MU-NLPC/Calc-aqua_rat huggingface dataset:
Car W and Car Y traveled the same 80-mile route. If Car W took 2 hours and Car Y traveled at an average speed that was 50 percent faster than the average speed of Car W, how many hours did it take Car Y to travel the route?  Choose the correct choice from the following
A) 2/3	B) 1	C) 4/3	D) 8/5	E) 3

Solution: The speed of car W is (distance)/(time) = 80/2= 
80/2 = 40
40 miles per hour.   The speed of car Y= 3/2*40= 
3/2*40 = 60
60 miles per hour --> (time)= (distance)/(speed) = 80/60 = 4/3 hours.   Answer: C.   Or: to cover the same distance at 3/2 as fast rate 2/3 as much time is needed --> (time)*2/3 = 2*2/3 = 4/3 hours.   Answer: 4/3.
Answer: C
Three candidates contested an election and received 1136, 7636 and 11628 votes respectively. What percentage of the total votes did the winning candidate get? Choose the correct option: A) 56% B) 57% C) 54% D) 53% E) 52%

Solution: Total number of votes polled = (1136 + 7636 + 11628)= 
(1136 + 7636 + 11628) = 20_400
20400.   Required percentage= (11628/20400 * 100)= 
(11628/20400 * 100) = 57
57%   Answer 57%)57%.
Answer: B
A 11% stock yields 8%.The market value of the stock is :	Choose the correct choice:  A) Rs. 72  B) Rs. 137.50  C) Rs. 112.50  D) Rs. 116.50  E) None of these

Solution: solution   To obtain Rs. 8, investment = Rs. 100.   To obtain Rs. 11, investment = Rs.(100/8X11)= Rs. 137.50   ∴ Market value of Rs. 100 stock = Rs.137.50   Answer Rs. 137.50
Answer: B
A train running at the speed of 60 km/hr crosses a pole in 6 seconds. Find the length of the train.  Choose one:
A) 100  B) 882  C) 772  D) 252  E) 121

Solution: Speed = 60*(5/18) m/sec = 50/3 m/sec   Length of Train (Distance) = Speed * Time     (50/3) * 6= 
  (50/3) * 6 = 100
100 meter.100nswer: 100
Answer: A
The lunch menu at a certain restaurant contains 4 different entrees and 5 different side dishes. If a meal consists of 1 entree and 2 different side dishes, how many different meal combinations R could be chosen from this menu?  Choose the most appropriate option:
A) 10
B) 20
C) 40
D) 80
E) 100

Solution: R=4401 * 5402   = 4 * (5*4*3!)/(3!2!)   = 4 * 10= 
4 * 10 = 40
40   Answer - 40
Answer: C
A pet store sells dog collars arranged on display such that each of the four different colors of dog collars, green, red, blue, and purple are consecutive in a repeating pattern. If five consecutively displayed collars are purchased, what is the probability that two of the collars are purple?
Options:
A) 1/4	B) 1/5	C) 1/16	D) 1/32	E) 4/5

Solution: After green, red, blue, and purple, the next dog collar can be of any given four colours, therefore we have total four possibilites, out of which we need purple i.e. 1 therefore prob is 1/4.
Answer: A
If n = 2.0823 and n* is the decimal obtained by rounding n to the nearest hundredth, what is the value of n* – n?	Choose one: A) -0.0053 B) -0.0023 C) 0.0007 D) 0.0047 E) 0.0153

Solution: n*=2.08   n*-n=2.08-2.0823   -0.0023   Answer: -0.0023
Answer: B
The ratio of 2 numbers is 4:6 and their H.C.F. is 8. Their L.C.M. is ? Choose the correct choice:
A) 20  B) 24  C) 52  D) 96  E) 60

Solution: Let the numbers be 4x and 6x   Their H.C.F. = 8   so the numbers are 4*8 , 6*8 = 32,48   L.C.M. = 96   Answer is 96
Answer: D
A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?
Options:
A) 3hr
B) 3hr 45min
C) 4hr 15min
D) 2hr 1min
E) 1hr 45min

