RikoteMaster/aqua-rat-mcqa huggingface dataset:
Question: For any integer p, *p is equal to the product of all the integers between 1 and p, inclusive. How many prime numbers are there between *5 + 3 and *5 + 5, inclusive?
A. None
B. One
C. Two
D. Three
Answer: A. None
Explanation: Generally *p or p! will be divisible by ALL numbers from 1 to p. Therefore, *5 would be divisible by all numbers from 1 to 5.
=> *5+3 would give me a number which is a multiple of 3 and therefore divisible (since *5 is divisible by 3)
In fact adding anyprimenumber between 1 to 5 to *5 will definitely be divisible.
So the answer is none (A)!
Supposing if the question had asked for prime numbers between *5 + 3 and *5 + 11 then the answer would be 1. For *5 +3 and *5 + 13, it is 2 and so on...
A
Question: If P and Q are positive integers, P - 2 is divisible by 4 and Q - 1 is divisible by 6, all of the following could be the value of P - Q Except?
A.3
B.1
C.15
D.4
Answer: D.4
Explanation: P-2 = 4Q
=> P = 4Q + 2 = 2,6,10,14,18,22,26,30,34,38,42,46,50
Q-1 = 6P
=> Q = 6P + 1 = 1,7,13,19,25,31,37,43,49,55,61,67,73
3=10-7
1=14-13
15=34-19
29=42-13
So , we can see that all the answer choices are possible here except D
Question: The length of a bridge in meters, which a train 120-meters long and traveling at 45 km/hr can cross in 30 seconds is?
A.255
B.285
C.315
D.345
Answer: A.255
Explanation: 45 km/h = 45000 m/3600 s = 12.5 m/s
In 30 seconds, the train can go 30(12.5) = 375 meters
Let x be the length of the bridge.
x+120 = 375 meters
x = 255 meters
The answer is A.
Question: Q is a set of five distinct prime numbers. If the sum of the integers in Q is even and the number x is a member of Q, then what is the least value that x can be?
A.2
B.3
C.4
D.5
Answer: A.2
Explanation: 2 + 3 + 5 + 7 + 11= 28 ( sum is even)
Least number = 2
A
Question: 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?
A.20
B.30
C.35
D.45
Answer: A.20
Explanation: 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
Question: Three coworkers P , T , R meet for Dinner. P had 5 sandwiches, T had 3 and R had none. Both PT share their sandwiches with R such that each one got the same amount. If R paid $ 8 to PT, how much of $ 8 should P get? Assume both sandwiches and Dollars can be split.
A.a) 5$
B.b) 3 $
C.c) 7 $
D.d) 4 $
Answer: C.c) 7 $
Explanation: we got total 8 sandwiches ....we are dividing equally in 3 people.
per person: 8/3
initially
p:8
T:3
r:0
now since r ate 8/3 so he is paying 8 dollars for 8/3 sandwiches.
now p also ate 8/3 of his 5 sandwiches = so he gave 5-8/3 to r ==>7/3 to r
ok we have 8 dollars for 8/3 sandwich
therefore for 7/3 sandwich we will get 7 dollars.
hence p should get 7 dollars.
hence C
Question: Which of the following could be the sum of the reciprocals of two prime numbers?
A.7/15
B.11/12
C.11/30
D.8/15
Answer: D.8/15
Explanation: 10/12 it is
8/15=(1/3) + (1/5)
Answer - D
Question: The average age of 6 persons living in a house is 23.5 years. Three of them are majors and their average age is 42 years. The difference in ages of the three minor children is same
A.6 years
B.5 years
C.3 years
D.8 years
Answer: B.5 years
Explanation: B
5 years
Total age of six persons = 23.5 × 6 = 141 years Total age of three major persons = 42 × 3 = 126 years
∴ Total age of three minor children = 141 – 126 = 15 years The difference in ages of the three minor children is same. Therefore, we take ages may be: 5, 5, 5; 3, 5, 7; 2, 5, 8 and 1, 5, 9 In all the cases, median will be 5 years
Mean age of minor children = 15/3 = 5 years
Question: A and B start a business jointly. A invests Rs.16000 for 8 months and B remains in the business for 4 months. Out of the total profit B claims 2/7th share. How much money is contributed by B?
