Review of Basic Mathematical Skills

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Chapter 2

Review of Basic Mathematical Skills

Objectives

• Add, subtract, multiply, and divide whole numbers

• Add, subtract, multiply, and divide fractions and reduce fractions to the lowest terms

• Add, subtract, multiply, and divide mixed numbers and reduce to the lowest terms

• Add, subtract, multiply, and divide decimals and round them to a specific number place value

• Convert fractions to decimal fractions

• Convert among fractions, decimals, and percentages

• Express numbers in ratio and proportion and solve for unknowns

Key Words

Pretest

Complete the following answers, and show all of your work. Round all decimals to the closest thousandth unless otherwise stated.

1. image = __________

2. 154 + 1063 + 25 + 2376 = __________

3. 163 − 69 = __________

4. 256 × 42 = __________

5. 256 ÷ 16 = __________

6. 3655 − 29 = __________

7. 25.6 + 1456 + 35.67 = __________

8. 354.29 − 45.390 = __________

9. 12.56 × 65.031 = __________

10. 655.08 ÷ 1.2 = __________

11. image = __________

12. image = __________

13. image = __________

14. image = __________

15. image = __________

16. image = __________

17. image = __________

18. image = __________

19. image = __________

20. image = __________

21. image = __________

22. image = __________

23. image = __________

24. image = __________

25. image = __________

26. image = __________

27. image = __________

28. 100.2 × 100.03 = __________

29. 721 − 0.01 = __________

30. 12.02 ÷ 6.01 = __________

31. A roll of adhesive is image inches long. The physician asks you to be sure the person has sufficient amount to apply bandages daily for 2 weeks using approximately image yards per bandage. How many rolls of adhesive should you be sure the patient buys?

32. A solution contains image cups of water. How many milliliters would be in image of the solution?

    Round the following to the nearest one hundredth.

33. 75.0023 __________

34. 12.015 __________

35. 6.12453 __________

36. 126.444 __________

    Change the following to decimal fractions.

37. image __________

38. image __________

39. image __________

40. image __________

    Change the following to ratios and reduce to the lowest terms possible.

41. image __________

42. image __________

43. image __________

44. image __________

    Solve the following proportions.

45. x : 3 :: 15 : 30 = __________

46. 5 : 15 :: 8 : x = __________

47. 3 : x :: 11 : 33 = __________

48. 1 : 50 :: x : 40 = __________

49. A pharmacist is able to fill six prescriptions in 10 minutes with the assistance of a pharmacy tech. How many prescriptions can be completed in 1 hour?

50. A prescription calls for 15 mL of water to reconstitute a medication in a 20-mL vial. How many milliliters of water would be needed proportionally to reconstitute 50-mL vial of the same medication?

    Interpret the following abbreviations.

51. mL __________

52. tsp __________

53. mg __________

54. g __________

55. gr __________

56. mcg __________

57. tbsp __________

58. image __________

59. image __________

60. pt __________

Whole Numbers

What Is a Whole Number?

A whole number is the basic mathematical symbol that we use on a daily basis. In the math world, whole numbers represent numerals consisting of one or more digits. The digits used in math are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numbers may also be written in words, such as one, two, ten, one thousand, ten million one hundred three, and the like. The word “and” should not be used between digits in a whole number. In the medical field, commas are not placed in numbers with more than four digits or seven digits, such as 1275 or 2356302. A comma might be misinterpreted as a decimal when used in whole numbers, causing a medication error.

A whole number does not have a decimal, fraction, or percentage. When you add, subtract, or multiply whole numbers, the answer will always remain a whole number such as 10, 25, or 100. For example, 45 + 12 = 57 or 5 × 45 = 225 because all of the numbers are whole numbers. In some cases if dividing whole numbers, the answer may be found as a whole number, a decimal number, or a whole number with a decimal number. For example, 10 divided by 2 is 5 or 11 divided by 3 would be image; all the numbers in this example contain whole numbers, but they are not evenly divisible and may have decimal remainders.

Example 2-1

image

A whole number is always to the left of a decimal point. The numeral 0 is a whole number with the decimal understood to be at the right side of the zero. In pharmaceutical mathematics, if no number is to the left of a decimal, a zero (0) is placed in the whole number place.

Tech Note

Whole numbers are numbers found to the left of the decimal point and may have one digit or more. Fractions and decimals may be added to a whole number, but the whole number is to the left in each of these cases. Examples: 25, 25.1, image.

Tech Note

Whole numbers are used in strengths of medications, doses of medications, inventory, and costs of medications, to give just a few examples of the need to understand this concept.

Adding and Subtracting Whole Numbers

When adding or subtracting whole numbers, align the digits from the right and then add or subtract one set of digits at a time starting from the right (from ones to tens to hundreds, etc.). Addition is finding a sum or total of two numbers. Subtraction is finding the difference between two numbers.

Example 2-2

image

Practice Problems A

Calculate the following problems, being sure to show your work and stating measurement designations as appropriate.

1. 25 + 56 = __________

2. 105 + 235 = __________

3. 1243 + 356 = __________

4. 50 g + 100 g = __________

5. 25 mg + 90 mg = __________

6. 240# + 133# = __________

7. 125,000 units + 250,000 units = __________

8. 7500 mL + 1000 mL = __________

9. 78 − 56 = __________

10. 285 − 168 = __________

11. 1549 − 1374 = __________

12. 725 mL − 650 mL = __________

13. 525 g − 175 g = __________

14. 156# − 48# = __________

15. 65 mL − 30 mL = __________

16. 350 mg − 125 mg = __________

17. 1250 mg − 615 mg = __________

18. 266 kg − 15 kg = __________

19. A bottle of antibiotic needs to be reconstituted with 100 mL of distilled water. You have a calibrated graduate that holds 50 mL. How many times will you have to fill the container with water to reconstitute the medication? __________

20. A vial of anesthetic contains 10 mL. The pharmacist asks that you add 30 mL of distilled water to prepare an anesthetic compound for mouth ulcers. How many milliliters will be in the bottle of prepared medication? __________

Multiplying Whole Numbers

Multiplication of whole numbers is actually a shortcut to repeated addition of numbers. When multiplying whole numbers, align the numbers or factors as you would for addition or subtraction. The use of “×” denotes that the numbers should be multiplied to find a product. Either number may be the multiplier or multiplicand and the product will not be affected. However, it requires less calculating if the multiplier has the fewer digits. You should also remember that if a number is multiplied by zero, the product is zero. First multiply the number in the multiplicand (number that is to be multiplied or is multiplied by another or the number on top of the problem) by the number at the far right of the multiplier (number that is used to multiply another number or that on the bottom of the equation).

Example 2-3

< ?xml:namespace prefix = "mml" />3 (multiplicand)×4 (multiplier)=12 (product)

image

Multiply the remaining numbers in the subsequent digits, moving from right to left. Be sure to keep the numbers to the far right in alignment with the numbers in the multiplier. Finally, add the numbers that have been multiplied to obtain the product.

Example 2-4

image

Example 2-5

image

Please notice the zero or “0”s are now in the first product line and then multiply the numbers following the zero or “0” in the multiplicand in the usual manner. Be sure to keep the alignment correct.

image Tech Alert

If the multiplier is divisible by ten (or ends in “0”) align the multiplier so the “0” is to the right of the multiplicand to shorten the computation of the problem. The number of “0s” to right of first number above zero in the multiplier will then be found to the right of the first line of numbers multiplied.

Dividing Whole Numbers

Division is a way of determining how many times one number can be found in another. It is actually repeated subtraction steps. Division is represented by either “÷” or “image.” The number being divided is called the dividend, and the number used to divide is the divisor. The result of the division is the quotient.

Example 2-6

35 (Dividend)÷5 (Divisor)=7 (Quotient) 

image

OR

image

Any number that is remaining when numbers are not exactly divisible is called a remainder. In math, when rounding numbers, any amount in the remainder (the number that remains after subtraction) that is less than half of the divisor is usually discarded and the number in the quotient remains the same. Conversely, when rounding, if the remainder is more than one half of the divisor, the number in the quotient is increased by one number. For example, 30 divided by 9 is 3, with 3 as a remainder. In this instance, the quotient would remain at 3. But if 33 is divided by 9, the quotient is 3 with a remainder of 6. Six is more than half of 9, so the quotient will become 4.

Learning Tip

Remember that any number divided by itself equals 1 and any number divided by 1 equals that same number. Zero divided by any number remains 0. (Zero can never be the divisor; i.e., 9 ÷ 0 has no meaning, and the answer remains 9.)

When dividing whole numbers, the number in the divisor must be smaller than the number in the dividend to be able to do the calculation. When the divisor is a single digit, the first number of the dividend must be divisible by the divisor or the second digit in the dividend is also used.

