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- 1. Week 1 Roman Numerals and Fractions
- 2. Uses of Roman Numerals in Pharmacy • Roman numerals were once used frequently in pharmacy; now you will only see them occasionally • Sometimes prescribers will use them on written prescriptions to indicate a quantity
- 3. Make sure you know these • I = 1 (you may also see “i”) • V = 5 (you may also see “v”) • X = 10 (you may also see “x”) • L = 50 • C = 100 • D = 500 • M = 1000
- 4. The symbol for 1, 10, 100 or 1000 can be repeated up to 3 times to make larger numbers. • C = 100 • CC = 200 • CCC = 300 • M = 1000 • MM = 2000 • MMM = 3000
- 5. Smaller valued symbols AFTER a larger valued symbol are ADDED • X = 10 • XI = 10 + 1 = 11 • XII = 10 + 1 + 1 = 12 • XIII = 10 + 1 + 1 + 1 = 13 • XV = 10 + 5 = 15 • XVI = 10 + 5 + 1 = 16 • XXXIII = 10 + 10 + 10 + 1 + 1 + 1 = 33 • MCX = 1000 + 100 + 10 = 1110
- 6. Smaller valued symbols BEFORE a larger valued symbol are SUBTRACTED • V = 5 • IV = 5 – 1 = 4 • X = 10 • IX = 10 – 1 = 9 • L = 50 • XL = 50 – 10 = 40 • C = 100 • XC = 100 – 10 = 90
- 7. Most of the Roman numerals you encounter in pharmacy will be the basic ones Example from a prescription: Percocet 5/325 XX Sig: ii bid This means: fill a prescription for 20 (XX) Percocet tabs, with 2 (ii) tabs to be taken twice daily (bid). Percocet is a combination drug; each tab contains 5mg oxycodone and 325mg acetaminophen (5/325).
- 8. To Remember • Memorize the basic symbols: I, V, X, L, C, D, M • The following symbols can be repeated up to 3 times to make numbers: I, X, C, M • Smaller valued symbols that come AFTER larger valued ones are ADDED, for example VI = 5 + 1 = 6 • Smaller valued symbols that come BEFORE larger valued ones are SUBTACTED, for example IV = 5 – 1 = 4
- 9. Test Yourself • C = ? • 100 • XVIII = ? • 10 + 5 + 1 + 1 + 1 = 18 • IV = ? • 5 – 1 = 4 • XLII = ? • (50 – 10) + 1 + 1 = 42 • MCXXX = ? • 1000 + 100 + 10 + 10 + 10 = 1130
- 10. Uses of Fractions in Pharmacy • Fractions are used all day, every day in pharmacy and are very important to master • For example, drug concentrations are expressed as fractions • On this drug label “40 mg/ml” is a fraction that tells you the concentration of the drug tobramycin in the vial
- 11. Fractions • Fractions express parts of a whole and can be written many different ways. • 3 parts out of 4 = 3 / 4 = 3 out of 4 = 3 : 4 • = 3 per 4 • The first (top) number in a fraction is called the numerator • The second (bottom) number in a fraction is called the denominator
- 12. Fractions • ANYTIME zero is the numerator of a fraction, the fraction is equal to 0 • For example 0 / 2 = 0 0 / 11,143 = 0 • ANYTIME the number one is the denominator of a fraction, the fraction is equal to the numerator • For example 3 / 1 = 3 0 / 1 = 0 6.6 / 1 = 6.6 • Zero cannot be the denominator of a fraction • For example 450 / 0 is “undefined”
- 13. Fractions • Any fraction where the numerator and denominator are the same is equal to 1. • For example, 9 / 9 = 1 4 / (3+1) = 1 • If two fractions are equal, they are called equivalent fractions • These 4 fractions are equivalent and are all ways of writing 1 / 2 or one half.
- 14. Test Yourself • Express 5 per 6 as a fraction • 5 / 6 • 16 / 0 = ? • This is undefined (division by 0 is not possible) • 55.5 / 1 = ? • 55.5 (any number over 1 = itself) • 0 / 734 = ? • 0 (zero over any number = zero) • 3/3 = ? • 1 (any number over itself is = 1) • Are these equivalent? ¾ and 1/3 • No, because they are not equal
- 15. Simplifying Fractions • When you do calculations with fractions, the answer should be “in simplest terms”. To simplify fractions, you need to first factor the numerator and the denominator, then cross out like terms (because any number over itself = 1) 4 2 x 2 2 x 2 2 _ = ___ = ___ = _ 6 2 x 3 2 x 3 3
- 16. Simplifying Fractions • Other examples of simplifying fractions: 15 3 x 5 3 x 5 5 __ = _____ = ______ = ___ = 5 / 9 27 3 x 3 x 3 3 x 3 x 3 3 x 3 250 5 x 5 x 5 x 2 5 x 5 x 5 x 2 5 __ = _______ = ________ = _ = 5 50 5 x 5 x 2 5 x 5 x 2 1
- 17. Test Yourself Simplify: 4 _ = ? 24 4 2 x 2 _ = ________ 24 2 x 2 x 2 x 3
- 18. Test Yourself Simplify: 4 2 x 2 1 1 _ = ________ = ___ = __ 24 2 x 2 x 2 x 3 2 x 3 6
- 19. To Remember • Any number over itself = 1 • Any number over 0 is undefined • Any number over 1 = itself • Any fraction with 0 in the numerator = 0 • Answers to fractions calculations should be simplified • To simplify, factor the numerator and denominator completely and cross out like terms on top and bottom.