Solution: Time taken by one tap to fill the half tank = 3 hr   part filled by the 4 taps in 1hour = 4*1/6 = 2/3   Remaining part = 1-1/2 = 1/2   2/3:1/2::1:x   x = 3/4hr = 45min   Total time taken = 3hr 45min   Answer is 3hr 45min
Answer: B
A, B and C rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing and the rent of the pasture is Rs. 105, then how much amount should C pay as his share of rent? Pick: A) 35 B) 27 C) 25 D) 15 E) 55

Solution: A : 27 : C   =10×7:12×5:15×3   =2×7:12×1:3×3   =14:12:9   Amount that C should pay   =105×9/35=3×9=27   Answer is 27.
Answer: B
In a certain apartment building, there are one-bedroom and two-bedroom apartments. The rental prices of the apartment depend on a number of factors, but on average, two-bedroom apartments have higher rental prices than do one-bedroom apartments. Let R be the average rental price for all apartments in the building. If R is $5,600 higher than the average rental price for all one-bedroom apartments, and if the average rental price for all two-bedroom apartments is $10,400 higher that R, then what percentage of apartments in the building are two-bedroom apartments?
Choose the correct choice from the following choices:
A) 26%	B) 35%	C) 39%	D) 42%	E) 52%

Solution: Ratio of 2 Bedroom Apartment: 1 Bedroom Apartment = 5400 : 104000 -----> 7 : 13   Let total number of Apartments be X   No. of 2 35%edroom Apartment = (7 / 20) * X   percentage of apartments in the building are two-bedroom apartments ---->   (7/20) * 100 ---> 35%   option 35% is correct
Answer: B
At 1:00 pm, a truck left city P and headed toward City Q at a constant speed of 35 km/h. One hour later, a car left City Q and headed toward City P along the same road at a constant speed of 40 km/h. If the distance between City P and City Q is 335 km, at what time will the truck and the car meet each other?
Options:
A) 5:00
B) 5:30
C) 6:00
D) 6:30
E) 7:00

Solution: At 2:00 pm, the truck and the car are 300 km apart.   The truck and the car complete a distance of 75 km each hour.   The time it takes to meet is   300/75= 
  300/75 = 4
4 hours.   They will meet at 6:00 pm.   The answer is 6:00.
Answer: C
A rectangular lawn of dimensions 80 m * 60 m has two roads each 10 m wide running in the middle of the lawn, one parallel to the length and the other parallel to the breadth. What is the cost of traveling the two roads at Rs.5 per sq m?  Select the correct option:  A) 6500  B) 2779  C) 2779  D) 3900  E) 2781

Solution: 6500rea = (l + b â€“ d) d   (80 + 60 â€“ 10)10 => 1300 m2   1300 * 5 = Rs.6500   6500nswer:6500
Answer: A
What was the day of the week on 28th May, 2006? Pick one.
A) Monday  B) Friday  C) Saturday  D) Tuesday  E) Sunday

Solution: 28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006)   Odd days in 1600 years = 0   Odd days in 400 years = 0   5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2) 6 odd days   (31[jan] + 28 [Feb]+ 31[Mar] + 30[April] + 28[May] ) = 148 days   148 days = (21 weeks + 1 day) 1 odd day.   Total number of odd days = (0 + 0 + 6 + 1)= 
(0 + 0 + 6 + 1) = 7
7 0 odd day.   Given day is Sunday   Answer is Sunday.
Answer: E
If x and y are different integers, both divisible by 7, then which is not necessarily true?
Choose the most appropriate option: A) x^2 + y^2 is divisible by 7 B) x - y is divisible by 7 C) xy is divisible by 7 D) x + y is divisible by 7 E) x + y is divisible by 14

