number. We can also use estimation to see if the decimal point is in the right place. 14 is slightly over half of 25, so the answer should be a little bigger than one-half (0.5).

Occasionally we divide decimals by powers of 10 (10, 100, 1000, etc.) Remember when you multiply by a power of 10 the decimal point simply moves to the right. It is similar with division. If you divide by a power of 10, you move the decimal point to the left the same number of places as the number of zeros in your power of 10. For example $1.49÷1000$ moves the decimal point 3 places to the left so 0.00149 is the answer. A good way to remember the left and right rules is to think of it as bigger and smaller. When we multiply by a number like 1000, we are making the number larger, so the decimal point has to move to the right. When we divide by a number like 1000, we are making the number smaller, so the decimal point has to move to the left.

Let’s look at an example with a repeating pattern. Look at $0.052÷0.99$. First we need to make the divisor a whole number. Remember the divisor is the number you are dividing by (0.99). Notice we will need to move the decimal point two places to the right to make 0.99 into a whole number (99). If we move the decimal point two places in divisor we also have to move the decimal point two places in the dividend (first number in the division). Moving the decimal point two places gives us 5.2. Notice the dividend does not have to be a whole number. Once the divisor 0.99 becomes a whole number, the decimal point of the dividend will be in the correct place for our division. Note: Notice the process of moving the decimal is really the same as multiplying the numerator and denominator of a fraction by the same number (100). This is a common practice with fractions and always gives an equivalent fraction. So $0.052÷0.99$ is the same as $5.2÷99$. Be careful in the order of division. Remember the first number goes inside the long division and the second number goes outside. Notice that the remainders repeat, 52 then 25, then 52, then 25. This will go on infinitely. When this happens, we put a bar over the repeating part. So our answer is $0.0\overline{52}$. Notice that we do not need to move the decimal back. It is in the correct place. So $0.052÷0.99$ is the same as $5.2÷99$. It is also good to note that we only need to put the bar over the first set of numbers that repeat. (For example, we do not need to write the answer as $0.0\overline{52525252}$).

$$\begin{array}{cc}& 0.0525252\\ 99& 5.2000000\\ & -495\\ & 250\\ & -198\\ & 520\\ & -495\\ & 250\\ & -198\\ & 52\end{array}$$

Now you try some examples with your instructor.

Fractions can also be converted to decimals simply by dividing the numerator by the denominator. Remember the denominator is the divisor and has to go on the outside of the long division. For example the fraction 5/8 is really $5÷8$, so if we add zeros to the 5 and divide, we get the following:

$\begin{array}{cc}& 0.625\\ 8& 5.000\end{array}$ Notice we did not have to move the decimal as our divisor 8 was already a whole number.

Try some more examples with your instructor.

Practice Problems Section 2C

Divide the following and write your answer as a decimal or whole number.

1. $$1.98÷100$$

   Check answer

   0.0198

2. $$0.036÷10$$

3. $$0.51÷1000$$

   Check answer

   0.00051

4. $$0.238÷10$$

5. $$317÷10000$$

   Check answer

   0.0317

6. $$4.23÷1000$$

7. $$13.8÷100$$

   Check answer

   0.138

8. $$0.00041÷1000$$

9. $$7.2÷10$$

   Check answer

   0.72

10. $$0.0334÷1000$$

11. $$55.7÷100$$

    Check answer

    0.557

12. $$6.644÷10000$$

13. $$799÷1000$$

    Check answer

    0.799

14. $$44.81÷10$$

15. $$7851÷100000$$

    Check answer

    0.07851

16. $$138.4÷{10}^2$$

17. $$0.00558÷{10}^3$$

    Check answer

    0.00000558

18. $$947÷{10}^4$$

19. $$0.43÷1$$

    Check answer

    0.43

20. $$0÷5.61$$

21. $$0.317÷0$$

    Check answer

    Undefined

22. $$0÷1.0987$$

23. $$0.0361÷1$$

    Check answer

    0.0361

24. $$5.97÷0$$

25. $$32.55÷1$$

    Check answer

    32.55

26. $$0.7709÷0$$

27. $$0.007÷1$$

    Check answer

    0.007

28. $$0÷24.6$$

29. $$99.428÷1$$

    Check answer

    99.428

30. $$0÷0.00581$$

31. $$0.035÷5$$

    Check answer

    0.007

32. $$0.356÷4$$

33. $$0.414÷3$$

    Check answer

    0.138

34. $$0.0054÷20$$

35. $$6.51÷12$$

    Check answer

    0.5425

36. $$0.841÷9$$

37. $$0.0045÷0.05$$

    Check answer

    0.09

38. $$4.16÷0.2$$

39. $$0.418÷0.03$$

    Check answer

    $$13.9\overline{3}$$

40. $$13.4÷0.008$$

41. $$0.247÷1.6$$

    Check answer

    0.154375

42. $$7÷0.06$$

43. $$0.444÷0.24$$

    Check answer

    1.85

44. $$0.0032÷1.2$$

45. $$3000÷0.25$$

    Check answer

    12,000

Convert the following fractions into decimals. Put a bar over any repeating parts.

46. Convert $\frac{1}{4}$ into a decimal.

47. Convert $\frac{2}{3}$ into a decimal.

    Check answer

    $$0.\overline{6}$$

48. Convert $\frac{3}{5}$ into a decimal.

49. Convert $\frac{5}{9}$ into a decimal.

    Check answer

    $$0.\overline{5}$$

50. Convert $\frac{7}{99}$ into a decimal.

51. Convert $\frac{3}{8}$ into a decimal.

    Check answer

    0.375

52. Convert $\frac{7}{24}$ into a decimal.

53. Convert $\frac{3}{25}$ into a decimal.

    Check answer

    0.12

54. Convert $\frac{7}{33}$ into a decimal.

55. Convert $\frac{1}{16}$ into a decimal.

    Check answer

    0.0625

56. Convert $\frac{19}{30}$ into a decimal.

57. Convert $\frac{17}{60}$ into a decimal.

    Check answer

    $$0.28\overline{3}$$

58. Jim works in construction and gets paid $1847.30 every four weeks. How much does he make each week?

59. Miko needs $14623.40 for a down payment on a home. If she can save $860.20 each month, how many months will it take her to get the money she needs? (Not counting interest).

    Check answer

    17 mo

60. Elena owns a jewelry store and made $4865.77 in total sales by selling diamond bracelets. If each diamond bracelet costs $374.29, then how many bracelets did she sell?

61. Wanda is a chemist working for a pharmaceutical company. She has a total of 880.46 mg of a new medicine. She needs to make pills where each pill has 7.5 mg in it. How many pills can she make? Will she have any left-over medicine?

    Check answer

    117 pills, yes, 2.96 mg left over

Apply the z-score formula to the following practice values. Use the formula $z=\frac{(x-\mu )}{\sigma }$

62. Find the z-score if $$x=17.6\text{ pounds},\mu =13.8\text{ pounds},\text{ and }\sigma =1.6\text{ pounds}$$.

63. Find the z-score if $$x=73.1\text{ inches},\mu =69.2\text{ inches},\text{ and }\sigma =2.5\text{ inches.}$$

    Check answer

    1.56

The following formula calculates the volume of a cone in geometry. $V=\frac{\left(\pi r^{2}h\right)}{3}$

64. Find the volume of a cone with a radius of 4 cm and a height of 1.2 cm. (Use $\pi \approx 3.14$)

65. Find the volume of a cone with a radius of 0.5 yards and a height of 2 yards. (Use $\pi \approx 3.14$)

    Check answer

    $$0.52\overline{3}$$

    cubic yards

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