PharCal Lecture

19/8 = (19 / 8) fl. Oz = 2 3/8 fl. oz 1st step: Multiply the whole number by the denominator: 2 x 8 = 16 2nd step: add the product we got in step 1 to the numerator: 16 + 3 = 19 3rd step: divide the sum from Step 2 by the denominator: 19 / 8 = 2.375

* change the fraction to a mixed number:

PROPORTION

- Expression of the equality of two ratios

Concrete or Denominate Number ex: 10 grams + 5 grams = 15 grams 10 milliliters - 5 milliliters = 5 milliliters

- may be added to or subtracted from any other number of the same denomination

Concrete or Denominate Number ex: 300 milligrams x 2 = 600 milligrams 12 ounces / 3 = 4 ounces

- may be multiplied and divide only by a pure number.

Decimal Fractions

-DECIMAL FRACTION/DECIMAL -The denominator of a decimal fraction is never written because the decimal point indicates the place value of the numerals. -The numerator is sufficient to express the fraction.

ROMAN NUMERALS

-Expresses fairly range of numbers by the use of a few letters of the alphabet in a simple "positional" notation. -Merely record QUANTITIES, not use in computation.

ex: -The ratio of 20:4 or 20/4 has a value of 5; -The ratio becomes 10:2 or 10/2, again the value of 5 *if both are divided by 2

-If the two terms of a ratio are multiplied or are divided by the same number, the value is unchanged, the value being the quotient of the first term divided by the second. -if both are divided by 2

Arabic Numerals

-Is properly called a decimal system -With only 10 figures - a ZERO and nine digits (1,2,3,4,5,6,7,8,9) -Different values are assigned to the digits according to the place they occupy in a row

Variation

-Most pharmaceutical calculations deal with simple, direct relationships, twice the cause, double the effect, and so on. -when you decrease the strength of a solution by increasing the amount of diluent.

Multiplying Fractions

-Multiply the numerators -Write the product over the product of the denominator ex: If the adult dose of a medication is 2 teaspoonfuls (tsp), calculate the dose for a child if it is ¼ of the adult dose. (always has a invisible 1) 1/4x2/1 = 2/4 or 1/2

Percent %

-PERCENT "in a hundred" -So 50% means 50 parts in each one hundred of the same item.

Dividing Fractions

-RECIPROCAL of a number is 1 divided by the number. -If you apply Rule no. 2 and regard 3 as 3/1, its reciprocal equals the inversion of this fraction. ex: The reciprocal of 3 is 1/3

Adding Fractions

-Reduce them to a common denomination -Add the numerators -Write the sum over the common denominator ex: In preparing batches of a formula, a Pharmacist used ¼ oz, 1/12 oz, 1/8 oz, 1/6 oz of a chemical. Calculate the total quantity of chemical used: (find the LCD first) 24 THE LCD... 1/4x6/24 = 6/24 6+2+3+4/24 = 15/24 or 5/8 1/12x2/24 = 2/24 1/8x3/24 = 3/24 1/6x4/24 = 4/24

Subtracting Fractions

-Reduce them to a common denomination -Subtract the numerators -Write the difference over the common denominator. ex: A hospitalized patient received 7/12 liter of a prescribed IV Infusion. If he had not received the final 1/8 liter, what fraction of a liter would he have received? 24 THE LCD... 7/12x2/24 = 14/24 14-3/24 = 11/24 1/8x3/24 = 3/24

RATIO

-Relative magnitude of two like quantities -Quotient of two like numbers

EQUIVALENT RATIO

-when two ratios have the same value -The product of the numerator of the one and the denominator of the other always equals the product of the denominator of the one and the numerator of the other; the cross product are equal. -Because: 2/4 = 4/8 2/4 4/8 = 16 -It is also true that if two ratios are equal, their reciprocals are equal. -Because: 2/4 = 4/8 then 4/2 = 8/4

100 units : 1 mL : 40 u : mL = 40 u (1 mL) = 0.4 mL 100 u

1.A If an insulin injection contains 100 units of insulin in each milliliter, how many milliliters should be injected to receive 40 units of insulin?

