# Quantitative Reasoning

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**Created:**2017-02-08

**Last Modified:**2017-04-23

**By adding 10 additional data points at the mean, you are “compressing” the data set around the mean. This decreases the standard deviation.**

**Possible combinations if the cards are identical:**

**RBRB, RBBR, BRBR**

**There are 3 possible combinations if the red and black cards are identical. If the red cards are different, we can double the number of combinations, so there are 6 total combinations. Since both the red and black cards are different, we will double it again and find that there are 12 possible configuration**s (I will enumerate them here, but on the test it is faster to not to do this!).

**Take the largest number minus the smallest number. This will give the range, The set with the largest range equate to the set with the largest standard deviation.**

**slope = rise over run**

**rise is y**

**run is x**

**so change in y over change in x**

**and if the lines are perpendicular to each other then the slop is inverse and reciprocated**

**Recognize 5-12-13 angle so you don't need to spend time figuring out the pythagorean theorem**

**cos = A/H**

**csc= inverse of sin = H/O**

**since it's a 5-12-13 angle, figure out what is H O, AND A then apply it**

__2ft/hr__= 6 sq. ft/hr

**Have to figure out probability of picking blue first and red second without replacements. The two probably will be multiplied. Then have to find probability of getting red first then blue second , the probability will be added to the first probability.**

*** Multiply probability when 1 event occur.**

*** Add probability when the same event occur but in different order.**

**First dice: 1, 3, 5 so probably getting those would be 3/6 , which is also = 1/2**

**Second dice = 3 so probability of getting this would be 1/6**

**then x the two probability together (1/2 * 1/6) = 1/12**

**68% with 3 keys % 5 keys**

**100% - 68%= 32% will not have 3 keys & 5 keys**

**so since there's symmetry to the normal curve, half the 32% will be the probability of less than 3 keys , and the other half is the probability of more than 5 keys**

**68% of the data lies within one standard deviation from the mean. In other words, 68% of people have between 3 and 5 keys on their keychain. Conversely, we know that 32% of people do not have between 3 and 5 keys on their keychain. Due to the symmetry of a normal curve, we say that 16% have less than 3 keys and 16% have more than 5 keys.**

**Take probability of A happening x probability of be not happening**

**1/4 x 8/9 = 8/36**

**=2/9**

**https://s3.amazonaws.com/datbootcamp/math/math-2-35-1.png**

**Recognize 5-12-13 right triangle.**

**and use Area of triangle = (1/2) base * height**

**Recognize it's 45-45-90 angle. x, x , x√2**

**SINCE x in this is case is 7, the hypothenuse will be 7√2**

**Volume of cube = s^3**

**Volume of sphere = (4/3)π r^3**

**The diff would be volume cube- volume sphere**

**tan = sin/cos should know**

**sin/cos (sin - cos) replace the tan with sin/cos since they r equivalent.**

**sin^2/cos - sin cos/cos factor them**

**sin^2/cos - sin**

**sin (sin/cos -1 ) factor sin out**

**sin (tan - 1 ) change sin/cos to tan since they r equiv.**

__rad__

__π__

__Credits earned____(4*3)+ (1*3) + (4*4) + (1* 2)__=__33__3.67**Credits taken 3+1+4 + 1 9**

**credit earned = how much it's worth x how much get + all the classes up using this method**

**credit taken = just add up the total credits**

**You can think to estimate this problem. 10% every month for 6 months. 10% of 1,000 is 100 so after 6 months it's going to be a little bigger than 600 so the closest to 600 is 700.**

**Don't do math, it will be time consuming.**

**First, figure out the distance Train A travel in 3 hours using the formula below**

**D= velocity * time**

**100 mi/hr * 3 hr**

**D= 300 miles ( this is the amount of distance Train A is ahead by)**

**Now you have the distance traveled by train A in 3 hours. Figure out the distance for train A in x hour and train B in x hour and equate them to find the time in hour.**

**Distance A in x hour**

**D= 100 * t + 300 (since it traveled 300 miles ahead)**

**Distance B in x hour**

**D= 150 * t**

**Now set them to equate each other**

**100t + 300 = 150t**

**300 = 50 t**

**6 = 6**

**Kelly is twice as old as her friend Jess.**

**K= 2J**

**In ten years, Kelly will be four more years older than three times Jess’ current age.**

**K + 10 = 3J + 4**

**Then substitute:**

**2J + 10 = 3J + 4**

**6 = J**

**Then substitute to find K age**

**K= 2J since J= 6**

**2(6)**

**= 12**

**3 (50%) + 7 (13%) = 10 (x%)**

**1.5 + . 91 = 10 (x%)**

**2.41 = 10 x**

**x = .241**

**% i s .241 x 100 = 24.1 %**

**Since area shaded is below straight line, y ≤ a with a being some positive number.**

**Since area is shaded above the parabola, y ≥ x^2**

**know y = x^2 has that curve**

**We can immediately see that the equation for the line would be y=a,**with a being some positive constant. We can

**also determine that the inequality in this problem would be y ≤ a, since the shaded area is below the straight line.**

**. We can see that the apex of the parabola is at (0, 0), meaning that the equation of the parabola is most likely of form: y = x^2.**We can