A.Rs.12829
B.Rs.12800
C.Rs.12293
D.Rs.12804
Answer: B.Rs.12800
Explanation: 16*8: x*4 = 5:2
x = 12.8 =>
Rs.12800
Answer: B
Question: Machine X takes 50 hours longer than machine Y to produce 1080 Widgets. Machine Y produces 20 percent more widgets in an hour than machine x does in an hour. How many widgets per hour does machine X produce
A.100
B.3.6
C.25
D.11
Answer: B.3.6
Explanation: Machine Y produces 20 percent more widgets in an hour than machine x does in an hour.
So if machine X produces 100 widgets, then machine Y produces 120 widgets.
Ratio of 120/100 = 6/5.
This is their speed of work (Y:X).
i.e. speed of their work (X:Y) = 5/6
Now, time is inversely proportional to speed.
Hence the ratio of the time spent (X:Y)= 6/5
Let us assume that they spend 6x and 5x hours.
Given that 6x-5x = 50
So, x = 50.
Hence 6x = 6*50 = 300 hours.
Hence X takes 120 hours to produce 1080 widgets.
So, in 1 hour, it can produce (1 * 1080)/300
= 3.6
Hence option (B).
Question: The average salary of all the workers in a workshop is $8000. The average salary of 7 technicians is $12000 and the average salary of the rest is $6000. The total number of workers in the shop is?
A.15
B.18
C.21
D.25
Answer: C.21
Explanation: Let the total number of workers be x
8000x = 12000*7 + 6000(x-7)
x = 21
Answer is C
Question: Four brothers Adam, Bill, Charles and David together contributed certain amount of money and purchased a car. The contribution made by Adam is half that of the total contribution made by the other three brothers, contribution made by Bill is one-fourth of the total contribution made by the other three brothers and the contribution made by Charles is two-third of the total contribution made by the other three brothers. If the total price of the car is $9300, find the contribution made by David.
A.$540
B.$580
C.$600
D.$620
Answer: D.$620
Explanation: Given that A:(B+C+D)= 1:2.
Hence A contributed 1/3rd of the total price.
Given B:(A+C+D)=1:4
Hence B contributed 1/5th of the total price.
Given C:(A+B+D)=2:3
Hence C contributed 2/5th of the total price.
Thus the contribution made by A,B and C= 1/3+1/5+2/5=14/15
So, the remaining 1/15th of the price is contributed by David.
Hence David's contribution= 1/15×9300=$620.
The answer is D.
Question: The average weight of 8 person's increases by 1.5 kg when a new person comes in place of one of them weighing 75 kg. What might be the weight of the new person?
A.81 kg
B.85 kg
C.90 kg
D.100 kg
Answer: A.81 kg
Explanation: Total weight increased = (8 x 1.5) kg = 6 kg.
Weight of new person = (75 +6) kg = 81 kg.
Answer : A
Question: What is the sum of all even numbers from 1 to 901?
A.122821
B.202950
C.281199
D.122850
Answer: B.202950
Explanation: Explanation:
900/2 = 450
450 * 451 = 202950
Answer: B
Question: if price of t.v set is reduced by 20%, then its sale increases by 80%, find net effect on sale value
A.44
B.45
C.46
D.47
Answer: A.44
Explanation: - a + b + ((-a)(b)/100)
= -20+80+(-20*80)/100
= -20+80-16
=44
ANSWER:A
Question: 96% of the population of a village is 23040. The total population of the village is?
A.28882
B.28883
C.24000
D.29887
Answer: C.24000
Explanation: X * (96/100) = 23040
X = 240 * 100
X = 24000
Answer: C
Question: A woman sells a horse for Rs.800 and loses something, if he had sold it for Rs.980, his gain would have been double the former loss. Find the cost price of the horse?
A.120
B.560
C.860
D.567
Answer: C.860
Explanation: Option C
Explanation:
CP = SP + 1CP = SP - g
800 + x = 980 - 2x
3x = 180 => x = 60
CP = 800 + 60 = 860
Question: The area of a square is equal to five times the area of a rectangle of dimensions 125 cm * 64 cm. What is the perimeter of the square?
A.273
B.800
C.277
D.251
Answer: B.800
Explanation: Area of the square = s * s = 5(125 * 64)
=> s = 25 * 8 = 200 cm
Perimeter of the square = 4 * 200 = 800 cm.
Answer: B:
Question: If a/b = 6/5, Then (5a + 4b) / (5a - 4b) = ?