is written as 0.1

1/10

Rule 1: Before performing any arithmetical operation involving fractions, reduce every mixed number to an improper fraction. ex: 1 1/5 = (1x5) + 1 = 6/5 RULE 1 continuation... If the final result of a calculation is an improper fraction, you may if you like, reduce it to a mixed number. 6/5 = 6*5 = 1 1/5 Rule 2: When performing an operation involving a fraction and a whole number, express (or at least visualize) the whole number as a fraction having 1 for its denominator. ex: Think of 3 as 3/1 42 as 42/1

2 Rules In computing fractions

1. A letter repeated once or more repeats its values. ex: X = 10 XX = 20 CCC = 300 MMMM = 4000 2. One or more repeated letters placed after a letter of greater value INCREASES the value of the greater number. ex: VI = 6 XII = 12 LX = 60 3. A letter placed before a letter of greater value DECREASES the value of the greater letter. ex: IV = 4 XL = 40 CM = 900 4. A bar placed above a letter or letters increases the value by 1000 times. ex: XV = 15 ; but, ___ XV = 15,000 _ XV= 1,005

4 GENERAL RULES for reading Roman Numerals

is written as 0.45

45/100

is written as 0.065

65/1000

3/8 = 3 ÷ 8 = 0.375 1/3 = 1 ÷ 3 = 0.3333.....

A common fraction may be changed to a decimal by dividing the numerator by the denominator (note that result may be a repeating or endless decimal fraction)

0.125 = 125/1000 = 1/8

A decimal fraction may be changed to a common fraction by writing the numerator over the denominator and (if desired) reducing to lowest terms.

A number in the form 1/8, 3/16 and so on is called a COMMON FRACTION, or often a fraction. Numerator- number of parts we have Denominator- total parts in whole ex: 2/5

COMMON FRACTIONS

1. a:b = c:d 2. a:b ∙∙ c:d 3. a/b = c/d

Can be written in any one of three standards form:

ROMAN NUMERALS ex: ss = 1/2 I or i = 1 V or v= 5 X or x= 10 L or l= 50 C or c= 100 D or d= 500 M or m= 1000

EIGHT LETTERS of FIXED Values (no letter or ZERO)

-Numbers and Numerals -Kinds of Numbers -Arabic Numerals -Roman Numerals -COMMON and DECIMAL Fractions -RATIO, PROPORTION,and Variation -SIGNIFICANT FIGURES

Fundamental of Pharmaceutical Calculations

a = c b d 12 tabs = 3900 mg = 3 tabs : x x = 3900 mg (3 tabs) = 975 mg 12 tabs

If 12 tablets contain 3900 milligrams of aspirin, how many milligrams should 3 tablets contain?

a = c b d 3 tabs : 975 mg = x : 3900 mg x = 3 tabs (3900 mg) = 12 tabs 975 mg

If 3 tablets contain 975 milligrams of aspirin, how many tablets should contain 3900 milligrams?

1. Volume 2. Weight / Mass 3. Temperature 4. Length

Important Measurements in Pharmacy

numerator > denominator ex: 7/3

Improper Fraction

If a = c ,then a = bc/d , b = ad/c , c = ad/b ,and d = bc/a

In any proportion, the product of the extremes is equal to the product of the means.

1. Metric System 2. Base Units

International System of Units

1. Natural or Whole Number 2. Fraction - division of whole number 3. Mixed Number - whole number + fraction 4. Abstract or Pure Number- how many times the unit 1 is contained in it w/o implying anything else is being counted/measured. 5. Concrete or Denominate Number- designated the total quantity of whatever has been measured.