**also see that the inequality in this problem would be y ≥ x^2, since the shaded area is above the parabola.**

**Look at MC options and pick out which option will factor into the given factor.**

**choice c**

**3 + x - 4 - 2x - 14 = x - 10**

**-15 - x = x - 10**

**-2 x = 5**

**x = 5/-2**

**log(675)= 2x + 3y.**

**what two numbers , with the( first number being squared ) times (the second number being cubed) will equate to 675**

**Look at the MC options and calculate from there**

**2x indicates that number is going to get squared**

**3y indicates that number is going to get cubed**

**5^2 * 3^3**

**25 * 27 = 675**

**First, square both sides to get rid of the square root.**

**Second do a 5/4 power on both sides to cancel out the power on the left side of the equation.**

**Then substitute 16 with 2^4 , then the power times each other power 4*5 / 4 = 5**

**so the final would be 2^5 = 32**

**what g(x) is equal to, substitute that for x in f(x)**

**if being asked for g(x) then just do the same thing, what f(x) is equal to, substitute for x in g(x)**

**Looking at the formula given, we can eliminate A, B and D since the slope will have something to do with 2/3**

**Our choices are between C & E**

**Plug in the points given in the the two choices to see which one is correct**

**6/10 (500) = x (250)**

**300= x (250)**

**30/25 = x**

**x = 120**

**Price of video game console after the 20% increase**

**with x being the price of the original game console**

**x + (x * .2) = 1.2x**

**Now that we have the price of the game after the 20% increase, figure out what the price of the game will be after the 10% coupon. USING THE formula we got earlier.**

**1.2x - ( .1 (1.2x)**

**1.2 x - .12 x = 1.08**

**The question is what is the net % change from the original price of the console. so use 100 as reference point.**

**108% - 100% = 8%**

**210**

**21 10**

**7 3 5 2**

**Now add 1 to each of the exponents and multiply together**

**7^1 3^1 5^ 1 2^1**

**so total of four exponents**

**(1+1) * (1+1) * (1+1) * ( 1+ 1)**

__Extended Explanation:__

__Why do we add 1 to each exponent? The reason why we add 1 to each exponent is because there are two possibilities for that exponent. The exponent of each prime factor can be either 0 or 1 in order to get all possible integer factors.__**Simplify**

**A. 3/9**

**B. 3/8**

**C. 8/4**

**D. 80/4**

**E. 10/1**

**Looking at the choices, D will be the largest.**

**Beer (5%) + Wine (12%) + Liquor (37%)**

**Now given the different parts, express them in mL terms. so**

**3 parts is 300 mL, 4 parts is 400 mL etc**

**300 (.05) + 400 (.12) + 300 ( . 37)**

**(15) + (148) + 36 = 199 mL**

**So, tax increase 25% .. What is the extra amount paid on $70 from this sales increase?**

**Let's just say the jacket cost $100, if tax increase 25%, that means you have to pay additional $25**

**Use that as reference point, since 70 is smaller than 100, the amount increase for 70 will be a little less than 25 , from the answer choices only 18 is a little less than 25**

**2 FT= 24 IN**

**24 in**

__2.54 cm__= 60.96 cm__10 mm__= 609.6 mm**1 in 1 cm**

**1 standard deviation 68%**

**400 - 500**

**2 standard deviation 95%**

**350=550**

**3 standard deviation 99.7%**

**300- 600**

**Since the given standard deviation of 50 secs. that means 1 standard deviation is 50 secs. so it will fall in the 68% which will contain 400-500. if we want 500 or more, you have to take 100%- 68%, you will get 32 % but this 32% contains both 2nd standard & 3 standard deviation. Since the curve is symmetrical, just divide this in half so 32/2 = 16**

