WordProbs - Module 5 – Word Problems Module 5 – Word...

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Unformatted text preview: Module 5 – Word Problems Module 5 – Word Problems Prerequisite: Complete Modules 2 and 4 before starting Module 5. Timing: Begin Module 5 as soon as you are assigned word-problem calculations. ***** Introduction In this module, you will learn to identify given quantities and equalities in word problems. You will then be able to solve nearly all of the initial problems assigned in chemistry with the same conversion-factor method that you used with success in Module 4. In these lessons, you will be asked to take steps to organize your data before you begin to solve a problem. Most students report that by using this method, they then have a better understanding of what steps to take to solve science calculations. ***** Lesson 5A: Answer Units -- Single or Ratio? Types of Units In these lessons, we will divide the units used in measurements into three types. • Single units have numerators but no denominators. Examples include meters, cubic centimeters, grams, and hours. • Ratio units have one unit in the numerator and one in the denominator. Examples include meters/second and g/mL. • Complex units are all other units, such as 1/sec or (kg·meters2)/sec2. Most of the calculations encountered initially in chemistry involve single units and ratios, but not complex units. Rules for single units will be covered in this module. The distinctions between single and ratio units will be covered in Module 11. Rules for complex units will be added in Modules 17 and 19. Rule #1: Know Where You Are Going In science calculations: To solve word problems, begin by writing “WANTED: Example: ? ” and the unit of the answer. WANTED: ? hours The first time you read a word problem, look only for the unit of the answer. Writing the answer unit first will • help you choose the correct given to start your conversions, • prompt you to write DATA conversions that you will need to solve, and • tell you when to stop conversions and do the math. © 2009 www.ChemReview.Net v.m9 Page 74 Module 5 – Word Problems Rules for Answer Units When writing the WANTED unit, it is important to distinguish between single units and ratio units. 1. An answer unit is a ratio unit if a problem asks you to find a. “unit X over one unit Y,” or b. “unit X /unit Y” or “unit X•unit Y─1” or c. “unit X per unit Y” where there is no number after per. All of those expressions are equivalent. All are ways to represent ratio units. Example: grams , also written grams/mL or g •mL─1, is a ratio unit. mL If there is no number in the bottom unit, or after the word per, the number one is understood. Example: “Find the speed in miles/hour (or miles per hour)” is equivalent to “find the miles traveled per one hour.” A ratio unit means something per ONE something. 2. An answer unit is a single unit if it has a numerator (top term) but no denominator. Example: If a problem asks you to find miles, or cm3, or dollars, it is asking for a single unit. 3. If a problem asks for a “unit per more than one other unit,” it WANTS a single unit. Example: If a problem asks for “grams per 100 milliliters,” or the “miles traveled in 27 hours,” it is asking for a single unit. A ratio unit must be something per one something. Writing Answer Units 1. If you WANT a ratio unit, write the unit as a fraction with a top and a bottom. Example: If an answer unit in a problem is miles/hour, to start: Write: WANTED: ? miles hour Do not write: WANTED: ? miles/hour or ? mph The slash mark ( / ), which is read as “per” or “over,” is an easy way to type ratios and conversion factors. However, when solving with conversions, writing ratio answer units as a fraction, with a clear numerator and denominator, will help in arranging the conversions to solve. 2. If a problem WANTS a single unit, write the WANTED unit without a denominator. WANTED: ? miles or WANTED: ? mL Single units have a one as a denominator and are written without a denominator. © 2009 www.ChemReview.Net v.m9 Page 75 Module 5 – Word Problems Practice Cover the answers below with a sticky note or cover sheet. Then, for each problem, write “WANTED: ? ” and the unit that the problem is asking you to find, using the rules above. After that WANTED unit, write an equal sign. Do not finish the problem. Write only the WANTED units. 