# ATAT 110 Basic Mathematics # Week 9 ##### What In this lesson we will shift somewhat from mathematical operations to the application of them to the aviation environment, as was our goal all along. ##### Why The systems that surround the physics and their representations in mathematics in aviation are many and varied. They are however based on systems designed to make things easier. ##### Testing You will be tested on this material on Graded Assignment 3, and the final test. ## Approach and Objectives By understanding the following topics, you will have achieved the learning outcome for this lesson. Consult your course outline for the learning outcomes and other details of this course. ### Course Learning Objectives CLO 1. Perform operations with whole numbers and fractions. CLO 2. Perform series of operations using the appropriate order of operations. CLO 3. Perform arithmetic operations with real numbers, including those in scientific notation. CLO 4. Compute and simplify powers and roots of signed numbers. CLO 7. Use imperial and metric units and unit conversions as they relate to physical quantities involved in problem-solving. ### Main Topics - [[T110T SSGW09#Units of Measurement|Units of Measurement]] - [[T110T SSGW09#Ratios|Ratios]] --- ## Units of Measurement We have learned all the required arithmetic to begin using it around the hangar. We will measure things, or make calculations before we measure things, or measure things to see what calculations may be necessary. In order to ensure accuracy and precision, we must know the units of measurement, and to which quantities they apply. >A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Units of measurement answer the question "what?". Q: How far is that? A: Four Q: Four what? Without the unit of measurement, the math is useless. Let's try a more scary aviation example. What if you were to ask me when the final test for this course was, and I said 3. What if you were thinking days, or minutes, or months? You might have a hard time in 3 [[T110T Intro#Testing and Grades|weeks!]] What follows is simply a list of quantities that you will encounter in the aviation department, and the units that apply. In several cases there are alternate systems of measurement. We will simply include them here, and you can consult the extras for more information: > [!info]- Extras >More information on the [metric system](https://en.wikipedia.org/wiki/Metric_system) and the [Imperial systems](https://en.wikipedia.org/wiki/Imperial_units). > Some background on the history of [metric in the US](https://www.npr.org/sections/thetwo-way/2017/12/28/574044232/how-pirates-of-the-caribbean-hijacked-americas-metric-system). We will not even get into the relationships between these units yet. That will come in the following weeks. Your goal is simply to associate units with the quantities they measure. This is more or less a memorization exercise, but you know many of them already. ### Length Distance is another way to express length, so be prepared to use these interchangeably. #### Imperial - Inch ($in$) - Foot ($ft$) - Yard ($yd$) - Mile, aka statute mile ($mi$) #### Metric - Metre ($m$) #### Other - Nautical mile ($nm$) ### Weight and Mass Mass is the amount of matter an object contains, weight is calculated when mass interacts with gravity and is a measure of force. Weight and mass are not the same thing, although you may hear them interchangeably. #### Imperial - Ounce ($oz$) - Pound ($lb$) - Ton ($t$) #### Metric - Gram ($g$) - Tonne, aka metric ton ($t$) ### Time - Second ($s$) - Minute ($min$) - Hour ($hr$) #### Longer Time units - Day - Week - Month - Year - Decade ury - Millenium ### Velocity and Speed Even though these two terms are not exactly the same, you will hear them used interchangeably in aviation, and the distinction doesn't matter too much to us as technicians. $ \text{Speed} = \frac{\text{Distance}}{\text{Time}}$ Because we are now dealing with distance and time, we have to combine units to describe these things. #### Imperial - Miles per hour ($mph$) - Feet per second ($ft/s$) #### Metric - Metres per second ($m/s$) - Kilometres per hour ($km/hr$) #### Other - Knots ($kn$) ### Volume Volume is the amount of space that a substance or object occupies. #### Imperial - Fluid ounce ($fl. oz$) - Pint ($pt$) - Quart ($qt$) - Imperial Gallon ($gal$) #### Metric - Litre ($L$) - Cubic centimetres ($cc$) - Meters cubed ($m^3$) #### Other Because the US has its own variation of the imperial units... - US ounce ($fl.oz.