Solution: Let x = 7k and let y = 7j.   Then x+y = 7(k+j) and k+j could be even or odd.   Thus 7(k+j) is not necessarily divisible by 14.   The answer is x + y is divisible by 14.
Answer: E
The ratio of investments of two partners P and Q is 7:5 and the ratio of their profits is 7:13. If P invested the money for 5 months, find for how much time did Q invest the money? Choose the most appropriate option:
A) 19 B) 17 C) 13 D) 10 E) 12

Solution: 7*5: 5*x = 7:13   x = 13   Answer: 13
Answer: C
Krishan and Nandan jointly started a business. Krishan invested three times as Nandan did and invested his money for double time as compared to Nandan. Nandan earned Rs. 4000. If the gain is proportional to the money invested and the time for which the money is invested then the total gain was?  Choose the correct choice from the following answers:
A) 28,009  B) 28,007  C) 28,006  D) 28,000  E) 28,004

Solution: 3:1   2:1   ------   6:1   1 ----- 4000   7 ----- ? => Rs.28,000   Answer:28,000
Answer: D
What is the area inscribed by the lines y =2, x = 2, y = 10-x on an xy-coordinate plane?	Choose the correct choice from the following answers:
A) a) 8
B) b) 10
C) c) 12
D) d) 14
E) e) 18

Solution: First, let's graph the lines y = 2 and x = 2   At this point, we need to find the points where the line y = 10-x INTERSECTS the other two lines.   For the vertical line, we know that x = 2, so we'll PLUG x = 2 into the equation y = 10-x to get y = 10-2= 
10-2 = 8
8   Perfect, when x= 2, y = 8, so one point of intersection is (2,8)   For the horizontal line, we know that y = 2, so we'll PLUG y = 2 into the equation y = 10-x to get 2 = 10-x. Solve to get: x = 8   So, when y = 2, x = 8, so one point of intersection is (8,2)   Now add these points to our graph and sketch the line y = 10-x   At this point, we can see that we have the following triangle.   The base has length 6 and the height is 6   Area = (1/2)(base)(height)   = (1/2)(6)(6)   = 18   Answer:   e) 18
Answer: E
What is the ratio whose term differ by 25 and the measure of which is 2/7?  Choices.
A) A)32:23 B) B)16:56 C) C)71:85 D) D)32:39 E) E)10:35

Solution: Let the ratio be x:(x+25)   Then, x/(x+25) = 2/7   x=10   required ratio = 10:35   Answer is 10:35
Answer: E
A set consists of 18 numbers, all are even or multiple of 5. If 6 numbers are even and 12 numbers are multiple of 5, how many numbers is multiple of 10?  Choose the correct choice from the following options: A) 0 B) 1 C) 2 D) 3 E) 5

Solution: {Total} = {Even} + {Multiple of 5} - {Both} + {Nether}.   Since{Neither}=0(allare even or multiple of 5) then:   18 = 6 + 12 - {Both}   + 0= 
  + 0 = 0
0(so 1 number is both even AND multiple of 5, so it must be a multiple of 10).   0nswer: 0.
Answer: A
How many seconds will a train 100 meters long take to cross a bridge 150 meters long if the speed of the train is 72 kmph?	Pick.
A) 22 sec B) 77 sec C) 25 sec D) 18 sec E) 12.5 sec

Solution: D = 100 + 150= 
100 + 150 = 250
250   S= 72 * 5/18= 
72 * 5/18 = 20
20 mps   T= 250/20= 
250/20 = 25/2 = around 12.5
12.5 sec   Answer:12.5 sec
Answer: E
If the population of a certain country increases at the rate of one person every 30 seconds, by how many persons does the population increase in 1 hour? Options
A) 100  B) 120  C) 150  D) 180  E) 160

Solution: Answer = 2 * 60= 
2 * 60 = 120
120   Answer is 120
Answer: B
A reduction of 35% in the price of oil enables a house wife to obtain 5kgs more for Rs.800, what is the reduced price for kg?  Choose the correct option
A) 42
B) 49
C) 56
D) 63
E) 70