Example 2-7

360÷9

image

In this example, 3 cannot be divided by 9, so you must move to 36 to be able to use the divisor 9. Nine will divide into 36 four times, and 9 will divide into 0 zero times, so 360 ÷ 9 = 40.

When there are multiple digits in the divisor, the divisor is rounded to the nearest 10 to estimate the quotient digit. The entire divisor is then used to complete the calculation.

Example 2-8

275÷25

image

Estimate to the nearest 10 digit or go from 25 to 30. If you divide 275 by 30, the answer is 9 (3 into 27), but when you multiply 25 by 9, the answer is 225 with 50 as a remainder. If you divide by 10, the answer still has a remainder of 25. So then try 11, and the answer is correct. 11 × 25 = 275 or 275 ÷ 25 = 11.

Learning Tip

When dividing by two, three, or more digit numbers, estimate the numbers before doing the calculation. This will cut down on the errors by helping to discover whether the calculated answers are near to the exact number needed after calculation. Round to the nearest multiple ending in zero. For example, in Example 2-8, the estimation of the divisor would become 25 to show that the quotient should first be close to 10. Thus, your answer of 11 is basically correct.

Practice Problems B

Calculate the following problems, being sure to show all your work. Estimate your work first. Round to the nearest hundreth as appropriate.

1. 56 × 10 = __________

2. 76 × 11 = __________

3. 100 × 87 = __________

4. 768 × 2 = __________

5. 3500 × 265 = __________

6. 345 × 70 = __________

7. image = __________

8. image = __________

9. image =

10. image =

11. 105 ÷ 5 = __________

12. 840 ÷ 40 = __________

13. 296 ÷ 4 = __________

14. 1125 ÷ 15 = __________

15. 150 ÷ 15 = __________

16. 162 ÷ 18 = __________

17. image = __________

18. image = __________

19. image = __________

20. 25 mg × 40 mg = __________

21. 275 mL × 4 mL = __________

22. 24# × 16# = __________

23. 150 mg × 15 mg = __________

24. 30 kg × 400 kg = __________

25. 21 qt × 28 qt = __________

26. 1250 mg ÷ 125 mg = __________

27. 250,000 units ÷ 150 units = __________

28. 2400 kg ÷ 48 kg = __________

29. 300 mg ÷ 15 mg = __________

30. You are asked to fill a prescription that has 10 tablets containing 250 mg each. What is the total actual ingredient of the medication? __________

31. The physician orders 100 tablets of a medication that is to be taken as 1 tablet four times a day until all the medication has been taken. The physician asks that these medications be placed in separate bottles for each week. How many tablets would you put in each bottle? How many bottles of medication should you have available for the pharmacist to check? __________

32. A prescription costs $15, and the customer gives you $50. How much money should you refund to the person? How many $5 bills would you refund to the patient? __________

33. The manufacturer’s label tells you to add 375 mL of water to a medication for reconstitution. You have a container that will hold 125 mL of water. How many times will you need to fill the container to reconstitute the medication correctly? __________

34. Mrs. Jones comes to the pharmacy and tells you that her child has lost her allergy medication. You look at the prescription and read that the original prescription is for 30 tablets to be taken once daily. Fourteen days have passed since Mrs. Jones filled this medication. After consulting with the pharmacist, you are told to replace the lost medication. How many tablets do you need to give Mrs. Jones so that her child can complete her treatment? __________ How many weeks of medication did the child take? __________

35. You are asked to provide the pharmacist with vials of medication to total 1000 mg. Each vial of medication holds 250 mg. How many vials of medication does the pharmacist need? __________

Fractions

What Is a Fraction?

When a whole number or unit is divided into parts, the parts are called fractions. Fractions are designated as proper and improper. A proper fraction is one in which the numerator is less than the denominator or is part of a whole. An improper fraction is defined as a fraction in which the numerator is equal to or greater than the denominator, a fraction that is equal to or greater than one, or a fraction that can be changed to a whole number or a whole number plus a proper fraction. Complex fractions, found later in the text, are fractional expressions in which the numerator, denominator, or both are expressed as fractions such as image.

Fractions may be expressed in the forma/bsuch as3/4orabsuch as34.

image

The “a” is called the numerator (top number found in a fraction). The “b” is the denominator (bottom number of a fraction). Both numbers must be whole numbers, and the denominator cannot be 0 or the resultant answer becomes 0. The denominator also tells how many times the whole unit has been divided.

image

The circle pictured shows a proper fraction. The shaded area shows the part of a whole. Can you tell the number in the numerator and denominator? If you said image you are correct.

Try with this example:

image

If you said the numerator is 5 and the denominator is 6, you are correct.

Tech Note

Fractions are used in pharmacy in both household and apothecary quantities as well as in business math calculations.

Practice Problems C

In the following examples, please indicate the numerator and the denominator.

1. image Numerator _____________________ Denominator _____________________
2. image Numerator _____________________ Denominator _____________________
3. image Numerator _____________________ Denominator _____________________
4. image Numerator _____________________ Denominator _____________________
5. image Numerator _____________________ Denominator _____________________

image

A fraction can also be thought of as indicating division—the numerator may be divided by the denominator. So when image is written, the fraction actually means 2 ÷ 3. The following illustrates this in pharmaceutical terms:

If a patient has 25 tablets in a prescription and takes 13 tablets, what is the fractional use?

Answer: image

A pharmacist has a bottle of medication that contains 90 tablets. He uses 50 tablets. What is the fractional use?

Answer: image or image. (When image is divided by 10 in both the numerator and the denominator, the result is image.)

Tech Note

Whole numbers are actually fractions in which the numerator is an exact multiple of the denominator. When whole numbers are shown in a graph, the entire area of the graph is shaded versus the shading of parts of the entire area in fractions.

image

A proper or simple fraction is a fraction in which the numerator is less than the denominator, such as image and image. An improper fraction is one in which the numerator is equal to or greater than the denominator, making the fraction a number of one or more such as image or image and image or 2. If the numerator and denominator are the same, the number becomes 1, such as image or image. An improper fraction may be written as a mixed number (a whole number and a fraction) or as an improper fraction (a fraction that has a numerator equal to or greater than the denominator). To change an improper fraction to a mixed number, the numerator is divided by the denominator and any remainder is written over the denominator, such as image (an improper fraction) = image (a mixed number). (When 43 is divided by 8, the answer is 5 with a remainder of 3, so image as a mixed number is image.) The remainder is the amount that is left over after the whole number has been divided into the numerator the maximum number of times (in this problem, 3 is the remainder that now is seen as image). An improper fraction should always be reduced to a mixed number after calculations have been completed.

A mixed number is a whole number with a proper fraction, such as image and image. To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the amount found in the numerator (e.g., image actually means that 4 is multiplied by 8 to provide the whole number of 32 and the 1 found in the numerator of the fraction is added to the total, or 32 + 1 = 33). The denominator remains the number found as the denominator of the mixed number, or image = image. Another example of changing a mixed number to an improper fraction is image: 2 × 3 + 2 with the same denominator of 3 or image.

Practice Problems D

Change the following improper fractions to mixed numbers or whole numbers. Show all of your work. Be sure to indicate measurement as appropriate.

1. image = __________

2. image = __________

3. image = __________

4. image = __________

5. image = __________

6. image = __________

7. image = __________

8. image = __________

9. image = __________

10. image = __________

11. image = __________

12. image = __________

    Change the following mixed numbers to improper fractions. Be sure to indicate measurement as appropriate.

13. image = __________

14. image = __________

15. image = __________

16. image = __________

17. image = __________

18. image = __________

19. image = __________

20. image = __________

21. image = __________

22. image = __________

23. image = __________

24. image = __________

    Indicate the fraction or mixed number for each of the following shaded areas.

    image

25. = __________

    image

26. = __________

    image

27. = __________

    image

28. = __________

    image

29. = __________

    image

30. = __________

31. A pharmacist has a medicine cup that is divided into eight equal sections. She fills to the line to indicate five parts. What is the fractional amount that the pharmacist has filled? __________

32. A customer asks you to show him how to divide a tablet (i.e., a scored tablet that is divided into four equal parts) into image of a tablet. Fill in the following tablet to show this fraction.

image

33. A medication requires that the customer mix granules in image of a glass of water. On the glass, show image of the glass of water.

image

34. A customer is supposed to take a dose of image tablets four times a day. The prescription drug is in whole tablets. Show what the customer should do.

image

35. A customer is supposed to give her child image of a medicine dropper of medication. Indicate on the medicine dropper what you would show the customer as a dose.

image

Reducing Fractions to the Lowest Term

Some fractions, called equivalent fractions, have the same equivalency without having the same portions of a whole. The fraction image could be image, image, or even image. For example, with image, the numerator and denominator can both be divided by 2 to give image; or with image, both parts of the fraction can be divided by 3 for image; finally, with image, both components of the fraction can be divided by 50 for image. Each of these fractions is actually one half of the total number of equivalent parts of a whole.