A.4
B.7
C.5
D.3
Answer: C.5
Explanation: Answer
Dividing numerator as well as denominator by b, we get
Given Exp. = (5a + 4b) / (5a - 4b) = (5a/b + 4) / (5a/b - 4)
Since a/b = 6/5
this implies that
=[(5*6)/5 +4]/[(5*6)/5 -4)]
= (6 + 4) / (6 - 4)
= 5
Option: C
Question: Working simultaneously and independently at an identical constant rate, 4 machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 2x units of product P in 3 days?
A.24
B.18
C.16
D.12
Answer: C.16
Explanation: 4 machines do x units in 6 days
we have x/6 => rate of the 4 machines
we know that we need to have 2x units in 3 days
therefore, we need to get to 2x/3 rate of the machines.
rate of one machine is x/6*1/4 = x/24.
now, we need to know how many machines need to work simultaneously, to get 2x done in 3 days.
2x/3 work needs to be done by machines that work at x/24 rate.
let's assign a constant Y for the number of machines:
(x/24)*y = 2x/3
y = 2x/3 * 24/x
cancel 3 with 24, and x with x and get -> 16. Answer choice C
Question: If x > 7, which of the following is equal to (x^2 - 6x + 9)/(x^2 - 9) ?
A.(x + 4)/(4(x - 4))
B.(x - 4)/(x + 4)
C.(x - 3)/(x + 3)
D.(x + 4)/(x - 4)
Answer: C.(x - 3)/(x + 3)
Explanation: (x^2 - 6x + 9)/(x^2 - 9)
= (x - 3)(x - 3) / (x + 3)(x - 3)
=(x - 3)/(x + 3)
C. (x - 3)/(x + 3)
Question: At what price must an article costing Rs.50 be marked in order that after deducting 10% from the list price. It may be sold at a profit of 25% on the cost price?
A.69.44
B.62.0
C.62.6
D.62.1
Answer: A.69.44
Explanation: CP = 50
SP = 50*(125/100) = 62.50
MP*(90/100) = 62.50
MP = 69.44
Answer: A
Question: A 300 meter long train crosses a platform in 39 seconds while it crosses a signal pole in 18 seconds. What is the length of the platform?
A.350
B.289
C.208
D.270
Answer: A.350
Explanation: Speed = [300 / 18] m/sec = 50/3 m/sec.
Let the length of the platform be x meters.
Then, x + 300 / 39 = 50/3
3(x + 300) = 1950 è x
= 350m.
Answer: A
Question: What is the least number of squares tiles required to pave the floor of a room 15.17 m long and 9.02 m broad?
A.814
B.266
C.287
D.999
Answer: A.814
Explanation: Explanation:
Length of largest tile = H.C.F. of 1517 cm and 902 cm = 41 cm.
Area of each tile =
Required number of tiles = = 814
Answer: A) 814
Question: In a friendship gang Andrew has 8 gang, in how many ways can he invite one or more of the gang to his house?
A.234
B.255
C.265
D.267
Answer: B.255
Explanation: Andrew can select one or more than one of his 8 gang.
=> Required number of ways = 28 – 1= 255.
B
Question: The probability of a lottery ticket being a prized ticket is 0.2. When 4 tickets are purchased, the probability of winning a prize on atleast one ticket is?
A.0.5922
B.0.5904
C.0.5287
D.0.1288
Answer: B.0.5904
Explanation: Explanation:
P(winning prize atleast on one ticket)
= 1 - P("Losing on all tickets")
= 1 - (0.8)4 = (1 + (0.8)2)(1 - (0.8)2)
= (1.64)(0.36) = 0.5904
Answer: B
Question: A can do a piece of work in 10 days, B in 15 days. They work together for 5 days, the rest of the work is finished by C in two more days. If they get Rs. 3000 as wages for the whole work, what are the daily wages of A, B and C respectively (in Rs):
A.300, 200, 252
B.300, 200, 250
C.300, 200, 259
D.300, 200, 251
Answer: B.300, 200, 250
Explanation: Explanation:
A's 5 days work = 50%
B's 5 days work = 33.33%
C's 2 days work = 16.66% [100- (50+33.33)]
Ratio of contribution of work of A, B and C =
= 3 : 2 : 1
A's total share = Rs. 1500
B's total share = Rs. 1000
C's total share = Rs. 500
A's one day's earning = Rs.300
B's one day's earning = Rs.200
C's one day's earning = Rs.250
Answer: B) 300, 200, 250
Question: 60 percent of movie theatres in Town X have 3 screens or less. 20% of those theatres sell an average of more than $300 worth of popcorn per showing. 50 percent of all the movie theatres in Town X sell $300 or less of popcorn per showing. What percent of all the stores on the street have 4 or more screens and sell an average of more than $100 worth of popcorn per day?
