Mathematics questions will account for 18% of the ASP exam.

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2 Mathematics questions will account for 18% of the ASP exam. This lesson will help prepare you for those questions and includes several sample questions for practice. 2

3 Ok, before we start this question, I want to remind you that back in Unit I, there was a second part to a calculator exercise. And, that calculator exercise demonstrated the use of the HP B2+ Calculator. Now, I am going to show you how that calculator can be useful in working these engineer and economy questions. I think once you see how the calculator does this with ease, you are going to want to spend the $40 to get one of these calculators. If anything, it is a huge time savings. If $1,000 is invested in an investment account paying 8% interest. How much will be accumulated in the account at the end of 10 years? Now, in these engineer and economy questions, you will always find that there is an interest rate, which is an I, a number of periods, which is the n, and then there is two kinds of money. There is a money at the beginning of the question, and there is a money that you are actually looking for. But, in reality, there are three kinds of money. So, just in review, if you go back to Unit I, part II, I believe this explains this very well. Ok, present money, p money, think of it as money in your pocket. If you can reach in your pocket and pay for something right now, that is present money. A money, an annuity, an annuity sounds like present money because you are going to reach in your 3

4 pocket to pull out some money, but the key is, you are going to do it more than once. And, when you do it m ore than once, it is now called a payment. So, the secret word for an annuity is payment. The third type of money, future money, is money that you placed in an institution. And, while in the institution, it became more money. And, it became more money because interest was applied. Now, what I suggest you do with these engineer and economy questions, is on that white board you are going to get when you take the exam, is you put the four components you are going to see in a formula. And, that is the 2 kinds of money, the interest rate, and the number of periods. So, in this case, the $1,000 represents present money because we are going to pull out of our pocket. So, next to P, put 1,000. The interest rate is a gimme, it said 8%. So, next to I, put an 8. The n was also kind of a gimme, it said 10 years. So, put a 10 next to n. So, the last and most difficult part of this question is identifying the type of money you want to convert to. And, in this case it says how much will be accumulated, which reflects future money. So, next to the f put your question mark. So, at this point, you have to determine which calculator you want to use. Now, for example, if you are going to stick with the TI 30 Calculator, you are actually going to have to pick a formula. And, the formula you want to pick is the one that says F=, and then it has the P with the I and the n. Then, you are going to make the substitutions and run the formula. Now, in this first formula, this is fairly straight forward. This is really not that difficult, and you should be able to get $2,159. Now, if you went ahead and you did the HP 10, the payoff is you don t pick a formula. The HP 10 will actually pick the formula for you. The order that you put the information in does not matter. In an HP 10 you simply cannot ask it for an answer until you have inputted your data. So, in this case, you put in a number and then you hit the key that correlates to what it is. In this case you put in 1,000 and you hit the pv key. And, once again, in the review of Unit I, part II, it tells you that you are only going to use the number pad and the first line of keys on the HP 10. So, the very first line of keys, the third key says pv. Then you are going to put in the 8 and you are going to hit the interest per year key, your going to put in the 10 and your going to hit the n key. And then, you will never hit equals, instead, you are going to hit the key that correlates to the answer, in this case fv. So, simply strike the fv key and you get your final answer. And, of course, in this case it sounds something like 21 or 2,258. So, it is that easy. Now, the deal is, when you go to do another question, you have to clear the HP 10 calculator s memory. And, again, that is in the review of Unit I, part II, it says go second function clear. 3

5 In three years, $400,000 will be required for Environmental Protection Agency (EPA) modifications to your plant. How much money should you invest at 10% to have the required amount when needed? Again, you identify this as an engineer and economy question. And, on your white board, you are going to put your components for the formula. So, our interest rate, the easy one, is 10. So, next to the interest you are going to put 10. The number of periods is three years. So, next to the n you are going to put 3. Now you have to identify the $400,000. Because we are talking about in 3 years the $400,000 represents future money. So, next to an F, you are going to put $400,000. Now, the hard question is, what kind of money are you looking for? And it says, how much money should you invest? In other words, reach in your pocket, and pull out some money. So, in this case you are going to put a question mark next to the P. If you use the TI 30 calculator, you have to pick a formula. And, the formula you want says P=. And, of course, it has an F, and your interests and number of periods. Again, because this is a single layer formula, it is not overly complicated to run the formula. Now, if you simply get the HP 10, it is this simple. You put in 400,000, you hit the fv key. You put in 10, you hit the I key, You put in 3 you hit the n key. And, for an answer you hit 4

