# ATAT 110 Basic Mathematics # Week 10 ##### What In this lesson we learn about a very common yet critically important category of applied mathematics: Unit conversions. ##### Why As we will see, conversions occur in many areas of aviation, many of them critical to flight safety. ##### Testing You will be tested on this material on Graded Assignment 4, and the final test. This lesson prepares you to demonstrate your competence on MP110-1. ## 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 5. Solve linear equations in one variable. %%whatever the fuck that's supposed to mean. I think solving for a variable qualifies, so there.%% - CLO 6. Apply percent and percent conversion in practical problems. - CLO 7. Use imperial and metric units and unit conversions as they relate to physical quantities involved in problem-solving. - CLO 8. Solve aviation related practical problems involving ratios and proportions. ### Main Topics - [[T110T SSGW10#Unit Conversions|Unit Conversions]] - [[T110T SSGW10#Conversion Practice|Conversion Practice]] --- ## Unit Conversions When working in aviation, you will often be forced to work in both metric and imperial systems of measurement. While Canada uses the metric system as it's official standard, most American built aircraft use the imperial system in their documentation. To add to the inconsistency, Eurpean aircraft often use the metric system in their documents. Unit conversion is not exclusive to working with two systems, but even within the same system. Converting between different length units (ie. feet to inches) is one example in the imperial system. Converting between different metric prefixes, km to m for example, can be considered another. Unit conversion may happen between different properties while given a conversion factor. For example, a technican may be tasked to convert the weight of fuel into volume of fuel using the fuel density. Hence, technicians must be mindful of the units they are working in and be diligent about converting back and forth between two units. The famous Gimli glider incident was caused by improper conversion during the refueling calculations which lead to the aircraft running out of fuel half way into the flight. [🔗](https://en.wikipedia.org/wiki/Gimli_Glider) To do a proper unit conversion, we need 3 pieces of information. 1. The original value and its unit 2. The desired unit. 3. The conversion steps. #### Example 1: Convert 37 ft. to inches. (1 ft. = 12 in.) 1. Original value is 37 and the unit is in feet. 2. We want the value to be inches. 3. We have one conversion from feet to inches. First, express the conversion equation into a fraction . $1ft = 12in \quad \rightarrow\quad \frac{1ft}{12in} \text{ or } \frac{12in}{1ft}$ There are two fraction that are created from this process and we'll figure out which one to use in the following steps. Next lay out the conversion such that you're multiplying the original value and unit with the conversion fraction to achieve the desired unit. $37ft \times \frac{12in}{1ft} = \text{? }in$ Notice how the equation has a $ft$ in the denominator? Similar to how we can eliminate a term if it exists in both in the numerator and the denominator, we can eliminate the $ft$ from the left side to give us: $37 \times \frac{12in}{1} = \text{? }in$ With the $ft$ gone, we can evaluate this equation as $ 37\times 12in = 444in$ A common mistake is to mix up which operation to use. When converting from feet to inches, you must multiply the value by 12 (as seen in the example above). But unless you have an intuition that inches are a smaller unit than feet, you may accidentally divide by 12. Let's see what happens if we select the wrong conversion fraction : $37ft \times \frac{1ft}{12in} = \text{? }in$ Notice how the $ft$ is in the numerator of the conversion fraction and we can't eliminate the $ft$ in the original because both are in the numerator. >When selecting the conversion fraction, pick the one where the desired unit is in the numerator and the original unit is in the denominator #### Example 2: Convert 37 ft. to m. (1 ft. = 12 in.) (1m = 39.37in) 1. Original value is 37 and the unit is in ft. 2. We want the value to be m. 3. We have two conversion: ft. to in. then in. to m We can separate this into a two step conversion. $37ft \times \frac{12in}{1ft} = 37 \times \frac{12in}{1} = 444in$ $444in \times \frac{1m}{39.37in} = 444 \times \frac{1m}{39.37} = 11.27m$ Alternatively, you can do it in one combined step by stringing the conversion fractions together. $37ft \times \frac{12in}{1ft} \times \frac{1m}{39.37in} = \text{? }m$ Notice how the $ft$ in the original term gets eliminated by the $ft$ in the denominator of the 1st conversion fraction. Similarly, the $in$ term in numerator of the 1st fraction gets eliminated by the $in$ in