mathewhe/aqua_rat huggingface dataset:
Question: It was calculated that 75 men could complete a piece of work in 30 days. When work was scheduled to commence, it was found necessary to send 25 men to another project. How much longer will it take to complete the work?
(A) 20 days.
(B) 30 days.
(C) 25 days.
(D) 28 days.
(E) 45 days.
Rationale: One day work = 1 / 30
One man’s one day work = 1 / ( 30 * 75)
Now:
No. Of workers = 50
One day work = 50 * 1 / ( 30 * 75)
The total no. of days required to complete the work = (75 * 30) / 50 = 45
Answer:E
Answer: (E) 45 days.
Question: The sector of a circle has radius of 21 cm and central angle 135o. Find its perimeter?
(A) 91.5
(B) 91.4
(C) 91.7
(D) 91.3
(E) 91.1
Rationale: Perimeter of the sector = length of the arc + 2(radius)
= (135/360 * 2 * 22/7 * 21) + 2(21)
= 49.5 + 42 = 91.5 cm
Answer:A
Answer: (A) 91.5
Question: A student has 7 trousers and 9 shirts. In how many different ways can the student select a trouser and a shirt?
(A) 45
(B) 51
(C) 57
(D) 63
(E) 69
Rationale: The student can select one trouser in 7 ways.
The student can select one shirt in 9 ways.
The number of ways to select one trouser and one shirt is 7 * 9 = 63 ways
The answer is D.
Answer: (D) 63
Question: A train 100 meters long completely crosses a 300 meters long bridge in 45 seconds. What is the speed of the train is?
(A) 32
(B) 545
(C) 88
(D) 66
(E) 09
Rationale: S = (100 + 300)/45 = 400/45 * 18/5 = 32
Answer: A
Answer: (A) 32
Question: A man has Rs. 496 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has ?
(A) 80
(B) 93
(C) 95
(D) 98
(E) 100
Rationale: Let number of notes of each denomination be x.
Then x + 5x + 10x = 496
16x = 496
x = 31.
Hence, total number of notes = 3x = 93.
B
Answer: (B) 93
Question: During a two-week period, the price of an ounce of silver increased by 25 percent by the end of the first week and then decreased by 20 percent of this new price by the end of the second week. If the price of silver was x dollars per ounce at the beginning of the two-week period, what was the price, in dollars per ounce, by the end of the period?
(A) 0.8x
(B) 0.95x
(C) x
(D) 1.05x
(E) 1.25x
Rationale: The price by the end of the period = (x*1.25)*0.8 = x.
Answer: C.
Answer: (C) x
Question: The cost per pound of milk powder and coffee were the same in June. In July, the price of coffee shot up by 200% and that of milk powder dropped by 60%. If in July, a mixture containing equal quantities of milk powder and coffee costs $5.10 for 3lbs, how much did a pound of milk powder cost in July?
(A) $4
(B) $0.4
(C) $1
(D) $3
(E) $1.65
Rationale: Lets assume price of Coffee in June =100x
Price of Tea in June =100x
Price of Coffee in July = 300x (because of 200% increase in price)
Price of Tea in July = 40x (because of 60% Decrease in price)
Price of 1.5 pound of Coffee1.5 pound of Tea in July will be = 450x + 60x = 510x
As per question
510x= 5.10$
x = 0.01S
So the Price of Tea in July = 40x = 40 x 0.01 = 0.4$/pound
Answer B
Answer: (B) $0.4
Question: The security gate at a storage facility requires a four-digit lock code. If the lock code must consist only of digits from 1 through 7, inclusive, with no repeated digits, and the first and last digits of the code must be odd, how many lock codes are possible?
(A) 120
(B) 240
(C) 360
(D) 720
(E) 1440
Rationale: X-X-X-X-X --> there are 4 odd digits from 1 to 7, inclusive thus options for the first and the last X's are: 4-X-X-3. Other X's can take following values: 4-5-4-3 --> 4*5*4*3=240.
Answer: B.
Answer: (B) 240
Question: A can do a piece of work in 20days, and B can do it in 15days. How long will they take if both work together?
(A) 7days
(B) 8.6days
(C) 13days
(D) 5.6days
(E) 10days
Rationale: A+B can do the work in 20*15 / 20+15 = 20*15/35 = 8.6days
Answer is B
Answer: (B) 8.6days
Question: The number of students in each section of a school is 24. After admitting new students, three new sections were started. Now, the total number of sections is 16 and there are 21 students in each section. The number of new students admitted is:
(A) 12
(B) 42
(C) 24
(D) 28
(E) 26
Rationale: Original number of sections = 16 - 3 = 13
Original number of students = 24 x 13 = 312
present number of students = 21 x 16 = 336
Number of new students admitted = 336 - 312 = 24
so the answer is option C) 24.