Solution: 800*(35/100)= 
800*(35/100) = 280
280 ---- 5   ? ---- 1=> Rs.56   Answer: 56
Answer: C
A tank is filled in 5 hours by three pipes A,B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?	Choose the correct choice from the following answers:
A) 35 hours
B) 25 hours
C) 20 Hours
D) Cannot be determined
E) None of these

Solution: Let pipe A alone takes to fill the tank = x hours.   Then, pipes B take twice fast as A = x/2   And pipes C take twice fast as B = x/4   Therefore, Tank filled in 5 hours by A,B,C pipes => 1/x + 2/x + 4/x = 1/5   ==> 7/x = 1/5   ==> x = 5*7= 
5*7 = 35
35 hours   x= 35 hrs.   35 hoursNSWER:35 hours
Answer: A
A family consists of two grandparents, two parents and three grandchildren. The average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of family?	Choose the correct choice
A) 31 5/7years
B) 31
C) 30
D) 33
E) 35

Solution: Required average   = (67 x 2 + 35 x 2 + 6 x 3 )/(2 + 2 + 3)   = (134 + 70 + 18)/7   = 222/7   = 31 5/7years.   31 5/7yearsnswer is 31 5/7years
Answer: A
A car covers a certain distance at aspeed of 55 kmph in 6 hours. To cover the same distance in 2 hrs, it must travel at a speed of?  Choose the correct choice
A) 567 km/hr  B) 678 km/hr  C) 165 km/hr  D) 789 km/hr  E) 720 km/hr

Solution: Distance = (55 x 6) = 330 km.   Speed = Distance/Time   Speed = 330/2= 
330/2 = 165
165 kmph   Answer : 165 km/hr
Answer: C
If m and n are positive integers of J such that m is a factor of n, how many positive multiples of m are less than or equal to 2n ?  Choose the correct option:	A) 2m/n + 1	B) 2n/m + 1	C) 2n/(m+1)	D) 2m/n	E) 2n/m

Solution: Lets say N=10, M=5   2N=20. so the answer should be 4 (20/5)   lets try to plug in the answers:   A-not an integer   B-not an integer   C-not an integer   D-1 (not the answer)   E-4 - the answer. (the only one).   I would choose E.   Method 2   N=M*A (A is an integer)   So - A=N/M   therefore in 2N A will be 2N/M   Again - Answer is 2n/m.
Answer: E
A 300 m long train crosses a platform in 45 sec while it crosses a signal pole in 18 sec. What is the length of the platform?
Select the correct option:
A) 287
B) 350
C) 828
D) 450
E) 122

Solution: Speed = 300/18 = 50/3 m/sec.   Let the length of the platform be x meters.   Then, (x + 300)/45 = 50/3   3x + 900 = 2250 => x = 450 m.   Answer: 450
Answer: D
How many different subset of the set {A, B, C, D, E. F, G) do not contain G.  Select:
A) 79  B) 78  C) 75  D) 80  E) 74

Solution: This is a Combination Solution   Since we have 6 different letters other than the letter G we can take any letter from the set of 6 to make a subset.   6C1 +6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 78   Plus the one set that is null that is the Set having no element in it which is 6C0 =1   = 78 + 1 = 16   79 = 79
Answer: A
The value of a machine depreciates at 20% per annum. If its present value is Rs. 1,50,000, at what price should it be sold after two years such that a profit of Rs. 24,000 is made? Choose the correct answer.
A) Rs. 120,002	B) Rs. 1,20,000	C) Rs. 1,20,00	D) Rs. 1,20,028	E) Rs. 1,20,011

Solution: Explanation:   The value of the machine after two years = 0.8 * 0.8 * 1,50,000 = Rs. 96,000   SP such that a profit of Rs. 24,000 is made = 96,000 + 24,000 = Rs. 1,20,000   Answer: Rs. 1,20,000
Answer: B
A box contains 80 balls, numbered from 1 to 80. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?  Choose the correct choice from the following choices.
A) 1/4
B) 3/8
C) 1/2
D) 5/8
E) 3/4

