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Estimating Circle the correct answer 1. If you have a can of soda which is 12 oz and you want to fill a 20 oz glass, you will need a) 1 can b) less than 1 can c) more than 1 can 2. If you have tablets with a strength of 0.25 mg and you must give 0.125 mg, you will need a) 1 tablet b) less than 1 tablet c) more than 1 tablet 3. Gas is $5.00 for 2 gallons and you want to fill your almost full gas tank which can hold an additional 1.5 gallons, you will spend a) $5.00 b) less than $5.00 c) more than $5.00 4. You have 2 tablets, one is labeled 0.15 mg and the other is labeled 0.3 mg. What is the total dosage of these tablets? a) 0.35 mg b) less than 0.35 mg c) more than 0.35 mg Some Problems 1. You are making sandwiches for a large crowd. All must be identical and must contain 2 slices of bread, 2 teaspoons mustard, 3 slices salami, 2 slices of cheese, 2 lettuce leaves, and 1 slice tomato. You have on hand 2 loaves of bread, 20 slices each 3 pounds of cheese, each pound gives 16 slices 1 quart of mustard 60 slices of salami An unlimited supply of tomatoes from your garden 3 head of lettuce, approximately 15 leaves in each How many sandwiches can you make?___________________________ For which ingredients is (are) there no leftovers? ____________________ In case you’ve forgotten your kitchen conversions: 1 quart = 4 cups; 1 cup = 16 tablespoons; 1 tablespoon = 3 teaspoons Math Review Page 1 of 10 2. You like 3 teaspoons of sugar and 1/4 cup of milk in each cup of coffee (3/4 cup coffee, 1/4 cup milk) you drink. Before leaving for a trip you prepare a thermos that holds 2 quarts of liquid. What quantities of coffee, sugar and milk will you use? (Assume the sugar occupies no volume.) 3. A kilogram (kg) is a little more than 2 pounds (lb). Which is heavier? 30 kg or 60 lb? Which is a better buy: bananas for 39 cents / lb or bananas for $ 1.00 / kg? 4. A kilometer is somewhat more than half a mile in length. Which is shorter? a mile or a kilometer? Which is faster? a car going 70 miles per hour or 70 kilometers per hour? Who is faster: the runner who runs a mile in 8 minutes or the runner who runs a kilometer in 4 minutes? 5. A gallon is 4 quarts. The volume of a liter is slightly larger than a quart. If you put 13.5 gallons of gasoline in your car, about how many liters is this? Which is cheaper? $ 2.40 for a gallon or 60 cents per liter? Math Review Page 2 of 10 Exponential Numbers Exponential numbers are a way of writing very small or very large numbers, a situation that occurs frequently in scientific measurements. For number larger than 1 Number form Exponential form 1 100 10 101 100 = 1 x (10)(10) 102 1000 = 1 x (10)(10)(10) 103 10000 = 1 x (10)(10)(10)(10) 104 Notice that the exponent indicates how many times the number 1 is multiple by 10, and how many places the decimal point is moved to the right. Example: 1. x 103 = 1000. Exercise 1. Write the following numbers in exponential form: 10, 000, 000 = _________________ 100, 000, 000, 000, 000, 000 = ____________ 1 million = 10, 000, 000, 000, 000 = _________________ _________________ Exercise 2. Write the following exponentials in number form 103 = _________________ 1019 = _____________________________________ 108 = __________________________ 1012 = ___________________________ For number smaller than 1 Number form Exponential form 0.1 = 1 10 10 -1 0.01 = 1 [(10)(10)] 10 -2 0.001 = 1 [(10)(10)(10)] 10 -3 Notice that a negative exponent indicates how many times the number is divided by 10, and how many places the decimal point is moved to the left. Example: 1. x 10 -3 = 0.001 Exercise 1. Write the following numbers in exponential form: 0.00001 = _________________ 0.0000000000000000000001 = ____________ Exercise 2. Write the following exponentials in number form 10 -5 = _________________ 10 -13 = ____________________________________ 10 -7= __________________________ Math Review Page 3 of 10 10 -11 = __________________________ Multiplying and Dividing with Exponential Numbers Multiplication example: 103 x 105 = In case you’ve forgotten the general rule, remember that exponential can be expanded: 103 x 105 = (10) (10) (10) x (10) (10) (10) (10) (10) = 108 103 x 105 = 103+5 = 108 General rule: to multiply exponential numbers, ADD the exponents. 