A.12
B.18
C.32
D.40
Answer: A.12
Explanation: Lets take numbers here.
Assume that the total number of movie theaters in the town = 100
Then number of movie theaters with 3 screens or less = 60
=> Number of movie theaters with 4 screens or more = 40
Movie theaters with 3 screens or less selling popcorn at more than $300 = 20% of 60 = 12
Number of movie theaters selling popcorn at $300 or less = 56
=> Number of movie theaters selling popcorn at more than $300 = 100-56 = 44
Of these 44 theaters, 12 are those with 3 screens or less
Therefore 12 (44-12) must be those with four screens or more
A is the answer
Question: The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is
A.21.5
B.11.5
C.61.5
D.41.5
Answer: B.11.5
Explanation: Middle numbers = [(10.5 x 6 + 11.4 x 6) - 10.9 x 11]
= (131.4 - 119-9) = 11.5.
Answer: B
Question: The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
A.3
B.4
C.5
D.6
Answer: B.4
Explanation: Let the ten's digit be x and unit's digit be y.
Then, (10x + y) - (10y + x) = 36
9(x - y) = 36
x - y = 4.
Answer: Option B
Question: The average amount with a group of seven numbers is Rs. 20. If the newly joined member has Rs. 40 with him, what was the average amount with the group before his joining the group?
A.s.17
B.s.16.66
C.s.15.50
D.s.29.33
Answer: B.s.16.66
Explanation: Total members in the group = 7
Average amount = Rs. 20
Total amount with them = 7 * 20 = Rs. 140
One number has Rs. 40. So, the amount with remaining 6 people = 140 - 40 = Rs. 100
The average amount with them = 16.66
= Rs.16.66.
Answer:B
Question: In a LAB experiment, a 100 ml compound of element X and element Y needs to be increased to 180 ml by adding some quantity of element X, and some quantity of element Y. If the original compound contains 30% of element X, how much (in ml) of element Y must be added so that element X will be exactly one third of the resulting mixture ?
A.50 ml
B.100/3 ml
C.20 ml
D.20/3 ml
Answer: A.50 ml
Explanation: There are 0.3*100=30 ml of element X in the initial solution.
We need X to be 1/3rd of 180 ml in the resulting solution, so 60 ml. Thus out of 80 ml of solution added, 30 must be X (in order it to become 60 ml) ans the remaining 50 must be Y.
Answer:A
Question: The annual interest rate earned by an investment increased by 10 percent from last year to this year. If the annual interest rate earned by the investment this year was 15 percent, what was the annual interest rate last year?
A.14.04%
B.15.04%
C.13.64%
D.12.04%
Answer: C.13.64%
Explanation: let i = interest rate
i(this year) = i(last year) + 0.1i(last year) = 1.1i(last year)
15 = 1.1 x i(last year)
i(last year) = 15/1.1 = 150/11 = 13.64 %
Answer : C
Question: If the average of a sequence of consecutive multiple of 18 is 153, and the greatest term is 270, how many terms F in the sequence are smaller than the average?
A.5
B.6
C.7
D.8
Answer: C.7
Explanation: Let the sequence have N terms starting 18*K
--> 18K, 18(K+1), 18(K+2), .... , 18(K+N-1) is the sequence.
The sum of the sequence is
---> 18K + 18(K+1) + 18(K+2) + .... + 18(K+N-1) =18*(K+(K+1)+...+(K+N-1)) = 18*(N*(2K+N-1)/2).
So the average is
153=[18*(N*(2K+N-1)/2)]/N = 18*(2K+N-1)/2 = 9 *(2K+N-1) ---> by cancelling 9 we get 2K+N-1=17 --> 2K+N=18 ---*).
On the other hand, since the greatest term is 270, 18(K+N-1)=270 --> by cancelling 18 we get K+N-1=15 --> K+N=16 ---**).
By *), **) we have K=2, N=14.
Since 153/18 = 8.5, 18*8<153<18*9. So the number of terms F which are smaller than the average is 18*2, 18*3, ..... ,18*8 ---> 7. The answer is C.
Question: A store reduced the price of all items in the store by 10% on the first day and by another 10% on the second day. The price of items on the second day was what percent of the price before the first reduction took place?