**16 will belong to 2nd standard deviation**

**16 will belong to the 3rd standard deviation.**

**since there are 6 different cards and the order does matter . use permutation**

**6*5*4*3*2*1 = 720**

**Since order DOESN'T MATTER, use combination.**

__available choice !__**(actual choice !) *( available choice ! - actual choice!)**

__10!__**7! ( 10!-7!)**

__10*9*8*7*6*5*4*3*2*1__**7*6*5*4*3*2*1 (3*2*1)**

__10*9*8____720__**(3*2*1) 6**

**=120**

**Notice that the set is not in order so have to put them in order**

**10, 13, 14, 16, 19**

**Standard deviation = how far the numbers are from the mean divided by the given numbers**

**Mean in this set is 14 since it's the middle number**

**10 is 4 away from 14**

**13 is 1 away from 14**

**14 is 0 away from 14**

**16 is 2 away from 14**

**19 is 5 away from 14**

**Given numbers are five ( 10, 13, 14, 16, 19)**

__4 + 1 + 0 + 2 + 5__=__12__= 2. 4 which is closest to mc C**5 5**

**APPALOOSA**be arranged?

__9!__

__since there are 9 options available__**3 duplicates of letter A, 2 duplicates of letter O, 2**

**duplicates of letter P**

**9*8*7*6*5*4***3*2*1**= 15,120**

__60480__**(2*1) (2*1)**

**4**

**USE LOGIC**

**whatever number you add to the number will be added to the denominator. Best approach, guess & check , start with picking choice C**

__5+9__=__14__= 53% not 60 %...**17+9 26**

**Since adding 9 pieces give 53%, to get to 60%, the answer should be a more than 9 and answer choice D is most legit!**

**if you want to solve mathematically**

__5+R__= . 6**17+R**

**5+ R = (.6) 17 + R**

**5+ R = 10.2 + .6 R**

**.4R = 5.2**

**R= 13**

*** BUT USE THE FIRST METHOD OF GUESS & CHECK BY KNOWING THAT WHATEVER YOU ADD TO THE NUMERATOR HAS TO BE ADDED TO THE DENOMINATOR.**

**realize that each added piece of red paper is also increasing the denominator of the equation.**Right now, there are 5 pieces of red paper and 17 total pieces of paper, so any added pieces of red paper will have to be added to both the numerator and denominator. The

**easiest way to approach this problem is to use the answer choices given to you and “guess and check.”**

**start with answer choice [C] because it is the middle value given as a possible answer.**If we added 9 pieces of red paper, the probability of choosing a piece of red paper would be:

**There are 6 ways to get 7**

**There are 2 ways to get 11**

**so**

__6__+__2__=__8__=__2__**36 36 36 9**

**TABLE ANSWERS**

**1 2 3 4 5 6**

**1 ***

**2 ***

**3 ***

**4 ***

**5 * ***

**6 * ***

**To get 7**

**(1,6) (2,5) (3,4) (4,3) (5,2) (6,1) total 6/36 since the table is 6x6**

**to get 11**

**(5,6) (6,5) total is 2/36 since the table is 6x6**

**just add them two probability together**

**6/36 + 2/36 = 8/36 which is reduced to 2/9 if you divide the numerator & denominator by 4**

**200 (20%) SPOTS 40 SPOTS**

**200 (35%) BROWN 70 BROWN**

**200 ( 10%) BOTH 20 BOTH**

**10% OF 200 IS 20, IF 20% IT'S DOUBLE SO 40**

**WE KNOW FROM THE LAST THAT 20% IS 40**

**NOW, ADD ANOTHER 15% TO THAT**

**10% OF 200 IS 20**

**5% OF 200 IS 10 ( SINCE 10% IS 20, 5 % WOULD BE HALF OF THAT) SO HALF OF 20 IS 10**

**10% OF 200 IS 20**

**IF CA'NT CALCULATE IN HEAD, USE CALCULATOR.**

**OVERALL NUMBER**

**40 SPOTS 70 BROWN 20 SPOT &B**

__-20 SPOTS & BROWN____- 20 SPOTS &BROWN__**20 BROWN 50 SPOTS**

**70**

**The mean for this problem is 13**

**10 is 3 away from 13**

**12 is 1 away from 13**

**13 is 0 away from 13**

**14 is 1 away from 13**

**16 is 3 away from 13**

**there are 5 numbers given**

**take the sum of how far all the numbers are away from the mean and divide by the total numbers given**

__3 + 1 + 0 + 1 + 3__= 1.6 which is close to 2 choice A**5**

**First, figure out the circumference of each of the WHEEL. then use that figure and create a ratio**

**C= 2 π r or π d**

**Circumference of front wheel 1 ft= 12 inches**

**C= 2(π)(12)**

**C= 24π**

**Circumference of back wheel**

**C= 2(π)r**

**C= 2(π) (3)**

**C= 6π**

**Now create a ratio using the circumferences of both wheels. to find the revolutions, use the distance traveled by the wheel divide the circumference.**