$) - US quart ($qt$) - US pint ($pt$) - US gallon ($gal$) ### Force Force is the measure of a mass that is being accelerated (or decelerated). The famous Newton's second law describes it as $\text{Force} = \text{Mass}\times \text{Acceleration}$ #### Imperial - Pound ($lb$) #### Metric - Newton ($N$) ### Energy and Work Energy is the quantifiable measure of how much work a system can do. Energy comes in many different forms such as thermal energy, gravitational potential energy, [[kinetic energy]], chemical energy, electrical energy, etc. Work is done by a system when it converts from one form of energy to another. #### Imperial - British Thermal Units ($BTU$) #### Metric - Joules ($J$) ### Power Power is the rate at which energy is being produced (or consumed) $ \text{Power} = \frac{\text{Work}}{\text{Time}}$ #### Imperial - Horsepower ($hp$) #### Metric - Watt ($W$) ### Torque Torque is the measure of the force that can cause an object to rotate about an axis. It is measured as an amount of force applied at a distance away from the pivot $ \text{Torque} = \text{Force}\times \text{Distance}$ #### Imperial - Foot pound ($ft \cdot lb$) - Inch pound ($in \cdot lb$) #### Metric - Newton metre ($N\cdot m$) ### Pressure Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. $ \text{Pressure} = \frac{\text{Force}}{\text{Area}}$ #### Imperial - Pounds per square inch ($psi$) #### Metric - Pascal ($Pa$) #### Other - Bar - Inches of mercury ($in \text { }Hg$) ### Temperature See a more detailed explanation of Temperature in [[T101 Week 1#Temperature|ATAT101]] #### Imperial - Fahrenheit ($^{\circ}F$) - Rankine ($^{\circ}R$) #### Metric - Celsius ($^{\circ}C$) - Kelvin ($K$) ### Electricity #### Voltage More in [[T105 Week 2#Volts|ATAT105]] - Volts ($V$) #### Current More in [[T105 Week 2#Current|ATAT105]] - Amperes or Amps ($A$) #### Resistance More in [[T105 Week 2#Resistance|ATAT105]] - Ohms ($\Omega$) #### Power - Watts ($W$) #### Energy - Kilowatt-hours ($kWh$) ### Memory A digital bit of memory can hold a value of one or zero. More on that later. - bit - byte ## Ratios Ratios define a quantitative relationship. It compares values. A ratio says how much of one thing there is compared to another. Ratios can be called proportions, as they describe the proportions of parts of a total. The word ratio is also related to the word rate. We will see why that is helpful shortly. Two numbers are written in the following forms: 3 : 2 3 to 2 or even 3/2 While ratios have certain things in common with fractions, they are not quite the same thing. Be careful with assumptions in this area. #### Example: An air to fuel mixture ratio of 13:2 indicates that there is 13 parts of air for every 2 part of fuel. From this we can infer a few different pieces of information. We can find the normalized proportion by evaluating the quotient of the two terms. $13\div 2 = 6.5$ This 6.5 can be written into a ratio of 6.5:1. If a ratio does not have a second term that follows, then it is assumed the second term is 1. We can also find a fractional representation of each of these terms in context of the whole. $\begin{align*} \text{Fuel:} \qquad \frac{\text{Fuel}}{\text{Fuel + Air}} = \frac{13}{15}\\ \\ \text{Air:} \qquad \frac{\text{Air}}{\text{Fuel + Air}} = \frac{2}{15}\\ \end{align*}$ This shows that of the 15 total parts involved, 13 parts are fuel and 2 parts are air. ### Scalability of Ratios Ratios are scalable, meaning that a ratio can be expressed in many ways as long as the quotient of the two terms remain unchanged. Similar to how we were able to generate endless different [[T110T SSGW02#Equivalent Fractions|equivalent fractions]] by multiplying the denominator and the numerator by a multiplier, the same technique can be applied to scale ratios up or down. #### Example: Epoxy has a resin to curing agent ratio of 3:2. Note that the quotient of this ratio is 1.5. If 3 