MU-NLPC/Calc-aqua_rat huggingface dataset:
What is the largest power of 3 contained in 200!  Select the correct option:  A) 88  B) 48  C) 66  D) 97  E) 39

Solution: In real basic terms, we're asked to find all of the '3s' in 200!   We can figure out that 200/3 = 66, so we know that there are at least 66 '3s' in 200! While that answer is among the 5 choices, it seems a bit too 'easy', so let's do a bit more work and list out the first few numbers that we know have a '3' in them:   3 = 3x1   6 = 3x2   9 = 3x3   Notice how both 3 and 6 have just one 3 in them, but 9 has TWO 3s (there's an 'extra' 3 that we have to account for). This implies that there are probably other numbers that include 'extra 3s' that we have to figure out:   To find those extra 3s, we have to look at numbers that contain 'powers of 3'...   3^2 = 9   3^3 = 27   3^4 = 81   3^5 = 243, but that's too big (we're only going up to 200). Keep in mind that a multiple of 81 is also a multiple of 9 and 27, so we don't want to count any of those values more than once.   200/9 = 22, so we know that there are at least 22 extra 3s (and certainly more because of the 27 and 81). With the 66 3s that we already have, those 22 extra 3s increase the total to 88. With the other extra 3s, we'll end up with MORE than 88 3s. There's only one answer that fits that logic...   ANSWER:97
Answer: D
A bank offers 5% C.I. calculated on half-yearly basis . A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is?  Select.
A) 126	B) Rs.121	C) 165	D) 134	E) 123

Solution: Amount = [1600 * (1 + 5/(2 * 100)2 + 1600 * (1 + 5/(2 * 100)]   = [1600 * 41/40(41/40 + 1)   = [(1600 * 41 * 81)/(40 * 40)]   = Rs. 3321.   C.I. = 3321 - 3200 = Rs.121.   Answer:Rs.121
Answer: B
12,31,68,129,?,347,516 Choose the most appropriate option: A) 129 B) 240 C) 347 D) 220 E) 520

Solution: 2*2*2 + 4= 
2*2*2 + 4 = 12
12   3*3*3 +4= 
3*3*3 +4 = 31
31   4*4*4 + 4= 
4*4*4 + 4 = 68
68   5*5*5 + 4= 
5*5*5 + 4 = 129
129   7 *7 *7 + 4= 
7 *7 *7 + 4 = 347
347   8 *8 *8 + 4= 
8 *8 *8 + 4 = 516
516.   6*6*6 + 4= 
6*6*6 + 4 = 220
220   ANSWER : 220
Answer: D
Each week, Harry is paid x dollars per hour for the first 30 hours and 1.5x dollars for each additional hour worked that week. Each week, Annie is paid x dollars per hour for the first 40 hours and 2x dollars for each additional hour worked that week. Last week Annie worked a total of 47 hours. If Harry and Annie were paid the same amount last week, how many hours did Harry work last week?  Answers.
A) 40
B) 42
C) 44
D) 46
E) 48

Solution: Annie earned 40x + 7(2x) = 54x   Let H be the number of hours that Harry worked.   Harry earned 30x + 1.5x(H-30) = 54x   (1.5x)(H) = 69x   H = 46 hours   The answer is 46.
Answer: D
In a garden man has to pick some flowers ans has to pass thru 3 doors. Each door has a guard. Man has to give half of the flowers he had to the guard and in return guard returns the man him one flower back. how many flower should man pick so that at the end he is left with same number of flowers he has picked?
Pick one
A) 1  B) 2  C) 3  D) 4  E) 5

Solution: suppose man pick up x flowers.   He gives half flowers he had to guard and guard in return gives 1 flower to man.   so,   after 1st door remained flowers towards guard=(x/2)+1   i.e(x+2)/2   after 2nd door remained flowers towards guard=[(x+2)/4]+1   after 3rd door remained flowers towards guard=[(x+6)/8]+1   i.e.(x+14)/8   at the end he is left with same number of flowers he has picked.   so(x+14)/8=x   8x-x=14   x=2   Hence he should pick up 2 flowers.   ANSWER:2
Answer: B
A boy wants to go abroad by boat and return by flight. He has a choice of 4 different boat to go and 5 flight to return. In how many ways, can the boy perform his journey?	Choose the most appropriate option:  A) 20  B) 25  C) 30  D) 40  E) 50