**first wheel ratio**

__84 π__= 3.5 revolutions**24 π**

**second wheel ratio**

__84 π__= 14 revolutions**6 π**

**now add the two up 3.5 + 14 = 17.5**

**take the distance traveled by wheel divide the circumference of the wheel**

**Draw the 3D BOX**

**REALIZE THAT IT'S A 45-45-90 angle so the missing side will be x√2. since x in this case is 1 , the missing side is just √2**

**then use Pythagorean to find the other missing side**

**(√2)^2 +(1)^2 = c^2**

**2+1=c^2**

**3 = c^2**

**√3 = c**

**KNOW THIS FORMULA**

**(n-2) 180 WITH N BEING THE NUMBER OF SIDES**

**IN THIS PARTICULAR PROBLEM N = 4 SINCE IT'S A QUADRILATERAL , NOT THAT THERE ARE 4 MEASUREMENTS DEGREES GIVEN.**

**(4-2) 180 = 360**

**2 * 180 = 360 ( this means that all the angles add up to 360**

**the ratio given 1:3:5:9**

**x + 3 + 5x + 9x = 360**

**18x = 360**

**x= 20**

**Since it has three sides, the angles add up to 180 so set the summation of all three angles.**

**x + x^2 + 7x= 180**

**8x + x^2 = 180**

**Rearrange**

**x^2 + 8x - 180 = 0**

**Use factors**

**(x + 18) (x- 10) = 0**

**x-10 = 0 or x+18= 0**

**x= 10 or x= -18**

**Area of square = s^2 s in this case is 4 since each square has a side length of 2.**

**area of square= 4^2**

**= 16**

**Area of circle = π (r^2) r in this case is 1 since the circle intersects each of the square at half point (stated in the problem)**

**area of circle = π (1^2) since 1 squared is = 1**

**=**

**π**

**SET UP THE RATIO MAKE SURE TO HAVE WHAT'S BEING ASKED WHICH IS AREA OF CIRCLE TO AREA OF FOUR SQUARE TOGETHER.**

**Area of circle =**

__π__**area of 4 square= 16**

**KNOW THIS FORMULA**

**(n-2) 180 with n= number of sides . in this case it's 18**

**(18- 2) 180 = 16(180) = 2880 =internal angles summation**

**since there are 80 sides, you have the / 18 to get angle of each**

__2880__= 160**18**

**csc^2 is = inverse of sin**

**csc= 1/ (sin) csc^2 =**

____1______1_____1__**(-√4)^2 √16 4**

**30 is given for the hypothenuse**

**14 is given for the opposite side**

**Looking at the answers, you don't need to figure out the adjacent side.**

**Know that to figure out the angle, arc has to be included in the answer.**

**sin is = O/ H = 14/30 so the choice C is arcsin = 14/30**

**Pear Mac**

**initial 5 k 7500**

**↓% ≡ ↑ %**

**5,000 - (5,000 * x) + 7500- (7500 *x) = 13,000**

**12500 - 2500 x = 13,000**

**-2500 x= 500**

**x = 5/-25**

**x= 1/-5 = -.20 = - 20%**

**since this is the decrease, the % increase is just the opposite**

**Richard is three times as old as Bryan.**

**R= 3 B**

**In six years, Richard will be twice as old as Bryan will be in six years.**

**R + 6 = 2 (B +6) HAVE to do 2 ( B + 6) because have to figure**

**out what age of B is in 6 years and double it since Richard is twice as old as Bryan in 6 years.**

**Figure out the amount of time Mike spent riding while doing 10 miles at 30mi/hr. use equation Distance= velocity * t**

**10= 30 * t**

**t= 10/30 = 1/3**

**AND**

**also the amount of time Mike spent riding while doing 20 miles at 10mi/hr . use equation Distance= velocity * t**

**20= 10 * t**

**t= 2**

**Add the two times together calculate the average velocity and use equation Distance= velocity * t**

**the distance traveled in this problem is 30 miles.**

**30 = v * (2+1/3) this is the time combined**

**30 = (1+6/3) v**

**30 = (7/3) v**

**v= 30 (3)/ 7**

**v= 90/7**

**v= 12. 85 which is close to choice B 13**

**Connie 100 papers/ 1 hour V= 100**

**Eric 80 papers / 1 hour V= 80**

**Rate is additive so combine the rate 180**

**Distance = rate * time distance in this case is 900 papers**

**900 = 180 * t**

**t = 5 hours**

**SHORTCUT: don't need to use formula**

**initial is 100,000 so after first year the amount of interest get is 10% of 100,000 which is = 10,000**

**so after three years , the interest will be around 30,000**

**so just add the interest to the initial value of 100,000 so total should be around 130,000**

**Since we estimated less than the real number, the real number should be around a little greater than 130,000 choice C fits the best**