Tech Note

Any fraction that has a denominator of 1 is equal to the number in the numerator, such as image = 6.

Reducing or simplifying fractions to the lowest term is actually the process of finding the lowest equivalent fraction through division of the numerator and denominator by the same number (e.g., the fraction image can be reduced by dividing by 2 [2 ÷ 2 = 1 and 4 ÷ 2 = 2] = image, or image can be reduced by 3 [3 ÷ 3 = 1 and 6 ÷ 3 = 2] to = image). In each case the reduced fraction contains the same amount of space in each whole. The difference is that the number of divided pieces of the whole has changed. The fractions are comparable, although the actual numerators and denominators have been altered by division with a number that is common to both parts of the fraction. This is dividing by a common factor of the numerator and denominator (e.g., with the fraction image, both the numerator and denominator can be divided by 5, so the fraction can be reduced to image).

Learning Tip

Failure to reduce fractions to lowest terms can be cause for the answer to be incorrect because the higher the numbers used, the more difficult the calculations.

Tech Note

If the lowest common denominator is not readily apparent, finding this number may require other steps. One way is to multiply the two denominators together and use that number as the common denominator, or the denominators may be multiplied by 2, 3, or 4 to find a common denominator (e.g., if the denominators are 3 and 4, a common denominator could be 12 because 3 × 4 = 12). If the denominators were 6 and 9, multiplying the 6 by 3 would provide denominator of 18 and multiplying 9 × 2 would also provide the denominator of 18. So in this case the common denominator would be 18, and the math could be calculated using the common denominator.

Practice Problems E

Reduce the following fractions to the lowest term. As always, show your calculations.

Solve the following problems and reduce to the lowest term. Be sure to indicate measurement as appropriate.

  Answer Lowest term
9. image = __________ __________
10. image = __________ __________
11. image = __________ __________
12. image = __________ __________
13. image = __________ __________
14. image = __________ __________
15. image = __________ __________
16. image = __________ __________
17. image = __________ __________
18. image = __________ __________
19. image = __________ __________
20. image = __________ __________

image

Adding and Subtracting Fractions

When adding or subtracting fractions, the denominators must be the same numeral. If the denominators are the same number, the numerators may either be added or subtracted (e.g., image + image can be added by adding the numerators, or 3 and 1 [3 + 1] and placing the answer over the denominator of 5. This provides an answer of image. To subtract image − image, subtract 1 from 3 [3 – 1]/5 for an answer of image).

Finding Common Denominators

For denominators to be the same for adding or subtracting fractions, the lowest common denominator, must be found. The smallest whole number that can be divided evenly by the denominators within the problem is the common denominator. Sometimes the lowest common denominator is easily found because one denominator is readily divided by the other denominator. For example, a common denominator for image + image could easily be found because 15 is divisible by 5. To change image to the common denominator, divide the denominator 5 into the denominator 15 (3); then multiply the 3 (numerator) times 3 to calculate image. Therefore image + image would provide an answer of image.

Tech Note

When estimating fractions, use 0, image, image, image, 1, and so on to estimate the answer with any calculations with fractions. Estimate either slightly larger or slightly smaller numbers. Remember that this is an estimate not a specific answer. It is for comparison for your exact answer.

Learning Tip

Estimation, even if mentally calculated, is important in accuracy with calculation of doses and dosages later. Please practice this as you go through this section as means to be sure your answer is reasonable. First estimate, then compute, and then check your answer to the estimation.

To change fractions to their equivalent fractions for adding or subtracting of fractions, first find the lowest common denominator and divide the denominator of the fraction to be changed into the lowest common denominator. Then multiply the numerator of the fraction being changed by the quotient found in the first step. Finally, place the product or answer found in this step over the lowest common denominator. In the previous example, in image + image, the lowest common denominator is 15. The problem with the common denominator is image (15 ÷ 5 = 3; 3 × 3 = 9; so image = image) + image = image.

Example 2-9

310+35=?

image

The fraction image is okay. The common denominator for both fractions will be 10.

10 is divisible by 5; 2 × 5 = 10

So, the fraction image

image

Learning Tip

Numbers ending in even numbers are divisible by 2; numbers ending in 5 or 0 are divisible by 5; and numbers ending in 0 are divisible by 10.

If a mixed number is in the expression, the mixed number should be changed to an improper fraction before finding the common numerical denominator. Then change both fractions to the lowest common denominator and add or subtract as indicated. For example, image, so now the problem is simply image. The next step is to find the lowest common denominator; in this problem it is 8; 8 ÷ 4 = 2; 2 × 3 = 6; leading to the fraction image. The problem is now image. This improper fraction can then be converted back to the mixed number of image (35 ÷ 8 = 4 with a remainder of 3 or image). Here is one more example: image. First, change the mixed number to an improper fraction (12 × 2 +11 = image). The problem now becomes image. The lowest common denominator is 12. To convert image using the lowest common denominator would be image (12 ÷ 6 = 2; 2 × 5 = 10; so image). Next, image, or image (45 ÷ 12 = 3 with a remainder of 9, or image or image). In this problem the fraction may be reduced.

Tech Note

When reducing fractions or when finding a common denominator, the numerator and the denominator must be divided by the same nonzero number. For example, image can be reduced to image by dividing the numerator and denominator by 3.

Practice Problems F

Solve the following problems and reduce fractions to the lowest term as appropriate. Show all of your calculations. Be sure to indicate measurement as appropriate.

1. image = __________

2. image = __________

3. image = __________

4. image = __________

5. image = __________

6. image = __________

7. image = __________

8. image = __________

9. image = __________

10. image = __________

11. image = __________

12. image = __________

13. image = __________

14. image = __________

15. image = __________

16. image = __________

17. image = __________

18. image = __________

19. image = __________

20. image = __________

21. image = __________

22. image = __________

23. image = __________

24. image = __________

25. image = __________

26. image = __________

27. image = __________

28. image = __________

29. image = __________

30. image = __________

31. image = __________

32. image = __________

33. image = __________

34. image = __________

35. image = __________

36. image = __________

37. image = __________

38. image = __________

39. image = __________

40. image = __________

41. A bottle of medication granules contains image oz of the desired medication. A newer medication container holds image oz of the medication. A customer asks you to calculate the difference in the amount of medication in the two bottles. What is your answer? __________

42. You are using a gallon bottle of distilled water to reconstitute dry antibiotics. The first medication requires image gallon. The second medication uses image gallon. The third medication needs image of a gallon. What is the total amount of distilled water used? __________

43. How much distilled water is left in the container? Hint: Remember that a fraction for a whole number has the same numerator and denominator. __________

44. The pharmacist asks you to find the common denominator for medications expressed in grains. You have prescriptions for gr image, gr image, and gr image. Express the medication doses in a common denominator. __________ What is the total amount that has been used? __________ If the total medication is gr v, how many grains are left following the completion of the prescription preparations? __________

45. You have image oz of a specific medication. In a day, you have used image oz of that medication. How much medication will you have left for use to fill prescriptions the next day? __________

Multiplying Fractions

Multiplication of fractions is straightforward. To find the product of fractions, multiply the numerators and then the denominators. The product should then be reduced to the lowest term. The calculations are easier than with addition or subtraction, which require a common denominator. For example, image. In this example, 15 is a number that can be used to reduce to lowest terms (15 ÷ 15 = 1; 30 ÷ 15 = 2), giving the final answer of image.

When multiplying or dividing fractions, you may shorten the steps by simplifying the numbers before finding the final product. If the numbers in the numerator and denominator can be divided by a common number, this step should be accomplished before multiplying the fractions. When completing this step, the numerator and denominator are on opposite sides of the “×”. An example follows:

image

Tech Note

Remember that to multiply (or divide) mixed numbers, you must change the mixed number to an improper fraction (e.g., image would be changed to image by multiplying 3 × 6 and adding the 1 in the numerator of the fraction, with the denominator remaining the same).

Dividing Fractions

To divide fractions or mixed numbers, you must change the mixed numbers to improper fractions. The next step in the multiple-step process is inversion (turn the fraction upside down) of the divisor (or the second number in the expression, which is the number to the right of the division sign) so that the numerator now becomes the denominator. For example, if image is inverted, the fraction would become image. In the problem image (image is the number to be inverted), the problem becomes image after the inversion of the numbers. After the inversion is complete, the remainder of the problem is actually multiplication of fractions. Remember to cancel any numbers that can be simplified to make the problem easier to solve. (Canceling numbers was shown earlier with the multiplication example.) To cancel a number, divide both the numerator and the denominator by a common number that will divide into both numbers evenly, or without a remainder. This is a shortcut used to reduce the value of the numerals in the fractions for ease of calculation. (Cancellation is used when fractions are being multiplied. Canceling cannot be accomplished if sums or differences are in numerators or denominators. If you question whether you can cancel, do not do it.) Then reduce the fraction further if this is possible. In the previous example, solving the problem gives an answer of image. Another example is image In this example, the problem is now image or simplified to 4 × 2 (10 ÷ 5) = 8, and 1 (5 ÷ 5) × 5 = 5, so the fraction is image. This can be changed to a mixed number of image.