1. If a car is traveling at 25 miles per hour, how many hours will it take to go 350 miles? 2. If 1.12 liters of a gas at STP has a mass of 3.55 grams, what is the molar mass of the gas, in grams/mole? 3. If a car travels 270 miles in 6 hours, what is its average speed? 4. A student needs 420 special postage stamps. The stamps are sold 6 per sheet, each stamp booklet has 3 sheets, and the cost is $14.40 per booklet. a. What is the cost of all of the stamps? b. How much is the cost per stamp? ANSWERS 1. The question asks for hours. Write WANTED: ? hours This problem is asking for a single unit. If the problem asked for hours per one mile, that would be a ratio unit, but hours per 350 miles is asking for a single unit. 2. Write WANTED: ? grams mole This is a ratio unit. Any unit that is in the form “unit X / unit Y” is a ratio unit. 3. In this problem, no unit is specified. However, since the data are in miles and hours, the easiest measure of speed is miles per hour, written WANTED: ? miles hour 4a. WANTED: ? $ which is a familiar unit of speed. This problem is asking for a ratio unit. or WANTED: ? dollars The answer unit is a single unit. 4b. WANTED: ? $/stamp or cents/stamp The cost per one stamp is a ratio unit. ***** Lesson 5B: Mining The DATA The method we will use to simplify problems is to divide solving into three parts. WANTED: DATA: SOLVE: This method will break complex problems into pieces. You will always know what steps to take to solve a problem, because we will solve all problems with the same three steps. © 2009 www.ChemReview.Net v.m9 Page 76 Module 5 – Word Problems Rules for DATA To solve word problems, get rid of the words. By translating words into numbers, units, and labels, you can solve most of the initial word problems in chemistry by chaining conversions, as you did in Module 4. To translate the words, the key is to write, in the DATA section on your paper, every number you encounter as you read the problem, followed by its unit and a label that describes what the number and its unit are measuring. This supplied DATA will identify what unstated additional conversions you will need to solve the problem. In the initial problems of chemistry, it is important to distinguish numbers and units that are parts of equalities from those that are not. To do so, we need to learn the many ways that quantities that are equal or equivalent can be expressed in words. Rules for Listing DATA in Word Problems 1. Read the problem. Write “WANTED: ?” and the WANTED unit. 2. On the next line down, write “DATA:” 3. Read the problem a second time. • Each time you find a number, stop. Write the number on a line under “DATA:” • After the number, write its unit plus a label that helps to identify the number. • Decide if that number, unit, and label is paired with another number, unit, and label as part of equality. 4. In the DATA section, write each number and unit in the problem as an equality a. Every time you read per. Per is written in DATA as an equal sign (=) . If a number is shown after per, write the number in the equality. Example: If you read “$8 per 3 pounds” write in the DATA: “$8 = 3 lb.” If no number is shown after per, write per as “ = 1 “ Example: If you read “12 sodas per carton,” write “12 sodas = 1 carton.” b. Every time you see a slash mark (/). A slash is the same as per. Example: If you see “25 km/hour” write “25 km = 1 hour” c. Every time you see unit x• unit y─1, which means the same as slash (/) and per. Example: If you see “75 g• mL─1” write “75 grams = 1 mL” d. Every time you see a conversion factor or a ratio unit. Write 25 miles or 25 miles/hr under DATA as “25 miles = 1 hour” hour e. Every time the same quantity is measured using two different units. If a problem says, “0.0350 moles of a gas has a volume of 440 mL,” write in the DATA: “0.0350 moles of gas = 440 mL” © 2009 www.ChemReview.Net v.m9 Page 77 Module 5 – Word Problems If a problem says that a bottle is labeled “2 liters (67.6 fluid ounces),” write in your DATA: “2 liters = 67.6 fluid ounces ” In both of the above are examples, the same physical quantity being measured in two different units. f. Any time two measurements are taken for the same process. If a problem says, “burning 0.25 grams of candle wax releases 1700 calories of energy,” write in your DATA section, “0.25 grams candle wax = 1700 calories of energy” Both sides are measures of what happened as this candle burned. 