KINDS OF NUMBERS

Base Units of the SI System -Liters -Grams -Celius -Meters

Measurement: -Volume (liquid) -Weight / Mass (solid) -Temperature -Length

example: 2 1/3

Mixed Number

numerator < denominator ex: 2/5

Proper Fraction

(a)To designate the number of dosage units prescribed ex: Capsules no. C (b) To indicate the quantity of medication to be administered ex: teaspoonfuls ii (c) In rare instances, with the common apothecaries' systems of measurement ex: grains iv

ROMAN Numerals are used in pharmacy only occassionally on prescriptions:

1 : 2 ● ○○ 1 2 3 3

Ratio

1 International System of Units (SI) -metric system 2. Avordupois - imperial system (USA) 3. Apothecary - imperial system (USA)

System of Measurements used in Pharmacy

1. Multiplying the numerator increases the value of a fraction, and multiplying the denominator decreases the value, but when BOTH NUMERATOR and DENOMINATOR are multiplied by the same number, the value does not change. ex: 2 = _3 x 2 = _6_ 7 3 x 7 21 2. Dividing the numerator decreases the value of a fraction, and dividing the denominator increases the value, but when NUMERATOR and DENOMINATOR are divided by the same number the value does not change. ex: 6 = _6 / 3 = _2_ 21 21 / 3 7

TWO PRINCIPLES

ex: 5,083.623 means: -5,000.000 or 5 thousands -+000.000 plus 0 hundreds -+080.000 plus 8 tenths -+003.000 plus 3 ones -+ 000.600 plus 6 tenths -+000.020 plus 2 hundredths -+ 000.003 plus 3 thousandths

The total value of any number expressed in the Arabic (decimal) system, then, is the sum of the values of its digits as determined by their position

Simply inverts its terms and multiply. When a fraction is to be divided by a whole number: 1st interpret the whole number as a fraction, having 1 for its denominator. 2nd Invert to get its reciprocal. 3rd Multiply

To divide common fractions:

The science of calculating with positive, real numbers.

What is Arithmetic?

Is a total quantity, or amount, of units,

What is a NUMBER?

Is a word, or sign, or a group of words or signs expressing a number.

What is a NUMERAL?

10 pints = 10 % = 40 pints (10 %) = 40 % 40 pints x 10 pints

ex: If 10 pints of a 10% solution are diluted to 40 pints, what is the percentage strength of the solution?

a/b = c/d to a:b = c:d 3 tabs : 975 mg = 12 tabs : x x = 975 mg (12 tabs) = 3,900 mg 3 tabs

ex: If 3 tablets contain 975 milligrams of aspirin, how many milligrams should be contained in 12 tablets?

1st step: 4 as 4/1 1/2 ÷ 4/1 = 1/2 x 1/4 = 1/8 0z.

example 1: If ½ ounce is divided into 4 equal parts, how much will each part contains?

1st step: 16 as 16/1 16/1 ÷ 1/32 = 16/1 x 32/1 = 512 samples

example 2: A manufacturer wishes to prepare samples of an ointment in sealed foil envelopes, each containing 1/32 ounce of ointment. How many samples may be prepared from 1 pound (16 ounces) of ointment?

1 + 6 + 8/24 = 15/24 = 5/8 3 = 3x8 = 24 1 1x8 8 continuation... 24/8 5/8 24-5/8 = 19/8 fl. oz 19/8 = (19 / 8) fl. Oz = 2 3/8 fl. oz

example: -If 3 fluidounces of a liquid mixture are to contain 1/24 fl. Oz of ingredient A, ¼ fl.oz of ingredient B, ad 1/3 fl. Oz of ingredient C, how many fluidounces of ingredient D are required? - If 3 fluidounces of a liquid mixture are to contain 1/24 fl. Oz of ingredient A, ¼ fl.oz of ingredient B, ad 1/3 fl. Oz of ingredient C, how many fluidounces of ingredient D are required? INTERPRET THE 3 fl.oz as 3/1, and reduce it to a fraction with 8 for a denominator. continuation... -If 3 fluidounces of a liquid mixture are to contain 1/24 fl. Oz of ingredient A, ¼ fl.oz of ingredient B, ad 1/3 fl. Oz of ingredient C, how many fluidounces of ingredient D are required?

0.125 x 100 = 12.5 %

example: Decimal Fractions- may be converted to percent by multiplying by 100. -Convert 0.125 to percent

3 ÷ 8 x 100 = 37.5 %

example: Common fractions- may be converted to percent by dividing the numerator by the denominator and multiplying 100. -Convert 3/8 to percent.

Systems of Measurement

metric, U.S. customary, SI system and apothecary