Solution: Ans.(A)   Sol. Number of choices to go by ship = 4 Number of choices to return by airline = 5 From, fundamental principle, total number of ways of performing the journey = 4 x 5 = 20 ways
Answer: A
Tane can make a handcrafted drum in 4 weeks. Zane can make a similar handcrafted drum in 6 weeks. If they both work together, how many weeks will it take for them to produce 15 handcrafted drums?  Options:
A) 30  B) 36  C) 70  D) 80  E) 150

Solution: Method I: the rates solution   “Tane can make a handcrafted drum in 4 weeks. Zane can make a similar handcrafted drum in 6 weeks.” Tane’s rate is (1 drum)/(4 weeks) = 1/4. Zane’s rate is (1 drum)/(6 weeks) = 1/6. The combined rate of Tane + Zane is   R = 1/4 + 1/6 = 3/12 + 2/12 = 5/12   That’s the combined rate. We need to make 15 drums — we have a rate and we have an amount, so use the “art” equation to solve for time:   T = A/R = 15/(5/12) = 15*(12/5) = (15/5)*12 = 3*12= 
3*12 = 36
36   BTW, notice in the penultimate step, the universal fraction strategy: cancelbeforeyou multiply (Tip #3:http://magoosh.com/gmat/2012/can-i-use- ... -the-gmat/. Tane and Zane need 36 weeks to make 15 drums.   Answer= 36.
Answer: B
The difference between the ages of two person is 10 years. Fifteen years ago, the elder one was twice as old as the younger one. The present age of the elder person is Select the correct option
A) 15	B) 25	C) 35	D) 45	E) 30

Solution: x>y;   x-y=10;   x-15=2*(y-15);   Finally, x=35.   ANSWER:35
Answer: C
How many 4 digit numbers are there, if it is known that the first digit is odd, the second is even, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?	Choose one
A) 20  B) 150  C) 225  D) 300  E) 380

Solution: 4 options for the first digit:1, 3, 5, 7, 9;   5 options for the second digit: 0, 2, 4, 6, 8;   4 options for the third digit: 2, 3, 5, 7;   4 options for the fourth digit: 0, 3, 6, 9.   Four digit # possible without the restriction (about the digit 2): 5*5*4*4=400   Numbers with two 2-s, 2X2X 5*1*1*4=20.   Thus there are 400-20= 
400-20 = 380
380 such numbers.   Answer: 380.
Answer: E
A cricket team played 200 games and won 80 percent of them. How many games did it win?
Select the correct option:
A) 80	B) 94	C) 160	D) 114	E) 124

Solution: 80% of 200 =x   0.80 x 200 = x   160=x   answer :160
Answer: C
Teams A, B, C,D and E were in a baseball tournament. Each team played each other team once. Team A beat team D. Teams A, B and C won the exact same number of games. How many games did team D lose?	Select the correct option.
A) 5 games  B) 4 games  C) 3 games  D) 2 games  E) 1 game

Solution: There are 4 teams and each team played each other team once. That tells us 2 facts:    Each team played 3 games    4 teams played a total of 6 games.   Since there are only 6 games, only 6 total wins are possible. Since A, B and C won the exact same number of games, they either all won 1 game or all won 2 games. Let’s examine both scenarios:   Scenario #1: A, B and C each won 1 game. In this scenario, D must have won all 3 games it played. That contradicts with the fact that A beat D. So, scenario 1 is not possible. So, it has to be scenario #2.   Scenario #2: A, B and C each 2 games. In this scenario, D won 0 games. That means D lost all 3 games it played.   correct answer 3 games
Answer: C
At Company X, senior sales representatives visit the home office once every 12 days, and junior sales representatives visit the home office once every 8 days. The number of visits that a junior sales representative makes in a 2-year period is approximately what percent greater than the number of visits that a senior representative makes in the same period?	Select the correct option.
A) 10%
B) 25%
C) 33%
D) 50%
E) 67%

Solution: Each 24-day period, senior representatives visit the home office 2 times while junior representatives visit 3 times, thus 50% more.   The answer is 50%.
Answer: D
Pipe A can fill a tank in 15 hours, pipe B in 10 hours and pipe C in 3 hours. If all the pipes are open, in how many hours will the tank be filled? Choose the correct choice from the following choices.
A) 2  B) 2.5  C) 3  D) 5  E) None