Tech Note

In division of fractions, after inverting the numerator and denominator of the fraction to the right of the division sign, continue with the steps as for multiplication—multiply the numerators and multiply the denominators, placing the products with the numerators over the denominators. Finally, reduce the resultant fraction to the lowest possible terms.

Practice Problems G

In the following problems, perform the multiplication or division and then reduce the fractions to the lowest terms. Show all of your calculations. Indicate measurement as appropriate.

1. image = __________

2. image = __________

3. image = __________

4. image = __________

5. image = __________

6. image = __________

7. image = __________

8. image = __________

9. image = __________

10. image = __________

11. image= __________

12. image = __________

13. image = __________

14. image = __________

15. image = __________

16. image = __________

17. image = __________

18. image = __________

19. image = __________

20. image = __________

21. image = __________

22. image = __________

23. image = __________

24. image = __________

25. image = __________

26. image = __________

27. image = __________

28. image = __________

29. image = __________

30. image = __________

31. image = __________

32. image = __________

33. image = __________

34. image = __________

35. image = __________

36. image = __________

37. image = __________

38. image = __________

39. image = __________

40. image = __________

41. image = __________

42. image = __________

43. image = __________

44. image = __________

45. image = __________

46. image = __________

47. image = __________

48. A medication order is for gr image of a drug to be given tid. You have gr image tabs in stock. How many tablets would you give with each dose? __________ How many tablets would you give in a day? __________ How many tablets would you need to fill this prescription for a 30-day supply? __________

49. Your stock bottle of medication contains 500 tablets. You have used image of the bottle. How many tablets do you have left in stock for filling future prescriptions? __________

50. A bottle of liquid antibiotic contains 150 mL of medication or 30 teaspoons. If each dose is image tsp, how many doses are in the bottle? __________. If the medication is given four times a day, how many days will this medication last? __________

Decimals

What Is a Decimal?

Decimals, special shorthand for fractions of powers of 10, are actually decimal fractions or fractions that have a denominator of 10, 100, 1000, or any multiple of the power of 10. Decimals appear to be whole numbers, but actually the decimal value is always less than 1. Decimal numbers may include a whole number, a decimal point, and the decimal fraction. The metric system, the most frequently used measurement system in the medical field, is based on the use of decimals. Using a decimal is simpler to read than fractions and is easier to use when making mathematical computations. Examples of decimals include 0.1 = image, 0.01 = image, or 0.001 = image.

When using the decimal system, think of the decimal point as the division between whole numbers that appear to the left of the decimal point and the decimal fraction that is found to the right of the decimal point. In the example 25.65, 25 is the whole number and 65 is 65 parts of 100 or image. (The decimal number is made into a fractional mixed number of 25 65/100.) With decimals, the numerator is the number following the decimal point. This number is placed over a number consisting of 1 followed by the number of zeros found in the number of the numerator. Thus 3.7 is image. With decimals, zeros may be added to or deleted from the end of the decimal without changing the actual value of the number. Thus 3.7 is the same value as 3.70 or 3.700. Likewise 5.050 is the same value as 5.05.

image

To write decimals using word names, the number to the left will be a whole number and the decimal point becomes the word “and,” with the number to the right of the decimal place followed by the place values of the decimal ending with “th.” For example, 3.47 would be three and forty-seven hundredths. Another example: 1.250 would be one and two hundred fifty thousandths; 4.5006 would be four and five thousand six ten thousandths.

Three the number to the left of the decimal point
and the decimal point
Forty-seven the number to the right of the decimal point
Hundredths the place value of the decimal

AND

One the number to the left of the decimal point
and the decimal point
Two hundred fifty the number to the right of the decimal point
Thousandths the place value of the decimal

Tech Note

The decimal line may be extended to the right of the decimal point as needed to meet the needs for the decimal places.

Practice Problems H

Write the following decimals into word names.

1. 4.34 _______________________________________________________________________________

2. 3.5 ________________________________________________________________________________

3. 6.750 _______________________________________________________________________________

4. 90.54 _______________________________________________________________________________

5. 954.6 _______________________________________________________________________________

6. 0.0035 _____________________________________________________________________________

7. 4.02 _______________________________________________________________________________

8. 0.26 _______________________________________________________________________________

9. 0.780 _______________________________________________________________________________

10. 0.175 _______________________________________________________________________________

To convert fractions to decimals, divide the numerator by the denominator. For example, image is 1 ÷ 2, or 1.0 ÷ 2, because 2 will not divide into 1. Thus 1.0 ÷ 2 = 0.5. Likewise, image is 3 ÷ 4, or 3.0 ÷ 4. Thus 3 ÷ 4 = 0.75 is a decimal answer.

Tech Note

Remember that in both of these problems you must place a zero (“0”) in front of the decimal point to prevent errors in reading the answer.

To convert a decimal to a fraction, the decimal number becomes the numerator of the fraction and the denominator is a 1 followed by zeros for the number of places to include the decimal and the numbers behind the decimal. For example, 0.25 would be 25 (the decimal number)/100 (a 1 followed by 2 zeros or a zero for the number of places behind the decimal point). Thus 0.25 can be converted to the fraction image.

Tech Note

Decimals are used in calculations of doses and dosage in the metric system and for calculation of dollars/cents in business math.

Practice Problems I

Change the following decimals to fractions. Do not reduce.

Tech Note

Do not be afraid of decimals, because you use these daily when you count money (e.g., $1.25 actually means one dollar and twenty-five hundredths of a dollar, $10.50 is 10 dollars and fifty hundredths of a dollar). These are examples of how often we use decimals without even thinking about the fact that the numbers are actually decimal fractions of a dollar.

Comparing Decimals

To compare a decimal, the decimal amounts must be aligned at the decimal point and zeros should be added so that the numbers behind the decimal point contain the same number of decimal places. This is important in the comparison of medications in the metric system of measurement. Compare 0.125, 0.25, and 0.5. At first glance 0.125 appears to be the largest number, but when the numbers are aligned and the proper zeros are added for numbers to have equal decimal places to the right of the decimal point, this proves to be incorrect.

Tech Note

Adding zeros to the right of the decimal does not change its value. Removing zeros from the end decimal places for the comparison of numbers also does not change the value as long as the decimal places to the right are properly aligned.

0.125 one-hundred twenty-five thousandths
0.250 two-hundred fifty thousandths
0.500 five hundred thousandths

The largest number is really 0.5 and not 0.125, which might not seem apparent without the added zeros. Also note that the decimals do not have a whole number to the left of the decimal point, so a zero was added to ensure that the decimal point was not overlooked. The addition of the zero is important to prevent possible confusion and to avoid errors in calculating medication doses.

Rounding Decimals

When figuring dosage calculations, decimals may need to be carried only to a certain number value following the decimal point. Accuracy in calculating dosages to certain decimal places may be necessary in some acute care settings, while, at the same time, rounding or approximating a decimal to a certain placement on the value line may be acceptable (see the decimal line in the “What Is a Decimal?” section). Rounding of numbers must never include the whole number to the left of the decimal point. Usually numbers are rounded to tenths or hundredths, either by adding to or subtracting from the decimal. Rounding to a specific place value makes it easier to measure a drug or to compare decimals in stock medicines to medication orders. For example, a calculation for a medication shows that 3.99 mg of a medication should be given to the patient and the drug is available in 4-mg tablets. In this case the patient would receive a tablet by rounding the decimals; 3.99 would be rounded to the whole number 4.

Tech Note

Step 1: Underline the digit in the place for rounding.

Step 2: Look at the digit to the right of the underlined digit. If it is greater than or equal to 5, round to the next highest number. If it is less than 5, leave all digits to the right of the underlined digits.

Step 3: Drop all digits to the right of the underlined digit.

To round hundredths to tenths, the same rule of less than 5 and 5 or over applies, but the number being changed is the tenth value. In the previous example, 1.14 would be rounded to 1.1 while 1.15 would become 1.2.

Decimals may also be rounded to the next whole number by rounding from the tenths value (e.g., 3.4 would remain 3, whereas 3.5 would become 4). However, due to the degree of specificity often needed in medical dosing, rounding up or down in this manner is rarely done unless the number is within a small percentage of the next whole number, or, if a product’s specific labeling indicates such rounding is preferred for ease of choosing a dosage form or for a measurable amount. It is more common to round dosages to the nearest tenth or hundredth.

