Ratio & Proportion PDF

CHAPTER 5 PROBLEMS SOLVING METHODS The essence of mathematics is not to make simple things complicated, but to make complicated things simple....

CHAPTER 5 PROBLEMS SOLVING METHODS The essence of mathematics is not to make simple things complicated, but to make complicated things simple. --Stanley Gudder Most calculations in pharmacy can be solved using either ratios and proportions or dimensional analysis (a.k.a., factor labels). Ratios We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to" -- we say "the ratio of something to something else" Example : The ratio of squares to triangles in the illustration below. Ratios can be written in several different ways: 3 as a fraction 4 using the word “to” 3 to 4 using a colon 3:4 Using the above images, make a comparison of triangles to all shapes written the following ways: 1) as a fraction 2) using the word “to” 3) using a colon Pharmaceutical Calculations RATIO & PROPORTION 1 GPL v3.0 or later, 2012 by Sean Parsons Multiplying or dividing each term by the same nonzero number will give an equal ratio. Example The ratio 2:4 is equal to the ratio 1:2. To tell if two ratios are equal, use a calculator and divide. If the division gives the same answer for both ratios, then they are equal. 1 2 4 6 1:2 = 2:4 = 4:8 = 6:12 or 2 = 4 = 8 = 12 Are 3:9, 1:3, and 9:27 all equal? Example Janine has a bag with 3 flash drives, 4 marbles, 7 books, and 1 orange. 1) What is the ratio of books to marbles? Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4. Two other ways of writing the ratio are 7 to 4, and 7:4. 2) What is the ratio of flash drives to the total number of items in the bag? There are 3 flash drives, and 3 + 4 + 7 + 1 = 15 items total. The answer can be expressed as 3/15, 3 to 15, or 3:15. Complete the following practice problems. 1) What is the ratio of squares to total? 2) What is the ratio of circles to triangles? 3) What is the ratio of triangles to squares? What is the ratio of circles to all? What is the ratio of triangles to squares to circles? Pharmaceutical Calculations RATIO & PROPORTION 2 GPL v3.0 or later, 2012 by Sean Parsons Attempt the following practice word problems. Practice Problems 1) Write a ratio comparing the oxygen tension of arterial blood (100 milliliters) to that of venous blood (40 milliliters). 2) If 2 oz of boric-acid are added to 598 oz of water, how many ounces are in the total solution? What is the ratio of ounces of boric acid to ounces of solution? 3) If 35 oz of a chemical are combined with 65 oz of water, how many oz are in the solution? What is the ratio of oz of chemical to oz of solution? Pharmaceutical Calculations RATIO & PROPORTION 3 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.1 Express the following as ratios reduced to lowest terms. 1) 1 to 5 5) 3 feet to 27 feet 9) 3 meters to 9 meters 2) 2 to 11 6) 25 liters to 5 liters 10) 100 g to 1,000 g 3) 2 grams to 9 grams 7) 36 cm to 6 cm 4) 15 cm to 23 cm 8) 24 inches to 9 inches Solve the following problems. 11) If 3 oz of boric-acid solution are added to 897 oz of water, how many ounces are in the total solution? What is the ratio of ounces of boric acid to ounces of solution? 