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SCIENTIFIC MEASUREMENT AND Chapter 2

Memorise Understand Importance * SI units and prefixes * Dimensional analysis High level: 11% of GAMSAT Section 3 maths-based questions released by ACER are related to content in this chapter (in our estimation). * Note that approximately 90% of such questions are related to just 3 chapters: 1, 2, and 3.

Introduction

It is extremely important to know the SI system of measurement for the GAMSAT. The metric system is very much related to the SI system. The British system, though familiar, does not need to be memorised for the GAMSAT (related questions would only be asked if relevant conversion factors were provided). 'Dimensional analysis' is a technique whereby simply paying attention to units and applying the basic algebra we covered in GM Chapter 1, you will be able to solve several real exam questions in GAMSAT Physics, Chemistry and Biology – even without previous knowledge in those subjects.

Additional Resources

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GAMSAT MATHS GM-37 2.0 Yes, GAMSAT Has a Need for !

A ‘unit’ is any standard used for Physics easier. It is normal if you cannot fill making comparisons in measurements. out the tables now, but please return to this Units form an important part of the founda- page after you have studied this chapter so tion of science and GAMSAT Section 3. you can complete all entries below.

For non-science students, this is a Science students? Do your best, be new language to learn but the vocabulary quick, note that some entries that appear is incredibly small. The units below can all simple may surprise you! ACER has exam be found in this chapter, and throughout questions directly dependent on your GAMSAT Physics and Chemistry. Your famil- understanding of SI units and prefixes, so

High-level Importance iarity with these units will make your study of let’s get started…

Section Units

2.1.3 length: electric current: Table 1: SI Base mass: thermodynamic temperature: Units time: amount of substance:

Section Units 2.1.3 area: speed, : Table 2: Examples volume: acceleration: of SI Derived Units

Section Units

2.1.3 frequency: power: Table 3: SI electric charge, quantity force: of electricity: Derived Units with Special Names electric potential difference, pressure, stress: electromotive force: and Symbols energy, work, quantity of heat:

Section Base 10 tera: deci: 2.1.3 giga: centi: Table 4: Important mega: milli: SI prefixes for GAMSAT, A closer kilo: micro: look hecto: nano: deca: pico:

GM-38 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

2.1 Systems of Measurement High-level Importance

2.1.1 British Units (Imperial System of Measurement)

You are probably already familiar with Minutes There are 60 seconds in several of these units of measurement, but we every minute. 1 min. = 60 s. recommend reviewing them at least once. Hours There are 60 minutes in every A. Length: These units are used to hour. 1 h. = 60 min. describe things like the length of physi- cal objects, the displacement of a physical Days There are 24 hours in every object, the distance something has traveled day. 1 day = 24 h. or will travel, etc. Area and volume are also measured as the square and cube (respec- Years The year is the largest unit tively) of these units. of time in the British System. There are 365 days in every Inches The inch is the smallest year. 1 yr. = 365 days measurement of length in the British System. C. Mass/Weight: These terms are not Feet There are 12 inches in every technically the same and we will discuss the foot. 1 ft. = 12 in. differences in Physics. The following units describe the amount of matter in an object. Yards There are 3 feet in every yard. 1 yd. = 3 ft. Ounces The ounce is the smallest unit of mass in the British System. Miles The mile is the largest unit of length in the British System. Pounds There are 16 ounces (oz.) in There are 5,280 feet in every every pound (lb.). 1 lb. = 16 oz. mile. 1 mi. = 5,280 ft. Tons The ton is the largest unit of mass in the British System. B. Time: These units describe the passage There are 2,000 lbs. in an of time. American ton, and 2,240 lbs. in a British tonne. Neither Seconds The second is the smallest unit should be committed to of time in the British System. memory.

GAMSAT MATHS GM-39 2.1.2 Metric Units

Measuring with Powers of 10: Unlike As you go down, you divide by 10 and the British System, the Metric System has as you go up, you multiply by 10 in order to only one unit for each category of measure- convert between the units. ment. In order to describe quantities that are much larger or much smaller than one EXAMPLE of the base units, a prefix is chosen from a How many metres is 1 kilometre? variety of options and added to the front of

High-level Importance the unit. This changes the value of the unit 1 km = 1000 m by some power of 10, which is determined by what the prefix is. The following are the From general knowledge, we know most common of these prefixes (with the that kilo means one thousand. This means representative symbols in brackets): there are 1000 metres in a kilometre. But just in case you get confused, you can also - use the clue from the mnemonic. Now we Milli (m) One thousandth (10 3) of the know that Kilo is three slots upward from base unit the Unit base. Hence we multiply 3 times by

- 10: 10 × 10 × 10 = 1000. Centi (c) One hundredth (10 2) of the base unit An even less confusing way to figure out how to do the metric conversions quickly and One tenth (10-1) of the base Deci (d) accurately, is to use a metric conversion line. unit This is quite handy with any of the common units such as the metre, litre, and grams. Deca (da) Ten (101) times the base unit