6 pv. It is that simple. Now just remember, before you started the problem, you had to clear the memory. And you simply do that by doing second function clear. And, you must do that every time you start a new problem. 4

7 It is anticipated that in five years from now, you will need a new piece of instrumentation. Assuming that the device can be purchased for $30,000 and that money is worth 15%, how much must be deposited yearly in an investment account to have the required amount? Now, again, you are going to put your components on your white board. You need to identify the starting money and the ending money. Because your interest rate is straight forward, %15, and your number of periods is 5 years. Again, if you are going to use the TI 30, you have to pick a formula. And, the formula you want starts with an F and ends with an A. Or, in other words, it says A equals. So, this is a little more complicated formula. You actually have to put an extra parentheses if you are going to use the TI 30 calculator to avoid a hierarchy mistake. So, again, you insert your components, you run the formula, and with any luck, you get your $4,449. If you use the HP 10, once again, always start by clearing the calculator, second function clear, and then you simply put in what you know. So, in this case, 30,000 hit the fv key, 15 hit the interest key, 5 hit the n key. And, for an answer, hit pmt. Remember, an annuity is a payment. And, on the HP 10 calculator, instead of an A key, there is a payment key. 5

8 You are proposing to replace an old piece of equipment with a newer model because you are having a high accident rate on the old equipment. The useful life of the new equipment is 10 years, and the annual cost of the accidents averages $1,000 per year. Considering only the cost of accidents and that the annual rate for monies is 8%, what is the maximum investment above the cost of the new equipment that would justify the purchase? Now again, on your white board, identify your components. Ok, interest 8%, number of periods 10. First question, what kind of money is $1,000? It says per year, which means it occurs more than once. So, that must be an annuity. What kind of money are you looking for? And, it says what is the maximum investment? In other words, reach in your pocket and pull out some money. So, that must be present money. 6

9 Again, if you are going to use the TI 30, you have to pick a formula. You want the one that says P=A. Again, it is a more complicated formula. Again, there are extra parenthesis you must put into the calculator to avoid a hierarchy mistake. But, if you actually practice that, you should be come proficient with that. Again, if you chose the HP 10, it is this simple, second function clear to clear the memory, 1,000=pmt, 8=interest, 10=n, and then for your answer simply hit the pv key. 7

10 What is the uniform annual payment that will amortize a loan of $400,000 in eight years with a 14% interest rate? Again, put your components on your white board. Interest is 14, n is 8 years. So, what kind of money is $400,000? It is called a loan. And, if you recall, when you go to a bank and you take out a loan, you actually leave with money. So, a loan is present money. So, once you got over that hurtle, what kind of money are you looking for? And, it uses the secret word, payment. It says, what is the uniform annual payment? So, we must be looking for an annuity. Again, if you are going to stick with the TI 30, you have to pick a formula. The formula will begin with an A, actually begin with a P and end in an A. You want the one that says A=. Again, it is a more complicated formula. It will take those extra parenthesis on the bottom to avoid that hierarchy mistake. Or, you can simply use the HP 10. And again, second function clear to begin, 400,000=pv, 14=interest, 8=n. And, for an answer hit pmt. 8

