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

Complex fraction

Fractions in which the numerator, denominator, or both are fractional units

Decimal

Number system based on the number 10 and progressions of 10s

Decimal place

Place values found to the right of the decimal point

Denominator

Bottom number of a fraction

Dividend

Number being divided in division

Divisor

Number by which another number is divided

Extremes

First and last number found in a

Fraction

Part of a whole number with a numerator and denominator

Improper fraction

Fraction in which the numerator is equal to or greater than the denominator; a fraction that is equal to or greater than 1

Inversion

To turn upside down, as with a fraction, or to interchange the terms of a ratio

Lowest term

Form of a fraction in which no common number will divide into both the numerator and denominator evenly

Means

Second and third numbers found in a

Mixed number

Number containing a whole number and a fraction

Multiplicand

Number to be multiplied by another

Multiplier

Number used to multiply another number

Numerator

Top number found in a fraction

Percentage

A means of expressing a portion of 100 parts

Product

Number obtained by multiplying two numbers together

Proper fraction

Fraction in which the numerator is less than the denominator

Proportion

Comparative relationship among the parts; one or more ratios that are compared

Quotient

Number obtained when one number is divided by another

Ratio

A means of describing the relationship between two numbers

Remainder

Number left after completing subtraction; in division the number left after division (number left after steps of division have been completed)

Rounding

To calculate a number to a desired place value when the exact number is not desired

Scored tablet

Tablet containing an indention for ease of breaking into equal parts

Whole number

Numeral consisting of one or more digits; number that is not followed by a fraction or decimal

Pretest

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

1. = __________

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 = __________

28. 100.2 × 100.03 = __________

29. 721 − 0.01 = __________

30. 12.02 ÷ 6.01 = __________

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

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

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

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.

Introduction

In calculation of medications, accuracy is extremely important for patient safety. Remember, the patient depends on your calculations and the pharmacist expects you to correctly make these calculations. Answers must be precise—partially correct answers are unacceptable and may endanger patient safety. Basic math skills are an important component in finding the correct answers. You learned many of these skills in early grade school, but you may not use them on a regular basis, so they may be forgotten.

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 ; all the numbers in this example contain whole numbers, but they are not evenly divisible and may have decimal remainders.

Example 2-1

Interesting Fact

The word digit used in numbers actually came from the Latin word digitus, meaning finger. Just as with our hands, each digit has its own value depending on where it is found in the number or numeral.

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, .

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

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)

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

Example 2-5

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.

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 “.” 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)

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

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

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. = __________

8. = __________

9. =

10. =

11. 105 ÷ 5 = __________

12. 840 ÷ 40 = __________

13. 296 ÷ 4 = __________

14. 1125 ÷ 15 = __________

15. 150 ÷ 15 = __________

16. 162 ÷ 18 = __________

17. = __________

18. = __________

19. = __________

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 .

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

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.

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 you are correct.

Try with this example:

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

A fraction can also be thought of as indicating division—the numerator may be divided by the denominator. So when 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:

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

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

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.

A proper or simple fraction is a fraction in which the numerator is less than the denominator, such as and . 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 or and or 2. If the numerator and denominator are the same, the number becomes 1, such as or . 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 (an improper fraction) = (a mixed number). (When 43 is divided by 8, the answer is 5 with a remainder of 3, so as a mixed number is .) 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 ). 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 and . 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., 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 = . Another example of changing a mixed number to an improper fraction is : 2 × 3 + 2 with the same denominator of 3 or .

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.

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

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

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 of a tablet. Fill in the following tablet to show this fraction.

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

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

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

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 could be , , or even . For example, with , the numerator and denominator can both be divided by 2 to give ; or with , both parts of the fraction can be divided by 3 for ; finally, with , both components of the fraction can be divided by 50 for . 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 = 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 can be reduced by dividing by 2 [2 ÷ 2 = 1 and 4 ÷ 2 = 2] = , or can be reduced by 3 [3 ÷ 3 = 1 and 6 ÷ 3 = 2] to = ). 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 , both the numerator and denominator can be divided by 5, so the fraction can be reduced to ).

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