**SHORTCUT: she spent x amount on the cans that last her for 30 days. If the cans are on sales, and if she spent the exact amount , then we can eliminate choices A B because the numbers will be greater than what she initially had. that leaves us choices C & D**

**DO CALCULATIONS**

**assume 30 days supplies cost 30 dollars so each can will cost 1 dollar**

__$30__sodas = $1 each soda**30 cans**

**If the cans are on sale 25% off , so she actually pay .75 cents for one can**

**if she spend exactly assumed $30**

__$30__= 40 days**.75can**

**Correct Answer: A. 8! / 5!**

**since order matters use PERMUTATION.**

**(8!)**SELECTION HAS

**(8-3)! selection has - positions available**

**http://datquestionoftheday.com/wp-content/uploads/2013/06/dat611.png**

**http://datquestionoftheday.com/wp-content/uploads/2013/06/dat611b.png**

__(1/4)__* 3600 seconds = 30 seconds

**Permutation equation:**

**http://datquestionoftheday.com/wp-content/uploads/2013/06/perm.gif**

**Combination equation:**

**http://datquestionoftheday.com/wp-content/uploads/2013/06/comb.gif**

**2 men can be picked from a group of 5 in (5|2) = 10 different ways, and 3 women can be picked from a group of 7 in (7|3) = 35 different ways. Therefore, the total number of ways a team can be formed is 10 × 35 = 350.**

**When order matter use permutation.**

**when order doesn't matter use combination**

**Permutation equation:**

**A father’s age is twice the sum of the ages of his four sons**

**let x= the four son's age @ present**

**F= 2 x**

**In 30 years, the total of the sons’ ages will be twice the age of their father.**

**In 30 years, the total son's age x+4 * 30 (because there are 4 of them) so you get x+120 (this will be the sum of the four sons ages in 30 years)**

**so in 30 years the father will be 2x+30**

**since the son's ages will be twice the age of the father you get 2(2x+30)**

**then set them equal to each other**

**2(2x+30) = x+ 120**

**Correct Answer: B. 10**

**This questions isn't asking to convert 18 F--> C. it's asking how much did it drop in degrees Celsius**

__change in degrees F__**change in degrees C**

**Correct Answer: A. 15**

**Mean is 15 so all the numbers add up divide by 4 should give 15**

__12+14+19+x__= 15**4**

**45+x = 60**

**x= 15**

**Correct Answer: A. 5’8”**

**1) Convert the following to inches**

**5’8 is 68 inches**

**5’2” is 62 inches**

**6’2” is 74 inches**

**2) Take avg (68+62+74)/3 = 68 inches**

**3) Convert 68 inches to feet**

**68 ft**

__12 inches__= 5 (2/3) ft**1 ft**

**so you have the first part , now convert 2/3 ft to inches**

__2__ft =__12 inches__= 8 inches**3 1 ft**

**so @ the end you get the average of 5 ft 8 inches**

**Correct Answer: C. (13×12)/(52×51)**

**The probability of picking a diamond card first time is**

**13/52**

**The probability of picking a diamond card first time is**

**12/51**

**this is because there is no replacement**

**and just take the two prob x each other**

__13____12__the answer choices has__13 * 12__**52 51 52 *51**

**Correct Answer: B. 5 to 3**

**Since**

**3****8 are women, then we know that**

__5__are men**8**

**to get ratio of men to woman it's 5 to 3**

**Correct Answer: B. 40y2/x**

__25x__=__25 y__reduced__5x__=__25y__?5x = 200y^2**40 y ? 8y ? ? =**

__200y^2__**5x**

**? =**

__40y^2__**x**

**Correct Answer: D. 720**

**order matters so use permutation.**

**This is clearly a permutation problem since the order of the books matters. The number of arrangements is 6P6 = 6! = 720.**

**Correct Answer: C. 68%**

**know standard deviations**

**1 standard deviation 68%**

**2 sd 95%**

**3sd 99.5%**

**Correct Answer: A. 5:4**

**prime # = a number greater than one that can divide by itself or 1 only. so it only has two factors, itself and 1. 1 is not a prime number because it only has 1 factor which is itself.**

