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Review of Mathematical Principles Overview Multiplication hint • Zero times any number is zero • Reduce all fractions to their lowest terms • Check the denominators before adding and subtracting fractions. IF they are different, find the lowest common denominator, then solve the problem. • Change all mixed numbers to improper fractions before working out the problems Fractions • For multiplication of fractions, just multiply the numerators and the denominators • For division of fractions, remember to invert the second fraction. Then multiply the numerators and the denominators. Arabic and Roman numerals • Arabic numeral system: used to express quantity and value e.g. 1, 2, 3, etc • Roman numeral system: used as units of apothecaries’ system of weights and measures in writing prescriptions. • Used to express dates in copyrights, formal manuscripts. Roman numerals • • • • • • • I= 1 V=5 X = 10 L = 50 C = 100 D = 500 M = 1000 Rules for using Roman numerals 1.Whenever a Roman numeral is repeated or a smaller numeral follows a larger one, the values are added together. – – – – III = 3 iii = 3 viii = 8 (5+3 = 8) LVII = 57 (50+5+2= 57) Rules for using Roman numerals 2.Whenever a smaller Roman numeral comes before a larger Roman numeral, subtract the smaller value: – iv = 4 ( 5-1= 4) – ix = 9 (10-1 = 9) – CD = 400 (500-100 = 400) Rules for using Roman numerals 3. Numerals are never repeated more than three times in a sequence: – iii = 3 In expressing dosages in apothecaries’ numeral system use lowercase rather than capital numerals. Dot placed over indicates lowercase. Rules for using Roman numerals 4. Whenever a smaller Roman numeral comes before two larger Roman numerals, subtract the smaller number from the numeral following it. -xix = 19 (10+ (10-1) = 19 -xxiv = 24 (10+10+ (5-1) = 24 FRACTIONS • All units of the apothecaries’ system are written as common fractions for all amts less than one. • Fractions for the foundation in dosage calculations when medication is in a different available dose form than that ordered. Basic Principles • Fraction: one or more equal parts of a unit. • Numerator: top number = how many parts are being used.f,m. • Denominator: bottom number • Fraction may be raised to higher terms by multiplying both terms of the fraction by the same number. – E.g. ¾ = multiply both numerator and denominator by 2 = 6/8 – 3x2=6 – 4x2=8 • Fractions are reduced to lower terms by dividing both the numerator and denominator by the same number. • E.g. to lower 3/9 to a lower term, divide both the numerator and the denominator by 3, converting it to 1/3; • 3/9 and 1/3 have the same value. • 1 • A proper fraction has a numerator smaller than the denominator. The number 4 is a proper fraction because it is less than 1; its numerator is less than its denominator • An improper fraction has a numerator the same as or larger than the denominator. The number 6/4 is an improper fraction because the numerator (6) is larger than the denominator (4). • In using fractions in calculations, the numerator and the denominator must be of the same unit of measure. • For example, if the numerator is in grains, the denominator must be in grains. • A mixed number is a whole number and a proper fraction. • Examples of mixed numbers are 4 1/3, 3 3/4, and 5 16/35. • • It is often necessary to change an improper fraction to a mixed number or to change a mixed number to an improper fraction when doing certain calculations. To change an improper fraction to a mixed number, divide the denominator into the numerator. The result is the whole number (quotient) and the remainder, which is placed over the denominator of the improper fraction • For example: 17/3 is an improper fraction. To convert to a mixed number: • 1. Divide the denominator (3) into the numerator (17): • 2. Move the remainder (2) over the denominator (3) • 3. Put the quotient (5) in front of the fraction: • • • • 5 = quotient 3/17 -17 2 = remainder • To change the mixed number 5 2/3 to an improper fraction, multiply the denominator of the fraction (3) by the whole number (5), add the numerator (2), and place the sum over the denominator. • A complex fraction has a fraction in either its numerator, its denominator, or both. • ½ /50 • 30/ 2/3 • 3½/¼ • Complex fractions may be changed to whole numbers or proper or improper fractions by dividing the number or fraction above the line by the number or fraction below the line. • change ½ /100 to a proper fraction • simply invert or reverse the numerator (100) and the denominator (1) and multiply by the result, or by 1/100. • = 2 X 100 Adding Fractions • If fractions have the same denominator, simply add the numerators, and put the sum above the common denominator. • If the fractions have different denominators, they must be converted to a number that each denominator has in common, or a common denominator. One can always find a common denominator by multiplying the two denominators by one another. Sometimes, both numbers will go into a smaller number