Maths For Pharmacy - JCU Australia
Mathematics forPharmacyThe module covers concepts such as: Maths refresher Fractions, Percentage and Ratios Unit conversions Calculating large and smallnumbers Logarithms
Mathematics forPharmacyContents1.2.3.4.5.6.7.8.9.10.11.Terms and OperationsFractionsConverting decimals and fractionsPercentagesRatiosAlgebra RefreshPower OperationsScientific NotationUnits and ConversionsLogarithmsAnswers
1. Terms and OperationsGlossaryπ₯π₯ 4 , 2π₯π₯, 3 & 17 are TERMS4 is an EXPONENT2 is a COEFFICIENT17 is the SUM of π₯π₯ 4 2π₯π₯ 3x4 2 π₯π₯ 3 17 is an EQUATIONπ₯π₯ 4 2π₯π₯ 3 17π₯π₯ is a VARIABLE3 is a CONSTANT is an OPERATORπ₯π₯ 4 2 π₯π₯ 3 is an EXPRESSIONEquation:A mathematical sentence containing an equal sign. The equal sign demands that theexpressions on either side are balanced and equal.Expression:An algebraic expression involves numbers, operation signs, brackets/parenthesisand variables that substitute numbers but does not include an equal sign.Operator:The operation ( , , , ) which separates the terms.Term:Variable:Parts of an expression separated by operators which could be a number, variable orproduct of numbers and variables. Eg. 2π₯π₯, 3 & 17Constant:Terms that contain only numbers that always have the same value.Coefficient:A number that is partnered with a variable. The term 2π₯π₯ is a coefficient withvariable. Between the coefficient and variable is a multiplication. Coefficients of 1are not shown.Exponent:A value or base that is multiplied by itself a certain number of times. Ie. π₯π₯ 4representsπ₯π₯ π₯π₯ π₯π₯ π₯π₯ or the base value π₯π₯ multiplied by itself 4 times. Exponents are alsoknown as Powers or Indices.In summary:A letter which represents an unknown number. Most common is π₯π₯, but can be οΏ½οΏ½οΏ½π₯32x 3 172Operator:Terms:Left hand expression:Right hand expression 3, 2π₯π₯ (a term with 2 factors) & 172π₯π₯ 317 (which is the sum of the LHE)1
The symbols we use between the numbers to indicate a task or relationships are the operators, andthe table below provides a list of common operators. You may recall the phrase, βdoing an operation.Symbol a Ξ MeaningAdd, Plus, Addition, SumMinus, Take away, Subtract,DifferenceTimes, Multiply, Product,Divide, QuotientPlus and MinusAbsolute Value (ignore βve sign)EqualNot EqualLess thanGreater thanMuch Less thanMuch More thanApproximately equal toLess than or equalGreater than or equalDeltaSigma (Summation)Order of OperationsThe Order of Operations is remembered using the mnemonic known as the BIDMAS or BOMDAS(Brackets, Indices or Other, Multiplication/Division, and Addition/Subtraction).Brackets{[(Indices or Otherπ₯π₯ 2 , sin π₯π₯ , ln π₯π₯ , ππππππMultiplication or DivisionAddition or Subtraction)]} ππππ or -The Rules:1. Follow the order (BIMDAS, BOMDAS or BODMAS)2. If two operations are of the same level, you work from left to right. E.g. ( ππππ ) ππππ ( ππππ )3. If there are multiple brackets, work from the inside set of brackets outwards. {[( )]}2
Example Problems:1. Solve:Step 1: 52 has the highest priority so:Step 2: 7 2 has the next priority so:5 7 2 52 5 7 2 25 5 14 25 Step 3: only addition left, thus left to right: 19 25 44Question 1: 5 7 2 52 44Here are some revision examples for practise:a. 10 2 5 1 b. 10 5 2 3 c. 12 2 2 7 d. 48 6 2 4 e. ππβππππ ππππ π‘π‘βππ οΏ½οΏ½π οΏ½οΏ½πππππ π π π π π π π π π π π π 18 3 2 2 143
2. Fractions β addition, subtraction,multiplication and divisionAdding and subtracting fractions draws on the concept of equivalent fractions. The goldenrule is that you can only add and subtract fractions if they have the same denominator, forexample,131323 .If two fractions do not have the same denominator, we must use equivalent fractions to find aβcommon denominatorβ before they can be added together.In the exampleto change1214ππππππππ12 , 4 is the lowest common denominator. Use the equivalent fractions concept241222by multiplying both the numerator and denominator by two: x 24Now that the denominators are the same, addition or subtraction operations can be carried out.In this example the lowest common denominator is found