**composite = a positive number that has factors in addition to 1 and itself.**

**1 is neither a prime or composite number**

**We need to figure out which numbers are primes and which ones are composite.**The following shows primes in blue and composites in red: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

__We have 5 reds and 4 blues. The correct answer is A.__

**Note that 1 is neither prime nor composite, it only has one factor**.**Ratio**

__composite number____5__**prime number 4**

**prime # = a number greater than one that can divide by itself or 1 only. so it only has two factors, itself and 1. 1 is not a prime number because it only has 1 factor which is itself.**

**2,3,5,7,**

**11, 13, 17, 19,**

**23, 29, 31, 37,**

**41, 43, 47, 53,**

**59, 61, 67, 71,**

**73, 79, 83, 89**

**97, 101, 103, 107**

**109, 113**

**http://datquestionoftheday.com/wp-content/uploads/2013/03/11.jpeg**

**y**over time,

**y**approaching an asymptote over time.

**asymptote**is a line or curve that approaches a given curve arbitrarily closely

**i dont understand how to solve this problem**

**http://datquestionoftheday.com/wp-content/uploads/2013/04/circle.png**

**know that the square root is always growing even when it reaches its upper limit.**

**It’s commonly known that logarithmic growth and s-curves can be used to represent populations, but these aren’t the only functions.**

**In our question we have two key features,**

__population growth and an upper limit__. We can quickly rule out choice D and E. They will not cap out, in fact they will grow even faster never reaching an upper limit.**Choice C is the negative absolute value. This is not growing but decreasing.**

**The cosine is also bounded by a certain amplitude, in this case it has an amplitude of 1. It is possible to use this function to model a bounded population, but the cosine function is a periodic function. This means there are places where the function grows and there are places where the function decreases.**

__The square root, on the other hand, is always growing even when it reaches its upper limit. This makes the square root function, choice B, the correct answer.__

**http://datquestionoftheday.com/wp-content/uploads/2013/03/Q11.png**

**Correct Answer: C. 128 cells**

**double every 15 mins .**

**2 hours contain 4 sections of 15 mins**

**2 hours**

**15 min doubled**

**15 min doubled**

**15 min doubled**

**15 min doubled**

**within 2 hours, bacteria double 8 times**

**so 500 x 2^8**

**5*10^2 *2^8**

**5 *100*256**

**= 128,000**

**.1% = .001 128,000* .001 = 128**

**If the bacteria double every 15 minutes, then that means over the course of 2 hours, they will double 8 times.**

**500 x 2^8 = 128,000 cells on the surface.**

**Then we apply the anti bacterial wash, which kills 99.9% of cells, meaning 0.1%, or 0.001 of the cells remain. 128,000 x 0.001 = 128 cells.**