For ease in adding decimals, use estimation to find the nearest tenth and add or subtract the rounded numbers. This will give an estimate of the desired answer. Then add the actual numbers; check that your answer is nearly equal to your estimate to be sure that it is reasonable.

Practice Problems J

Round to the nearest hundredth.

Adding and Subtracting Decimals

Adding and subtracting of decimals requires aligning whole numbers and decimal points, as you would do with whole numbers. After aligning the decimal points, add zeros at the end of the decimal fraction until all decimal numbers are to the same decimal place. Then just add or subtract as you would for whole numbers, remembering to correctly place the decimal point in the answer. The problem 3.4678 − 2.34, after alignment of the decimal, would look like the following:

image

Learning Tip

Zeros may be added as the last number behind a decimal point and in front of the whole number without changing the value of the number. Whole numbers are understood to have a decimal point to the right of the number.

Practice Problems K

In the following problems, add or subtract decimals as indicated. Estimate your possible answer. Show all of your work. Round answers to the nearest hundredth. Be sure to indicate measurement as appropriate.

1. 2.35 + 3.1 + 4.678 = __________

2. 5.7 + 18.25 + 95.37 = __________

3. 2.38 + 14.7 + 1346 = __________

4. 6.002 + 3.23 + 9.1 = __________

5. 9.7 + 5.68 + 3.3 = __________

6. 5.75 + 4.678 + 2 = __________

7. 12.5 mg + 220.25 mg + 2.75 mg = __________

8. 6.75 mg + 125 mg + 4.25 mg = __________

9. $12.50 + $0.42 + $140.67 = __________

10. $5.67 + $136.99 + $89.09 = __________

11. 2.76 − 1.98 = __________

12. 4.8 − 1.987 = __________

13. 1.567 − 0.986 = __________

14. 75.3 − 16.95 = __________

15. 18.2 − 4.762 = __________

16. 125 − 0.125 = __________

17. $15.75 − $5.65 = __________

18. $17.49 − $5.05 = __________

19. 0.2 g − 0.02 g = __________

20. 12.5 mg − 10.5 mg = __________

21. 275 mg − 225.5 mg = __________

22. 65 mg − 45.5 mg = __________

23. A prescription costs $25.50. The customer gives you two $20 bills. How much change do you owe the customer? __________

24. A stock bottle of medication contains 500 mg of drug used in compounding other medications. You used 125 mg for one prescription and 62.5 mg for a second prescription, while the third prescription was for a child and only 25.25 mg were necessary. What quantity medication has been used? __________ What quantity of the original medication is left? __________

25. A customer has three prescriptions—one costing $35, the second costing $17.50, and the third costing $23.60. To pay for the prescriptions, the person gives you four $20 bills. Is this enough money for the prescriptions? __________ What is the total cost of the prescriptions? __________ If this is enough money, how much change do you need to return to the person? If you do not have enough money, how much more money is necessary to cover the costs? __________ Indicate if money is returned to the person or if more money is needed __________.

Multiplying Decimals

Multiplying decimals is similar to multiplying whole numbers. The alignment of the numbers is identical without having a relationship to the placement of the decimal. The difference is the proper placement of the decimal place after the product of the multiplication has been found. To multiply decimals, multiply the numbers after aligning calculations the numbers to the right and disregard the placement of the decimal. After the product of the multiplication of the multiplicand and the multiplier have been found, count the number of places to the right of the decimal points in both the multiplicand and the multiplier. Finally, place a decimal point in the product by counting from the right to left the number of decimal places found in both elements of the problem. An example follows.

image

Zeros may be removed from the right end of a decimal number (to the right of the decimal point) without changing the value of the number; therefore if the multiplier has zeros at the end, remove the zeros and then multiply. This step simplifies the multiplication. For example, if multiplying 7.350 × 0.20, cancel the zeros image so that the problem now looks like 7.35 × 0.2 or an answer of 1.470. The zero in the answer should then be canceled (image) to give the final answer of 1.47.

If the multiplier is 10 or a multiple of 10 (e.g., 100, 1000, etc.), a shortcut in multiplying is to just move the decimal places as many places to the right as there are zeros in the multiplier, or image, or just move the decimal point one space to the right in the 12.2. or 122. If the multiplier is 100, then the decimal would be moved 2 places; with 1000, 3 places, etc.

Practice Problems L

Calculate the following, being sure to show your calculations. Estimate the answer prior to figuring the final answer. Cancel zeros as appropriate in the final answer and round to hundredths. Be sure to indicate measurement as appropriate.

1. 65.3 × 10 = __________

2. 13.2 × 100 = __________

3. 4.25 × 10 = __________

4. 0.004 × 100 = __________

5. 0.2 × 1000 = __________

6. 16.5 × 0.5 = __________

7. 23.52 × 0.5 = __________

8. 0.35 × 0.45 = __________

9. 59.5 × 39.99 = __________

10. 2.5 mg × 1.5 mg = __________

11. 1.25 mg × 5 mg = __________

12. 2.5 mL × 30 mL = __________

13. $0.50 × $10.00 = __________

14. 2.5 qt × 0.75 qt = __________

15. 18.5 mg × 15 mg = __________

16. 35 g × 12.5 g = __________

17. 0.005 mg × 1 mg = __________

18. A physician orders 2.5 mg of a drug that is to be taken daily for 10 days. What is the total amount of the drug the patient will be taking? __________

19. A mother will give her child 2.5 mL of an antipyretic every 4 hours for fever. If the child is given the medication six times during the day, how many milliliters of medication will the child receive in a day? __________

20. You have a stock bottle of medication that contains 12.25 mg of medication. You have five bottles of the medication available. How many total milligrams of medication are available for dispensing? __________

Dividing Decimals

Dividing decimals is much like dividing whole numbers. Write the problem as for long division, such as the following:

image

For example, if 2.50 is to be divided by 1.25, the problem should appear as image. If a decimal appears in the divisor, move the decimal to the right until the divisor is a whole number. Then move the decimal point in the dividend the same number of places to the right. For example, in the previous example, the problem would appear as image when the decimals in the divisor and dividend are both moved (image). Place a decimal point on the quotient (answer) line directly above the decimal point in the dividend and add zeros behind the decimal point in the dividend if more places are necessary for dividing. The answer is then 2. For example, 1 ÷ 5 would be image. In this problem, 5 cannot be divided into 1, so the problem must become image or 0.2 by placing the decimal point directly over the decimal point in the dividend and then dividing. Remember that if a whole number is not in front of the decimal point, a zero must be added to prevent medication errors. Another example would be 40.44 ÷ 0.4, which, when placed in the long division format, would be image. Next, move the decimal in the divisor to the whole number 4 and move the decimal in the dividend to 404.4. For image, the answer is 101.1. Be sure to fill in the 0 in the quotient when the number cannot be divided by the divisor, as in the “0” in 101.1.

In some cases the division does not come out even, such as when 3 is divided into 1. The answer will be image to the number of places that zeros are added to the dividend. In these cases, the number may be shown with a 3 with a line over the 3, meaning image is a repeating number. In this case in the medical field, the 3 should be rounded to the hundredth place or 0.33. Again, remember to place the 0 in front of the decimal point.

Practice Problems M

Calculate the following problems. Show your work. Be sure to indicate measurement as appropriate.

1. 268.4 ÷ 4 = __________

2. 125 ÷ 0.25 = __________

3. 1.5 ÷ 0.3 = __________

4. 19.95 ÷ 10.5 = __________

5. 33.03 ÷ 0.03 = __________

6. 15.66 ÷ 0.44 = __________

7. 25.5 mg ÷ 15 mg = __________

8. 2.5 g ÷ 0.5 g = __________

9. $24.68 ÷ $0.40 = __________

10. $24.65 ÷ $0.15 = __________

11. $10.50 ÷ $0.10 = __________

12. 120 mg ÷ 100 mg = __________

13. 2500 mL ÷ 100 mL = __________

14. 2.5 L ÷ 0.5 L = __________

15. 12.5 g ÷ 50 g = __________

16. 1.25 mg ÷ 5 mg = __________

17. $124.80 ÷ $4.40 = __________

18. 1.174 L ÷ 0.33 L = __________

19. A pharmacist has a stock bottle of medication containing 12.5 mL. To use this with prescriptions, the pharmacist will need to divide the medication into five equal parts. How many milliliters will be in each? __________

20. A customer comes to the pharmacy to obtain an expensive medication that contains 250 mg/25 mL. This medication can be divided into several prescriptions for ease of payment. She states that the total amount of medication, which costs $255.30, needs to be divided into five equal dispensing units. How many milligrams of medication will be dispensed each time? __________. How many milliliters of medication will be dispensed? __________. How much will the customer pay for each prescription? __________

Points to Remember

Decimals

• Always place a zero before a decimal if no whole number is present.

• If dividing decimals and the number is a repeating one in the quotient, round to the nearest hundredth (e.g., image would be 1.67).