by multiplying 3 and 5, and then thenumerators are multiplied by 5 and 3 respectively:(1 5) (2 3)1 25 6 11 3 51515(3 5)Compared to addition and subtraction, multiplication and division of fractions is easy to do,but sometimes a challenge to understand how and why the procedure works mathematically. For12example, imagine I have of a pie and I want to share it between 2 people. Each person gets aquarter of the pie.12Mathematically, this example would be written as:1214 .Remember that fractions and division are related; in this way, multiplying by a half is the same asdividing by two.121214So (two people to share) of (the amount of pie) is (the amount each person will get).23But what if the question was more challenging: pies.716 ? This problem is not as easy as splittingA mathematical strategy to use is: βMultiply the numerators then multiply the denominatorsβTherefore,23 716 (2 7)(3 16) 1448 724An alternative method you may recall from school is to simplify each term first. Remember, βWhatwe do to one side, we must do to the other.β23The first thing we do is look to see if there are any common multiples. For 716that 2 is a multiple of 16, which means that we can divide top and bottom by 2: ? we can see4
2 23 716 21378 1 73 8 724Division of fractions seems odd, but it is a simple concept:You may recall the expression βinvert and multiplyβ, which means we flip the divisor fraction (second12term fraction). Hence, 2 ππππ π‘π‘βππ π π π π π π π π ππππ 1This βflippedβ fraction is referred to as the reciprocal of the original fraction.2312Therefore, is the same asQuestion 21. Find the3.11622 92321 (2 2)(3 1)2reciprocal of 25 7. 4913Note: dividing by half doubled the answer.2.234.376.85. 2 5 3 9 7243 17 13 2 5 ( 25) ( 5)4 2 7 8. If we multiply 8 and the reciprocal of 2,what do we get? 9. Which is the better score in a physiology test; 17 out of 20 or 22 out of 25?10. What fraction of H2O2 is hydrogen?111. A patient uses a glass that holds 5 of a jugβs volume. The patient drinks eight full glasses duringthe course of the day. What fraction of the second jug is left at the end of the day?5
3. Converting Decimals & FractionsConverting Decimals into FractionsDecimals are an almost universal method of displaying data, particularly given that it is easier toenter decimals, rather than fractions, into computers. But fractions can be more accurate. For1example, is not 0.33 it is 0.33Μ3The method used to convert decimals into fractions is based on the notion of place value. The placevalue of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths,and so on Example problems:a) 0.5 has 5 in the tenths column. Therefore, 0.5 is510 12(simplified to an equivalent fraction).3751000114b) 0.375 has the 5 in the thousandth column. Therefore, 0.375 isc) 1.25 has 5 in the hundredths column and you have 125100 38The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator,you can use the fraction button. This button looks different on different calculators so read yourmanual if unsure.If we take375from1000Enter 375 thenexample 2 above:38followed by 1000 press and answer shows as .NOTE: The calculator does not work for rounded decimals; especially thirds. E.g, 0.333 This table lists some commonly encountered fractions expressed in their decimal form:Decimal0.1250.250.333330.375Question 3:Fraction18141338Decimal0.50.66667.750.2Convert to fractions (no Calculator first, then check).a) 0.65 b) 2.666 c) 0.54 13Fraction12233415d)3.14 e) What is 40 multiplied by 0.2 (use your knowledge of fractions to solve)6
Converting Fractions into DecimalsConverting fractions into decimals is based on place value. Using the concept of equivalent fractions,25we can easily convert into a decimal. First we convert to a denominator that has a 10 base:252 2into tenths 5 2 410 we can say that two fifths is the same as four tenths: 0.4Converting a fraction to decimal form is a simple procedure because we simply use the divide key onthe calculator.Note: If you have a mixed number, convert it to an improper fraction before dividing it on yourcalculator.Example problems: 23 2 3 0.66666666666 0.67173 17 3 5.6666666 5.67 38 3 8 0.3755 3 9 (27 5) 9 3.555555556 3.56Questions 4Convert to decimals. Round your answer to three decimal places where appropriate.a.1723b. 57223 c. 56 d.295 Watch this short Khan Academy video for further explanation:βConverting fractions to decimalsβ (and vice nverting-fractionsto-decimals7