A.80.0
B.80.9
C.81.0
D.81.1
Answer: C.81.0
Explanation: consider price of the all items as $100
after a initial reduction of 10% price becomes = 0.9 *100 = $ 90
after the final reduction of 10% price becomes = 0.9* 90 = $ 81
price of all items on second day is 81% of price on first day
correct answer option C
Question: The length of the train which is moving at 54km/hr, can crosses a bridge of length 245 m in 30 sec, is ?
A.260 m.
B.345 m.
C.130 m.
D.245 m.
Answer: C.130 m.
Explanation: Speed = [45 X 5/18] m/sec = [25/2] m/sec Time = 30 sec Let the length of train be x metres. Then, (245 + x)/30
= 25/2 => 2(245 + x)=750
130 = x
= 130 m.
Answer:C
Question: What is the greatest value of n such that 4^n is a factor of 25! ?
A.6
B.5
C.7
D.4
Answer: C.7
Explanation: Pretty simple, really.
If m = 7, then 4m = 28, which is 14x2, both of which are included in 25!
Since 7 is the largest number here, its the answer.
Answer is C
Question: Find the last term of a G.P whose first term is 9 and common ratio is ( 1/3 ) if the sum of the terms of the G.P is ( 40/3 )
A.1/3
B.2/5
C.1/4
D.2/3
Answer: A.1/3
Explanation: Sum of the G.P . = ( First term - r * last term)/1 – r
40/3 = 9 – 1/3 ( last term ) / 2/3
Last term = ( -40/3*2/3+9 ) * 3 = - 80/3 + 27 = 1/3
ANSWER:A
Question: Simplify:1397 x 1397
A.9157109
B.1956109
C.1951609.
D.1951709
Answer: C.1951609.
Explanation: 1397 x 1397	= (1397)2
= (1400 - 3)2
= (1400)2 + (3)2 - (2 x 1400 x 3)
= 1960000 + 9 - 8400
= 1960009 - 8400
= 1951609.
Answer is C
Question: The three sides of an acute angled triangle (all in cms) are 12, 19 and j. How many integer values can j take?
A.23
B.22
C.15
D.8
Answer: D.8
Explanation: In acute angle triangles all angles in the triangle should be less than 90.
The maximum length of j can be when it is the hypotenuse of the triangle.
So, j= sqrt(12^2 + 19^2) = 22.5 (approx)
The maximum length of j can be when side of length 12 is the hypotenuse of the triangle.
So, j = sqrt (19^2 - 12^2) = 14.7 (approx)
So, j>14.7 and j<22.5.
Hence, possible values for j can be 15,16,17,18,19,20,21 and 22. Total 8 values.
Answer D.
Question: One hour before John started walking from P to Q, a distance of 13 miles, Ann had started walking along the same road from Q to P. Ann walked at a constant speed of 3 miles per hour and John at 2 miles per hour. How many miles had Ann walked when they met?
A.6 miles
B.8,4 miles
C.9 miles
D.9,6 miles
Answer: C.9 miles
Explanation: Ann walks from Q to P at a speed of 3 miles/hr for one hour. She covers 3 miles in 1 hour and now distance between John and Ann is 13 - 3 = 10 miles.
Ann walks at 3 mph and John at 2 mph so their relative speed is 3+2 = 5 mph.
They have to cover 10 miles so it will take them 10/5 = 2 hours to meet.
In 2 hrs, Ann would have covered 2 hrs * 3 miles per hour = 6 miles.
Adding this to the 3 miles she covered before John, Ann covered a total of 3 + 6= 9 miles.
Answer (C)
Question: From a group of 5 managers (Joon, Kendra, Lee, Marnie and Noomi), 2 people are randomly selected to attend a conference in Las Vegas. What is the probability that Marnie and Noomi are both selected?
A.0.1
B.0.2
C.0.25
D.0.4
Answer: A.0.1
Explanation: (1/6)*(1/5)+(1/6)*(1/5)=2/30=1/15
ANS:A
Question: Working simultaneously and independently at an identical constant rate, 5 machines of a certain type can produce a total of x units of product P in 8 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?
A.30
B.18
C.16
D.12
Answer: A.30
Explanation: 5 machines do x units in 8 days
we have x/8 => rate of the 5 machines
we know that we need to have 3x units in 4 days
therefore, we need to get to 3x/4 rate of the machines.
rate of one machine is x/8*1/5 = x/40.
now, we need to know how many machines need to work simultaneously, to get 3x done in 4 days.