11 What is the lump sum of money required now to avoid an annual end of year expense of $2,000 for seven years with interest at 11.25%? Again, using your white board, put your components down. And, of course, the easy ones are the interest rate, and the 7 years for n. What kind of money is $2,000? They are calling it an annual year end expense. Again, it occurs more than once. So, that must be an annuity. What kind of money are we looking for? It says, what is the lump sum of money required now? In other words, reach in that pocket and pull out some money. So, it must be present money. And again, if you are going to use the TI 30, you have to pick a formula. You want the one that says P=A. And, again, it is a more complicated formula, it will require the extra parenthesis on the bottom to avoid that hierarchy mistake. Again, if you simply stick with the HP 10, second function clear, 2,000=pmt, is your interest rate, 7 is your n. And, for an answer, pv. 9

12 You are a safety professional in an industrial facility. During your inspections, you have determined some improvements are required that will cost $100,000. You decide to put this in a five year strategic plan and know that the cost of the project will be $150,000 in five years. How much money would you need to invest on a quarterly basis into an account that receives 10% annual interest compounded on a quarterly basis to cover the cost of the project in five years? This is probably the most complicated question you will get, if you get one like this on the exam. Because, you have a lot of extra stuff, and you have to do some converting. On your white board, when you write your components, the easy ones of course were the 10% interest rate and the 5 years. But, you actually have 2 starting monies ($100,000 and $150,000). The $100,000 is present money, and the $150,000 is future money. That intro doesn t apply to the question because you decided to a 5 year strategic plan. So, we are not going to use present money, we are actually going to use future money. And, because it wants to know what are investment is on a quarterly basis, we are talking about an annuity. Alright, now the twist to this question is, we have to fix our interest rate, and we have to fix our number of periods. Because, in the question it says our interest rate would be 10

13 compounded quarterly. Now, the simplest thing to do is the easy math, which is take your 10 and divide it by 4, just that simple. And, then that way you have the 2.5 interest rate per quarter. Because your periods are in years, and we are going to do this on quarters, we have got to convert the years to quarters. Again, do the simple math, just go 5x4, and get your 20 quarters. 10

14 Now, if you are going to use the TI 30, you have got to put in a complicated formula. You want the one that says A=F, there is extra parenthesis. And, of course, I probably haven t emphasized this for these questions, but when you put the interest rate into the formula in the TI 30 calculator, it has to be put in as a decimal value, not the percent value. In the case of the HP 10, you put it in just as it looks on paper, you put in just the number of the percent number. Ok, so if we use the HP 10, you will see how simple this is. Once we have got the converting taken place, we go 150,000, we hit the FV key, we put in the 2.5, and we hit the I key, we put in 20 and hit the n key, and for an answer, we hit pmt. 11

15 The company fire brigade is preparing to attack a large hydrocarbon fuel based fire at your plant. The approach involves several hose lines and a combination of foam and dry chemical suppression agents. All water fed lines are supplied from two pumpers arranged in parallel for reliability. The only two fire pumps available have individual probabilities of success of.9 and.82 respectively. Based on this information, what is your estimate of the overall probability of water supply failure during this operation? 12

16 Now, if you recall in probability, if you can make a fraction, you make a fraction. This case, we cannot make a fraction. Then, we have to work in the probability of failure. We have to draw a fault tree, and then we have to apply the Boolean logic. 13

17 Now, in this case, the two pumpers, they gave us success instead of failure. So, we have to derive failure from success. So, failure is 1 minus success. So, I go 1 knot minus the.9 for the first pump, that is a.1, and 1 minus a.82, which is.18 for the second pump. Because it said parallel., we are going to multiply. So, go.1 times.18, which will give you.018, which is a 1.8 percent probability of failure. 14

18 Given the sample distribution shown here, compute the standard deviation: On your TI 30 Calculator, the second function of the data key gives you two statistic variations. A one variable, and a two variable. The one variable is called standard deviation, because the only variable is the x. So, you are then going to go second function data, it is going to underline the one variable key. So, all you need to do is hit enter. Now you are in the stat function. You will now hit the data key, and it will ask you for your first data point. In this case, the first data point is the number 8. So, you are going to enter an 8. Now, the next function in your calculator is down arrow, not equals. So, down arrow. The navigation key is off to the right, under the window. You down arrow, and it says frequency. And, that is when you put in the 5. Because, you have the number 8 five times. So, the TI 30 actually short cuts this for you. Then, you are going to down arrow, and it is going to say what is your next data point. And, you are going to put in 11. And, then you are going to down arrow, and it is going to say frequency, and you are going to put in 2. Then, you will down arrow, and it says what is your next data point and you are going to put in a 15, and then you are going to down arrow and give it a frequency of 3. Now, once the data is in the calculator, you are going to hit the stat variable key, and when you hit the stat variable key, it is going to say calculate just for a moment, and then it is going to spit out a bunch of stuff. 15