Solution: Sol.   Part filled by (A + B + C) in 1 hour = (1/15 + 1/10 + 1/3) = 1/2.   âˆ´ All the three pipes together will fill the tank in 2 hours.   2nswer 2
Answer: A
If x/y= 4 and y is not '0' what % of x is 2x-y Choose the correct choice:
A) 150% B) 175percent C) 200% D) 250% E) 350%

Solution: let z be the percentt   x*z/100=2x-y   zx=7y*100   z=700y/4y   z7 *100= 
7 *100 = 700
700y/4y   z=350/2= 
350/2 = 175
175%   ANSWER:175percent
Answer: B
A sum of money invested for a certain number of years at 8% p.a. simple interest grows to Rs.180. The same sum of money invested for the same number of years at 4% p.a. simple interest grows to Rs.120 only. For how many years was the sum invested? Choose the correct answer
A) 25years	B) 15years	C) 20years	D) 22years	E) None of these

Solution: Explanation:   From the information provided we know that,   Principal + 8% p.a. interest on principal for n years = 180 …….. (1)   Principal + 4% p.a. interest on principal for n years = 120 ……… (2)   Subtracting equation (2) from equation (1), we get   4% p.a. interest on principal for n years = Rs.60.   Now, we can substitute this value in equation (2),   i.e Principal + 60 = 120   = Principal = Rs.60.   We know that SI = , where p is the principal, n the number of years and r the rate percent of interest.   In equation (2), p = Rs.60, r = 4% p.a. and the simple interest = Rs.60.   Therefore, 60 =(60*n*4)/100   => n = 100/4= 
100/4 = 25
25 years.   25yearsnswer: 25years
Answer: A
WHEN A PERSON SELLS TWO ITEMS $1150 each, ONE AT A GAIN OF 20% AND OTHER AT A LOSS OF 20%.THEN THE SELLER INCURES A LOSS OR GAIN OF: Select the correct option:	A) 2% gain	B) no loss no gain	C) 4% loss	D) 1% loss	E) 5% gain

Solution: When two items are sold at the same price, one at a profit of a% and other at a loss of a% (which means the cost price of one item was lower than the selling price and the cost price of the other item was higher than the selling price), there will always be a loss of (a^2)/100 %. e.g. Here a = 20, so loss % = (20)^2/100 % = 4%.   Answer : 4% loss
Answer: C
Two trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. In what time will they be clear of each other from the moment they meet?	Answers:
A) 12  B) 67  C) 26  D) 20  E) 23

Solution: Relative speed = (42 + 30) * 5/18 = 4 * 5= 
4 * 5 = 20
20 mps.   Distance covered in passing each other= 120 + 280= 
120 + 280 = 400
400 m.   The time required= d/s = 400/20= 
400/20 = 20
20 sec.   Answer:20
Answer: D
E and F work at a hospital with 4 other workers. For an internal review, 2 of the 6 workers will be randomly chosen to be interviewed. What is the probability that E and F will both be chosen?  Choose the correct choice from the following choices.
A) A)1/3	B) B)1/4	C) C)1/15	D) D)3/8	E) E)2/3

Solution: Total number of people = 6   Probability of selecting E first and then F is 1/6∗1/5=1/30   Probability of selecting F first and then E is 1/6∗1/5=1/30   Therefore probability of selecting E and F for the review is 1/30+1/30=1/15   Answer is 1/15
Answer: C
If the sides of a triangle are 78 cm, 72 cm and 30 cm, what is its area? Choose the correct option:  A) 1030 cm^2  B) 1380 cm^2  C) 105 cm^2  D) 1000 cm^2  E) 1080 cm^2

Solution: The triangle with sides 78 cm, 72 cm and 30 cm is right angled, where the hypotenuse is 78 cm.   Area of the triangle   = 1/2 * 72 * 30= 
1/2 * 72 * 30 = 1_080
1080 cm^2   Answer: 1080 cm^2
Answer: E
The labeled price of a table is Rs. 8,500. The shopkeeper sold it by giving 5% discount on the labeled price and earned a profit of 15%. What approximately is the cost price of the table?	Pick one:	A) Rs. 5758	B) Rs. 6195	C) Rs. 6425	D) Rs. 7021	E) None of these

Solution: Explanation:   Labeled price = Rs. 8,500   By giving 5% discount on labeled price, the selling price is   = 95 / 100 * 8500 = Rs. 8075   By earning 15% profit on the selling of price Rs. 8075 the cost price is   = 100 / 115 * 8075 = Rs. 7021.74   Therefore, approximate cost is 7021.74   ANSWER Rs. 7021
Answer: D
Two trains travelling in the same direction at 30 and 22 kmph completely pass off another in 1 minute. If the length of the first train is 125 m, what is the length of the second train? Select.
A) 7.33 m	B) 8.33 m	C) 25.63 m	D) 28.33 m	E) 8.93 m