5. Watch for words such as each and every that mean one. One is a number, and you want all numbers in your DATA table. If you read, “Each student was given 2 sodas, ” write “ 1 student = 2 sodas” 6. Continue until all of the numbers in the problem are written in your DATA. 7. Note that in writing the WANTED unit, you write “per one” as a ratio unit, and “per more than one” as a single unit. In the DATA, however, because per is written as an equality, “per one” and “per more than one” can be written in the same way. Practice For each phrase below, write the equality that you will add to your DATA based on the measurements and words. On each side of the equal sign, include a number and a unit. After each unit, if two different entities are being measured in the problem, add additional words that identify what is being measured by the number and unit. After every few, check your answers. 1. The car was traveling at a speed of 55 miles/hour. 2. A bottle of designer water is labeled 0.50 liters (16.9 fluid ounces). 3. Every student was given 19 pages of homework. 4. To melt 36 grams of ice required 2,880 calories of heat. 5. The cost of the three beverages was $5. 6. The molar mass is 18.0 grams H2O•mole H2O─1. 7. Two pencils were given to each student. 8. The dosage of the aspirin is 2.5 mg per kg of body mass. 9. If 125 mL of a gas at STP weighs 0.358 grams, what is the molar mass of the gas? 10. If 0.24 grams of NaOH are dissolved to make 250 mL of solution, what is the concentration of the solution? © 2009 www.ChemReview.Net v.m9 Page 78 Module 5 – Word Problems ANSWERS Terms that are equal may always be written in the reverse order. If there are two different entities in a problem, attach labels to the units that identify which entity the number and unit are measuring. Doing so will make complex problems much easier to solve. 1. 55 miles = 1 hour (Rule 2b) 2. 0.50 liters = 16.9 fluid ounces (Rule 2e) 3. 1 student = 19 pages (Rule 3) 5. 3 beverages = $5 (Rule 2f) 6. 18.0 grams H2O = 1 mole H2O 7. 1 student = 2 pencils (Rule 3) 8. 2.5 mg aspirin = 1 kg of body mass (Rule 2a) 9. 125 mL of gas at STP = 0.358 grams gas (Rule 2e) 10. 0.24 g NaOH = 250 mL of solution 4. 36 grams ice = 2,880 calories heat (Rule 2f: Equivalent) (Rule 2c) (Rule 2f) ***** Lesson 5C: Solving For Single Units The Law of Dimensional Homogeneity In science, the units on both sides of an equality must be equal. We will use this law to simplify problem solving by starting each calculation with an equality: ? WANTED unit = # given unit then convert the given to the WANTED unit. DATA Formats If a Single Unit is WANTED If a problem WANTS a single unit, one number and unit in the DATA is likely to be • either a number and its unit that is not paired in an equality with other measurements, • or a number and its unit that is paired with the WANTED unit in the format “? units WANTED = # units given” We will define the given as the term written to the right of the equal sign: the starting point for the terms that we will multiply to solve conversion calculations. If a problem WANTS a single-unit amount, by the laws of science and algebra, at least one item of DATA must be a single-unit amount. In problems that can be solved using conversions, one measurement will be a single unit, and the rest of the DATA will be equalities. If a single unit is WANTED, watch for the one item of data that is a single unit amount. In the DATA, write the single number, unit, and label on a line by itself. It is a good practice to circle that single unit amount in the DATA, since it will be the given number and unit that is used to start your conversions. Variations on the above rules will apply when DATA includes two amounts that are equivalent in a problem. We address these cases in Module 11. However, for the problems you are initially assigned in first-year chemistry, the rules above will most often apply. © 2009 www.ChemReview.Net v.m9 Page 79 Module 5 – Word Problems To SOLVE After listing the DATA provided a problem, below the DATA, write SOLVE. Then, if you WANT a single unit, write the WANTED and given measurements in the format of the single-unit starting template. ? unit WANTED = # and unit given • ________________ unit