19 Now, across the top of the screen will be answer choices. So, using your navigation keys, you are going to right arrow over to the sx, because the sx represents sample. And, you should have

20 If you did it the math way, you would have to build the table, you would have to summate some columns, and you would have to run the formula. Now all of that is found in your example. 16

21 Your company has an aggressive safety awards program for your over the road transport drivers. There are 20 drivers who are semi finalists for three monetary awards each $1,500 in value. How many ways can the awards be given? 17

22 Now, this is a combination. We have both combinations and permutations. A combination exists when the order doesn't matter. And, in this case, the three awards are all the same. So, there is not a definite first, second, or third place. We just need to give out these three awards amongst 20 drivers. A permutation is when order matters. So, there would actually be a first place, a second place, and a third place. And, that is actually our next question. So, our calculator has a short cut for this. 18

23 On the test, they will give you the two formulas one for combination and one for permutation. The formulas are somewhat complicated because of the factorial keys that you will have to use. Now if we go back to the PRB key on the TI 30, and you hit the PRB key, you will see the npr function and an ncr function and the factorial function. The npr or ncr is the order in which you are going to enter the data to solve the problem. So, the first thing we have to understand is what the n represents, the r represents. In this case, the n is number of drivers. The r is the rate. In this case, the rate is the three awards. So, we need to put the 20 in for the n, and the 3 in for the r. But, we have a choice of either doing a permutation or a combination. And, again, because order doesn t matter here, this is a combination. So, this is how it would work in the TI 30, you would put in the 20, you would then hit the PRB key. You would then arrow over to the ncr function, underline it, hit enter to collect that function. Then, you would put in the 3 for the three awards. So, your calculator now looks like 20 ncr 3. Now, simply hit equals, and you get your Otherwise, you have a little bit more of a challenge trying to enter the formula. Because, on the bottom of the formula, you have to enter extra parentheses to avoid the hierarchy mistake. 19

24 If in the preceding question, the awards had been $2,000 for first place, $1,500 for second place and $1,000 for third place, how may ways could the awards be given? Now, because we have a definitive first, second and third place, order matters. And, when order matters, it is called a permutation. Now, if we simply use the shortcut in our TI 30, we would go 20, which is our n, the 20 drivers, we hit the PRB key, and we select npr. We then put in the 3 for our rate, the 3 awards, and we hit equals. So, our calculator key strokes would be 20 npr, 3=. And there is our The formula for a permutation is a little easier than a combination, but you still need the extra parenthesis in the bottom of the formula to avoid that hierarchy mistake. 20

25 How many different ways can you sit the five members of your safety committee at a large round table? Now, because it said how many different ways, it means order matters. And, of course, since there is five members, there must be five chairs. So, again, if I use the short cut in the TI 30, I would go 5, for the five members, hit the PRB key, select npr, enter the five for the five chairs, and hit equals. And, that would give me my 120. Now, again, you could use the formula. But, you do need to put in that extra set of parentheses in order to avoid the hierarchy mistake. Now, there is another short cut when we are talking about permutations. When it says both the number and the rate, are the same number. In this case, the five committee members and the five chairs. And, that short cut is to simply do a factorial. So, that would have been 5 factorial. So, to put that in the calculator, you enter the number five, you hit the PRB key, you arrow all the way to the right to underline the symbol, and you hit enter. Then, you have the expression 5 factorial. Then, you simply hit equals again, and you get the same