Solution: RS = 30 – 22 = 8 * 5/18 = 2.22 mps   T = 60 sec   D = 2.22 * 60 = 133.33 m   125   --------   8.33 m   ANSWER:8.33 m
Answer: B
Find the odd man out. 2, 5, 10, 17, 26, 38, 50, 65	Choices.
A) 50	B) 38	C) 26	D) 65	E) 75

Solution: Explanation :   The pattern is (1×1)+1 , (2×2)+1 , (3×3)+1 , (4×4)+1 , (5×5)+1 , (6×6)+1 , (7×7)+1 , (8×8)+1   Hence, in place of 38, the right number was (6×6)+1 = 37. Answer : Option 38
Answer: B
A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?	Choose the correct choice from the following answers
A) 45(4/11) % B) 45 % C) 45(5/11) % D) 44(5/11) % E) None of these

Solution: Explanation :   Total runs scored = 110   Total runs scored from boundaries and sixes = 3 x 4 + 8 x 6 = 60   Total runs scored by running between the wickets = 110 - 60= 
110 - 60 = 50
50   Required %= (50/110) × 100 = 500/11 = 45(5/11)%   Answer : Option 45(5/11) %
Answer: C
(x —1)(x + 3) > 0
(x +5)(x—4) < 0
Which of the following values of x satisfy both inequalities shown?
I. -6
II. -4
III. 5
IV. 2
Choose one
A) I and II only B) I and III only C) II and III only D) II and IV only E) I, II, III, and IV

Solution: (x - 1)(x + 3) > 0 --> x<-3 or x>1.   (x + 5)(x - 4) < 0 --> -5<x<4.   Among choices only -4 and 2 satisfy both inequalities.   Answer: II and IV only.
Answer: D
A salesperson receives a base salary of $1000 per month and a commission of 7.5% of the selling price of each item sold above the first 50 item. If this month she sold 530 items for $150 each, what will her monthly salary be?	Answers.
A) $1,500	B) $1,800	C) $6,400	D) $2,500	E) $2,800

Solution: Ans : 1000+ 7.5 % * 150 * (530-50) = $ 6400   ANSWER:$6,400
Answer: C
In a day hands of clock will be together Answers:
A) 12 times B) 24 times C) 20 times D) 22 times E) 25 times

Solution: 22 times {The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock) Therefore in a day , It coincides 22 times }   ANSWER:22 times
Answer: D
Three people of different age groups are to be seated on a bench. How many different sitting arrangements are possible if oldest must always sit next to youngest ?
Answers: A) 2 B) 4 C) 6 D) 8 E) 10

Solution: Consider O and Y one. But, in this arrangement they can sit in two different combinations.   O first and Y second or Y first and O second.   Total we have (O and Y) and third person= 2 arrangements to do.   Total arrangements= 2*2= 
2*2 = 4
4   4 is the answer
Answer: B
During the last four years, a large truck company purchased diesel at rates of $1.2, $1.4, $1.6 and $2.1 per gallon. What was the mean cost of diesel over the four-year period if the company spends the same amount of dollars on diesel each year?  Choose the correct choice from the following answers
A) $2.5 B) $2.55 C) $1.51 D) $3.25 E) $3.0

Solution: finding the Harmonic mean here.   HM = 4/(1/12 + 1/14 + 1/16 + 1/21) = 15.1   $1.51
Answer: C
A man is 26 years older than his son. In two years, his age will be twice the age of his son. The present age of the son is?  Answers:
A) 11
B) 25
C) 24
D) 22
E) 91

Solution: Let the son's present age be x years.   Then, man's present age = (x + 26) years.   (x + 26) + 2 = 2(x + 2) x + 28 = 2x + 4 => x = 24.   Answer: 24
Answer: C
Jane makes toy bears. When she works with an assistant, she makes 44 percent more bears per week and works 10 percent fewer hours each week. Having an assistant increases Jane’s output of toy bears per hour by what percent?  Choose the correct choice from the following: A) 20% B) 60% C) 100% D) 180% E) 200%