given This template is a way of remembering the rule that your first conversion factor should cancel the given unit. The given measurement that is written after the = sign will be the listed in the DATA. circled single unit To convert to the WANTED unit, use the equalities in the DATA (and other fundamental equalities if needed). Summary: The 3-Step Method to Simplify Problem Solving 1. WANTED: When reading a problem for the first time, ask one question: what will be the unit of the answer? Then, write “WANTED: ?”, the unit the problem is asking for, and a label that describes what the unit is measuring. Then add an = sign. Write WANTED ratio units as fractions, and single units as single units. 2. DATA: Read the problem a second time. • Every time you encounter a number, under DATA, write the number and its unit. Include as part of the unit a descriptive label if possible. • Then see if that number and unit are equal to another number and unit. If a problem WANTS a single unit, most often one measurement will be a single unit and the rest will be equalities. Circle the single unit in the DATA. 3. SOLVE: Start each calculation with an equality: ? WANTED unit = # given unit. If you WANT a single unit, substitute the WANTED and given into this format. ? unit WANTED = # and unit given • _________________ unit given Then, using equalities, convert to the WANTED unit. Solve the following problem in your notebook using the 3-step method above. Q. If a car’s speed is 55 miles/hr., how many minutes are needed to travel 85 miles? * * * * * (the * * * mean cover the answer below, write your answer, then check it.) © 2009 www.ChemReview.Net v.m9 Page 80 Module 5 – Word Problems Your paper should look like this. WANTED: ? minutes = DATA: 55 miles = 1 hour 85 miles SOLVE: ? minutes = 85 miles • 1 hour • 55 miles 60 min. = 1 hour 93 minutes You can solve simple problems without listing WANTED, DATA, SOLVE, but this 3-part method works for all problems. It works especially well for the complex problems that soon you will encounter. By using the same three steps for every problem, you will know what to do to solve all problems. That’s the goal. Practice Many science problems are constructed in the following format. “Equality, equality,” then, “? WANTED unit = a given number and unit.” The problems below are in that format. Using the rules above, solve on these pages or by writing the WANTED, DATA, SOLVE sections in your notebook. If you get stuck, read part of the answer at the end of this lesson, adjust your work, and try again. Do problems 1 and 3, and problem 2 if you need more practice. Problem 1. If 2.2 pounds = 1 kg, what is the mass in grams of 12 pounds? WANTED: ? (Write the unit you are looking for.) DATA: (Write every number and unit in the problem here. If solving for a single unit, often one number and unit is unpaired, and the rest are in equalities, Circle the unpaired single unit.) SOLVE: (Substitute the above into “? unit WANTED = # and unit given • _____________ “ unit given then chain the equalities to find the unit WANTED.) ? ***** © 2009 www.ChemReview.Net v.m9 Page 81 Module 5 – Word Problems Problem 2. If there are 1.6 km/mile, and one mile is 5,280 feet, how many feet are in 0.50 km? WANTED: ? DATA: SOLVE: ? ***** Problem 3. If there are 3 floogles per 10 schmoos, 5 floogles/mole, and 3 moles have a mass of 25 gnarfs, how many gnarfs are in 4.2 schmoos? (Assume single digit whole numbers are exact.) WANTED: DATA: SOLVE: © 2009 www.ChemReview.Net v.m9 Page 82 Module 5 – Word Problems ANSWERS 1. WANTED: ?g= 2.2 pounds = 1 kg DATA: 12 pounds SOLVE: ? g = 12 pounds • 1 kg • 103 g = 12•103 g = 5.5 x 103 g 2.2 pounds 1 kg 2.2 A single unit is WANTED, and the DATA has one single unit. Note that the SOLVE step begins with “how many grams equal 12 pounds?” Fundamental conversions such as kilograms to grams need not be written in your DATA section, but they will often be needed to solve. Be certain that you have mastered the metric system fundamentals. 