12) If 30 oz of a chemical are combined with 60 oz of water, what is the ratio of chemical to ounces of solution? 13) If 20 oz of alcohol are added to 40 oz of water, how many ounces are in the total solution? What is the ratio of ounces of alcohol to ounces of total solution? 14) If 10 grams of salt are added to 40 grams of water, what is the total weight of the water-and-salt solution? What is the ratio of grams of salt to grams of solution? (Ordinarily, you would not measure water by mass, but it is possible to do and therefore the problem could be reproduced.) 15) If 3 mL of glycerin are added to 87 mL of water, what is the ratio of the mL of glycerin to the mL of solution? 16) If 2 grams of a drug are added to 38 grams of petrolatum, what is the ratio of the grams of drug to the grams of the mixture? 17) If 5 oz of a drug are added to 95 oz of water, how many ounces are in the total solution? What is the ratio of ounces of drug to ounces of solution? Pharmaceutical Calculations RATIO & PROPORTION 4 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.2 Express the following as ratios reduced to lowest terms. 1) 1 to 4 5) 5 feet to 19 feet 9) 1,000 m to 10 m 2) 3 to 17 6) 23 liters to 25 liters 10) 12 grams to 48 grams 3) 3 grams to 15 grams 7) 3 cm to 18 cm 4) 2 cm to 7 cm 8) 35 inches to 7 inches Solve the following problems. 11) To perform a test, a lab technician adds 5 oz of a test liquid to 395 oz of water. How many ounces are in the solution? What is the ratio of ounces of test liquid to ounces of solution? 12) If 2 oz of chemical are combined with 8 oz of water, how many ounces are in the total solution? What is the ratio of ounces of chemical to ounces of solution? 13) A marathon runner has a normal heart rate of 68 beats/min. After completing a marathon, her heart rate is 110 beats/min. Express the ratio of her normal heart rate to her heart rate after a marathon. 14) If 3 oz of boric-acid 1 solution are added to 741 oz of water, how many ounces are in the total solution? What is the ratio of boric acid to ounces of solution? 15) In 1964, the U.S. Public Health Service reported that the risk of developing lung cancer is 10 times greater for moderate smokers and 20 times greater for heavy smokers than for nonsmokers. Express ratio of risk of cancer for nonsmokers to moderate smokers to heavy smokers. (Hint, even if you don't smoke you still have a chance of developing lung cancer.) 1 Since it has been mentioned repeatedly for various problems, you may be asking, “What's boric-acid?”. Boric-acid, also called boracic acid or orthoboric acid, is a mild acid often used as an antiseptic, insecticide, flame retardant, in nuclear power plants to control the fission rate of uranium, and as a precursor of other chemical compounds. It exists in the form of colorless crystals or a white powder and dissolves in water. It has the chemical formula B(OH)3. When occurring as a mineral, it is called sassolite. Pharmaceutical Calculations RATIO & PROPORTION 5 GPL v3.0 or later, 2012 by Sean Parsons Proportions A proportion is a statement of equality between two ratios. Example 1 2 = 2 4 This proportion can also be expressed as “1:2 = 2:4” or “1 is to 2 