Solution: Let's assume just Jane 40 bears per 40/hrs a week, so that is 1 bear/hr. With an assistant she makes 57.6 bears per 36 hours a week or 1.6 bears/hr ([40 bears * 1.44] / [40 hrs * .90]).   [(1.6 - 1)/1] * 100% = 60%   ANSWER:60% (1.6 - 1)/1  * 100= 
 (1.6 - 1)/1  * 100 = 60
60%   ANSWER:60%
Answer: B
John must arrange 4 different physics books and 3 different mathematics books on one shelf. If the first book on the leftmost side is a mathematics book and no physics book is next to another physics book. How many different arrangements exist? Choose the correct choice from the following:  A) 6  B) 9  C) 36  D) 144  E) 720

Solution: First Place, is reserved for Math book. There are 3 Math books which can be arranged in3 ways   Restriction comes into playhysics book cannot be together. Hence, the maththe physics books take alternate places   Second Place, Physics Book - in 3 ways   Third Place, Math Book - in 2 ways (only 2 books left)   Fourth Place, Physics - in 2 ways (only 2 Physics book left)   Rest two places can be arranged in one way only.   Total no of ways = 4*3*3*2*2*1*1= 
4*3*3*2*2*1*1 = 144
144 ways   144
Answer: D
A picture 30" × 20" has a frame 2½" wide. The area of the picture is approximately how many times the area of the frame?
Pick:	A) 4	B) 2 1⁄2	C) 2	D) 5	E) None of these

Solution: Length of frame = 30 + 2.5 × 2= 35 inch   Breadth of frame = 20 + 2.5 × 2 = 25 inch   Now, area of picture = 30 × 20 = 600 sq. inch   Area of frame = (35 × 2.5) + (25 × 2.5)   ∴ Area of frame = 875 – 600 = 275 sq. inch   4nswer 4
Answer: A
An athlete runs 200 meters race in 56sec. His speed is? Choices
A) 20km/hr	B) 15km/hr	C) 12.8km/hr	D) 25km/hr	E) 40km/hr

Solution: speed = 200/56 = 25/7 m/sec   = 25/7 * 18/5 km/hr = 12.8km/hr   Answer is 12.8km/hr
Answer: C
A factory producing tennis balls stores them in either big boxes, 25 balls per box, or small boxes, with 17 balls per box. If 94 freshly manufactured balls are to be stored, what is the least number of balls that can be left unboxed?
Choose the most appropriate option:
A) 0 B) 1 C) 2 D) 3 E) 4

Solution: There is no way to store 94 balls without leftovers: 94−0∗25=94- 94−2∗25=44 ,94−3∗25=19 are not divisible by 17   .93 balls can be stored successfully: 93−1∗25=68 is divisible by 17. Thus, 93=1∗25+4∗17 and we need 1 big box and 4 small boxes.   Answer: 1
Answer: B
Each month a retailer sells 100 identical items. On each item he makes a profit of $50 that constitutes 10% of the item's price to the retailer. If the retailer contemplates giving a 5% discount on the items he sells, what is the least number of items he will have to sell each month to justify the policy of the discount?  Select the correct option: A) 191 B) 213 C) 221 D) 223 E) 226

Solution: For this question, we'll need the following formula:   Sell Price = Cost + Profit   We're told that the profit on 1 item is $20 and that this represents 10% of the cost:   Sell Price = Cost + $50   Sell Price = $500 + $50   Thus, the Sell Price is $550 for each item. Selling all 100 items gives the retailer...   100($50) = $2,000 of profit   If the retailer offers a 5% discount on the sell price, then the equation changes...   5%(550) = $27.5 discount   $522.5 = $500 + $22.5   Now, the retailer makes a profit of just $22.5 per item sold.   To earn $2,000 in profit, the retailer must sell....   $22.5(X) = $2,000   X = 2,000/22.5   X = 222.222222 items   You'll notice that this is not among the answer choices.... 221 and 223 are.   Selling 221 items would get us 9(221) = $1989 which is NOT enough money. To get back to AT LEAST $2,000, we need to sell 223 items.   Final Answer:   223
Answer: D
Local kennel has cats and dogs in the ratio of 3:4. If there are 8 fewer cats than dogs, how many dogs are in the kennel? Choose the correct choice from the following choices:
A) 28 B) 26 C) 24 D) 32 E) 30