3x/4 work needs to be done by machines that work at x/40 rate.
let's assign a constant Y for the number of machines:
(x/40)*y = 3x/4
y = 3x/4 * 40/x
cancel 4 with 40, and x with x and get -> 30. Answer choice A
Question: During the last four years, a large truck company purchased diesel at rates of $1.1, $1.4, $1.8 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?
A.$2.5
B.$2.55
C.$1.51
D.$3.25
Answer: C.$1.51
Explanation: finding the Harmonic mean here.
HM = 4/(1/12 + 1/14 + 1/18 + 1/21) = 15.07
C
Question: How many revolutions will a circular wheel make to travel a distance of 3 km, given that the radius is 2.4 m?
A.11085.7
B.15085.7
C.35085.7
D.15015.7
Answer: A.11085.7
Explanation: 2 * 22/7 * 2.4 * x = 15085.7
x = 1000
Answer: A
Question: The cost price of a radio is Rs.1500 and it was sold for Rs.1260, find the loss %?
A.16%
B.18%
C.17%
D.78%
Answer: A.16%
Explanation: 1500 ---- 240
100 ---- ? => 16%
Answer:A
Question: If 50 apprentices can finish a job in 4 hours, and 30 journeymen can finish the same job in 5 hours, how much of the job should be completed by 10 apprentices and 15 journeymen in one hour?
A.1/9
B.29/180
C.26/143
D.3/20
Answer: D.3/20
Explanation: 50 apprentices can finish the job in 4 hours,thus:
10 apprentices can finish the job in 4*5 = 20 hours;
In 1 hour 10 apprentices can finish 1/20 of the job.
30 journeymen can finish the same job in 4,5 hours,thus:
15 journeymen can finish the job in 5*2 = 10 hours;
In 1 hour 15 journeymen can finish 1/10 of the job.
Therefore, in 1 hour 10 apprentices and 15 journeymen can finish 1/20+1/10=3/20 of the job.
Answer: D.
Question: Find the compound interest on Rs. 10,000 in 2 years at 4% per annum, the interest being compounded half-yearly.
A.Rs 824.32
B.Rs 822.32
C.Rs 700
D.Rs 900
Answer: A.Rs 824.32
Explanation: Principal = Rs. 10000; Rate = 2% per half-year; Time = 2 years = 4 half-years.
Amount =
Rs [10000 * (1+(2/100))4] = Rs(10000 * (51/50) * (51/50) * (51/50) * (51/50))
= Rs. 10824.32.
:. C.I. = Rs. (10824.32 - 10000) = Rs. 824.32.
ANSWER A Rs 824.32
Question: There are six different models that are to appear in a fashion show. Two are from Europe, two are from South America, and two are from North America. If all the models from the same continent are to stand next to each other, how many ways can the fashion show organizer arrange the models?
A.72
B.48
C.64
D.24
Answer: B.48
Explanation: Let's say that Models of Europe are represented by E1 and E2 respectively
Let's say that Models of South America are represented by S1 and S2
Let's say that Models of North America are represented by Na and N2
So we have to define the total ways of entries of E1 E2 S1 S2 N1 N2
We have to keep E1 E2 together and similarly N1 N2 together and similarly S1 S2 together so make three groups of two models from same region each
Arrangement of these three Group can be done in 3! ways
Arrangement of Models within Group can be done in 2!x2!x2! ways
i.e. Total Ways of Arranging the models so that they are grouped as per region together = (3!)*(2!x2!x2!) = 6*8 = 48
Answer: Option B
Question: In N is a positive integer less than 200, and 14N/60 is an integer, then N has how many different positive prime factors S?
A.2
B.3
C.5
D.6
Answer: B.3
Explanation: I like to put the numbers in prime factors so it is easier and faster to visualize.
14*n/60
If we write the factors of 14 --> 2, 7, and the factors of 60 --> 2, 2, 3, 5, we have
(2*7*n)/(2^2*3*5)
Simplifying
7*n/(2*3*5)
The only way the equation above has an integer value is if n has AT LEAST the factors 2, 3 and 5, so we can simplify again and we have the number 7.
The number could be 2*3*5, or 2*3*5*2, or 2*3*5*.....
However to be less than 200 we can not add any prime number.
2*3*5 = 120
If we added the next prime factor 7, we would have S=2*3*5*7 = 840
Thus, answer B