- 20. Adding and Subtracting Fractions • If the denominators are the same, simply add or subtract the numerators and keep the same denominator • For example 1 / 4 + 2 / 4 = (1 + 2) / 4 = 3 / 4 • Remember to simplify your answer if needed (in the above example, 3 / 4 is already in simplest terms). • If your answer has a numerator greater than the denominator, the answer can also be simplified • For example 2 / 3 + 2 / 3 = 4 / 3 which is the same as 1 1/3.
- 21. Adding or Subtracting Fractions • If the denominators are NOT the same, one or more of the fractions in the problem will have to be converted into an equivalent fraction so that all of the fractions have the same (common) denominator. • For example: 1 1 _ + _ = ? You can’t simply add numerators, 2 4 since the denominators are different. For the new denominator, choose a number that both old denominators will divide into.
- 22. Adding Fractions 1 1 _ + _ = ? 2 4 Both 2 and 4 will divide into 4, so choose 4 for the new denominator. To convert ½ to fourths, multiply both top and bottom by 2. 1 (2) 1 2 1 3 ___ + _ = _ + _ = _ 2 (2) 4 4 4 4
- 23. Adding and Subtracting Fractions • Another example: 3 2 _ - _ = ? 4 3 4 and 3 can both be divided into 12 Multiply ¾ by 3/3 to convert to 12ths. Multiply 2/3 by 4/4 to convert to 12ths. 3 2 3 (3) 2 (4) 9 8 1 _ - _ = ___ - ___ = _ - _ = _ 4 3 4 (3) 3 (4) 12 12 12
- 24. Test Yourself 2 1 _ + _ = ? 5 2 5 and 2 can both be divided into 10 Multiply 2/5 by 2/2 to convert to 10ths. Multiply ½ by 5/5 to convert to 10ths. 2 (2) 1 (5) 4 5 9 ____ + ___ = _ + _ = _ 5 (2) 2 (5) 10 10 10
- 25. To Remember • When adding or subtracting fractions, if the denominators are the same, you can simply add or subtract the numerators and keep the same denominator. • If the denominators of the two fractions you are adding or subtracting are not equal, you must convert them to fractions with the same common denominator. • Find a number that both denominators will divide into, and use that as the new denominator.
- 26. Multiplying Fractions • Multiplying fractions is actually easier than adding or subtracting them! • Simply multiply the numerators together, and then then the denominators, then simplify if needed. Example: 1 1 _ x _ = ? 2 3 1 1 1 x 1 1 _ x _ = ____ = _ 2 3 2 x 3 6
- 27. Multiplying Fractions • Another example: 4 1 4 x 1 4 2 x 2 2 x 2 2 _ x _ = ____ = _ = ____ = ___ = _ 5 6 5 x 6 30 2 x 15 2 x 15 15
- 28. Test Yourself 3 1 _ x _ = ? 7 4 3 1 3 x 1 3 _ x _ = ____ = _ (already simplest terms) 7 4 7 x 4 28
- 29. Dividing Fractions • To divide two fractions, you take the reciprocal of the second fraction, then multiply. • What is a reciprocal? • You simply “flip” the fraction over (swap numerator and denominator) • For example, the reciprocal of 2/3 is 3/2. • The reciprocal of ¼ is 4/1 or 4. • The reciprocal of 61 is 1/61 (since 61 = 61/1).
- 30. Dividing Fractions • Division example: 1 1 1 8 1 x 8 8 2 x 2 x 2 _ ÷ _ = _ x _ = ___ = _ = _____ = 2 8 2 1 2 x 1 2 2 2 x 2 x 2 4 ______ = _ = 4 2 1
- 31. Dividing Fractions • Another example: 3 2 3 3 3 x 3 9 _ ÷ _ = _ x _ = ___ = _ 5 3 5 2 5 x 2 10
- 32. Test Yourself The reciprocal of 1/12 is ? 12/1 or 12 The reciprocal of 7/8 is ? 8/7 To divide two fractions, multiply the first fraction by the reciprocal of _____? The second fraction
- 33. Test Yourself 1 3 _ ÷ _ = ? 17 2 1 3 1 2 1 x 2 2 _ ÷ _ = _ x _ = ____ = _ 17 2 17 3 17 x 3 51
- 34. Remember • To multiply fractions, multiply the numerators and the denominators together and simplify the answer. • To divide fractions, multiply the first fraction by the reciprocal of the second fraction. • To find the reciprocal of a fraction, swap the numerator and the denominator.
- 35. Using Fractions in Pharmacy • Look closely at this drug label • Note the concentration of the drug highlighted in the red bar: “ 500mg PE/10 ml”. The concentration is expressed as a fraction. • Underneath that you will see another fraction --- “50mgPE/ml”. This fraction is equivalent to the first and is the simplified version of the first fraction. 500 5 x 5 x 5 x 2 x 2 5 x 5 x 5 x 2 x 2 5 x 5 x 2 ___ = ___________ = __________ = ______ = 50 10 5 x 2 5 x 2 1
- 36. Next steps • Do the homework problems and check your answers using the back of the textbook. • Complete your discussion board assignment. Also post any observations or questions you may have about the powerpoint and the homework. • Review this powerpoint as well as the homework problems before taking the weekly quiz.