4. PercentageThe concept of percentage is an extension of the material we have already covered about fractions.To allow comparisons between fractions we need to use the same denominator. As such, allpercentages use 100 as the denominator. The word percent or βper centβ means per 100. Therefore,27% is27.100To use percentage in a calculation, the simple mathematical procedure is modelled below:For example, 25% of 40 is25 10040 10Percentages are most commonly used to compare parts of an original. For instance, the phrase β30%off sale,β indicates that whatever the original price, the new price is 30% less. However, the questionmight be more complex, such as, βHow much is left?β or βHow much was the original?βExample problems:a.An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chickenbreasts are normally 9.99 per kilo. What is the new per kilo price?Step 1: SIMPLE PERCENTAGE:22 1009.99 2.20Step 2: DIFFERENCE: Since the price is cheaper by 22%, 2.20 is subtracted from the original:9.99 β 2.20 7.79b.A new dress is now 237 reduced from 410. What is the percentage difference? As you cansee, the problem is in reverse, so we approach it in reverse.Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus weneed to subtract 237.00 from 410 to see what the discount was that we received. 410 β 237 173Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of 410 was 173, and soπ₯π₯ 410 173we can use this equation:100π₯π₯ 100π₯π₯173 100410We can rearrange the problem in steps:dividing 410 from both sides to getmultiply both sides by 100.get 42.Now we have π₯π₯ 173100 4101410 410 173 410 this step involvedNext we work to get the π₯π₯ on its own, so weNext we solve, so 0.42 multiplied by 100, 0.42 100 and we The percentage difference was 42%.Letβs check: 42% of 410 is 173, 410 - 173 237, the cost of the dress was 237.00 .Question 5a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST?b) When John is exercising his heart rate rises to 180 bpm. His resting heart rate is 70 % of this. Whatis his resting heart rate?c) Which of the following is the largest?35ππππ1625ππππ 0.065 ππππ 63%? (Convert to percentages)8
5. RatiosA ratio is a comparison of the size of one number to the size of another number. A ratio representsfor every determined amount of one thing, how much there is of another thing. Ratios are usefulbecause they are unit-less. That is, the relationship between two numbers remains the sameregardless of the units in which they are measured.Ratios use the symbol : to separate quantities being compared. For example, 1:3 means 1 unit to3 units.There is 1 red square to 3 blue squares1:31 to 3Ratios can be expressed as fractions but you can see from the above diagram that 1:3 is not the13same as . The fraction equivalent isExample:14A pancake recipe requires flour and milk to be mixed to a ratio of 1:3. This means one part flour to 3parts milk. No matter what device is used to measure, the ratio must stay the same.So if I add 200 mL of flour, I add 200 mL x 3 600 mL of milkIf I add 1 cup of flour, I add 3 cups of milkIf I add 50 grams of flour, I add 150 grams of milkScaling ratiosA ratio can be scaled up:1:4 2:8Or scaled down:3:15 1:51:5 is the same as2:10 is the same as3:15 is the same as4:20 and so on9