fl. oz of resin and 2 fl. oz of curing agent was combined, then a total of 5 fl. oz of epoxy is produced. We can scale this ratio up as much as needed to suit our needs. The ratio can be tripled to give us 15 fl.oz of epoxy using 9 fl.oz of resin and 6 fl. oz of curing agent. This gives us a resin to curing agent ratio of 9:6. Note that every term in this ratio was multiplied by three and the quotient of the two terms (9÷6) remains unchanged from the original ratio. Similarly, the ratio can be halved to give us 2.5 fl.oz of epoxy using 1.5 fl.oz of resin and 1 fl. oz of curing agent. This gives us a resin to curing agent ratio of 1.5 :1. Note that every term in this ratio was divided by 2 and the quotient of the two terms (1.5÷1) is still the same as before. To neatly put it into an equation: $3:2 = 9:6 = 1.5:1$ All of these ratios share the fact that the quotient of the two numbers is 1.5 which make them equivalent ratios. ### Calculating Missing Variables in a Ratio Let's stay with our example but change it up a bit. Let's say I had 15 fl. oz. of resin and I wanted to use it all for my project. How much curing agent would I need? Let's describe it a little differently and apply math that we already know. We know that the required ratio is 3 : 2. We want to know what the ratio is if we have 15 on the left, so 15 : ? At this point we can use an [[T110T SSGW05#Equation Manipulation|equation manipulation]] technique called cross multiplication: Here are the equivalent ratios expressed as equivalent fractions: $\frac{3}{2} = \frac{15}{x}$ Then we can proceed to isolate the $x$ from this equation which is the unknown value we wish to solve. 1. We bring the $x$ out of the denominator on the right hand side. This is achieved by multiply both sides by $x$. On the right side of the equation, the $x$ in the numerator and the denominator cancels out, effectively eliminating it from that side. $\frac{3}{2} \times x = \frac{15}{x}\times x\qquad \rightarrow\qquad \frac{3x}{2} = 15$ 2. Similarly, we bring the 2 out of the denominator on the left hand side by multiplying both sides by 2. $\frac{3x}{2} \times 2 = 15 \times 2 \qquad \rightarrow\qquad 3x = 30$ > Step 1 and 2 can be combined by bringing the denominator on the right side into the left side and vise versa. ![[Pasted image 20211004205028.png|200]] 3. We must further isolate the $x$ on the left side by dividing both sides by 3. $ \frac{3x}{3} = \frac{30}{3} $ 4. Which gives us our final result: $ x = 10 $ So, if we have 15 fl. oz of resin, to preserve the correct 3:2 ratio, we would have to mix in 10 fl. oz of curing agent. Work the problem backwards, that is, divide both sides of the 15:10 ratio to see that it does indeed simplify to 3:2. Be prepared to work these kinds of problems in all their permutations. Another example: Of 200 students, 50 decide to follow the avionics program. What is the ratio of avionics students to maintenance students? Since our total is 200 and 50 of the 200 are avionics, we calculate that the maintenance students number 150 (200 - 50). We now have our avionics to maintenance ratio of 50:150. Simplify this ratio to 1:3 Notice that the ratio of avionics students to maintenance students is 1:3 but if expressed as a fraction, avionics students are 1/4 of the student population, and the avionics student body is 1/3 the size of the maintenance student body. ### Percentages Similar to ratios and fractions, percentages are also equivalent fractions that are scaled to a denominator of 100. For example: 30 students in a class of 40 passed the ATAT110 midterm. This can be expressed as 30/40. To convert this to a [[percentage]] we must scale it such that the denominator is 100 but the ratio remains the same $\frac{30}{40} = \frac{x}{100}$ Using the cross multiplying technique, we can isolate $x$ as: $ \frac{30\times 100}{40} = x $ Which gives us $x = 75\%$ Another way [[percentage]] is used is $x\%$ of $y$. This expression indicates a multiplication operation: $ y \times x\% \quad \text{or}\quad y\times\frac{x}{100}$ Examples: What is 20% of 50. $ 50\times\frac{20}{100} = 10$ What is 50% of 20 $ 20\times\frac{50}{100} = 10$ > Notice how the two examples evaluate to the exact same value.