2. WANTED: ? feet = 1.6 km = 1 mile DATA: 1 mile = 5,280 feet 0.50 km SOLVE: ? feet = 0.50 km • 1 mile • 5,280 feet = 0.50•5280 feet = 1,600 feet 1.6 km 1 mile 1.6 3. WANTED: ? gnarfs = 3 floogles = 10 schmoos DATA: 5 floogles = 1 mole 3 moles = 25 gnarfs 4.2 schmoos SOLVE: At the SOLVE step, first state the question, “how many gnarfs equal 4.2 schmoos?” Then add the first conversion, set up to cancel your given unit. ? gnarfs = 4.2 schmoos • ________________ schmoos Since only one equality in the DATA contains schmoos, use it to complete the conversion. ? gnarfs = 4.2 schmoos • 3 floogles 10 schmoos On the right, you now have floogles. On the left, you WANT gnarfs, so you must get rid of floogles. In the next conversion, put floogles where it will cancel. ? gnarfs = 4.2 schmoos • © 2009 www.ChemReview.Net v.m9 3 floogles • ____________ floogles 10 schmoos Page 83 Module 5 – Word Problems Floogles is in two conversion factors in the DATA, but one of them takes us back to schmoos, so let’s use the other. ? gnarfs = 4.2 schmoos • 3 floogles 10 schmoos • 1 mole 5 floogles Moles must be gotten rid of, but moles has a known relationship with the answer unit. Convert from moles to the answer unit. Since, after unit cancellation, the answer unit is now where you WANT it, stop conversions and do the arithmetic. ? gnarfs = 4.2 schmoos • 3 floogles • 1 mole • 25 gnarfs = 4.2•3•25 = 2.1 gnarfs 3 moles 10•5•3 10 schmoos 5 floogles ***** Lesson 5D: Finding the Given Ratio Unit Givens In chemistry, the initial quantitative topics generally involve solving for single units, so that will be our initial focus as well. Conversion factors may also be used to solve for ratio units, as we did in Lesson 4E. However, we will defer the most of the rules to use conversions to solve for ratio units until Lesson 11B, when ratio units will be needed to solve for the concentration of chemical solutions. If you need to solve word problems that have ratio-unit answers, now or at any later point, Lesson 11B may be done at any time after completing this lesson. Single-Unit Givens When solving for single units, the given quantity is not always clear. Example: A student needs special postage stamps. The stamps are sold 6 per sheet, each stamp booklet has 3 sheets, 420 stamps are needed, and the cost is $43.20 per 5 booklets. What is the cost of the stamps? Among all those numbers, which is the given needed as the first term when you SOLVE? For a single-unit answer, finding the given is often a process of elimination. If all of the numbers and units are paired into equalities except one, that one is your given. In your notebook, write the WANTED and DATA sections for the stamps problem above (don’t SOLVE yet). Then check your work below. ***** Answer: Your paper should look like this. WANTED: ? $= or DATA: 1 sheet = 6 stamps ? dollars = (you could also solve in cents) 3 sheets = 1 booklet 420 stamps $43.20 = 5 booklets © 2009 www.ChemReview.Net v.m9 Page 84 Module 5 – Word Problems Since you are looking for a single unit, dollars, your data has one number and unit that did not pair up in an equality: 420 stamps. That is your given. To SOLVE, the rule is If you WANT a single unit, start with a single unit as your given. Apply the above rule, and SOLVE the problem. ***** Answer SOLVE: If you WANT a single unit, start with the single-unit starting template. ? $ = 420 stamps • ___________ stamps If you needed that hint, adjust your work and then finish. ***** ? $ = 420 stamps • 1 sheet • 1 booklet • $ 43.20 = 6 stamps 3 sheets 5 booklets $ 201.60 Practice For each problem below, use the WANTED, DATA, SOLVE method. If you get stuck, peek at the answers and try again. Do at least two problems. If you plan on taking physics, be sure to do problem 3. On each of these, before you do the math, double check each conversion, one at a time, to make sure the conversion is legal. 1. A bottle of drinking water is labeled “12 fluid ounces (355 mL).” What is the mass in centigrams of 0.55 fluid ounces of the H2O? (Use the metric definition of one gram). 2. You want to mail a large number of newsletters. The cost is 18.5 cents each at special bulk rates. On the post office scale, the weight of exactly 12 newsletters is 10.2 ounces. The entire mailing weighs 125 lb. There are 16 ounces (oz.) in a pound (lb.). a. How many newsletters are being mailed? b. What is the cost of the mailing in dollars? 