Scaling ratios is useful in the same way that simplifying fractions can be helpful, for example, incomparing values. For ratios the same process as simplifying fractions is applied β that is, scalingmust be applied to both numbers.For example, a first year physiology subject has 36 males and 48 females, whereas theendocrinology subject has 64 males and 80 females. You are asked to work out which cohort has thelargest male to female ratio.The male: female ratios can be expressed as:36:48 β physiology subject64:80 β endocrinology subjectBoth numbers of the ratio 36:48 can be divided by 12 to leave the ratio 3:4Both numbers of the ratio 64:80 can be divided by 16 to leave the ratio 4:5These two ratios cannot be easily directly compared, but they can be rescaled to a common value of20 for the females, against which the males can be compared if they are rescaled equivalently.3 (x5):4 (x5) 15:20 β physiology subject4 (x4):5 (x4) 16:20 β endocrinology subjectComparing the ratios now shows that the endocrinology subject has a slightly higher ratio of malesto females.Questions 6:a) For a 1:5 concentration of cordial drink, how much cordial concentrate do I have to add towater to make up a 600 mL jug?b) Jane reads 25 pages in 30 minutes. How long does it take her to read 200 pages?c) A pulse is measured as 17 beats over 15 seconds. What is the heart rate per minute?d) Which of the following ratios is the odd one out? 9:27. 3:9, 8:28, 25:7510
Ratios and PercentagesRecall from the figures above, that the numbers in the ratio represents parts of a whole. To converta ratio to percentage values, simply add the two parts of the ratio, to give the whole (total) and foreach part, divide by the total. Then use the normal procedure to calculate the percentage bymultiplying by 100.Using the example from above, the table below calculates the percentage values of males andfemales for each subject from the ratios. Percentages allow for a quantified comparison.Physiology 3:4TotalMaleEndocrinology 4:573 100 42.9%74 100 57.1%7Female94 100 44.4%95 100 55.6%9We are correct; there are more males than females (in percent) in the endocrinology subject.Drug calculations using the expression solute in diluentSometimes drug ratios are written in the form of solute in diluent; for example 1 in 4.This means 1 part of every 4 of the final volume is concentrated solution and this is mixed with 3parts of the diluent. When expressed in this way the total parts is 4.diluentConcentrated solutionQuestion 7:a) Write 2:3 in the form β2 in ?βb) Write a solution of 1 in 8 in the form of a ratioc) How much concentrate do you need to make the following dilutions?i.iii.500 mL of a 1 in 4 solutionii.600 mL of a 1:5 solution4L of 1:2000 solutiond) Heparin, an anticoagulant, is available in a strength of 5000 units/mL. If 3000 units is required,what volume of heparin will be injected?11
6. Algebra RefreshAddition and Multiplication PropertiesMaths PropertyRuleCommutativeThe number order foraddition or multiplicationdoesnβt affect the sum orproduct1 3 3 1ππ (ππ ππ) (ππ ππ) ππ1 (2 3) (1 2) 3ππππ ππππAssociativeSince the Number orderdoesnβt matter, it may bepossible to regroup numbersto simplify the calculationExampleππ ππ ππ ππ2 4 4 2Distributiveππ(ππππ) (ππππ)ππππ(ππ ππ) ππππ ππππ2 (2 3) (2 2) 3Zero Factorππ 0 0If ππππ 0, then eitherππ 0 or ππ 02 0 0A factor outside the bracketcan be multiplied withindividual terms within abracket to give the sameresultRules for Negatives ( ππ) ππ( ππ)( ππ) ππππ ππππ ( ππ)ππ ππ( ππ) (ππππ)( 1)ππ ππRules for Division οΏ½οΏ½π:ππ ππππ ππ ππππ ππππ ππ ππ ππ ππ ππππ ππ π‘π‘βππππ ππππ ππππππ οΏ½οΏ½π ππ(multiply everything by b)ππ ππ ππ ππππ ππππ2(3 1) 2 3 2 1 ( 3) 3( 2)( 3) 2 3 2 3 ( 2) 3 2 ( 3) (2 3)( 1) 2 2 πΌπΌπΌπΌ4 44 22 2 6 6 3 31 3 π‘π‘βππππ 1 42 4 2 3ππππππ(multiply by d)ππππ ππππ12
Question 8:a) Simply the followingi.3(π₯π₯ 1) 2(π₯π₯ 5)ii.2π₯π₯ 8 (43 π₯π₯)iii.2π₯π₯ 2 3π₯π₯ 3 π₯π₯ 2 2π₯π₯ iv.2(4 6)2 312 b) Expand 2π¦π¦ 2 (3π₯π₯ 7π¦π¦ 2)c) What is the molecular mass of H2O2 Given O 16 and H 1d) Calculate the molecular mass of CuSO4 7H2O hydrated copper sulphate. Given Cu 64, S 32, O 16 and H 113