Solution: I don't think order matters in this case, because 2 + 2 + 1 = 2 + 1 + 2   My answer is: 1/4   1/4
Answer: A
Two pipes A and B can fill a tank in 30 and 60 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?  Choices:
A) 10
B) 20
C) 30
D) 24
E) 25

Solution: Part filled by A in 1 min. = 1/30   Part filled by B in 1 min. = 1/60   part filled by (A + B) in 1 min. = 1/30 + 1/60 = 1/20.   Both the pipes can fill the tank in 20 minutes.   Answer: 20
Answer: B
If x is a positive odd integer and y is a negative even integer, which of the following must be true?  Choose the correct choice:	A) x^3 + y is a positive odd integer	B) x^2 + y^2 is a negative odd integer	C) x^0 + y^11 is a negative odd integer	D) x + y is a positive odd integer	E) x + y is a negative odd integer

Solution: Anything power 0 is always 1 and given that Y is even and negative, then   1 + (-2) will be negative with odd power values for Y.   Answer: x^0 + y^11 is a negative odd integer
Answer: C
In what time will a train 100 metres long cross an electric pole, if its speed be 144 km/hr? Select:
A) 2.5 sec	B) 4.25 Sec	C) 5 sec	D) 12.5 sec	E) None

Solution: Solution   Speed	= (144 x 5 /18) m/sec   = 40 m / sec   Time taken	= (100 /40)= 
(100 /40) = 5/2 = around 2.5
2.5 sec.   2.5 secnswer 2.5 sec
Answer: A
A delegation of 5 members has to be formed from 3 ladies and 5 gentlemen. In how many ways the delegation can be formed, if 2 particular ladies are always included in the delegation?	Choose the correct choice from the following
A) 20	B) 27	C) 24	D) 25	E) 12

Solution: There are three ladies and five gentlemen and a committee of 5 members to be formed.   Number of ways such that two ladies are always included in the committee   = 6C3 = (6 * 5 * 4)/6= 
(6 * 5 * 4)/6 = 20
20.   20nswer:20
Answer: A
Five people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by: Choose the correct choice from the following:	A) 1/4	B) 2/7	C) 3/4	D) 4/7	E) None of them

Solution: Original share of 1 person = 1/5   New share of 1 person = 1/4   Increase = (1/4- 1/5	= 1/20)   Therefore, Required fraction = (1/20)/(1/5)	= (1/20)x (5/1)	= 1/4   1/4nswer is 1/4.
Answer: A
What will come in place of the x in the following Number series? 5,28,57,88,125,166
Choose the correct option
A) 209 B) 159 C) 166 D) 170 E) 180

Solution: 28=23+5   57=29+28   88=31+57   125=37+88   166=41+125   209=43+166= 
43+166 = 209
209   209
Answer: A
If x is the sum of p and q, then which one of the following must equal p – q? Select the correct option
A) x+2q
B) x-2q
C) x+q
D) x-q
E) xq

Solution: Given x= p+q if you subtracts both side with -2q   x-2q = p-q   x+2qnswer : x+2q
Answer: A
Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is: Pick:	A) 3 : 1	B) 3 : 2	C) 3 : 8	D) 3 : 25	E) 3 : 4

Solution: Let the speeds of the two trains be x m/sec and y m/sec respectively. Then, length of the first train = 27 x meters, and length of the second train = 17 y meters. (27 x + 17 y) / (x + y) = 23 ==> 27 x + 17 y = 23 x + 23 y ==> 4 x = 6 y ==> x/y = 3/2.   Answer: Option 3 : 2
Answer: B
A certain roller coaster has 5 cars, and a passenger is equally likely to ride in any 1 of the 5 cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller coaster 5 times, what is the probability that the passenger will ride in each of the 5 cars?  Choose the correct choice.
A) 0
B) 24/625
C) 2/9
D) 1/3
E) 1