Answer: (C) 24
Question: Kelvin is expected to pay a total of $1968300 on his mortgage with his bank. He is expected to make an initial monthly payment of $150 after which his monthly payment will be triple the amount of the previous month. How many months will it take him to fully repay the mortgage?
(A) 10
(B) 13
(C) 10
(D) 9
(E) 11
Rationale: First payment--$150
Total amount to be repaid--$285000
First month payment--$150
Second month payment( triple the previous month payment)--$150*3=$300
Third month payment( triple the previous month payment)--$300*3=$900
Fourth month payment( triple the previous month payment)--$900*3=$2700
Fifth month payment( triple the previous month payment)--$2700*3=$8100
Sixth month payment( triple the previous month payment)--$8100*3=$24300
Seventh month payment( triple the previous month payment)--$24300*3=$72900
Eight month payment( triple the previous month payment)--$72900*3=$218700
Ninth month payment( triple the previous month payment)--$218700*3=$656100
Tenth month payment( triple the previous month payment)--$656100*3=$1968300
Answer is C
Answer: (C) 10
Question: What is 10-8+6-4+...+(-12)?
(A)  8
(B)  10
(C)  12
(D)  14
(E)  16
Rationale: The expression considers all even numbers between 10 and -12 with alternate addition and subtraction of the numbers.
The numbers to be used are: 10, 8, 6, 4, 2, 0, -2, -4, -6, -8, -10, and -12
Now, the first term is positive and the next term is subtracted. So, the required expression becomes,
10 - 8 + 6 - 4 + 2 - 0 + (-2) - (-4) + (-6) - (-8) + (-10) - (-12)
= 10 - 8 + 6 - 4 + 2 - 0 - 2 + 4 - 6 + 8 - 10 + 12
= 42 - 30
= 12
Hence the correct answer choice is C.
Answer: (C)  12
Question: Bobby bought 3 shares, and which he sold for $96 each. If he had a profit of 20% on the sale of 2 of the shares each but a loss of 20% on the sale of the other share, then on the sale of both shares Bobby had...
(A) a profit of 10
(B) a profit of 8
(C) a loss of 8
(D) a loss of 10
(E) neither a profit nor a loss
Rationale: Cost of the first stock: 96/1.2=80, so profit from it 96-80=16 == > 16*2 = 32 profit on 2 shares
Cost of the second stock: 96/0.8=120, so loss from it 96-120=-24
Overall profit 32-24=+8
Answer: B.
Answer: (B) a profit of 8
Question: Bhanu spends 30% of his income on petrol on scooter 21% of the remaining on house rent and the balance on food. If he spends Rs.300 on petrol then what is the expenditure on house rent?
(A) 2287
(B) 140
(C) 128
(D) 797
(E) 147
Rationale: Given 30% (Income ) = 300 ⇒⇒ Income = 1000
After having spent Rs.300 on petrol, he left with Rs.700.
His spending on house rent = 21% (700) = Rs.147
Answer:E
Answer: (E) 147
Question: A and B started a business in partnership investing Rs.20,000 and Rs.15,000 respectively. After six months, C joined them with Rs.20,000. What will be A's share in the total profit of Rs.25,000 earned at the end of 2 years from the starting of the business?
(A) Rs.7500
(B) Rs.9000
(C) Rs.9500
(D) Rs.10,000
(E) None
Rationale: Solution
A : B : C	=(20000×24):(15000×24):(20000×18)
=4 :3 :3
A's share	=Rs.(25000×4/10)
= Rs.1000.
Answer D
Answer: (D) Rs.10,000
Question: If two projectiles are launched at the same moment from 1455 km apart and travel directly towards each other at 470 km per hour and 500 km per hour respectively, how many minutes will it take for them to meet?
(A) 90
(B) 85
(C) 80
(D) 75
(E) 70
Rationale: The projectiles travel a total of 970 km per hour.
The time to meet is 1455/970=1.5 hours = 90 minutes
The answer is A.
Answer: (A) 90
Question: Boy sells a book for Rs. 450 he gets a loss of 10 %, To gain 10 %, what should be the SP?
(A) Rs.450
(B) Rs.500
(C) Rs.550
(D) Rs.640
(E) Rs.650
Rationale: Find Selling Price to gain 10 %.
Now, we are asked to find selling price to gain 10% profit.
Hint:
Selling Price =	(100 + Gain%)	× C.P.