26 A process enclosure design retrofit is being evaluated at one of the plants over which you (as the Health & Safety Manager) have responsibility. The following formula has been devised to determine the cost effectiveness of several options. Using the formula below, determine which reduction offers the greatest cost benefit ratio per db. So, we are given these two formulas. One to solve for C, and one to solve for A. And of course, we can t solve for A until we have solved for C. Now the x s in the formula, the unknowns, are answer choices. So, this is literally a plug and chug type question. We are going to plug each answer choice in to both formulas to determine what the costbenefit ration is for the retrofit. 22

27 Ok. So, we begin with the 5dB. So, with the 5dB, we plug it into the C formula, and then that will give us the C value. And, then we take the C value, and the 5dB, and we plug it into the A formula. And, we get a cost per db of 282,500. Now, we have to do this for each answer choice. Now, when we do it for the 15, we end up with $164,167 per db. So actually, as the db value increased, the cost per db went down. 23

28 So, we now do it for 25. Now, when we do it for 25 db, we continue to go down because we get a 152,500 per db. Now, if you choose at this time to short cut it, because your next answer is 35 db, even a larger number, that can be terminal. Because, what you actually find is 35 db when you run it through the formula, your cost per db actually slightly increased to $156,071 per db. So, it wasn t the 35 db as you best cost reduction, it was 25 db. 24

29 A safety professional has been assigned to the same project for the last five years and has made a site wide safety inspection each week. Assuming a normal distribution, what is the best estimate of the number of inspections that had a safety scores within ±2 standard deviations of the mean? Now, the first thing we have to recall here is, what is the definition of ±2 standard deviations. And of course, ±2 represented 95.45% of the bell curve. So, we are talking about 95%. So, the simple math is the 52 weeks in a year times the 5 years, for a total of 260 inspections. We are going to take the 260 inspections and then apply the ±2 standard deviations, which is the 95%. So, in other words, we are going to go 260 times the 95% and that should give us 248 inspections failed within ±2 standard deviations. 25

30 A company decides to reflect the worker s compensation losses against the profit function and to determine how many units must be sold to offset these costs. Indirect costs are five times direct costs, there is a profit margin of 15% on each unit sold, and the worker s compensation for the last year totaled at $90,000. What is the volume in sales needed to offset the worker s compensation losses, when the losses are treated as production cost? Now, even if you don t like the math, you should be able to look at your answer choices, and eliminate a couple of answers. For example, 600,000. If it only took 600,000 in sales to offset a 90,000 expense I don t think so. And, then of course we have the d answer, which is way out there. It takes $30,000,000 in sales to offset a $90,000 cost? Yeah, I don t think so. So, you should be able to eliminate a couple of choices here. And then, if you don t like the math, it is either b or c. And so, you know, you have a 50/50 shot. Ok. So, if we do it the math way, we first have to figure out how many parts make up a unit. And, it was hinted in the question. It says indirect costs are five times direct. In other words there are five parts of the cost that are indirect to each one part that is direct. So, it actually takes six parts to make the unit. Now if the profit, total profit, is 26

31 15%, I am going to take the 15% and divide it by the 6 parts to get the profit per unit. In this case, 15 divided by 6 is 2.5. So, I have a 2.5% profit margin per unit. Now, I am going to go back in my workers compensation loss, which was $90,000,and I am going to divide it by the profit per unit, which is my 2.55%. That would give me a volume of $3,600,000 in sales to offset that cost. 26

32 When a project has been proposed, it must first go through a preliminary analysis in order to determine whether or not it has a positive net present value using the minimum acceptable rate of return (MARR) as the discount rate. The MARR is the target rate for evaluation of the project investment. This is accomplished by creating a cash flow diagram for the project and moving all of the transactions on that diagram to the same point, using the MARR as the interest rate. If the resulting value at that point is zero or higher, then the project will move on to the next stage of analysis. Otherwise, it is discarded. The MARR generally increases with increased risk. What is the definition of MARR? Ok. So, MARR, this is basically a know it or you don t. MARR is the minimum attractive rate of return. Which is our B answer. 27

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