Kilo (k) One thousand (103) times the K H D U D C M base unit

There is a mnemonic that may be m l used to identify these prefixes: g King Kilometre Kilo Henry Hectometre Hecto (h) Figure GM 2.1: The Metric Conversion Died Decametre Deca Line. The letters on top of the metric line Unexpectedly Unit Base Unit stands for the “King Henry” mnemonic. On Drinking Decimetre Deci the other hand, the letters below the metric line - m, l, g – stand for the unit bases, metre, Chocolate Centimetre Centi litre, or gram, respectively. Milk Milimetre Milli

GM-40 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

To use this device, draw out the Now, let’s try converting centimetre to metric line as shown in Fig GM 2.1. From kilometre: What is 6.3 cm in km?

the centremost point U, the prefixes going High-level Importance to the left represent those that are larger 1. Place your pen on the c (centi) point than the base unit (kilo, hecto). These also in the metric line. correspond to the decimal places that you will be moving from the numerical value of 2. Moving from c to k (kilo) takes five the unit to be converted. Those going to places going to the left. This also the right are for the ones smaller than the means moving five places from the unit (deci, centi, milli). decimal point of the number 6.3. 6.3 cm = .000063 km

EXAMPLE Using this method definitely makes doing the metric conversions so much How much is 36 litres in millilitres? faster than the fraction method! Step 1: Place your pen on the given unit, in this case L (litre). Then count the There are other prefixes that are often number of places it takes you to reach the used scientifically: unit being asked in the problem (millilitre). Tera (T) 1012 times the base unit Giga (G) 109 times the base unit 6 k h d u d c m Mega (M) 10 times the base unit Micro (μ) 10-6 of the base unit Nano (n) 10-9 of the base unit Pico (p) 10-12 of the base unit L A. Length: As with British length units, : Converting litre to millilitre Fig GM 2.2 these are used to measure anything that has using the metric conversion line. to do with length, displacement, distance, etc. Area and volume are also measured as the Step 2: Because it took you three places square and cube (respectively) of these units. going to the right to move from the litre to the millilitre units, you also need to add Metres The metre is the basic unit of three places from the decimal point of the length in the Metric System. number 36.0. Other millimetre, centimetre, 36 L = 36,000 ml Common kilometre Forms

GAMSAT MATHS GM-41 B. Time: These are units that quantify the C. Mass: These are units that describe passage of time. the amount of matter in an object.

Seconds Just as in the British System, Grams The gram is the basic metric the second is the basic unit unit of mass. of time in the Metric System. Minutes, hours, and the other Other milligram, kilogram British units are not technically Common part of the Metric System, but Forms they are often used anyway in High-level Importance problems involving metric units.

Other millisecond Common Forms

2.1.3 SI Units

SI units is the International System of Units (abbreviated SI from the French NOTE Le Système International d'Unités) and is a modern form of the metric system. SI units Typically, because of Chemistry, are used to standardise all the scientific students think that the litre (L) is an SI calculations that are done anywhere in the unit. It is not. The cubic metre is the SI world. Throughout this book, and during the unit for volume. It is important that you real exam, you will see the application of know that 1 L = 1000 cubic centimetres (= cc or mL) = 1 cubic decimetre. base SI units and the units derived from the

base SI units.

GM-42 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

Table 1: SI Base Units Table 2: Examples of SI Derived Units Base quantity Name Symbol SI derived unit

SI base unit area square metre m2 High-level Importance length metre m volume cubic metre m3 mass kilogram kg speed, velocity metre per second m/s time second s acceleration metre per second m/s2 electric current ampere A squared thermodynamic temperature kelvin K amount of substance mole mol

Table 3: SI Derived Units with Special Names and Symbols SI base unit frequency hertz Hz - s-1 force newton N - m·kg·s-2 pressure, stress pascal Pa N/m2 m-1·kg·s-2 energy, work, quantity of heat J N·m m2·kg·s-2 power W J/s m2·kg·s-3 electric charge, quantity of electricity coulomb C - s·A electric potential difference, volt V W/A m2·kg·s-3·A-1 electromotive force

NOTE

We will see all the units from these 3 tables in the Physics and Chemistry chapters. Do not try to memorise the last 2 columns in Table 3. However, if this is your second time studying from this page, you should be able to derive all the units displayed in the last 2 columns of Table 3. In fact, the derivation of units through dimensional analysis is a regular type of GAMSAT ques- tion. You will be tested on this point with the GAMSAT-level practice ques- tions in Physics and Chemistry.