100
Selling Price =	(100 + 10)	× 500
100
Selling Price =	(110)	× 500
100
Therefore, selling Price = Rs.550
C
Answer: (C) Rs.550
Question: At a certain organisation, the number of male members went up by 15% in the year 2001 from year 2000, and the number of females members went down by 6% in the same time period. If the total membership at the organisation went up by 1.2% from the year 2000 to 2001, what was the ratio of male members to female members in the year 2000?
(A) 1:2
(B) 1:3
(C) 2:3
(D) 3:2
(E) 2:1
Rationale: men increase by 15% ==> 1.15M = males in 2001
women decrease by 6% ==> 0.94F = women in 2001
total employees increase by 1.2% ==> 1.012*(M + F) = total number of employees in 2001
Obviously
(males in 2001) + (females in 2001) = total number of employees in 2001
1.15M + 0.94F = 1.012*(M + F)
1.15M + 0.94F = 1.012M + 1.012F
1.15M - 1.012M = 1.012F - 0.94F
0.138M = 0.072F
M/F = (0.072)/(0.138) = 72/138 = 1/2
Answer =(A)
Answer: (A) 1:2
Question: In the first 25 overs of a cricket game, the run rate was only 5.4. What should be the run rate in the remaining 25 overs to reach the target of 400 runs?
(A) 16.5
(B) 13.2
(C) 10.6
(D) 11.2
(E) 7.25
Rationale: Required run rate = 400 - (5.4 x 25)	/25	=	265/25 = 10.6 Option C
Answer: (C) 10.6
Question: The president of a country and 4 other dignitaries are scheduled to sit in a row on the 5 chairs represented above. If the president must sit in the center chair, how many different seating arrangements are possible for the 5 people?
(A) 4
(B) 5
(C) 20
(D) 24
(E) 120
Rationale: If the president is sitting in the center, you only have 4 chairs left. There are 4 possibilities for 1st chair, 3 possibilities for 2nd chair, 2 possibilities for 3rd chair, and 1 for last chair. Multiply all those and you get 24.
ANS:D.
Answer: (D) 24
Question: A standard Veggiematik machine can chop 40 carrots in 4 minutes. How many carrots can 4 standard Veggiematik machines chop in 5 minutes?
(A) 50
(B) 100
(C) 150
(D) 200
(E) 250
Rationale: Direct Relationship:-
1 standard Veggiematik machine - 40 Carrots - 4 minutes,
1 standard Veggiematik machine - 10 Carrots - 1 minute,
Now
4 standard Veggiematik machine - ? Carrots - 5 minutes,
Hence = 10X4X5 =200 Carrots
Answer D
Answer: (D) 200
Question: {10,9,3,4,2}
Of the following value which value should be inserted in the set of 5 values to get a median of 4?
(A) 3
(B) 5
(C) 6
(D) 8
(E) 7
Rationale: Median is the middle number of a sorted distribution. In case of even number of items, Median will be the average of middle two values.
The question asks fora new number that will cause the median of new set at 4.
Mentionable that given set is already sorted and number of items of new set will be 5+1 = 6.
To get median 4,the sum of 3th4th Term should be 8.
So to keep the median 4,the new item should be >=5.
The answer is B
Answer: (B) 5
Question: Set A contains three different positive odd integers and five different positive even integers. Set B contains two different positive odd integers and three different positive even integers. If one integer from set A and one integer from set B are chosen at random, what is the probability that the product of the chosen integers is even?
(A) 3/5
(B) 7/10
(C) 11/20
(D) 17/20
(E) 33/40
Rationale: There can be an odd product only if both integers are odd.
P(odd product) = 3/8*2/5 = 3/20
P(even product) = 1 - 3/20 = 17/20
The answer is D.
Answer: (D) 17/20
Question: Right triangle XYZs the base of the prism in the figure above. If XY=XZ=âˆš9 and the height of the prism is 6, what is the volume of the prism?
(A) 3
(B) 27
(C) 13
(D) 12
(E) 15
Rationale: Volume of prism = Area of base * height = 1/2* (square root of 9) * (square root of 9)*6=27
Answer: B
Answer: (B) 27
Question: Missing number in the series
4 9 20 43 90 185 376 759 1526 3061 ?
(A) 6132
(B) 6185
(C) 6190
(D) 6195
(E) 6180
Rationale: 4*2+1=9
9*2+2=20
20*2+3=43
43*2+4=90
185*2+6=376
376*2+7=759
759*2+8=1526
1526*2+9=3061
3061*2+10=6132
ANSWER:A
Answer: (A) 6132
Question: The product of all the prime numbers less than 15 is closest to which of the following powers of 10 ?