the denominator of the 2nd fraction. This gives us: $37 \times \frac{12}{1} \times \frac{1m}{39.37} = \text{? }m$ $\frac{37 \times 12}{39.37} \times 1m = 11.27m$ ### Common Conversions Here are a list of common conversions between units. Don't panic. You're not expected to memorize this for your test. Note that not every possible conversion rate is given. For example, if you are asked to convert between inches and miles, you would have to do a few conversions, or better, do the calculations to determine the single conversion rate. ![[Faculty/Student/Content/ATAT/T110/assets/unknown.png|]] ## Conversion Practice This exercise brings together a number of skills, and is preparation for MP 110-1. Work along with all of the examples and MP110-1 will be a simple matter of answering 10 similar questions. ### Length - Feet, inches, statute miles, nautical miles, metres We will build up slowly, and show that a comprehensive list of all the conversions possible would be large and hard to follow. To streamline this, we start with some basics, and then rely on them to make more complicated conversions later.  Here is an example: ![[Pasted image 20221028111149.png|200]] Given the above reference, how could you determine how many inches are in a yard?  You must do one of two things: Do two conversions: first, convert the yard to feet, and then convert 3 ft into inches, or Make a new conversion formula by multiplying feet by inches. This gives you:  1 yard = 36 inches. Either way works, try both and find what you are more comfortable with. Let’s see why it’s best to keep our conversion reference table as simple as possible. ![[Pasted image 20221028111333.png|200]] We’ve now added a new unit, and based it on yards. Simple enough. But what if we needed to know how many feet were in a mile?  We use one of the same two methods above. But what if we wanted to know how many inches were in a mile?  Same story. But if we were to include all of these in the list, it would get very much larger very quickly. Since the math here is simply multiplication, it is simpler to just list the basics and do some quick calculations when required. This does mean that a single conversion could require a few equations, but a little practice will make them quite easy. Notice the advantages of the metric system. Once you know the prefixes, you really don’t need to have a reference formula to convert to another metric unit.  For example, it is easy to see that 1 km = 1,000 metres. This also means that for converting from other systems, we only really need to know the conversion to the basic metric unit, e.g. grams, metres, etc. Let’s complete our length reference with other measurements that you may run into on the job. For the MP and the test, ensure you are including any of the substeps we have just learned about. ![[Pasted image 20221028112149.png|200]] Note that you now can convert many different lengths.  For example, even though it is not explicitly in our reference list, it is a simple matter to convert: - Nautical miles to feet (nautical miles to yards, yards to feet) - Miles to kilometres (mile to yards, yards to feet, feet to inches, inches to metres, metres to kilometres) This may seem a little complicated, but it is really still just basic arithmetic. ### Velocity - feet/sec., mph, knots, km/hr For these kinds of conversions, you build on previous knowledge. All of the distance units have been covered, and we are assuming you know how to convert seconds, minutes and hours. For example, to convert 65 mph into feet/sec you can take this strategy: 65 miles multiplied by 5280 will give us feet.  65 x 5280 = 342,200 ft. This is feet/hour though, so we convert the hours to seconds by dividing by 3600 (60 secs x 60 min = 1 hour) 342,200/3600 = 95.3 ft - 65 MPH = 95.3 ft/sec Another example converting knots to km/hr: 300 knots = 300 /.54 = 555.55 km.  This one is simple because the time unit is the same.  - 300 knots/hr = 555.55 km/hr What would it be if you were asked to convert it to metres/sec? 555.55 km/hr is equal to 555,550 m/hr .  Then, divide by 60 to give us per minutes, and divide by 60 again for seconds: - 555,550 /3600 =154.32 m/sec ### Weight and Mass - pounds, ounces, grams, ton, metric ton The varying systems of measurement of mass make this area tricky, and you must take care to ensure you are dealing with the correct units. The ton is a complicated matter.  First of all, there are at least three types, and secondly, it is also used as a measurement of volume rather than mass. Be sure of your units when faced with these kinds of problems. We will keep things simple for our purposes, and compare Imperial tons with metric tons and used as a measurement of mass only. Once again, the metric system simplifies our operations.  