GAMSAT MATHS GM-43 Table 4: Important SI prefixes for GAMSAT, A closer look Prefix Base 10 Decimal English Word Name Symbol tera T 1012 1000000000000 trillion giga G 109 1000000000 billion mega M 106 1000000 million kilo k 103 1000 thousand hecto h 102 100 hundred deca da 101 10 ten High-level Importance BASE UNIT - 100 1 one deci d 10-1 0.1 tenth centi c 10-2 0.01 hundredth milli m 10-3 0.001 thousandth micro µ 10-6 0.000001 millionth nano n 10-9 0.000000001 billionth pico p 10-12 0.000000000001 trillionth

There exists more SI prefixes than quality digital image = megabytes, storage those presented in Table 4 but, if needed, space on a smartphone = gigabytes, but additional prefixes will be defined during storage space on a computer is increasingly the exam. However, Table 4 is considered measured in terabytes. ‘assumed knowledge’ and therefore must be memorised. The following practice questions do not require previous knowledge in Physics. As we have seen in GM 2.1.2, each For ‘beginners’, please feel free to consult prefix name has a symbol (Table 4) that is Tables 1-4 to assist in solving the questions. used in combination with the symbols for units of measure (Tables 1-3). For example, NOTE the symbol for kilo- is 'k', and is used to produce 'km', 'kg', and 'kW', which are the As mentioned in Chapter 1, whenever SI symbols for kilometre, kilogram, and possible, consider using a sheet of paper kilowatt, respectively. Even if you do not or Post-It note to cover the worked solu- have a science background, you have likely tion while you try to answer the question heard of many SI prefixes because of your yourself. You will benefit from doing so use of computers (a byte is a unit of digital for most of the practice questions in this memory): very small files like the text of an chapter and subsequent chapters. email = kilobytes, larger files like a good

GM-44 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

EXAMPLE 1 EXAMPLE 2 The power (P) of an electrical What is the sum of 5.00 mV and appliance can be calculated from the 10 µV? High-level Importance current (I) that flows through it and the A. 15 µV potential difference (V) across it, such that B. 5.1 mV P = IV (PHY 10.2). C. 5.01 mV D. 5.001 mV Determine the power if the potential difference is 5 mV and the current is 5 mA. The common base unit is the volt (V, see Table 3). A. 25 µW B. 25 MW We must ensure that the prefixes are C. 2.5 × 10−3 W −3 the same in order to perform the addition D. 2.5 × 10 mW (or subtraction if that had been the case). Thus 5.00 mV + 10 µV = 5.00 mV + 0.01 P = IV = 5 mA × 5 mV mV = 5.01 mV. Answer C. = 5 × 10−3 A × 5 × 10−3 V P = 25 × 10−6 A•V = 25 µW When units are presented as exponents, for example, in square and ● the latter is equivalent to 25 × 10−6 W cubic forms, the multiplication prefix must = 25 × 10−3 mW, NOT 2.5 × 10−3 mW. be considered part of the unit, and thus ● Ref. Tables 1-4; PHY 10.2. Answer: A. included as part of the exponent.

Note that the preceding question and Consider the following cases: those to follow have no science assumed 2 knowledge other than the understanding of ● 1 km means one square kilometre, SI units. Also note that if the base units have or the area of a square of 1000 m by prefixes, all but one of the prefixes must be 1000 m and not 1000 square metres. expanded to their numeric multiplier, except ● 2 Mm3 means two cubic megametres, when combining values with identical units or the volume of two cubes of 1 000 (i.e. ‘you can add apples to apples but you 000 m by 1 000 000 m by 1 000 000 can’t add apples to oranges’). m or 2 × 1018 m3, and not 2 000 000 cubic metres (2 × 106 m3). Consider the following (if you are stuck, please consult any of the preceding tables – 1 to 4 – to try to work out the answer before looking at the worked solution).

GAMSAT MATHS GM-45 EXAMPLE 3 ● 3 MW = 3 × 106 W = 3 × 1 000 000 W = 3 000 000 W Which of the following is NOT 2 3 2 3 2 2 equivalent? ● 9 km = 9 × (10 m) = 9 × (10 ) × m = 9 × 106 m2 = 9 × 1 000 000 m2 2 A. 3 MW = 3 × 1 000 000 W = 9 000 000 m 2 6 2 B. 9 km = 9 × 10 m ● 5 cm = 5 × 10−2 m = 5 × 0.01 m C. 5 cm = 5 × 0.01 m = 0.05 m D. Each option above represents an equivalence. ● Answer: D. High-level Importance

2.2 Mathematics of Conversions (Dimensional Analysis)

2.2.1 Dimensional Analysis with Numeric Calculations

You are about to learn, or revise, a tion to assess the money that you have topic of enormous importance for your earned: 5 × Q. Easy, but in science, that is exam. In fact, the trick or skill of dimen- a dangerous calculation. sional analysis will help you solve prob- lems during the real GAMSAT even when Every day, in major hospitals around you might not understand the question! the world, the health of patients is at risk How is this possible? ACER will give you due to medication errors. One major issue the units in the question and the units for is dosage. Patients sometimes receive the answer choices. Even if the question is the wrong dose because of human error: contorted, maths speaks. There is a step- someone did not pay careful attention to by-step, logical technique to take you from the units. one set of units to another. Basically, a dimension is a measure- The irony is that you have been using ment of length in one direction. Examples this technique all of your life for simple include width, depth and height. A line has problems without necessarily applying rigid one dimension, a square has two dimen- maths. Let’s say that you had been working sions (2D), and a cube has three dimen- for Q dollars/pounds/euros per hour. After sions (3D). In Physics, ‘dimensions’ can 5 hours, you want to remind yourself of also refer to any physical measurement your progress so you do a quick calcula- such as length, time, mass, etc.