**Correct Answer: D. 198**

**KNOW VOLUME OF CYLINDER**

**volume of cylinder = πr^2h**

**(3.14) (9)(7)**

**= 198**

**just substitute**

**Correct Answer: C. 0 to 55 mph in 3.5 seconds**

**the options provide the distance and seconds. To find the speed, take distance/time**

**D= v * t**

**d/ t = v**

**A, B, D all = 15**

**C = 15.7**

**E= 14**

**SO C has the fastest speed**

**Correct Answer: C. 1/4**

**probability of team A wins 1 game is 1/2**

**probability of team a wins 3 games**

**1/2 * 1/2 * 1/2 = 1/8**

**probability of Team B win's 1 game = 1/2**

**probability of team B wins 3 games**

**1/2 * 1/2 * 1/2 = 1/8**

**so add 1/8 + 1/8 = 2/8 = 1/4**

**KNOW sin^2 + cos^2 = 1**

**Eliminate choices A & E & B**

**CHOICE C & D**

**E.**

__sin^2x__+__cos x__**cos x sin ^2x**

__sin^2x sin^2x__+__cos ^2 x____cosx cosx____cosx__**sin^2x cosx sin^2x cos x sin^2 x cosx sin^2x**

**D. know that tan= sinx/cosx**

**cosx (tan * sinx + cosx)**

**cosx(**

__sinx__sinx + cosx)**cosx**

**cosx (**

__sin^2__x +__cosx__* cosx)**cosx 1**

**cosx (**

__sin^2 x__+__cos^2x)__**cos x cosx**

**cosx**

__(sin^2 x + cos^2x)__the cosx cancel out so you get

__cosx__(sin^2 x + cos^2x)

**Correct Answer: A. 1.11 x 10^-2**

**10^-2 is the same thing as 1/10^2**

**10^-3 is the same thing as 1/10^3**

**10^-4 is the same thing as 1/10^4**

__1__+__1__+__1__find common denominator 10^4**10^2 10^3 10^4**

__10__+__100__+__1__111__111 x 10^-4 = 1.11 x 10^-2__**10^4 10^4 10^4 10^4**

**http://datquestionoftheday.com/wp-content/uploads/2013/08/dat818.png**

**Correct Answer: C. 6 x 10^7**

**know when multiplying exponents, the exponents add up**

**know when taking exponent of exponent, the exponents multiply each other.**

**(2.4 * 10^-6) * (5 * 10^6) ^2**

**(2.4 * 10^-6) * (25 * 10^12)**

**(2.4 * 10^-6) * (2.5 * 10^13) since move one number left 1 decimal , move the other numbers to the right 1 decimal**

**2.4 * 2.5 = 6 and the exponents add up so = -6 + 13= 7**

**you get 6 * 10^7**

__water bottle__

__bathtub__

__bathtub__

**Correct Answer: A. 15.925**

**area of triangle = 1/2 * b * h**

**know base has a length of 7, now need to find height of the triangle**

**the main diagonal slope of 13/20 since slope is change in y over change in x**

__13____x__20x= 91**20 7 x= 91/20 = 4.55**

**so 1/2 * 7 * 4.55 = 15.925**

**or 1 metric ton = 2,100 lbs**

**1 English ton = 2000 lb.**

**so the difference is 100**

**Correct Answer: B. -4**

**4y + 6x = 26 substitute 4(2) + 6x = 26**

__3y -6 = -12__8 + 6x = 26**7y = 14 6x = 18**

**y = 2 x = 3**

**Solve for 2x-5y 2(3)-5(2) = -4**

**6-10**

**Given**

__1__= 9**y**

**Solving for y**

**1= 9y**

**1/9 = y**

**Given**

__1__+ y =__2____1__+__1__=__2__**x 3 x 9 3**

__1__=

__2__-

__1__

__1__=

__6__-

__1__=

__1__=

__5__x= 9/5

__9__

**J**

**udging from the huge jumps in percent over the short time in the beginning we can rule out Linear and S-curve, as those functions do not rapidly increase.**

**(if its exponentially decreasing). This leaves us with choice D.**

__Finally let’s look towards the end of the data. It appears to be converging to 95. This allows us to rule out the exponential and the quadratic, as these functions continue off towards infinity or zero__**Correct Answer: B. 34.375 π mm3**

***Challenge question***

**There are several non-calculus ways to solve this problem. 1st let’s look at the shape; it’s a cylinder that has a slice cut off. One way to calculate the volume is to see that you have a cylinder that has a height of 3 mm, and a half of a cylinder that has a height of 5 mm.**

**Split into 2 cylinder and calculate individual volume and add them together**

**top cylinder**

**V= π r^2 h**

**π (5/2)^2 * 3**

**π (25/4)*3**

**V= 18.75 π**

**Bottom cylinder**

**V= π r^2 h**

**π (5/2)^2 * (5/2) height is 5 but divide 2 because half is π (25/4)* (5/2) missing**

**π (25/4)*3**

**15.62 π**

**18.75 π + 15.62 π = 34.375 π**

**∩**L refers to the intersection of sets K and L. In other words, K ∩ L includes

**only numbers that are parts of both sets**

_{a}(8x

^{2}) – log

_{a}(2x) = log

_{a}12

_{a }(8x

^{2}) – log

_{a }(2x) = log

_{a}12

_{a}(8x

^{2}) – log

_{a}(2x) = log

_{a}12

^{2 }/ 2x = 4x

^{2 }+ 3, g(x) = √ (x-3 )

**√ f ( g(x) * g(f(x) (square root the whole thing)**

^{2}

^{2}- 3

^{2.5}/ √x )

^{2}= 9, what is the value of x

^{a}/ x

^{b}= x

^{(a – b)}and (x

^{a})

^{b }= x

^{(a * b).}Rewriting square root as a half exponent can help in simplifying:

^{2.5}/ √x )

^{2}= 9

^{5 }/x = 9

^{4}= 9

^{4})

^{1/4}= (3

^{2})

^{1/4}

^{}

^{1/2 }= ±√3

^{2}-xy +3) / (xy)

^{2})

^{18}y

^{4}z

^{12)}) / (x

^{5}y

^{9}z)

^{2}/ y

^{4}z

^{4}y

^{2}/ z

^{3}

^{3}z

^{5}/ y

^{4}z

^{5}/ y

^{7}

^{a})

^{b}= x

^{a*b:}

^{a}/ x

^{b }= x

^{(a-b)}:

^{2}and G(x)= 2 + x, find the value of x given that F(G(x)) = 10

**we can plug in a test point in the shaded area and see which inequality remains true. The easiest point to plug in would be (0,0):**

**x+3x + 2(x+2) = 106**

**Let x be the first number**

**The third number is twice more than the first number plus 2**

**2(x+2)**

**The second number is three times the first number**

**3x**

^{4}+ 27

^{2}) / 9

^{3}

^{a})

^{b}= x

^{a*b}, so we can break down the above equation further.