• When adding or subtracting decimals, always align decimal points and add zeros behind the decimal point if necessary to have decimal places the same length. Adding zeros behind the last decimal number does not change its value.

• When multiplying decimals, multiply the numbers together and then count the number of decimal places in the numbers multiplied and place the decimal point in the product (answer) to the left of the total number of places counted.

• To divide decimal fractions, form a whole number in the divisor by moving the decimal point to the right, then move the decimal point in the dividend (number to be divided) the same number of decimal places to the right that has been moved in the divisor. Place the decimal point in the quotient (answer) directly over the decimal point in the dividend and divide as if the numbers are whole numbers.

• When multiplying by 10 or multiples of 10 (e.g., 100 or 1000), the only step necessary is to move the decimal point in the product (answer) to the right by the number of zeros in the multiplier.

• To divide by 10 or a multiple of 10, the decimal point in the quotient (answer) may be moved to the left by the number of zeros in the divisor.

Percentages

Tech Note

In pharmacology, percentages are used to find the volume of solute in the solvent when preparing liquid medications, to find the amount of medication that has been administered over a given amount of time, discounts, and inventory control.

Converting a Decimal to a Percentage

A percentage is actually a part of 100 in fractions or hundredths in decimals. To express a fraction as a percent (%), the denominator is 100 and the numerator is the part of 100 found within the percent. For example, 5% would be image or 5 parts of 100.

To convert a decimal to a percentage, multiply by 100 (which results in the decimal place being moved). The decimal point is moved two places to the right, and a percent sign is added. To convert 0.50 to a percentage, move the decimal two places to the right (image) by multiplying by 100 and add the % sign. So 0.50 is equal to 50%. If the decimal does not have two places to the right of the decimal point, you must add a zero so that the decimal point can be moved two places because two decimal places are needed to multiply by 100; 0.5 would need a zero added (0.50) and then the decimal point should be moved two places (image) so that 0.5 becomes 50%. Finally, if the number is a whole number such as 5, two zeros must be added to the number to find the percentage. Thus 5, when multiplying by 100, two zeros would be added (image), and 5 would become 500%.

Learning Tip

To remember the direction in which to move the decimal point when changing from a decimal number to a percentage or vice versa think of the alphabet: A-Z. D stands for decimal point while P stands for percent. To get from D to P in the alphabet you move to the right (D → P). So to change a decimal number to a percent, move the decimal point to the right. If you want to change a percentage number to a decimal number move from P to D (D ← P). To change from a percent to a decimal move decimal point from right to left.

Practice Problems N

Convert the following decimals to percentages. Show your calculations.

Converting Percentage to a Decimal

Remember that % actually means parts per 100. To convert a percentage to a decimal, drop the percent sign and divide the number by 100 (the dividend being the number in the percentage and the divisor being 100). This can actually be accomplished by moving the decimal point two places to the left. (This was discussed in dividing by multiples of 10 in the section on dividing decimals.) For example, 25% would become 0.25 by removing the % sign and moving the decimal (image). An example is 25% = image (or 25 ÷ 100) = 0.25. Remember, to prevent errors, always place a zero in front of the decimal place if no whole number is present. As with changing a decimal to a percentage, zeros may be added in front of the number as needed so that the two decimal places may be moved (e.g., 8% would become 0.08 [image] in decimals or 8/100).

If the number is already a decimal percentage, adding two zeros in front of the decimal point will allow the change to a decimal; for instance, 0.8% would need two zeros to become a decimal. In this problem, 0.8% would be 0.008 (image or 0.8 divided by 100).

Finally, if the percentage is more than 100%, a whole number will be apparent in front of the decimal places (e.g., 255 is actually 255.0 in the decimal system, so moving the decimal to percentage would be 25500% [25500.0]). The moving of the decimal point is the same as with numbers less than 100%. If 155% is converted to a decimal, drop the percent sign and move the decimal two places to the left, or 155% = 1.55 (image).

Practice Problems O

Convert the following percentages to decimals. Show your calculations.

Converting a Fractional Percentage to a Decimal

To change a fractional percentage to a decimal, first convert the fraction to a decimal and round to the nearest hundredth, leaving the percent sign as part of the number. Then convert the percentage to a decimal as you would with other percentages to decimal numbers. For example, image would first need to be converted to 0.5% (1 ÷ 2 = 0.5). Then convert the 0.5% to a decimal by moving the decimal two places to the left while adding the necessary zeros, or image = 0.005 (image). Remember that the calculation is actually dividing by 100.

Practice Problems P

Convert the following fractional percentages to decimals. Show your calculations. Round to the nearest 10000th.

Converting a Fraction to a Percentage

Remember that the denominator in a percentage is always 100 (because percent is a part of 100) and the number beside the % sign becomes the numerator. To convert a fraction to a percentage, multiply the fraction by 100 or by image (the fraction for 100) and add the percent sign (%). This is the same step that was used to convert a fraction to a decimal. If the division is not even, round to ten-thousandths before converting the decimal to a percentage.

Example 2-10

image

If the fraction is a mixed number, change the mixed number to an improper fraction and then multiply by 100. For example, image or image. Then multiply by 100:

image

Now divide this answer by 100 to change to decimal: 260 = 2.60% (2.6% after rounding)

Remember that “0” at the end of a decimal is dropped.

Tech Note

Conversions of fractions to percentage may also be calculated using ratio and proportion such as a : b :: x (%) : 100. This is discussed in the next section of the chapter under ratio and proportion.

Practice Problems Q

Convert the following fractions and mixed numbers to percentages. Show your calculations. Round to hundreths as appropriate.

Converting a Percentage to a Fraction

To convert a percentage to a fraction, drop the % sign and write the number over 100 (or the percentage number becomes the numerator). The denominator will always be 100. Finally, reduce the fraction to its lowest terms. For example, to change 16% to a fraction, 16% becomes image when changed to a fraction. This will reduce to image. Remember if the fraction is not reduced to the lowest term, it may be considered incorrect.

If the percentage is written as a mixed number, the mixed number becomes the numerator and 100 is the denominator. The mixed number must be changed to an improper fraction placed over 100. For example, to change image to a fraction, first change the mixed number to the improper fraction image. Remember that when dividing in percentage, you are actually dividing by 100. The fraction is now image. To divide fractions, the fraction must be inverted so that the divisor becomes the dividend and the dividend becomes the divisor; thus the equation now is as follows:

image

Practice Problems R

Convert the following percentages to fractions. Reduce fractions. Show your calculations.

Ratios

Expressing Numbers as Ratios

Ratio indicates the relationship of one number to another or a whole. As with fractions, a ratio is expressed as numerators and denominators separated by a colon ( : ) rather than a division line found in fractions. Numerators are to the left of the colon with denominators to the right, such as 1 : 2. The colon is the traditional way to write a division sign in a ratio and is representative of “of,” “per,” “to,” or “in.” For example, image as a fractional expression would be 3 : 4 when written in ratio. Like a fraction, a ratio is the relative size of two quantities and should be reduced to lowest terms. Because of the relationship of the numbers in a ratio, the value of the ratio will not be changed if both the numerator and denominator are multiplied or divided by the same number. Multiplication and division are the only numeric operations that can be performed on a ratio without changing its value.

Tech Note

A ratio may be written as a fraction, and a fraction may be written as a ratio because each has a numerator and denominator (2 : 3 or image).

When numbers are expressed in ratios, numbers that designate a quantity must be expressed in the same units of measure. For example, if the numerator of the ratio is expressed in inches, the denominator must also be expressed in inches. So to establish a ratio of 3 inches : 1 foot, the 1 foot must be changed to 12 inches (12″ = 1 foot). Now the ratio has the same unit of measure in inches and the ratio is 3″ : 12″, or the ratio is reduced to 1 : 4. Thus 3 qts : 1 gal (1 gal = 4 qt) becomes 3 qt : 4 qt, or the ratio 3 : 4. Conversions within and among units of measure will be presented in Chapters 3 and 4.

To express a percentage as a ratio, the denominator will always be 100. For example, 30% = image = 30 : 100.

To change a decimal to a ratio, multiply the decimal by 100 or move the decimal point two places to the right and 100 will become the denominator. For example, 0.09 would be 9 : 100, 0.675 would be 67.5 : 100, and 0.0008 would be 0.08 : 100.

Ratios may also be shown as fractions, such as 3 : 4 would be image or 20 : 100 would be image, which will reduce to image by dividing both the numerator and the denominator by 20.

Tech Note

Ratios and ratio and proportion are used for dose/dosage calculations, such as number of tablets when the exact dosage for an order is not available or if the strength of drug per volume of medication is needed. This is also used for conversions between units of measure and in inventory control, but it may be very useful with other calculations and in checking math calculations.