as 2 is to 4”. Practice Problem What would be two other ways to express the following: 𝑎𝑎 𝑐𝑐 = 𝑏𝑏 𝑑𝑑 solving for a variable in a proportion You have three options: 1) basic algebra 2) cross multiplying 3) “means” and “extremes” Let's solve the following equation for a using all three methods 𝑎𝑎 𝑐𝑐 𝑏𝑏 = 𝑑𝑑 algebra Solving for a using basic algebra. First, we need to isolate a. We can do this by multiplying both sides of the equation by b. Then you can cancel out the bs on the left hand side of the equation, and this will leave you with your final answer. Multiply both sides by b. Now cancel out the bs on the left. Your final answer 𝑎𝑎 𝑐𝑐 𝑏𝑏×𝑎𝑎 𝑏𝑏×𝑐𝑐 𝑏𝑏×𝑎𝑎 𝑏𝑏×𝑐𝑐 𝑏𝑏 = 𝑑𝑑 𝑏𝑏 = 𝑑𝑑 𝑏𝑏 = 𝑑𝑑 𝑏𝑏×𝑐𝑐 𝑎𝑎 = 𝑑𝑑 Pharmaceutical Calculations RATIO & PROPORTION 6 GPL v3.0 or later, 2012 by Sean Parsons cross multiplying Solving for a using cross multiplication. You can multiply diagonally and set both multiplication problems as equal. Then we will want to isolate the a by dividing both sides by d. Cancel out the ds on the appropriate side and you will be left with the final answer. Cross multiply Divide by d. 𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑐𝑐 𝑎𝑎×𝑑𝑑 𝑏𝑏×𝑐𝑐 𝑎𝑎×𝑑𝑑 𝑏𝑏×𝑐𝑐 𝑏𝑏 = 𝑑𝑑 𝑏𝑏 = 𝑑𝑑 𝑎𝑎 × 𝑑𝑑 = 𝑏𝑏 × 𝑐𝑐 𝑑𝑑 = 𝑑𝑑 Cancel 𝑑𝑑 = 𝑑𝑑 Your final answer 𝑏𝑏×𝑐𝑐 𝑎𝑎 = 𝑑𝑑 “means” and “extremes” Solving for a using “means” and “extremes”. To do this, you need to rewrite your proportion using colons. Next, you multiply your means (your inside numbers) and set them equal to the value of your extremes (outside numbers) when they are multiplied. Then we will want to isolate the a by dividing both sides by d. Cancel out the ds on the appropriate side and you will be left with the final answer. Multiply your “means” and “extremes” Rewrite it using colons 𝑎𝑎 𝑐𝑐 𝑏𝑏 = 𝑑𝑑 𝑎𝑎: 𝑏𝑏 = 𝑐𝑐: 𝑑𝑑 𝑎𝑎: 𝑏𝑏 = 𝑐𝑐: 𝑑𝑑 𝑎𝑎 × 𝑑𝑑 = Your final answer Divide by d. 𝑎𝑎×𝑑𝑑 𝑏𝑏×𝑐𝑐 𝑎𝑎×𝑑𝑑 𝑏𝑏×𝑐𝑐 𝑏𝑏×𝑐𝑐 𝑏𝑏 × 𝑐𝑐 𝑑𝑑 = 𝑑𝑑 Cancel 𝑑𝑑 = 𝑑𝑑 𝑎𝑎 = 𝑑𝑑 Now that you've been shown three different ways to solve for a variable in a ratio proportion, let's try a practice problem. Solve for each of the variables in the statement “W is to X as Y is to Z”. 1) The value of W is: 2) The value of X is: 3) The value of Y is: 4) The value of Z is: Pharmaceutical Calculations RATIO & PROPORTION 7 GPL v3.0 or later, 2012 by Sean Parsons Practice Problems Solve for the variable in each of the following practice problems. 3.8𝐿𝐿 𝑁𝑁 1) 14𝐿𝐿 = 25𝐿𝐿 𝑥𝑥 3.0𝑔𝑔 2) 7.2𝑔𝑔 = 2.0𝑔𝑔 3) 5: 𝑁𝑁 = 7: 9 4) 3: 4 = 𝑥𝑥: 15 Pharmaceutical Calculations RATIO & PROPORTION 8 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.3 Solve the following ratio proportion problems. 1) In the proportion 5:8 = 25:40, a) The extremes are _____ and _____. b) The means are _____ and _____. Solve for N in each of these problems. 