^{5}= 32 = [D], but if not, you can calculate 2*2*2*2*2=32 on the DAT calculator from this point.

**First, we**

__must determine the base unit of repetition.__In the repeating decimal, 0.181818181818…, the base unit is “18.”In the repeating decimal given by the problem, the base unit is “13579.”**and so the 55th digit will be the 5th digit in the repeating unit, so the answer must be 9, [E].**

__The base unit of repetition is in a group of 5 digits. Therefore, the 5th digit in each repeating unit will always be the same,__**0.13579**1357913579

**the base unit of repetition is in a group of 5 digits. Therefore, the 5th digit in each repeating unit will always be the same. and so the 55th digit in the 5th digit in the repeating unit**

^{6}to (3 ⁄ 4)

^{8}?

^{a}/ x

^{b}= x

^{a-b}

__maximum # of combinations without repeats !__**[repeat A!] [repeat B!] etc**

**ie. how many diff combo of letters can make from word BENDER**

**6!/ (2!)**

**6 TOTAL letters/ 2 repeats of the letter E**

**Standard deviation**is a measure of spread in a data set & the variation of the values.

**data set**with larger standard deviation will have a larger spread of values &

**usually a larger range.**

**range**is the

**difference between the largest & smallest value in a data set.**

**dataset with a large range**will u

**sually have a large standard deviation**

__A& B have the same standard deviation. so if these were the the options in a multiple choice questions , they both CAN'T BE right because they are both the same.__**Repeating the ‘middle’ number twice decreases the spread of the data set,**while repeating the maximum number twice increases the spread of the data set;

**The mean, or average,**of a data set is the normal average of a data set in which you sum all of the numbers and then divide by the total number of points in the data set

**median**is the middle value of a data set.

**mode**is the value that occurs most often in the data set

__-10__

__-10__

**Pr(at least one event occur)= 1- pr(no events occur)**

^{9}

**Remember this Pythagorean**

**2-3- √13**

**3-4-5**

**Perimeter of Triangle = a+ b + c (sum of the three sides)**

**Since the entire triangle is made up of two of these right triangles, we can write the following equation for the total area:**

^{2}+9

^{2}= c

^{2}

^{}

^{9+81 = c^2 }

^{ c= √90}

^{}

^{Perimeter of a square = a + b + c + d}

^{since each there are four sides , perimeter will be}

^{ 4* √90 = 4√90}

^{}

^{}

^{2}(x) = 1/2 are there in the interval of 0 ≤ x ≤ 2π?

^{2}(x)= 1/2

**Because sin(x) could be either positive or negative, it could potentially appear in all 4 quadrants (positive in quadrants I and II, negative in quadrants III and IV). Therefore, the correct answer is 4 solutions.**

**Compound Interest**

**A= P(1 + r/t)**

^{n}^{t}^{}

^{A= the final amount}

^{P= beginning amount (principal) }

^{r= the interest rate (written as a decimal)}

^{n= the number of times interest compounded each year}

^{t= the number of years }

__time 1 * time 2__

__12*6__=

__72__

__8__

**"and" means multiplication**

**"or" means addition**

**parabola that opens upward (like letter U) & whose vertex is located @ (h,k) has the general formula of**

**(x-h)**

^{2}=4(y-k)**Constructing Pascal's Triangle. Start with the two top rows, which are 1 and 1 1. To find any number in the next row, add the two numbers above it, as shown in the following diagram. At the beginning and end of each row, where there is only one number above, write a 1.**

**1 1**

**1 2 1**

**1 3 3 1**

**1 4 6 4 1**

**1 5 10 10 5 1**

**In exponential functions, a fixed base is raised to a variable exponent.**

**ie. 3e**

^{x}=y+8

**the exponent is some sort of variable**

**In power functions, a variable is raised to a fixed exponent**

**i.e 2x**

^{2}+y=6

**the exponent is fixed while the variable is raised to the fixed exponent.**

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