We know that 1,000 grams is 1kg for instance. ![[Pasted image 20221028112820.png|200]] If you are asked to convert 3 tons to metric tons, how do you proceed? 3 tons = 6000 lb 6000 lb = 96,000 oz 96,000 oz = 2,721,600 g 2,721,600 g = 2,721.6 kg 2,721.6 kg = 2.7216 metric tons To create your own reference for this, if you needed it often for instance, you could do the following calculation: Ton = 2000 lb 2000 lb = 32,000 oz 32,000 oz = 907,200 g 907,200 g = 907.2 kg 907.2 kg - = .9072 metric tons Therefore your new reference would be: 1 ton = .9072 metric tons.  You could reverse this equation to state 1 metric ton = 1.1 tons ### Volume - Pints, quarts, imperial gallons, US gallons, litres Units of volume also have the complication of coming at us from different systems. The same logic and arithmetic applies to volume as it does to mass. Here is a real life example of how this works. Imagine you have heard that gasoline prices in Buffalo NY are at $2.10/gallon. Is it worth it to drive down for those prices? Let us examine what we need to do to compare these properly. First, the unit of measurement is in US gallons (because it is in New York State), so we must first do that conversion.  US Gallon = 128 US oz = 3.78 litres (128/33.8) To find out their price per litre, we must divide the cost by the number of litres, so: 1 litre = $2.10/3.78 So our American neighbours are paying about 55 cents per litre. BUT, their dollar is not the same as ours, so we are not done.  Take the exchange rate as $1.00 US = $1.33 CDN. So, 55 cents multiplied by 1.33 is about 74 cents Canadian per litre. So, are our neighbours paying less for their gasoline? If you are fragile, do not calculate how much less they pay per tank, or month, or year! Incidentally, fuel accounts for significant portion (18%) of an airline's operating costs. ![[Pasted image 20221028113621.png|200]] ### Weight instead of Volume? In aviation, it is more important to know the weight of our fuel than the space it may take up. The tank will take up space anyway, but the weight that the aircraft is carrying is of crucial importance to many aspects of performance and safety. Here is an example of converting volume to weight. Note that you must check the documentation for specifics (fuel type, temperature etc). These are only examples. ![[Pasted image 20221028113726.png|200]] By now you may not feel that the Gimli glider accident was based solely on stupidity. It is not necessarily evident what is happening when you do conversions such as these. It is useful to develop certain logical instincts. You know that a gram weighs less than an ounce (approx. 28 times less).  Therefore, if you are calculating mass, and you are going from grams to ounces, you would expect to see a much smaller number (approx. 28 times less). This is a first line defence against errors in this area. If you get a number that is larger, you may have multiplied when you should have divided, or you have misplaced the decimal point somewhere in your calculations. Misplacing the decimal point is a common error when converting entirely in the metric domain as well, so be aware. >You should also be aware that these errors can just as easily happen when you are using a calculator, or even an online conversion utility. If you state the question incorrectly, you will get an incorrect answer. ### Temperature - Fahrenheit, Celsius, Rankine, Kelvin The relationship between temperature scales introduces some complications. It is no longer simple arithmetic to convert between scales that have different size degrees.  So the formulas are more difficult: > C° = (F°-32) *5/9 > F° = C° *9/5 +32 For example:  Convert 100° C to F: 100 *9/5 +32 900/5 +32 180 +32 212°F >  Celsius to Kelvin: -273.15 We know that the only difference between Celsius and Kelvin is the 0° point.  They are exactly 273.15° apart. >  Fahrenheit to Rankine: -459.67 Fahrenheit to Rankine has a similar relationship, with the 0° points being 459.67° apart. ![[Pasted image 20221028120031.png|200]] You are not required to memorize these conversion formulas, and the following graphic will be provided to you while working on the MP and on the final exam: ![[Pasted image 20221028120119.png|250]] If you have followed along with this practice, you will have covered most of the concepts or problems that will be presented to on MP101-1, and on tests. If you had difficulty with any of this, you should concentrate your efforts there. Run through the examples provided above one more time. Make up other conversions, and check them via google. Contact your prof. This is not about getting a good score or getting the MP done, it's about being ready to continue your challenging road to becoming a technician. Dig in! ## MP 110-1 Complete this MP by doing the 10 question quiz under assessments in eCentennial.