GM-46 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

Dimensional analysis (also called Since both expressions are equal to ‘factor-label method’ or the ‘unit-factor 1, then both expressions are equal to each

method’) permits the solving of problems other: High-level Importance across the sciences simply by carefully 1 hour 60 minutes analysing and manipulating units. Unlike 1 = = the ‘normal’ equations which you have 60 minutes 1 hour seen in Chapter 1, equations developed during the process of dimensional analysis And, even more powerful is the idea require you to focus on numbers and units. that you can multiply any expression by 60 Dimensional analysis uses the fact that minutes / 1 hour, or, 1 hour / 60 minutes any number or expression can be multi- because it is the same as multiplying that plied by one without changing its value. expression by the number 1. But which expression should you use? It depends on The Process: In order to convert a quan- your assessment of the units you have and tity from one type of unit to another type the units you wish to end up with (this is the of unit, all you have to do is set up and ‘analysis’ in dimensional analysis). execute multiplication between ratios. Each conversion from the preceding subsections When you are performing a conver- of Chapter 2 is actually a ratio. sion, you should treat the units like numbers. This means that when you have a fraction Consider the following equation that with a certain unit on top and the same unit you would naturally feel is reasonable: on bottom, you can cancel out the units leaving just the numbers. 60 minutes = 1 hour You can multiply a quantity by any of Divide both sides by 1 hour and you your memorised conversions, and its value get: will remain the same as long as all of the 60 minutes units, but one, cancel out. = 1 1 hour EXAMPLE 1

But if we had divided both sides of the Given that there are 2.54 cm in an inch, first equation by 60 minutes, you get: and 12 inches in 1 foot, how many inches are there in 3 feet? 1 hour 1 = 60 minutes First, determine which conversion will help. We have been provided a conversion directly between feet and inches, so that is

GAMSAT MATHS GM-47 what we’ll use and we'll ignore the distractor EXAMPLE 2 (cm). {Yes, the real GAMSAT will deliver How many inches are there in 5.08 some distractors.} metres? {Try the conversion yourself before Next, determine which of the two looking at the solution. Please go back to possible conversion ratios we should use. the previous Example - or section - to find The goal is to be able to cancel out the an appropriate conversion factor.} ori­ginal units (in this case, feet), so we We cannot convert metres directly into want to use whichever ratio has the orig- inal units in the denominator (in this case, inches, but we can convert metres to centi- metres and then centimetres into inches. High-level Importance inches/feet). We can set up both these conversions at 12 in the same time and evaluate. 33 ft =× ft 1 ft 100 cm 1 in 5.08 m =×× 5.08 m Now perform the unit cancellation. 1 m 2.54 cm

12 in Next, cancel the units. =×3 1 100 1 in = 36 in =×5.08 × 12.54 In many instances, you will not have 508 in a direct conversion. All you have to do in =  2.54 such a case is multiply by a string of ratios instead of just one (we will be using this = 200 in important technique in EXAMPLE 3). EXAMPLE 3 Warning for science students: We know that some of you may feel this is all Calculate the number of seconds in a trivial, and you may have had a successful day. academic career looking at problems and just saying “divide by 60” or some such 1) Check the units of the answer: Convert thing. ACER has special traps for those who the English “seconds in a day” to the skip steps. Careful attention to units, irre- maths, seconds/day or s/day. spective of familiarity, will get you a higher GAMSAT score. 2) Next, assess what you already know. You know that there are 60 seconds in one minute, 60 minutes in an hour, and 24 hours in one day. But to convert to

GM-48 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

maths, should you use, for example, 60 Since all is equal, we are left with the seconds/minute, or, 1 minute/60 sec- following calculation:

onds? They are equivalent expressions High-level Importance so we can use either, but only one of 60 × 60 × 24 = X the two will help solve this problem. Try to quickly complete the calculation above. 3) Choose the conversion that takes you towards your answer. In our first step, If your method gets you the correct we determined that seconds was in the answer, well done! When you have numerator, so we must choose a match- completed hundreds of practice questions ing conversion, 60 seconds/minute has in GAMSAT sciences, you will begin to the correct unit in the numerator. Let’s notice number patterns. In the preceding begin to set up the equation: calculation, you can set the 2 zeros aside, and complete 6 × 6 × 24. In the next step, 60 seconds X seconds × A × B = rather than calculating 36 × 24, which is minute day perfectly fine, you might notice that 6 × 24 is 144. You might find 6 × 144 faster than 4) Now the new problem that we have is 36 × 24. 6 × 144 = 864, now return those 2 to get rid of “minute” in the denominator zeros, and we have 86 400 seconds/1 day and replace it with “day”. Should “A” be (8.64 × 104 s/day). 60 minutes/hour, or, 1 hour/60 minutes. Only if we choose a conversion with On the real exam, you will rarely need minutes in the numerator could it cancel to be so precise. You will first assess the the “minute” in the denominator. We do units of the answer choices, then assess the same analysis for 24 hours/day and how close the numbers are, then you would we get: have determined the degree of precision necessary for your calculation (how much 60 seconds 60 minutes 24 hours X seconds × × = can you safely estimate in order to increase minute hour day day speed?). We will examine these issues in the next example. Now we can cancel the units to confirm that our analysis was correct (i.e. that the units on the left of the equal sign are exactly NOTE equal to the units on the right): Make sure you check and see that all of 60 seconds 60 minutes 24 hours X seconds your units cancel properly! A lot of unnec- × × = minute hour day day essary errors can be avoided simply by paying attention to the units.