Solution: If he is to ride 5 times and since he can choose any of the 5 cars each time, total number of ways is   = 5 * 5 * 5 *5 * 5= 
5 * 5 * 5 *5 * 5 = 3_125
3125   Now the number of ways if he is to choose a different car each time is= 5 * 4 * 3 * 2 * 1= 
5 * 4 * 3 * 2 * 1 = 120
120   So the probability is= 120/3125 = 24/625   Answer : 24/625
Answer: B
The cost of an article is decreased by 20%. If the original cost is $40, find the decrease cost.	Select the correct option: A) 33 B) 32 C) 68 D) 36 E) 38

Solution: Original cost = $40   Decrease in it = 20% of $40   = 20/100 Ã— 40   = 800/100= 
800/100 = 8
$8   Therefore, decrease cost= $40 - $8 = $32   Answer:3240 -  8= 
40 -  8 = 32
$32   Answer:32
Answer: B
The average age of 15 students of a class is 15 years. Out of these, the average age of 5 students is 14 years and that of the other 9 students is 16 years, The age of the 15th student is	Choices:
A) 11 years B) 12 years C) 13 years D) 18 years E) 10 years

Solution: Age of the 15th student = [15 * 15 - (14 * 5 + 16 * 9)] = (225-214)= 
(225-214) = 11
11 years.   11 yearsnswer: 11 years
Answer: A
If the selling price of 6 toys is same as the cost price of 12 toys . Find the gain or loss percentage?
Choose the most appropriate option: A) 65% B) 75% C) 85% D) 68% E) 100%

Solution: Let the C.P of each toys be Re 1. Then,   S.P of 6 toys = C.P of 12 toys= Rs. 12   Now, C.P of 6 toys= Rs.6, S.p of 6 toys = Rs 12   Gain = Rs ( 12 -6 )= 
( 12 -6 ) = 6
Rs 6.   Gain%= ( 6/6× 100 ) % = 100%   Answer : 100%
Answer: E
If $0.30 is the commission for sales of $1,000, what percent of the sales amount is the commission? Pick one
A) 3% B) 0.3% C) 0.03% D) 0.003% E) 0.0003%

Solution: % of sales amount of commission   = (0.03%ommission/Total Value) *100   = (0.3/1000) * 100= 
(0.3/1000) * 100 = 0.03
0.03%   The answer is 0.03%.
Answer: C
A trader bought a car at 50% discount on its original price. He sold it at a 80% increase on the price he bought it. What percent of profit did he make on the original price?
Choose the most appropriate option:
A) 17%  B) 62%  C) 12%  D) 19%  E) 71%

Solution: Original price = 100   CP = 95   S = 95*(180/100) = 112   100 - 171= 71%   Answer: 71%
Answer: E
A train running at the speed of 90 km/hr crosses a pole in 9 sec. What is the length of the train?
Choose the correct choice from the following:
A) 225 m	B) 150 m	C) 187 m	D) 167 m	E) 197 m

Solution: Speed = 90 * 5/18= 
90 * 5/18 = 25
25 m/sec   Length of the train= speed * time = 25 * 9= 
25 * 9 = 225
225 m   225 mnswer: 225 m
Answer: A
The difference between the simple interest received from two different sources on Rs. 1500 for 3 years is Rs. 13.50. The difference between their rates of interest is?
Choose the correct option:  A) 0.8%  B) 8.3%  C) 0.3%  D) 2.3%  E) 0.4%

Solution: (1500 * R1 * 3)/100 - (1500 * R2 * 3)/100 = 13.50 4500(R1 - R2) = 1350   R1 - R2 = 0.3%   Answer:0.3%
Answer: C
GOOD is coded as 164 then BAD coded as 21.if ugly coded as 260 then HUMP?	Choose one.
A) 200 B) 210 C) 220 D) 232 E) 240

