Prerequisite Mathematics Review Lab 1 Prerequisite Mathematics Review

NUMBERS AND NUMERALS A number is a total quantity or amount, whereas a numeral is a word, sign, or group of words and signs representing a number.

ARABIC AND ROMAN NUMERALS Arabic Numerals Arabic numerals, such a 1, 2, 3, etc., are used universally to indicate quantities. These numerals, which are represented by a zero and nine digits. Roman Numerals Roman numerals are used with the apothecary’s system of measurement to designate quantities on prescription.

I. Roman numerals

Roman numerals To express quantities in the roman system, eight letters of fixed values are used : Letter Value ss ½ I 1 V 5 X 10 L 50 C 100 D 500 M 1000

Roman numerals There are four basic principles for reading and writing Roman numerals: A letter repeats its value that many times (XXX = 30, CC = 200, etc.). A letter can only be repeated three times. If one or more letters are placed after another letter of greater value, add that amount. VI = 6 (5 + 1 = 6)LXX = 70 (50 + 10 + 10 = 70)MCC = 1200 (1000 + 100 + 100 = 1200)

Roman numerals If a letter is placed before another letter of greater value, subtract that amount. IV = 4 (5 – 1 = 4) XC = 90 (100 – 10 = 90)CM = 900 (1000 – 100 = 900)

Roman numerals Several rules apply for subtracting amounts from Roman numerals: Only subtract powers of ten (I, X, or C, but not V or L)For 95, do NOT write VC (100 – 5). DO write XCV (XC + V or 90 + 5) Only subtract one number from another. For 13, do NOT write IIXV (15 – 1 - 1). DO write XIII (X + I + I + I or 10 + 3) Do not subtract a number from one that is more than 10 times greater (that is, you can subtract 1 from 10 [IX] but not 1 from 20—there is no such number as IXX.)For 99, do NOT write IC (C – I or 100 - 1). DO write XCIX (XC + IX or 90 + 9)

Roman numerals A bar placed on top of a letter or string of letters increases the numeral's value by 1,000 times. (XV = 15, but X̅V̅ = 15,000)

Roman numerals Example Write the following in Roman: 27 1876 126 999

Roman numerals Write the following in Arabic: MCMLIX xlviii Lxxxiv lxxii

Perform the following operations and indicate your answer in Arabicnumbers: XII + VII XXVI − XII XXIV ÷ VI XIX × IX

II. Fractions

Fractions A fraction is a portion of a whole number. Fractions contain two numbers: the bottom number (referred to as denominator) and the top number (referred to as numerator). The denominator in the fraction is the total number of parts into which the whole number is divided. The numerator in the fraction is the number of parts we have.

Fractions A proper fraction should always be less than 1, i.e., the numerator is smaller than the denominator. Examples: 5/8, 7/8, 3/8 A proper fraction such as 3/8 may be read as ‘‘3 of 8 parts’’ or as ‘‘3 divided by 8.’’

Fractions An improper fraction has a numerator that is equal to or greater than the denominator. It is therefore equal to or greater than one. Examples: 2/2 = 1, 5/4, 6/5 To reduce the improper fraction, divide the numerator by the denominator.

Fractions Examples: 8/8 = 8 ÷ 8 = 1 6/4 = 6 ÷ 4 = 1 2⁄4 9/4 = 9 ÷ 4 = 2 1⁄4

Fractions Simplifying the fraction: find the largest number (referred to as greatest common divisor) that will divide evenly into each term. Examples: 15/20 = 15 ÷ 5/20 ÷ 5 = 3/4 12/18 = 12 ÷ 6/18 ÷ 6 = 2/3 7/21 = 7 ÷ 7/21 ÷ 7 = 1/3

Fractions Adding fraction: To add fractions reduce them to common denomination, add the numerators, and the sum over the common denominator Example: 4/6 + 2/5 = 20/30 + 12/30 = 32/30

Fractions Some numbers are expressed as mixed numbers (a whole number and a fraction). To change mixed numbers to improper fractions, multiply the whole number by the denominator of the fraction and then add the numerator. Examples: 10 5⁄8 = 85/8 3 5⁄6 = 23/6

Fractions Subtracting of Fractions To subtract one fraction from another, reduce them to a common denomination, subtract, and write the difference over the common denominator. Example 7/12 – 1/8 = 14/24 – 3/24 = 11/24

Fractions Multiplying fractions To multiply fractions, multiply the numerators and write the product over the product of the denominators. Example 2/3 x 4/5 = 8/15 2/5 x 1/2 = 2/10 = 1/5 Reduce your answer to lowest terms, when possible.

Fractions Dividing fraction: To divide a whole number or a fraction by a proper or improper fraction, invert the divisor and multiply. Example: 4/5 ÷ 2/3 = 4/5 × 3/2 = 6/5 or 1 1⁄5

Fractions DECIMALS Decimals are another means of expressing a fractional amount. A decimal is a fraction whose denominator is 10 or a multiple of 10. Example: 0.8 = 8/10 0.08 = 8/100 0.008 = 8/1000 A decimal mixed number is a whole number and a decimal fraction. Example: 4.3 = 4 3/10

Fractions Example A bottle of Children’s Tylenol contains 30 teaspoonfuls of liquid. If each dose is 1⁄2 teaspoonful, how many doses are available in this bottle? A prescription contains 3/5 gr of ingredient A, 2/4 gr of ingredient B, 6/20 gr of ingredient C, and 4/15 gr of ingredient D. Calculate the total weight of the four ingredients in the prescription?

Fractions A pharmacist had 10 g of codeine sulfate. If he used it in preparing 5 capsules each containing 0.025 g, 10 capsules each containing 0.010 g, and 12 capsules each containing 0.015 g, how many g of codeine sulfate were left after he prepared all the capsules?

Logarithm

Logarithm The logarithm of a positive number N to a given base b is the exponent x to which the base must be raised to equal the number N. Therefore, if N = bx then logb N = x For example, with common logarithms (log), or logarithms using base 10, 100 = 102 then log 100 = 2, The number 100 is considered the antilogarithm of 2.

Conversion of temerature

Conversion of temerature Temperature is measured with a thermometer. Standard Scales: Use the freezing and boiling points of water at atmospheric pressure as basis. Fahrenheit oF (32 - 212) oF = (1.8 x oC) + 32 Celsius oC (0 -100) oC = (oF - 32)/1.8

A thermometer on the wall of a room reads 86 Fº A thermometer on the wall of a room reads 86 Fº. What is the room temperature in Cº.

Home work Write the following in Roman numerals: 28 65 17 1763 Convert the following Roman numerals to Arabic numerals: xlvi lxxiv xlvii xxxix A tablet contains 1/20 gr of ingredient A, 1/4 gr of ingredient B, 1/12 gr of ingredient C, and enough of ingredient D to make a total of 20 gr. How many grains of ingredient D are in the tablet? Convert 140 C° to F°