GAMSAT MATHS GM-49 The following are 2 typical GAMSAT- then a year, then a lifetime by estimating level dimensional analysis practice ques- the average lifespan - also from Figure 1. tions. You can easily estimate your heart rate by counting your pulse while watching a EXAMPLE 4 clock for a minute for comparison, but this question refers to Figure 1. Consider the following diagram. The heart rate for ‘Human’ on the 1000 Mouse graph is approximately 60-70 b.p.m. (= 500 Hamster beats per minute as explained by the

High-level Importance Monkey caption below the graph). Because the 200 Rat Cat answers are far enough apart, which Marmot 100 Dog Giraffe Human commonly occurs during the real exam, Tiger Horse 50 whether you estimate 60 or 70 (or even Lion Elephant 80), you will approximate the same answer. 20 Whale We will examine the log scale in GM 3.8

Heart rate (b.p.m, log scale) at which point you will better understand 0 10 20 30 40 50 60 70 80 90 why the heart rate is most likely less than Life expectancy (years) 75 b.p.m. From Figure 1, we can estimate Figure 1: Heart rate in beats per minute (b.p.m.) life expectancy of a person as 80 years/ on a log scale vs. life expectancy in years for lifetime. various mammals. Putting all of the preceding together, Estimate the average number of heart we get: beats over a lifetime for a person. 70 beats 60 minutes 24 hours 365 days 80 years • • • • A. 2.9 × 1011 1 minute 1 hour 1 day 1 year 1 lifetime B. 2.9 × 109 C. 2.9 × 107 What happened to the units? × 5 D. 2.9 10 70 beats 60 minutes 24 hours 365 days 80 years • • • • Please try the calculation before 1 minute 1 hour 1 day 1 year 1 lifetime looking at the worked solution. As a result of the cancellations, the This question is asking for the number final units must be beats/lifetime, or in of heart beats per lifetime for a human other words, heart beats over a lifetime. being. If we can determine the rate of heart beats per minute from Figure 1, we We have completed the dimensional could scale that quantity up to an hour, analysis, so what about the maths?

GM-50 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

Keeping in mind that the maths needs Once you had determined 365 × 8 × 106, to be done quickly and efficiently by hand, without having done any longhand calcula-

we should seek ways to simplify the calcu- tion, you could have observed that 365 is High-level Importance lation. Fortunately, in terms of the order of somewhere between 3 × 102 and 4 × 102. operations (GM 1.2.3), we can perform the Thus 365 × 8 × 106 = between 3 and 4 × multiplications in any order that we want. 102 × 8 × 106 = between 24 and 32 × 108 = In other words, we can choose to combine between 2.4 and 3.2 × 109. Answer B. terms which create relatively simple an- swers. By observing how precise (or im- precise) the answer choices are, you will For example: 70 × 60 is 4200 which be able to gauge to what degree you can is approximately 4000 (notice that we are estimate in order to save time. Because rounding down to simplify the calculation, suitable approximating is such a valuable GM 1.4.3B; if we get an opportunity to skill for the time-pressured GAMSAT Sec- round another calculation upward, it might tion 3, we will be highlighting where it ap- restore a bit of balance). The 365 does plies in the worked solutions for GAMSAT not seem to be clearly simplified; however, Maths and subsequent chapter-ending 24 × 80 is approximately 25 × 80 which is practice questions in GAMSAT Physics, 2000 (not only is it simpler than immediate- Chemistry and Biology. ly using the 365, but it has the added ad- vantage that we rounded upwards a small amount). NOTE

Thus, without performing any long- On the real exam, sometimes there could hand calculations – only through observa- be a question that is missing some data tion – we have converted 4 factors to 4000 because ACER believes that you should × 2000 = 8 000 000 = 8 × 106. Now we are be able to estimate the missing informa- left with 365 × 8 × 106. If you have not al- tion within a reasonable error margin, as ready done it, you should complete 365 × 8 examples: the rate of growth of a part of as a quick exercise. your body, change in height or weight over time, your heart rate, etc. After all, Now we have 2920 × 106 = 2.9 × 109, if Example 3 did not have Figure 1, most which is approximately 2.9 billion heart people would guess that a resting heart rate is between 60 and 100 b.p.m. Even beats in a lifetime. If you did the calculation if a person chose a number between 30 with a calculator (which is not permissible and 200, it would not matter because the for the real exam), the result would still be answer choices are 100 times apart, so approximately 2.9 billion heart beats per they would still approximate the same lifetime. Answer B. answer. Expect that you could be asked the unexpected but that you are equipped With experience, you will not have to with the tools to answer correctly. calculate 365 × 8 to get the correct answer.