Solution: GOOD=(7+15+15+4)*4= 
(7+15+15+4)*4 = 164
164   BAD= (2+1+4)*3= 
(2+1+4)*3 = 21
21   UGLY= (21+7+12+25)*4= 
(21+7+12+25)*4 = 260
260   coding follows   (sum of position of alphabets)*(no. of letters in the given word)   so,   HUMP=(8+21+13+16)*4= 
(8+21+13+16)*4 = 232
232   ANSWER:232
Answer: D
A certain telescope increases the visual range at a particular location from 100 kilometers to 150 kilometers. By what percent is the visual range increased by using the telescope ?  Choose the correct choice:  A) 30%  B) 33 1/2%  C) 40%  D) 60%  E) 50%

Solution: Original visual range = 100 km   New visual range = 150 km   Percent increase in the visual range by using the telescope   = (150-100)/100 * 100%   = 1/2 * 100= 
1/2 * 100 = 50
50 %   Answer 50%
Answer: E
If k is a positive integer, which of the following must be divisible by 21?	Choose the most appropriate option:	A) (k – 4)(k)(k + 3)(k + 7)	B) (k – 4)(k – 2)(k + 3)(k + 5)	C) (k – 2)(k + 3)(k + 5)(k + 6)	D) (k + 1)(k + 3)(k + 5)(k + 7)	E) (k – 3)(k + 1)(k + 4)(k + 6)

Solution: 24=8*3.   Note that the product of two consecutive even integers is always divisible by 8 (since one of them is divisible by 4 and another by 2). Only option B offers two consecutive even numbers for any integer value of k: k-4 and k-2, if k=even or k+3 and k+5 if k=odd.   Also from the following 3 consecutive integers:(k-4), (k-3),(k-2)one must be divisible by 3, if it's not k-4 or k-2 then it must be k-3 (if it's k-4 or k-2 option B is divisible by 3 right away). But if it's k-3 then (k-3)+6=k+3must also be divisible by 3.   So, option B: (k – 4)(k – 2)(k + 3)(k + 5) is divisible by 8 and 3 in any case.   (k – 4)(k)(k + 3)(k + 7)nswer: (k – 4)(k)(k + 3)(k + 7).
Answer: A
The sum of first three terms of the Geometric sequence is 13/12 and their product is -1. Find the common ratio  Choices:
A) 3/4  B) -3/4  C) 4/3  D) 1/2  E) 1/8

Solution: We make the first three terms is of a/r, a, ar   Whose sum=a(1/r+1+r)=13/12   Their product = a3=-1   Hence, (3r+4)(4r+3)=0   r=-3/4 or -4/3   ANSWER -3/4 -3/4
Answer: B
x percent of y percent of z is decreased by y percent. What is the result?
Choose the correct option
A) (100xyz - xy^2z) / 1,000,000
B) (xz - y) / 100
C) (xz - y) / 10,000
D) (xyz - 2y) / 100
E) (xyz - 2y) / 10,000

Solution: X = 10   Y = 50   Z = 100   First, 50% of 100 = 50   Next, 10% of 50 = 5   Finally, 5 decreased by 50% of itself is 2.5   So we're looking for an answer that equals 2.5 when X=10, Y=50 and Z=100.   Answer A. (100 xyz - xy^2z) / 1,000,000 = (5,000,000 - 2,500,000)/1,000,000 = 2,500,000/1,000,000 = 2.5 This is a MATCH   Answer B. (xz - y) / 100 = (1,000 - 50)/100 = 950/100= 
950/100 = 19/2 = around 9.5
9.5 NOT a match   Answer C. (xz - y) / 10,000= 950/10,000 = .095 NOT a match   Answer D. (xyz - 2y) / 100 = (50,000 - 100)/100 = 49,900/100 = 499 NOT a match   Answer E. (xyz - 2y) / 10,000 = 49,900/10,000 = 4.99 NOT a match   Final (100xyz - xy^2z) / 1,000,000nswer:(100xyz - xy^2z) / 1,000,000
Answer: A