Solution: Lets work with the data given to us. We know that there ratio of cats to dogs is 3:4 or   cats 3   dogs 4   we can write number of cats as 3x and number of dogs as 4x and we know that 4x-3x= 8 (therefore x = 8)   Then # of dogs = 4x8= 32   Answer is 32
Answer: D
The sum of digits of a two digit number is 8,The difference between the digits is 6. Find the number  Choices
A) 85	B) 94	C) 83	D) 71	E) None

Solution: Description :   => x+y=8, x-y=6   Adding these 2x =14   => x=7, y=1.   Thus the number is 71   Answer 71
Answer: D
The remainder obtained when a prime number greater than 9 is divided by 9 is Choose the correct choice from the following choices:
A) a)2 or 4 B) b) 1 or 5 C) c) 3 or 5 D) d) 4 or 5 E) e) 5

Solution: The easiest way to answer this question is probably to pick numbers. Use the next two prime numbers greater than 9, i.e. 13 and 11:   13/9 = 1 + remainder of 4   11 /9 = 1 + remainder of 2   This is already sufficient to see that answer choice a) is correct.   a)2 or 4
Answer: A
If 0 < q < 1, what is the median of the values q, q^-1, q^2, q^1/2 and q^3? Options:
A) q B) q^-1 C) q^2 D) q^1/2 E) q^3

Solution: A. To make the problem less confusing, I picked a number between 0 and 1 such as q = 1/4 and calculate:   q = 1/4   q^-1 = 4   q^2 = 1/16   q^(1/2) = 1/2   q^3 = 1/8.   From this, rearrange the number and the median is 1/4, which is q. qnswer q
Answer: A
If the difference between the length and breadth of a rectangle is 23 m and its perimeter is 266 m, what is its area?  Choose the correct choice from the following.
A) 4290
B) 2535
C) 2530
D) 2515
E) 2520

Solution: length = breadth +23. Therefore,   4× breadth +2×23=266 m   ⇒ breadth =55 m   length =55+23= 
55+23 = 78
78 m   4290rea=78×55=4290 m2   4290nswer is 4290.
Answer: A
What are the last two digits of (301*401*503*604*646*547*448*349)^2	Select
A) 96
B) 76
C) 56
D) 44
E) 16

Solution: ((301*401*503*604*646)*(547*448*349))^2   If you observe above digits, last digit are: 1,1,3,4,6,7,8,9; 1&5 are missing; so I have rearranged them so that multiplication will be easy for me as initial 4 digits have last two digits as 01,01,03,04,46 and final three as 47*48*49.   Solving for only last two digits and multiplying them we get:( (03*04*46)(56*49))^2 = (52*44)^2=88^2=44   Hence answer is 44
Answer: D
The simple interest on Rs.14000 at a certain rate of interest in five years is Rs.7200. Find the compound interest on the same amount for two years at the same rate of interest.	Choose the correct choice
A) 3052.82
B) 3052.51
C) 3028.31
D) 3028.11
E) 3028.81

Solution: R = 100 I / PT   => R = (100 * 7200)/ (14000 * 5) = 10.28%   CI = P{ [1 + R /100]n - 1}   = 14000 { [ 1 + 10.28 / 100]^2 - 1} = Rs.3028.11   Answer:3028.11
Answer: D
In the quadratic equation ax2 + 8x + 16 = 0, if the sum of two roots is -8, what is the product of the two roots?
Choose the correct option.
A) 16 B) 14 C) 20 D) 15 E) 13

Solution: Explanation:   The sum of the roots of the quadratic equation ax2 + bx + c = 0 are (-b/a) and the product of the roots are (c/a).   Thus, in the equation ax2 +8 x + 40 = 0, where a = a, b = 8 and c = 16.   we get, sum of the roots = -8 / a= -8   a= 1   Product of the roots = 16 / 1= 
16 / 1 = 16
16   16NSWER: 16
Answer: A
A group of four women and four men have tickets for seven adjacent seats in one row of a theatre. If the three men will not sit in three adjacent seats, how many possible different seating arrangements are there for these 7 theatre-goers?	Choose the correct choice from the following answers:
A) 8! – 5!4!
B) 7! – 4!3!
C) 7! – 5!3!
D) 7 × 2!3!2!
E) 2!3!2!

Solution: 8 people can sit in 8! different ways. But because 4 men cannot sit together, we take them as a unit.   This unit of men, among themselves can sit in 4! ways.   Hence, 8! - 4!.   This unit of men along with 4 women can sit in 5! different ways which also needs to be eliminated.   Hence 8! - 5!4!   8! – 5!4!
Answer: A