7. Power OperationsPowers are also called exponents or indices; we can work with the indices to simplify expressionsand to solve problems.Some key ideas:a)b)c)d)Any base number raised to the power of 1 is the base itself: for example, 51 5Any base number raised to the power of 0 equals 1, so: 40 1Powers can be simplified if they are multiplied or divided and have the same base.Powers of powers are multiplied. Hence, (23 )2 23 23 26e) A negative power indicates a reciprocal: 3 2 Certain rules apply and are often referred to as: Index Laws.132Below is a summary of the index rules:Index LawSubstitute variables for valuesππππ ππππ ππππ ππ23 22 23 2 25 32ππππ ππππ ππππ ππ(ππππ )ππ ππππππ(ππππ)ππ ππππ ππππ(ππ ππ)ππ ππππ ππππππππ ππ ππππππππππ ππ ππππππ ππ36 33 36 3 33 27(42 )5 42 5 410 1048 576(2 5)2 22 52 4 25 100(10 5)3 23 8; (103 53 ) 1000 125 84 2 81 311 24163 8 263 63 63 3 60 1; (6 6 1)EXAMPLE PROBLEMS:a) Simplifyb) Simplify65 63 62 72 64 65 3 2 72 64 6 6 72 6 4ππ5 β4 ππ 1 ππ5 ππ 1 β4 ππ 4 β4Watch this short Khan Academy video for further explanation:βSimplifying expressions with /simplifying-expressions-with-exponents14
Question 9:a) Apply the index laws/rules:5 2 5 4 52 lifyvi.Simplifyvii.Simplifyviii.Simplify ππ 1 ππ π₯π₯ 2 π₯π₯ 5 4 2 t 3 42 (54 )3 24 3634 32 3 5 9(π₯π₯ 2 )33π₯π₯π₯π₯2 b) What is the value of π₯π₯ for the following?i.ii.iii.iv.v.49 7π₯π₯14 2π₯π₯88 111 2π₯π₯480 2π₯π₯ 31 51Show that16ππ2 ππ33ππ 3 ππ 8ππ2 ππ9ππ 3 ππ5 6ππππ 515
8. Scientific NotationNumbers as multiples orfractions of tenNumberNumber as a power of ten10 x 10 x 1010 x 101010 x 10010 110 210 3Scientific notation is a convenient method of representing and working with very large and very smallnumbers. Transcribing a number such as 0.000000000000082 or 5480000000000 can be frustrating sincethere will be a constant need to count the number of zeroes each time the number is used. Scientific notationprovides a way of writing such numbers easily and accurately.Scientific notation requires that a number is presented as a non-zero digit followed by a decimal point andthen a power (exponential) of base 10. The exponential is determined by counting the number places thedecimal point is moved.The number 65400000000 in scientific notation becomes 6.54 x 1010.The number 0.00000086 in scientific notation becomes 8.6 x 10-7.1.)106(Note: 10-6 (Coefficient)(base 10)If n is positive, shift the decimal point that many places to the right.If n is negative, shift the decimal point that many places to the left.Question 10:Write the following in scientific notation:a. 450b. 90000000c. 3.5d. 0.0975Write the following numbers out in full:e. 3.75 102f.3.97 101g. 1.875 10 1h. 8.75 10 316
Multiplication and division calculations of quantities expressed in scientific notation follow theindex laws since they all they all have the common base, i.e. base 10.Here are the steps:MultiplicationA. Multiply the coefficientsB. Add their exponentsC. Convert the answer to scientificNotationExample: ππ. ππ ππππ ππ (ππ. ππ ππππ ππ )ππ. ππ ππ. ππ ππππ. ππππ (multiply coefficients)ππππ ππ ππππ ππ ππππ( ππ ππ) ππ (add exponents) ππππ. ππππ ππππ ππ β check itβs in scientific notation ππ. ππππππ ππππ ππ β convert to scientific notation Division1. Divide the coefficients2. Subtract their exponents3. Convert the answer to scientificNotationExample:(9 1020 ) (3 1011 )9 3 3 (divide coefficients)1020 1011 10(20 11) 9 (subtract exponents) 3 109 β check itβs in scientific notation Recall that addition and subtraction of numbers with exponents (or indices) requires that the baseand the exponent are the same. Since all numbers in scientific notation have the same base 10, foraddition and subtraction calculations, we have to adjust the terms so the exponents are thesame for both. This will ensure that the digits in the coefficients have the correct place value so theycan be simply added or subtracted.Here are the steps:Addition1. Determine how much the smallerexponent must be increased by so it isequal to the larger exponent2. Increase the smaller exponent by thisnumber and move the decimal point ofthe coefficient to the left the samenumber of places3. Add the new coefficients4. Convert the answer to scientificnotationExample: ππ ππππππ (ππ ππππππ )ππ ππ ππ increase the small exponent by 2 to equal thelarger exponent 4ππ. ππππ ππππππ the coefficient of the first term is adjustedso its exponent matches that of the second term ππ. ππππ ππππππ ππ ππππππ the two terms nowhave the same base and exponent and thecoefficients can be addedππ ππ. ππππ ππππ check itβs in scientific notation Subtraction1. Determine how much the smallerexponent must be increased by so it isequal to the larger exponent2. Increase the smaller exponent by thisnumber and move the decimal point ofthe coefficient to the left the samenumber of places3. Subtract the new coefficients4. Convert the answer to scientificnotationExample:(5.3 1012 ) (4.224 1015 )15 12 3 increase the small exponent by 3 to equalthe larger exponent 150. 