GAMSAT MATHS GM-51 2.2.2 Dimensional Analysis with Variables

There will be occasions on the dimension of the physical quantity speed real exam where you will need to apply or velocity (metres/second = m/s) is length/ dimensional analysis but there will not be time (L/T). any SI units, nor SI prefixes, nor numbers! This becomes a disciplined exercise to Which of the following represents the ensure that you understand how variables gravitational constant G in the fundamental can be manipulated using basic maths. dimensions of mass (M), length (L) and High-level Importance time (T)? EXAMPLE 5 A. M-1L3T-2 B. M2L3T-2 Consider the Law of Gravitation C. M-1L-3T-2 where the force of gravity F is: D. M2L3T2 2 F = G(m1m2/r ) This is a classic example of dimen- ● F is the force between the masses; sional analysis. First, you should know ● G is the gravitational constant; that the unit of force is a newton which is also a kg•m/s2 (see Tables 1-3, GM 2.1.3; ● m1 is the first mass; in PHY 2.2, we will see the very important ● m2 is the second mass; ● r is the distance between the centres Newton’s Law where F = ma so the units of the masses. of F must be mass ‘m’ in kg multiplied by acceleration ‘a’ which is m/s2). Of course, {We will discuss the preceding according to the preamble for the question, Physics equation in PHY 2.4. If you have the 2 m’s in the Law of Gravitation repre- no background in Physics, please carefully sent masses (M) and the ‘r’ represents a consult Tables 1-3, GM 2.1.3, in order distance or length (L). So we get: to remind yourself of the units for the 2 F = G(m1m2/r ) various variables that are presented in this problem. Aside from a basic comfort with Now transferring to the fundamental SI units, there is no assumed knowledge quantities except for G: necessary to solve this question.} MLT -2 = GM2L-2 The dimension of a physical quantity In order to isolate G (our unknown), can be expressed as a product of the 2 -2 basic physical dimensions of mass (M), divide both sides by M L : length (L) and time (T). For example, the MLT -2 / M2L-2 = G

GM-52 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

Remove the common M (GM 1.4.2): same question on the exam, the preceding final answer is the same as m3kg−1s−2 LT -2 / ML-2 = G

(Table 1, GM 2.1.3). High-level Importance

Since the answer choices have no Of course, if it is your first time symbol for division, we can remove the completing such a problem, it may seem denominator by following the rules for quite challenging. Ideally, you will see this exponents (GM 1.5.1, 1.5.3, 1.5.5): type of problem dozens of times before you sit the GAMSAT so that dimensional LT -2M-1L2 = G analysis, in its many forms, will become Thus: G = M-1L3T-2 routine. Regarding the step where we isolated G: we will be doing many more Answer A. Note that in SI units, which practice questions isolating variables in the is the other way you could be asked the next chapter.

GOLD STANDARD WARM-UP EXERCISES CHAPTER 2: Scientific Measurement and Dimensional Analysis

1. How many millimetres are there in 75 4. If a paperclip has a mass of one gram and metres? a staple has a mass of 0.05 g, how many staples have a mass equivalent to the A. 750 mm C. 7,500 mm mass of one paperclip? B. 75 mm D. 75,000 mm A. 10 2. Which of the following is the shortest B. 100 distance? C. 20 D. 25 A. 10 m C. 10 cm B. 1,000 mm D. 0.1 km 5. Which of the following is the number of 5 minutes equivalent to 17 hours? 3. A triathlon has three legs. The first leg is 6 a 12 km run. The second leg is a 10 km A. 1,080 swim. The third leg is a 15 km bike ride. B. 1,056 How long is the total triathlon in metres? C. 1,050 D. 1,070 A. 37,000 m C. 1,000 m B. 3,700 m D. 37 m