Points to Remember

Practice Problems S

Change the following to ratios, being sure that the ratio is expressed in the same unit of measure and is reduced to the lowest term. Show your calculations.

Expressing Ratios as Proportions

True proportion is the expression of equality between two ratios that have an equal relationship or an equation formed between two equal ratios such as 4 : 6 and 8 : 12. In these pairs of numbers, the relationship between 4 and 8 is that 8 is twice as much as 4 and 12 is twice as much as 6. So these numbers are equally proportional to each other, although the numbers are not the same. The proportion in each is 2, or the second number in the proportion is twice as much as the first number in the proportion. We can place this in a proportional equation, 4 : 8 :: 6 : 12 or a fractional equation image. When the fractions are cross-multiplied image both answers will be 48, or the products will be equal.

When writing the proportional equation of image, we will multiply the two outside numbers (numbers a and d) or the extremes, and the two inside numbers (numbers b and c), or the means, so 4 × 12 and 6 × 8, with both answers being 48. The product of the means will always equal the product of the extremes in a true proportion. The computation of the problem can be checked by placing the answer into the unknown (x) and multiplying the means and extremes, checking that the answers are the same.

Tech Note

The equality of the proportions after calculation is important for ensuring patient safety and should be calculated with each use of ratio and proportion.

image

Learning Tip

Multiply the outside numbers, “outsies,” and then the inside numbers, “insies,” to figure proportion. Think of “outsies” versus “insies” rather than extremes and means respectively, for ease of learning.

Points to Remember

Practice Problems T

Which of the following are true proportions? Mark with either a yes or a no.

In the health care field, the concepts of ratio and proportion are often used to calculate different quantities of medication or to calculate dosages. If an amount of medication in a ratio is known and a new quantity of a substance is necessary, proportion may be used to obtain the new amount of medication that is required by multiplying the means and extremes. Proportional equations are therefore used to find the missing or unknown amount, often signified by “x.” Knowing three of the four parts of the proportion is necessary for the proportion to be formed. When writing the proportion, like units of measure must be in positions a and c as well as in positions b and d when the proportion is designated in the following manner—

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Notice that positions a and c are in milliliters and b and d are in liters.

Remember that proportion can also be shown as a relationship between two fractions and then cross-multiply to solve for the unknown or to prove the computation is correct. The unknown (x) should always be placed on the left side of the equation.

Tech Note

Tips for working with proportions follow:

When three terms of a proportion are known, the fourth term may be found by making the fourth term an “x” and then simply calculating for the x in the equation by cross-multiplication or multiplying “insies” and “outsies.” The mathematical problem that is formed can then be solved.

Each side of an equation may be multiplied or divided by the same nonzero number to reduce or simplify the equation and to isolate “x”, such as in the problem 2x = 24 x can be isolated by using 2 as the denominator.

2x2=242orx=24÷2or12

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A proportion may be proved by adding the found answer into the “x” position in the previous problem and calculating the problem to prove both sides of the equation are equal in the proportional equation. In the example 1 : 2 :: x : 24 with the above calculation: 1 : 2 :: 12 : 24 or 24 = 24. So the calculation is correct. This is an important step for patient safety.

Practice Problems U

With the unknown as “x,” solve the following problems creating proportions as needed. Remember that the ratio components may be written as fractions and then you may cross-multiply. Indicate the measurement as appropriate.

1. 4 : x :: 3 : 12 = __________

2. x : 2 :: 14 : 7 = __________

3. 20 : x :: 5 : 10 = __________

4. 7 : x :: 6 : 12 = __________

5. x : 160 :: 7 : 140 = __________

6. 7 : x :: 35 : 125 = __________

7. x : 11 :: 2 : 2.2 = __________

8. 5 : x :: 2.5 : 125 = __________

9. 0.22 : x :: 0.33 : 6.6 = __________

10. $0.20 : x :: $1.00 : $25 = __________

11. 24 mg : x :: 16 mg : 12 mg = __________

12. 2 g : 24 g :: x : 36 g = __________

13. x : 36 tab :: 6 tab : 8 tab = __________

14. $20 : $15 :: $100 : x = __________

15. 18# : 12# :: 12# : x = __________

16. 20 mg : 24 mg :: 18 mg : x = __________

17. $1.20 : $4 :: x : $9 = __________

18. 2 qt : 6 qt :: x : 48 qt = __________

19. 5 tab : 35 tab :: x : 28 tab = __________

20. 5 mL : x :: 15 mL : 9 mL = __________

21. 0.16 g : x :: 0.4 g : 0.15 g = __________

22. 14 mL : 2 L :: 21 mL : x = __________

23. John knows that Mr. Smith needs 14 tablets for a week’s supply of an antiinflammatory drug. Mr. Smith is going on vacation and needs a 4-week supply. How many tablets does John need to fill the prescription? Show your work. __________

24. Periactin liquid is labeled as 5 mg/5 mL. How many mg would be in 25 mL? Show your work. __________

25. An antilipidemic agent contains 5 mg of the medication per tablet. How many tablets would be necessary to supply 35 mg per dose? Show your work. __________

26. A cough medication contains 50 mg of active ingredient per mL. The physician desires 100 mg per dose. How many mL would each dose contain? Show your work. __________

27. Dr. Jones writes a prescription stating that he wants the patient to take 75 mg of a medication three times a day. The stock bottle shows 37.5 mg per tablet. How many tablets should the patient take at a time? How many tablets should the patient take in a day? Show your work. __________

28. A physician orders a 250 mg dose of an antibiotic for a child. The pediatric liquid medication contains 125 mg per 5 mL. How many milliliters should the child take with each dose? __________

29. If the physician wants the child to take the previously mentioned medication three times a day, how many milligrams of the medication will the child take in 1 day? __________

30. How many milliliters of the medication will the child take in 1 day? __________

Quantities

Determining the Percentage of a Quantity

In dosage calculation, computation of a given percentage or part of a quantity may be figured to ascertain the part of a whole quantity that is in question. If a percentage of a whole quantity is in question, the known percentage is multiplied by the whole quantity to provide the needed information. The equation for finding the percentage follows:

Percentage (amount of the whole quantity)=Percent number×Whole number (base)

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To find a percentage of a whole number, the term “of” means to multiply the number by the percentage: What is 3% of 42? First change the % to a decimal by dividing by 100 or moving the decimal point 2 places to the left or 0.03. Then multiply 42 × 0.03 = 1.26. So 1.26 (what) is 3% of 42.

Dividing percentage answers the question of “what.” For instance, 15 is what percentage of 45? Set up the part number (15) over the whole number (45) and then complete the division.

1545=13

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Finally, convert 1/3 to 33% by dividing 1 by 3 and multiplying by 100%. Or use the following ratio and proportion:

x(%):15 (whole number)::100%:45 (whole number)

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Tech Note

Remember, 100 is always the denominator in %.

As a review of determining the percentage of one number to another, make a fraction with the numbers. The denominator will be the number following the word “of” with the other number in the problem being the numerator. Consider the question, 10 is what percentage of 75? Place the 75 as the denominator and the 10 as the numerator or 10/75 or 10 ÷ 75 = 0.13 or 13% after changing answer to percentage.

Tech Note

Percentage of a quantity is used in strengths of medications as well as finding discounts for customers and pricing of medications.

The fraction may be found in three ways:

Learning Tip

A few tips on working with percentages of quantities are as follows:

For example, what is 50% of 15?

x=50%×15,orx=0.5×15,or7.5

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Answer: 50% of 15 is 7.5.

For example: What percent of 15 is 5?

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For example: What is 75% of 200?

x = 200 × 75%

NOTE: 75% = 75/100 or 0.75

x = 200 × 0.75

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OR

image

75% of 200 = 150

As a short cut, remember that you may set up a problem containing zeros at the end of a number by making that number (containing zeros) the multiplier and adding the zeros to the end of the product prior to multiplying the remaining number.

Tech Note

When working with medications, the higher the percentage of the active ingredient in a medication, the greater the strength of the medication within the whole.

Practice Problems V

Solve the following problems by translating them using the hints described earlier. Show your work, and round answers as appropriate. Indicate measurement as appropriate.