2 4 𝑁𝑁 10 2) = 𝑁𝑁 N = _____ 11) 5 = 25 N = _____ 3 5 15 𝑁𝑁 10 3) = N = _____ 12) 6 = 12 N = _____ 7 𝑁𝑁 3 1 𝑁𝑁 6 4) 6 = 𝑁𝑁 N = _____ 13) 7 = 42 N = _____ 5 7 𝑁𝑁 2 5) = 𝑁𝑁 N = _____ 14) 15 = 5 N = _____ 10 2 3 𝑁𝑁 4 6) = 𝑁𝑁 N = _____ 15) 5 = 10 N = _____ 8 𝑁𝑁 4 3 𝑁𝑁 7) = 6 N = _____ 16) 5 = 100 N = _____ 3 𝑁𝑁 3 125 𝑁𝑁 8) = 24 N = _____ 17) 1,000 = 8 N = _____ 8 𝑁𝑁 5 3 375 9) 12 = 6 N = _____ 18) = N = _____ 8 𝑁𝑁 𝑁𝑁 6 2 𝑁𝑁 10) 4 = 8 N = _____ 19) = 12 N = _____ 3 Pharmaceutical Calculations RATIO & PROPORTION 9 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.4 Write the following problems in fractional-equation form. 1) 3:x = 6:7 3) K:8 = 4:16 5) 2:3 = x:9 2) 5:8 = N:10 4) 7:9 = 14:N 6) 7:N = 14:28 Solve the following proportions for N. 7) N:7 = 6:14 2.5 𝑁𝑁 13) 16 = 8 8) 8:10 = 4:N 4.2 2.1 14) 𝑁𝑁 = 100 9) 3:N = 6:18 6.8 13.6 15) 50 = 𝑁𝑁 10) 5:8 = N:24 12.0𝑔𝑔 8.4𝑔𝑔 11) N:9 = 6:18 16) 4.6𝑔𝑔 = 𝑁𝑁 12) 8:N = 16:32 Pharmaceutical Calculations RATIO & PROPORTION 10 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.5 Write the following proportions in fractional-equation form. 1) 4:X = 7:9 3) N:72 = 4.8:12.0 5) 5:17.1 = N:2 2) 3:5 = N:8 4) 15.0:N = 31.0:1 6) 315:32 = N:35 Solve the following proportions for N. 7) N:2 = 15:3 𝑁𝑁 1.7 13) 2.2 = 1 8) 4:5 = 8:N 14.0𝑔𝑔 12.0𝑔𝑔 14) 70𝑚𝑚𝑚𝑚 = 𝑁𝑁 9) 5:N = 10:17 2.1𝐿𝐿 𝑁𝑁 15) 8𝐿𝐿 = 12.0𝐿𝐿 10) 5:9 = N:10 14.1𝑜𝑜𝑜𝑜 4𝑜𝑜𝑜𝑜 3.5 70 16) = 11) 𝑁𝑁 = 100 5.8𝑜𝑜𝑜𝑜 𝑁𝑁 𝑁𝑁 $0.07 𝑁𝑁 12 17) = $1.00 12) 1.4 = 28 $712 Pharmaceutical Calculations RATIO & PROPORTION 11 GPL v3.0 or later, 2012 by Sean Parsons 5 Step Method You may also notice the way the example was broken down with five questions. This is known as the “5 Step Method”. The “5 Step Method” is simply a way to help you interpret the data in a problem and get the answer. While certainly the problems in this book can be solved without using the “5 Step Method”, you will find that many example problems throughout the book are broken down this way. The five steps are: 1) QUESTION (identify what the question is asking); 2) DATA (identify the data in the problem and any necessary conversion factors); 3) MATHEMATICAL METHOD/FORMULA (identify the method or formula needed to solve the problem); 4) DO THE MATH (exactly what the statement says), and 5) DOES THE ANSWER MAKE SENSE? (sometimes an answer will not make sense, and that may mean that there is an error). Now we should try a practice problem using dimensional analysis and the “5 Step Method”. How many tablets will you need to provide a 24 hour supply of metoprolol succinate 50 mg/dose, 1 dose/day, if you have 25 mg tablets available? QUESTION DATA MATHEMATICAL METHOD / FORMULA DO THE MATH DOES THE ANSWER MAKE SENSE? Pharmaceutical Calculations RATIO & PROPORTION 12 GPL v3.0 or later, 2012 by Sean Parsons  Practice 5.6 (5.9) Express the following as ratios reduced to lowest terms. 1) 1 to 9 2) 2 g to 12 g 3) 6 cm to 36 cm Solve the following word problems. 4) What is the ratio of chemical to ounces of solution if 20 oz of chemical are combined with 80 oz of water? 5) How many ml of a drug are needed to prepare 2,500 ml of a 1:20 solution? 6) How much hydrocortisone is needed to prepare 120 grams of a 1:50 ointment? Write the following proportions in fractional-equation form. 7) 5:x = 10:15 8) 6:11 = Y:12 9) 1:3 = N:18 Solve the following ratio proportions for N. 10) N:10 = 3:5 4.1𝑔𝑔 6.1𝑔𝑔 15) 12.3𝑔𝑔 = 𝑁𝑁 11) N:10 = 4:5 4.4𝑚𝑚𝑚𝑚 𝑁𝑁 16) 17.6𝑚𝑚𝑚𝑚 = 4𝑚𝑚𝑚𝑚 12) N:9 = 2:3 5𝑖𝑖𝑖𝑖. 𝑁𝑁 3.1𝑔𝑔 8.1𝑔𝑔 17) 12𝑖𝑖𝑖𝑖. = 8𝑖𝑖𝑖𝑖. 13) 15.5𝑔𝑔 = 𝑁𝑁 50𝑔𝑔 𝑁𝑁 18) 100𝑚𝑚𝑚𝑚 = 4𝑚𝑚𝑚𝑚 4.2𝑔𝑔 7.1𝑔𝑔 14) 16.8𝑔𝑔 = 𝑁𝑁 Pharmaceutical Calculations RATIO & PROPORTION 13 GPL v3.0 or later, 2012 by Sean Parsons