GAMSAT MATHS GM-53 6. The three children in a family weigh 10. A novel medication is found to have a 67 lbs., 1 oz., 93 lbs., 2 oz., and 18 lbs., density of 7.8 µg/mL. What is the mass of 5 oz. What is the total weight of all three 295 mL of the novel medication? children? {You may go back to section 2.1 to find an appropriate conversion factor.} A. 2.3 g C. 2.3 µg B. 2.3 mg D. 2.3 pg A. 178.8 lbs. B. 178.5 lbs. 11. What is the approximate number of C. 178.08 lbs. minutes in 1 year? D. 179.8 lbs. A. 5.3 x 103 B. 5.3 x 105 High-level Importance A lawyer charges clients $20.50 per hour 7. 7 to file paperwork, $55 per hour for time in C. 5.3 x 10 9 court, and $30 per hour for consultations. D. 5.3 x 10 How much will it cost for a 90-minute 8 12. The dimension of a physical quantity can consultation, 6 hours time filing paper- work, and 1 hour in court? be expressed as a product of the basic physical dimensions of mass (M), length A. $110.28 C. $88.25 (L) and time (T). For example, the dimen- B. $100.75 D. $127.33 sion of the physical quantity speed or velocity (metres/second = m/s) is length/ 8. If a car moving at a constant speed travels time (L/T). 20 centimetres in 1 second, approximately Given that F = at -1 + bt 2 where F is the force how many feet will it travel in 25% of a and t is the time, then the dimensions of minute? {You may go back to section 2.1 a and b must be, respectively (note: this to find an appropriate conversion factor.} is a challenging question but it is at the A. 10 C. 12 level of the real GAMSAT. You can look at B. 15 D. 9 the table in section 2.1 to guide you to the dimensions that should apply to the force 9. The Dounreay Nuclear Power Station F but that would not be given to you on has been in operation for quite some the real exam. Don’t worry if you could not time. Over the last six years, they have do this problem. We will revisit this ques- turned out a total of two megawatt-years tion type in our Physics practice problems of energy. Assuming that operations were and in the practice exam at the back of continuous over a six year period at a the book): constant rate, what was its power in A. LT -2, T-2 (W)? B. T, T -2 A. 3.3 × 105 W C. 3.3 × 102 W C. LT -1, T-2 B. 6.6 × 105 W D. 6.6 × 102 W D. MLT -1, MLT-4

GM-54 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES

NOTE

If you have completed all of the practice questions with worked solutions in GAMSAT Maths High-level Importance Chapter 2, consider logging into your GAMSAT-prep.com account, clicking on Videos, Physics, and then the following virtual-classroom video: Dimensional Analysis, Reviewing and Manipu- lating Equations. If necessary, you can have Tables 1-3 in GM 2.1.3 open to consult during the video. Good luck!

Chapter 2 Worked Solutions

Question 1 D Question 4 C See: GM 2.1, 2.2 See: GM 2.1 Construct a ratio comparing millimetres to metres using the Construct an equation that expresses an unknown number definition of the prefix “milli.” Remember that we want to of staples, times the weight of each, equals the weight of convert from metres to millimetres, so the denominator of one paperclip: the fraction we use for this ratio must contain the units of 00. 51x = 1000 mm metres: Now multiply this ratio and the given 1 1m . x = = 20 00. 5 value: 1000 mm  75m  = 75,000 mm. Question 5 D  1m  See: GM 2.1, 2.2 Convert the mixed number to an improper fraction (which Question 2 C you should be able to 'do in your head' because 17 times See: GM 2.1.2 6 can be broken down to 10 times 6 = 60 PLUS 7 times 6 Start with any of the choices and compare it to the rest: = 42 so SUBTOTAL = 102 PLUS 5 for a TOTAL = 107): 0.1km = 100m > 10 m 5 107 17 = 6 6 0.1 km > 10 m

10 cm < 10 m Convert using the fact that 60 minutes equals 1 hour (notice that the number 6 cancels so the problem is reduced to 107 1000mm = 1m > 10 cm times 10 = 1070): 107 60 minute Question 3 A  hours  1070 minutes    = See: GM 2.2  6  1hour  The total length of the triathlon is 12 km + 10 km + 15 km = 37 km. Express the ratio of kilometres to metres as a fraction, with kilometres in the denominator to Question 6 B See: GM 2.1, 2.2 1000 m cancel the units of 37 km: Now multiply: Add like units: 1 km . 1000 m  67 lbs. + 93 lbs. + 18 lbs. = 178 lbs. 37 km  = 37,000 m.  1 km  1 oz. + 2oz. + 5oz. = 8oz.

GAMSAT MATHS GM-55 Convert to pounds the part of the total weight that is in Notice that the equation is constructed to allow “years” to ounces and add to the rest of the weight: cancel (i.e. it is in the numerator and in the denominator). 2 × 106 watt-years/6 years = 0.33 × 106 W = 3.3 × 105 W  1 lbs.  ()8oz.   = 05. lbs. 16 0z.  Question 10 B 178 lbs. 05 lbs. 178 55lbs. +=.. See: GM 2.1.3, 2.2 Even if you have never heard of ‘density’, even if you Question 7 D have never known that density is mass divided by volume See: GM 2.1.1, 2.2 (PHY 6.1.1), you can just examine the units provided Given that the charges are: and dimensional analysis (GM 2.2) will lead you to the correct answer. All of the answers are in a form of grams $20.50 per hour to file paper, (GM 2.1.3), we are given µg/mL so in order to get g we