0053 1015 the coefficient of the firs t term isadjusted so its exponent matches that ofthe second term (0.0053 1015 ) (4.224 1015 ) the twoterms now have the same base andexponent and the coefficients can besubtracted.15 4.2187 10 check itβs in scientific notation 24
Questions 11:a) (4.5 10 3 ) (3 102 )b) (2.25 106 ) (1.5 103 )c) (6.078 1011 ) (8.220 1014 ) (give answer to 4 significant figures).d) (3.67 105 ) (23.6 104 )e) (7.6 10 3 ) ( 9.0 10 2 )f)Two particles weigh 2.43 X 10-2 grams and 3.04 X 10-3 grams. What is the difference in theirweight in scientific notation?g) How long does it take light to travel to the Earth from the Sun in seconds, given that theEarth is 1.5 X 108 km from the Sun and the speed of light is 3 X 105 km/s?25
9. Units and Unit ConversionMeasurement is used every day to describe quantity. There are various types of measurements suchas time, distance, speed, weight and so on. There are also various systems of units of measure, forexample, the Metric system and the Imperial system. Within each system, for each base unit, otherunits are defined to reflect divisions or multiples of the base unit. This is helpful for us to have arange of unit terms that reflect different scaleMeasurements consist of two parts β the number and the identifying unit.In scientific measurements, units derived from the metric system are the preferred units. The metricsystem is a decimal system in which larger and smaller units are related by factors of 10.Table 1: Common Prefixes of the Metric SystemPrefixAbbreviationRelationship to Unitmega-M1 000 000 x UnitExponentialRelationship toUnit106 x Unitkilo-k1000 x Unit103 x Unitdeci-dUnit10 1 x Unitcenti-cmilli-mmicro-Β΅nano-nUnits1/10 x Unitor0.1 x Unit1/100 x Unitor0.01 x Unit1/1000 x Unitor0.001 x Unit1/1 000 000 x Unitor0.000001 x Unit1/1 000 000 000 x Unitor0.000000001 x UnitExample2.4ML -Olympic sizedswimming poolThe average newbornbaby weighs 3.5kgmeter, gram, litre, sec2dm - roughly thelength of a pencil10 2 x UnitA fingernail is about1cm wide10 3 x UnitA paperclip is about1mm thick10 6 x Unithuman hair can be upto 181 Β΅m10 9 x UnitDNA is 5nm wide26
Table 2: Common Metric ConversionsUnitLarger UnitSmaller Unit1 metre1 kilometre 1000 meters100 centimetres 1 meter1000 millimetres 1 meter1 gram1 kilogram 1000 grams1000 milligrams 1 gram1 000 000 micrograms 1 gram1 litre1 kilolitre 1000 litres1000 millilitres 1 litreOften converting a unit of measure is required, be it travelling and needing to convertmeasurements from imperial to metric (e.g. mile to kilometres), or converting units of different scale(e.g. millimetres to metres). In the fields of science and pharmacy, converting measurement can be adaily activity.It helps to apply a formula to convert measurements. However, it is essential to understand the howand why of the formula otherwise the activity becomes one simply committed to memory withoutunderstanding what is really happening. Most mistakes are made when procedures are carried outwithout understanding the context, scale or purpose of the conversion.Unit Conversion rules:I.Always write the unit of measure associated with every number.II.Always include the units of measure in the calculations.Unit conversion requires algebraic thinking (see Maths Refresher Workbook 2).E.g. Converting 58mm into metres: 58 ππππ The quantity1 ππ1000 ππππ1 ππ1000 ππππ 0.058 ππis called a conversion factor; it is a division/quotient; in this case it has metreson top and mm on the bottom. The conversion factor simply functions to create an equivalentfraction, as 1m is equal to 1000m and any value divided by itself is equal to 1.We work with information given, a conversion factor, and then the desired unit.info givenconversionfactordesiredunit27