1. What percentage of 105 is 35? (Round to tenths.) __________

2. 25 is what percentage of 200? __________

3. 90% of 50 is what? __________

4. 105% of 0.9 is what? __________

5. 250 is what percentage of 75? (Round to a whole number.) __________

6. What percentage of 750 is 15? __________

7. What percentage of 49 is 11? (Round to hundredths.) __________

8. What is 45% of 180? __________

9. What percentage of 3 is image? (Round to a whole number.) __________

10. Four is what percentage of 240? (Round to hundredths.) __________

11. What is 15% of $25.40? __________

12. What is 64% of 8 oz? __________

13. If you have 25 tablets, what is 5%? __________

14. What percentage of 40 tablets is 22 tablets? __________

15. What is image of 48 mg? __________

16. If a discount of 15% is expected on a purchase of $25, what is the amount of the discount? __________

17. If a patient wants 60% of a prescription that is written for 60 tablets, how many tablets will be dispensed to the patient? __________

18. A physician orders 150% of a prescription that is written for 8 oz. How many ounces will be dispensed? __________

19. 40 tablets is what percentage of 90 tablets? (Round to hundredths.) __________

20. 35% of 70 tablets is what? __________

21. Six inches is what percentage of 24 inches? __________

22. image teaspoons are 20% of what total number of teaspoons? __________

23. 125 tablets is what percent of 1500 tablets? (Round to hundredths.) __________

24. 60% of 12 oz is what? __________

25. What is 3% of 1200 mL? __________

Posttest

Before taking the Posttest, retake the Pretest to check your understanding of the materials presented in this chapter.

Solve the following problems as indicated. Round to the nearest hundredth on the second line when appropriate. Indicate measurement as appropriate. As always, show your calculations.

1. 13 + 24.6 + 36.72 + 0.45 = __________  __________
2. 15.87 − 5.2 = __________  __________
3. 26.45 + 4.792 + 120.005 + 3.202 = __________  __________
4. 12.76 × 5.2 = __________  __________
5. 25.01 × 10.10 = __________  __________
6. 25 ÷ 0. 25 = __________  __________
7. 16.82 ÷ 4.02 = __________  __________
8. 2 − 1.75 = __________  __________

image

Solve the following fractional equations and reduce to the lowest term.

9. image = __________  __________
10. image = __________  __________
11. image = __________  __________
12. image = __________  __________
13. image = __________  __________
14. image = __________  __________
15. image = __________  __________
16. image = __________  __________
17. image = __________  __________
18. image = __________  __________
19. image = __________  __________
20. image = __________  __________

image

Convert the following fractions to decimals. Round to the nearest hundreth.

21. image = __________

22. image = __________

23. image = __________

24. image = __________

25. image = __________

    Convert the following decimals to fractions. If improper fractions are calculated, convert to a mixed number. Reduce as appropriate.

26. 0.44 = __________

27. 1.64 = __________

28. 5.33 = __________

29. 0.86 = __________

30. 10.8 = __________

    Express the numbers in ratio expressions.

31. A medication container has 50 tablets in it for dispensing. __________

32. A 4-oz medication bottle contains image oz cough syrup. __________

33. To reconstitute a liquid, 250 mL of water is necessary for 500 mg. __________

34. An antibiotic contains 250 mg in 5 mL of medication. __________

35. Two capsules contain 1000 mg of penicillin. __________

    Express the following in proportional equations and solve the unknown as appropriate. Be sure to show your calculations. Reduce as appropriate. Indicate measurement as appropriate.

36. A bottle of medication contains 75 tablets. How many tablets would be found in five bottles of medication? __________

37. A physician orders medication to be given at 250 mg per dose. What quantity of medication will be given in a day if the medication is given four times a day? __________

38. John has a prescription for medication to be taken one tablet twice a day. How many tablets of medication will John need to take as prescribed for 2 weeks? __________

39. The pharmacist wants a stock of medication for at least 10 patients available at all times. The medication usually requires a prescription of 24 tablets. How many tablets will need to be in stock each day? __________

40. If the medication mentioned in the previous problem comes in bottles of 50 capsules per bottle, how many bottles of medication are necessary for inventory requirements? __________

41. Dr. Smith calls and wants the dispensing of 3 months supply of medication for hypertension. The usual dispensing amount for 1 month is 60 tablets. How many tablets do you need to count for the physician’s order? __________

42. Mary comes into the pharmacy with a prescription for 90 tablets for a 30-day supply. She does not have the money to buy the complete prescription but wants to buy half today and will get the remainder next week when she has more money. How many tablets do you need to give her for a 2-week supply? __________

    Solve the following quantity percentage problems using equations.

43. Three tablets are what percent of 90? (Round to hundredths.) __________

44. 45% of 1500 mL is what? __________

45. What is 15% of 60 tablets? __________

46. 250 mg is what percent of 1000 mg? __________

47. A prescription is for 120 tablets. What is 40% of that prescription? __________

48. Jesse wants to know what 5% of his prescription of 125 mL is. __________

49. Juan is to take 25% of 24 tablets. How many tablets will Juan take? __________

50. What percent of 125 tablets is 25 tablets? __________

Review of Rules

Fractions

• To change an improper fraction to a mixed number, divide the numerator by the denominator.

• To change a mixed number to an improper fraction, multiply the denominator by the whole number and then add the numerator to the product obtained. This number then becomes the numerator for the fraction, and the denominator of the fraction retained is the original denominator.

• To reduce a fraction to the lowest term, divide both the numerator and denominator by the greatest common factor.

• To add or subtract fractions with a common denominator, add or subtract the numerators as indicated. Then write the sum or difference as the numerator for the fraction, and the common denominator becomes the denominator for the fraction. Reduce this fraction to the lowest term or change to a mixed number as indicated.

• For fractions with an uncommon denominator, the least common denominator must be found. The number that will divide exactly into the denominators must be found, and then the numerators of each fraction must be multiplied by the common number. After finding the common denominator, the mathematical calculation of adding or subtracting may be performed. When adding or subtracting fractions with uncommon denominators, add or subtract the numerators as indicated and then reduce to lowest terms or to the mixed number.

• To multiply fractions, multiply the numerators and multiply the denominators, with the product of the numerators being the new numerator and the product of the denominators being the new denominator. Reduce the fraction or change to a mixed number as indicated.

• To multiply mixed numbers, change the mixed number to an improper fraction and then multiply, as mentioned earlier with fractions.

• To divide fractions, invert the fraction that is the divisor and multiply the dividend by the inverted fraction—the remainder of the problem is actually like multiplying fractions. Reduce to the lowest term.

• To divide mixed numbers, change the mixed number to a fraction and then proceed as if the problem were a fractional unit. Reduce the fraction as needed or change to a mixed number as indicated.

Decimals

• To read a decimal, the number to the left of the decimal is a whole number, “and” is used to represent the decimal point, and the number to the right of the decimal point is read as a whole number value of the number farthest to the right including the decimal point as a number space, with “th” added to the number space. For example, 2.34 would be 2 and 34 hundredths.

• To round a decimal place to a certain place or determined number, begin at the farthermost right digit. If that number is 5 to 9, the second digit to the right will be increased by one (i.e., 1.567 would become 1.57). If the farthermost number is 0 to 4, the second number will remain the same. After determining the number of digits to remain, delete the excess numbers using the previous rule to the correct place (e.g., if 8.78934 is to be rounded to hundredths, the answer would be 8.79).

• To add or subtract decimal numbers, align the numbers, being sure that the decimal points are aligned. Add zeros at the end of the decimal fraction until all decimal numbers are the same length. Then add or subtract as for whole numbers, being sure the decimal point is correctly aligned in the answer. Whole numbers are understood to have a decimal point to the right of the number.

• To multiply decimal numbers, multiply the numbers as if the numbers are whole numbers, then count the numbers to the right of the decimal points and finally add a decimal point in the product so that there are as many decimal points in the product as in the numbers being multiplied.

• To divide a decimal number by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend.

• To divide a decimal number by a decimal number, count the number of digits to the right of the decimal in the divisor; place a decimal point at the end of the same number of places in the dividend, moving the decimal point in place and adding zeros as needed; and then place the decimal point at that place in the quotient directly above the decimal point in the dividend. Divide the numbers as if the divisor is a whole number.

• To change a fraction to a decimal, divide the numerator by the denominator.

• To change a decimal to a fraction, place the decimal number as the numerator with the denominator being 1 plus the number of zeros as found in the decimal number and the decimal point (i.e., 0.01 would be image). Reduce the fraction as appropriate.

Ratio, Proportion, and Percentages

• If two fractions are equivalent, the cross-products will be equal. If the cross-products in a proportional equation are equal, the fractions are equivalent or the proportion is true.

• If each member of an equality is multiplied by the same number, the products will be equal.

• If each member of an equality is divided by the same number, the quotients will be equal.

• To multiply or divide in %, first change the percentage to a decimal and then do the calculation.

• To change a percentage to a decimal form, multiply by 0.01 or move the decimal point two places to the left. To change decimal to percent, move the decimal point two places to the right. Think of the alphabet placement of “D” and “P.”

• To solve an unknown proportion, multiply the means or “insies” and the extremes or “outsies,” setting the equation with the unknown on the left being equal to the quotient numbers on the right. Then solve for the unknown.

• To solve a percent proportion, the “what” translates to the unknown or a letter, “of” translates to times or “×,” “is” translates to equal or “=,” and “%” may be either image or 0.01 for multiplication of the percentage. Read the problem and insert the translation, then solve the unknown.

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