High-level Importance $55 per hour for time in court, need to multiply µg/mL by mL and then mL cancels and $30 per hour for consultations, we will have µg. Then we can convert to mg by using our SI Unit Prefix knowledge. If you have not considered the a 90-minute consultation = $30 + $15 = $45. preceding then try to answer the question before looking at 8 /6 hours = 80 minutes time filing paper work = $20.50 the worked solution. + $6.83 = $27.33

Since 20 minutes = $6.83 7.8 µg/mL × 295 mL = (7.8 × 295) µg = approx. (8 × 300) µg = 2400 µg = 2.4 mg 1 hour in court = $55 Notice that the answer choices are at least 1000 times apart Total charges = $45 + $27.33 + $55 = $127.33 and so small approximations can be done with confidence. Notice that the first and third charge add to $100 making the calculation trivial. Question 11 B See: GM 2.1, 2.2 Question 8 A X minutes/year = 60 minutes/hour × 24 hours/day × 365 See: GM 2.1, 2.2; dimensional analysis days/year Multiply by all ratios necessary to convert centimetres to feet (via inches) and seconds to minutes, and divide by 4 to Cancel units on the right side (hours, days) and we are left calculate the speed for only 25% of a minute: with: 1 X minutes/year = 60 × 24 × 365 minutes/year = approx. 60 20 cm/sec.  in./cm  ()  × ×  25. 4  (25 400) m/y 1 1  ft./in.s 60 ec./min.   10 ft./min. × × 5  ()  ≈ X minutes/year = 60 10 000 = 600 000 m/y = 6 10 m/y 12   4  which is a reasonable approximation of answer choice B, especially when all of the other choices are 100 times apart. Question 9 A See: GM 2.1, 2.2; dimensional analysis Question 12 D See: GM 2.2; dimensional analysis This problem is strictly a matter of dimensional analysis. We will explore force in Physics. Force is in units called In the SI system, “mega” means 106 ‘newtons’ which is mass times acceleration, thus a newton 1 Megawatt = 103 kW = 106 W is equivalent to a kg(m/s2) [see section 2.1] and since kg is Therefore, power in watts = (Total number of watt-years)/ M (mass), m is L (length) and s is T (time), we get that the (Number of years) force is dimensionally equivalent to ML/T2. Now the terms

GM-56 Chapter 2: SCIENTIFIC MEASUREMENT AND DIMENSIONAL ANALYSIS GAMSAT MASTERS SERIES added on the right side of the equation must also be dimen- Cancel T: sionally equal to ML/T2. Let’s first solve for ‘a’ (note: most ML/T = a = MLT-1 of the steps that we will show are really mental manipula- tions but we’ll show the steps in case you are not used to it): Now we solve for ‘b’: High-level Importance ML/T2 = at-1 = aT-1 = a/T ML/T2 = bt2 = bT2 To isolate ‘a’, multiply through by T: To isolate ‘b’, divide both sides by T2: (T)ML/T2 = (T)a/T ML/T2/T2 = ML/T4 = b

Gold Standard has cross-referenced the content in this chapter to examples from ACER’s official GAM- SAT practice materials. It is for you to decide when you want to explore these questions since you may want to preserve some of ACER’s materials for timed mock-exam practice.

Examples – Dimensional analysis: Q3, Q5, Q12 of 1; Q31 and 36 of 3; Q29 and Q87 of 5. Note that “Q” is followed by the question number, and, for example, “of 1” refers to booklet number 1 which is ref- erenced in the Spoiler Alert table at the end of Chapter 1. The 10 full-length HEAPS GAMSAT practice tests (by Gold Standard and MediRed), exams 1 through 10, contain specific cross-references to this chapter within the worked solutions.

Chapter Checklist Reassess your ‘learning objectives’ for this chapter: Go back to the first page of this chapter and re-evaluate the top 3 boxes and the Introduction. Please be sure that you have completed the Need for Speed exercises at the beginning of this chapter. Complete a maximum of 1 page of notes using symbols/abbreviations to represent the entire chapter based on your learning objectives. These are your Gold Notes. Consider your multimedia options based on your optimal way of learning: Download the free Gold Standard GAMSAT app for your Android device or iPhone. Create your own, tangible study cards or try the free app: Anki. Record your voice reading your Gold Notes onto your smartphone (MP3s) and listen during exercise, transportation, etc. Try out the Gold Standard GAMSAT Physics online videos at gamsat-prep.com which have heaps of calculations, or you can try other maths options on YouTube like Khan Academy or Leah4sciMCAT playlist for Maths Without a Calculator (the latter was produced for MCAT preparation but it remains helpful for GAMSAT). Reassess your schedule for your full-length GAMSAT practice tests: ACER and/or HEAPS exams. Ensure that you have scheduled one full day to complete a practice test and 1-2 days for a thorough assessment of worked solutions while adding to your abbreviated Gold Notes. Reassess your progress in scheduling and/or evaluating stress reduction techniques such as regular exercise (sports), yoga, meditation and/or mindfulness exercises (see YouTube for suggestions).

GAMSAT MATHS GM-57 High-level Importance