10a x 10b = 10 a+b General rule: to divide exponential numbers, SUBTRACT the exponents. 10a 10b = 10 a-b Exercise 1: Complete the following 10 5 x 10 6 = 10 8 = 10 5 10 8 x 10 -4 = 10 23 = 10 -9 10 8 x 10 –6 x 10 -5 = 10 4 10 -3 x 10 -9 = 10 -11 = 10 5 10 96 = 10 54 10 5 x 10 9 x 10 -4 = 10 14 x 10 2 x 10 -6 Scientific Notation Any number can be expressed as a number with an exponent. For example Another example 7.5 x 10 2 = 7.5 x 100 = 750. 8.6 x 10 -3 = 8.6 x 0.001 = 0.0086 Notice that a positive exponent moves the decimal place to the right (makes the number bigger), and a negative exponent moves the decimal place to the left (makes the number smaller) When a number is expressed as a number between 1 and 10 times an exponent this is called scientific notation: Scientific notation: = number between 1 and 9.999… x 10 example: 6548 = 6.548 x 1000 = 6.548 x 10 3 Math Review Page 4 of 10 a Exercise 1. Write the following in expanded form (no exponents): 5.62 x 10 5 = _____________________ 8.6 x 10 –3 = ____________________ 8.923 x 10 11 = _____________________ 1.00234 x 10 –4 = ________________ Exercise 2. Write the following in scientific notation: 584, 000, 000 = ____________________ 0.00000003456 = ________________ 57 = ______________________________ 0.258 = ________________________ Exponential and Scientific Notation More Practice Exercise 1: Solve the following problems, using exponents and giving your final answer in scientific notation. No calculators for this. Example: (4 x 10 -3 ) (15 x 10 8 ) = 4 x 15 x (10 –3 ) (10 8 ) = 60 x 10 5 = 12 x 10 2 = 1.2 x 10 3 5 x 10 3 5 10 3 5 10 3 (3 x 10 9 ) (8 x 10 –7 ) = (2 x 10 –2 ) (6 x 10 –5 ) (0.030) (0.60)____ = (0.00075) (12000) (4.2 x 10 3 ) (9 x 10 5 ) = (7 x 10 –4 ) (3 x 10 4 ) (900) (5.4) (8.0 x 10 6 )_ = (0.00002) (0.027) Math Review Page 5 of 10 Conversions within the SI system (Metric System) 1. Important prefixes: The SI system uses a series of prefixes to indicate size. The prefixes indicated in BOLD below are used frequently in chemistry pico nano micro milli centi deci kilo mega giga Exercise: Below each prefix given above, write its numerical value and the abbreviation. For example: micro means 10-6 and is abbreviated as . (use textbook, chapter 2) 2. Using prefixes: It’s important to fully understand how to use prefixes. The prefixes are added to base units. Some common base units are listed below gram Exercise: meter liter second Below each base unit, write the abbreviation. (use your text, chapter 2) For example: The abbreviation for megameter is Mm with the M for mega and the m for meter. Exercise: Fill in the following blanks with the correct abbreviations for the units _____ picogram _______ milliliter _______ decimeter _____ nanosecond Try to think in concrete terms. For example 1 centimeter = 10 -2 m. This means 100 cm = 1 m or 1 cm = 0.01 m. Which is bigger the meter or the centimeter? Exercise: Fill in the following blanks with the correct numbers or units 1 pm = _________________ m 1 m = __________________ pm 1 nL = _________________ L 1 L = __________________ nL 1 g = __________________ g 1 g = __________________ g 1 mm = __________________ m 1 m = __________________ mm 1 cm = __________________ m 1 m = __________________ cm Math Review Page 6 of 10 1 dL = _________________ L 1 L = _________________ dL 1 kg = _________________ g 1 g = _________________ kg 1 ML = _________________ L 1 L = _________________ ML 1 Gg = _________________ g 1 g = _________________ Gg We can use the equivalences to make unit conversions. This means to change from 1 unit to another. For example: a medium-sized bug is 8.4 mm in length. Express this length in meters. First, consider if the number should be bigger or smaller than 8.4. Which is the bigger unit? The meter is the bigger unit so the answer should be smaller than 8.4. The equivalences are used as ratios to make the conversion. The units in the numerator (on top) will cancel with identical units in the denominator (on bottom ).There are usually multiply ways to do the same conversion. 8.4 mm x 10-3 m = 8.4 x 10 –3 m 1 mm or 8.4 mm x 1 m___ = 8.4 x 10 –3 m 10 3 mm The problem can be done either way. Both are correct. Exercise: Complete the following. Do each both ways. A rock is 2.3 cm. Express this length in m. A person has a mass of 70 kg. Express this mass in g. A dose of the flu vaccine in 0.0005 L. Express this value in mL. Math Review Page 7 of 10 Conversions between Metric and English systems The relationships between metric and English units are not simple numbers and you do not need to learn or memorize these conversion relationships. They will be given on exams and quizzes or you can look them up for homework. For example, several tables are found in chapter 2 and inside the back cover of your book. Example: 1 kg = 2.20 lb (pounds) or 0.454 kg = 1 lb Is 1 kg bigger or smaller than 1 pound? _________________ We can use the equivalences to make unit conversions. This means to change from one unit to another. For example: a full-term newborn baby weighs 3.26 kg. What is her weight in pounds? First, consider if the number should be bigger or smaller than 3.26. Which is the smaller unit? The lb is the smaller unit so the answer should be larger than 3.26. The equivalences are used as ratios to make the conversion. There are usually multiply ways to do the same conversion. 3.26 kg x 2.20 lb = 7.17 lb 1 kg or 3.26 kg x 1 lb___ = 7.17 lb 0.454 kg The problem can be done either way. Both are correct. Exercise: Complete the following. Do each both ways. A piano weighs 952 pounds. Express this weight in kg. A can of Pepsi is 12 fluid ounces. Express this volume in mL(1 fl oz = 29.6 mL) (one way only) A race is 7.5 km. Express this distance in miles. (use table in book) Math Review Page 8 of 10 Estimating Numbers – Using your Calculator When using your calculator, there are three things you must do to be sure you have the right answer: 1. First estimate the answer. The estimate should be rough. This must be done by hand without the calculator, to provide an independent check on your calculator result. 2. Then, use your calculator to get an answer. If your calculator result is not fairly close to your estimated answer, go over steps 1 and 2 until the two results agree. 3. Finally, report your answer with correct number of significant figures (chapter 2 ) Example: (6.293 x 10 8 )(8.31 x 10 –5 ) = ? Estimate: (6 x 10 8 )( 8 x 10 –5 )_ 4 x 10 12 (1546) (7.63 x 10 –12 ) (1.5 x 10 3)(7 x 10 –12) Calculator result : 4.43 x 1012 For the following, show your estimated and your calculator result. (87.5)(0.00000046) = (3.047 x 10 9) (4.832) (55.04) (439 x 10 -5) = (3.0062 x 10 15)(4.98 x 10 –3) (0.00068)3 = 8.74 + 13.6509 – 50.08 Three apples weigh: 49.75 g, 51.93 g, and 50.62 g. Calculate the average mass (weight) of the apples (show set-up) Math Review Page 9 of 10 Algebra The following problems will give you an idea of the sort of algebra you need to know for this course. For each of the following, solve for the unknown variable 2x = 34 z + 23 = 204 3x – 21 = 6 0.3y = 0.24 5x + 75 = 45 y2 = 49 3x2 = 48 516 = 6 x D=m v if m = 6 kg D=m v if D = 6 g / mL Math Review and v = 9 mL, Page 10 of 10 calculate D and v = 12 mL, calculate m