Example problem:Convert 125 milligrams (mg) to grams (g). There are 1000 mg in a gram so our conversion factor is1 ππ1000 ππππThe working is as follows: 125 ππππ 1 ππ1000 ππππ 0.125 ππHere the mg cancels out leaving g, which is the unit we were asked to convert to.It is helpful to have a thinking process to follow. This one comes from the book, IntroductoryChemistry (Tro, 2011, pp. 25-35). There are four steps to the process: sort, strategise, solve andcheck.1. Sort: First we sort out what given information is available.2. Strategise: The second step is where a conversion factor is created. You may need to look ata conversion table and possibly use a combination of conversion factor.3. Solve: This is the third step which simply means to solve the problem.4. Check: The last step requires some logical thought; does the answer make sense?5.Example problem: Convert 2 kilometres (km) into centimetres (cm). Sort: we know there are 1000 metres in one km, and 100cm in one metre. Strategise: So our conversion factors could be Solve: 2 ππππ 2 ππππ1000 ππ100 ππππ π₯π₯ ππππ1ππππ1ππ1000 ππ100 ππππ 1ππππ 2 1000 πππ100ππππ1ππ100 ππππ 2 ππππ 200,000 ππππCheck: is there 200,000cm in a kilometre? Yes, that seems sensible.Question 12:Convert the following:a) 0.25 g to milligramsb) 15 Β΅g to mgc) 67 mL to Ld) Using this information: 1 ππ2 10000ππππ2 1000000ππππ2Convert 1.5ππ2 ππππππππ ππππ21ππ 100ππππ 1000ππππ28
More Examples:a) Convert 0.15 g to kilograms andmilligramsBecause 1 kg 1000 g, 0.15 g can be convertedto kilograms as shown:ππ. ππππ π π π±π±ππ π€π€π€π€ππππππππ π π ππ. ππππ π π π±π±ππππππππ π¦π¦π¦π¦ ππππππ π¦π¦π¦π¦ππ π π ππ. ππππππππππ π€π€π€π€Also, because 1 g 1000 mg, 0.15 g can beconverted to milligrams as shown:b) Convert 5234 mL to litresBecause 1 L 1000 mL, 5234 mL can beconverted to litres as shown:ππππππππ π¦π¦π¦π¦ π±π±ππ ππππππππππ π¦π¦π¦π¦ ππ. ππππππ ππQuestion 13:a) Convert 120 g to kilograms and milligrams. Use scientific notation for your answer.b) Convert 4.264 L to kilolitres and millilitresc) Convert 670 micrograms to grams. Give your answer in scientific notation.d) How many millilitres are in a cubic metre?e) How many inches in 38.10cm (2.54cm 1 inchf)How many centimetres in 1.14 kilometres?g) How many litres are in 3.5 105 millilitres?Watch this short Khan Academy video for further explanation:βUnit conversion word problem: drug nversion/v/unit-conversion-example-drug-dosage29
10. Logarithms42i.ii.The 2 is called the exponent, index orpower.The 4 is called the base.3With roots we tried to find the unknown base. Such as, π₯π₯ 3 64 is the same as 64 π₯π₯; (π₯π₯ is thebase).A logarithm is used to find an unknown power/exponent. For example, 4π₯π₯ 64 is the same aslog 4 64 π₯π₯This example above is spoken as: βThe logarithm of 64 with base 4 is π₯π₯.β The base is written insubscript.The general rule is: ππ ππ π₯π₯ log ππ ππ π₯π₯i.ii.iii.In mathematics the base can be any number, but only two types are commonlyused:a. log10 ππ (ππππππππ 10) is often abbreviated as simply Log, andb. log ππ ππ (ππππππππ ππ) is often abbreviated as Ln or natural loglog10 ππ is easy to understand for: log10 1000 log 1000 3 (103 1000)log 100 2Numbers which are not 10, 100, 1000 and so on are not so easy. For instance,log 500 2.7 It is more efficient to use the calculator for these types ofexpressions.R eferback tothe IndexLaw sQuestion 14: Write the logarithmic form for:a) 52 25b) 62 36c) 35 243Use your calculator to solved) Log 10000 e) Log 350 f)Ln 56 g) Ln 100 Loga(x) y then ax ylogb(xy) logb(x) logb(y)Use the provided log laws to simplify the following:logb(x / y) logb(x) - logb(y)i) log5 log6logb(x y) y logb(x)j) log4 β log8a logb(x) logb(xa)Some logrules for yourreferencek) 2log4 β 2log230
11. AnswersQ 1. Order of Operationsd) 12e) a) 1b) 22c) 10Q2. Fraction Multiplication and Div
Converting Decimals into Fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33Μ The method used to convert decimals into
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