3. If the distance from an antenna on Earth to a geosynchronous communications satellite is 22,300 miles, given that there are 1.61 kilometers per mile, and radio waves travel at the speed of light (3.0 x 108 meters/sec), how many seconds does it take for a signal from the antenna to reach the satellite? © 2009 www.ChemReview.Net v.m9 Page 85 Module 5 – Word Problems ANSWERS 1. WANTED: ? cg = DATA: 12 fl. oz = 355 mL 0.55 fl. oz 1.00 g H2O(l) = 1 mL H2O(l) (metric definition of one gram) SOLVE: ? cg = 0.55 fl. oz. • 355 mL • 1.00 g H2O(l) • 1 cg = 1,600 cg 12 fl. oz 1 mL H2O(l) 10─2 g 2a. WANTED: ? newsletters DATA: 18.5 cents = 1 newsletter 12 exact newsletters = 10.2 ounces 16 oz. = 1 lb. 125 lb. (a definition with infinite sf) SOLVE: WANTED: ? dollars (Strategy: 2b. ? newsletters = 125 lb. • 16 oz. • 12 newsls = 2,350 newsletters 1 lb. 10.2 oz. Since you want a single unit, you can start over from your single given unit (125 lb.), repeat the conversions above, then add 2 more. Or you can start from your single unit answer in Part a, and solve using the two additional conversions. In problems with multiple parts, to solve for a later part, using an answer from a previous part often saves time. ) same as for Part a. DATA: SOLVE: WANTED: ? seconds = DATA: 3. ? dollars = 2,350 newsls • 18.5 cents • 1 dollar = $ 435 1 newsl 100 cents 22,300 miles 1.61 km = 1 mile 3.0 x 108 meters = 1 sec SOLVE: ? sec = 22,300 mi. • 1.61 km • 103 meters • 1 s = 22,300 • 1.61 • 103 sec = 0.12 s 8m 3.0 x 108 1 mile 1 km 3.0 x 10 (This means that the time up and back for the signal is 0.24 seconds. You may have noticed this onequarter-second delay during some live broadcasts which bounce video signals off satellites but use faster land-lines for audio, or during overseas phone calls routed through satellites.) ***** © 2009 www.ChemReview.Net v.m9 Page 86 Module 5 – Word Problems Lesson 5E: Some Chemistry Practice Listing Conversions and Equalities Which is the best way to write DATA pairs —as equalities, or in the fraction form as conversion-factor ratios? Mathematically, either form may be used. In DATA: the equalities 1.61 km = 1 mile can be listed as 3.0 x 108 meters = 1 sec. 1.61 km , 1 mile 3.0 x 108 meters 1 sec. In these lessons, we will generally write equalities in the DATA section. This will emphasize that when solving problems using conversions, you need to focus on relationships between two quantities. However, listing the data in the fraction format is equally valid. Data may be portrayed both ways in textbooks. Why “Want A Single Unit, Start With A Single Unit?” Mathematically, the order in which you multiply conversions does not matter. You could solve with your single unit given written anywhere on top in your chain of conversions. However, if you start with a ratio as your given when solving for a single unit, there is a 50% chance of starting with a ratio that is inverted. If this happens, the units will never cancel correctly, and you would eventually be forced to start the conversions over. Starting with the single unit is a method that automatically arranges your conversions right-side up. Practice Let’s do some chemistry. The problems below supply the DATA needed for conversion factors. In upcoming modules, you will learn how to write these needed conversions automatically even when the problem does not supply them. That small amount of additional information is all that you will need to solve most initial chemistry calculations. You’re ready. Solve the two problems below in your notebook. Don’t let strange terms like moles or STP bother you. You’ve done gnarfs. You can do these. Check your answer after each problem. 1. Water has a molar mass of 18.0 grams H2O per mole H2O. How many moles of H2O are in 450 milligrams of H2O? 2. If one mole of all gases has a volume of 22.4 liters at STP, and the molar mass of chlorine gas (Cl2) is 71.0 grams Cl2 per mole Cl2 , what is the volume, in liters, of 3.55 grams of Cl2 gas at STP ? © 2009 www.ChemReview.Net v.m9 Page 87 Module 5 – Word Problems ANSWERS 1. WANTED: ? moles H2O = DATA: 18.0 grams H2O = 1 mole H2O 450 mg H2O SOLVE: ? moles H2O = 450 mg H2O • 10─3 g • 1 mole H2O = 2.5 x 10─2 moles H2O 1 mg 18.0 g H2O Write chemistry data in 3 parts: Number, unit, formula. Writing complete labels will make complex problems easier to solve. 450 has 2 sf. 2. WANTED: ? L Cl2 DATA: 1 mole gas = 22.4 L gas 71.0 g Cl2 = 1 mole Cl2 3.55 g Cl2 SOLVE: ? L Cl2 = 3.55 g Cl2 • 1 mole Cl2 • 22.4 L Cl2 = 1.12 L Cl 2 71.0 g Cl2 1 mole Cl2 ***** © 2009 www.ChemReview.Net v.m9 Page 88 ...
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