(A)  10^9
(B)  10^8
(C)  10^7
(D)  10^6
(E)  10^4
Rationale: P = 2*3*5*7*11*13
= 21 * 11 * 13*10
~ 10^ 4
Answer : E
Answer: (E)  10^4
Question: The price of 2 sarees and 4 shirts is Rs. 1600. With the same money one can buy 1 saree and 6 shirts. If one wants to buy 12 shirts, how much shall he have to pay ?
(A) Rs. 2100
(B) Rs. 2200
(C) Rs. 2400
(D) Rs. 2500
(E) Rs. 2550
Rationale: Let the price of a saree and a shirt be Rs. x and Rs. y respectively.
Then, 2x + 4y = 1600 .... (i)
and x + 6y = 1600 .... (ii)
Divide equation (i) by 2, we get the below equation.
=> x + 2y = 800. --- (iii)
Now subtract (iii) from (ii)
x + 6y = 1600 (-)
x + 2y = 800
----------------
4y = 800
----------------
Therefore, y = 200.
Now apply value of y in (iii)
=> x + 2 x 200 = 800
=> x + 400 = 800
Therefore x = 400
Solving (i) and (ii) we get x = 400, y = 200.
Cost of 12 shirts = Rs. (12 x 200) = Rs. 2400.
C)
Answer: (C) Rs. 2400
Question: A and B invested some money such that capital of A is 20% more than that of B. If after one year they get profit of Rs.22000 then profit share of B is
(A) Rs.7500
(B) Rs.8000
(C) Rs.9000
(D) Rs.10,000
(E) None of the above
Rationale: A:B=120:100= 6:5
then B share =5/11*(22000)
=10,000
ANSWER:D
Answer: (D) Rs.10,000
Question: The diagonal of a cube is 8√3. find its volume and surface area.
(A) 354
(B) 384
(C) 324
(D) 344
(E) 364
Rationale: √3.a = 8 √3 => a = 8
Volume = a(power)3 => (8 × 8 × 8) Cm(power)3 => 512 Cm(power)3
Surface area => 6 a(power)2 => (6 × 8 × 8) Cm(power)2 => 384 Cm(power)2
Answer is B.
Answer: (B) 384
Question: In what time will a train 100 meters long cross an electric pole, if its speed is 126 km/hr
(A) 8.5 seconds
(B) 2.86 seconds
(C) 3.5 seconds
(D) 2.5 seconds
(E) 2.6 seconds
Rationale: First convert speed into m/sec
Speed = 126*(5/18) = 35 m/sec
Time = Distance/speed
= 100/35 = 2.86 seconds
Answer: B
Answer: (B) 2.86 seconds
Question: A shopkeeper has 280 kg of apples. He sells 45% of these at 20% profit and remaining 60% at 30% profit. Find his % profit on total.
(A) 24%
(B) 25%
(C) 27%
(D) 28%
(E) 35%
Rationale: If the total quantity was 100
Then 45X 20% + 60 X30%= 27
This profit will remain same for any total quantity unless the % of products remains the same. Hence 'C' is the answer
Answer: (C) 27%
Question: Sum of two numbers is 80. Greater number exceeds by 5 from four times of the smaller. Find the numbers?
(A) 65, 15
(B) 64, 16
(C) 62, 18
(D) 60, 20
(E) 61, 22
Rationale: Explanation:
x + y = 80
x – 4y = 5
x = 65 y = 15
ANSWER IS A
Answer: (A) 65, 15
Question: A water tank which could be filled in 5 hours takes one more hour to be filled as a result of a leak in its bottom. If the tank is full calculate the time it will take for the leak empty it?
(A) 32 hours
(B) 30 hours
(C) 34 hours
(D) 31 hours
(E) 33 hours
Rationale: Part filled without leak in 1 hour = 1/5
Part filled with leak in 1 hour = 1/6
Work done by leak in 1 hour
=1/5âˆ’1/6=30 hours
Answer:B
Answer: (B) 30 hours
Question: 117 x 217 + 83 x 83=?
(A) 79698
(B) 20578
(C) 80698
(D) 81268
(E) None of them
Rationale: =(117)^2 + (83)^2
=(100+17)^2 + (100-17)^2
=2[(100)^2+(17)^2]
=2[10000+289]
=2 x 10289 = 20578
Answer is B
Answer: (B) 20578
Question: When the integer x is divided by the integer y, the remainder is 60. Which of the following is a possible value of the quotient x/y?
I. 15.18
II.18.17
III. 17.16
(A)  I only
(B)  II only
(C)  III only
(D)  I and II only
(E)  I and III only
Rationale: Problem statement was difficult to parse. but some how I managed to understand what we can do.
1) x/y = 15 + 0.18
2) x/y = 18 + 0.17
3) x/y = 17 + 0.15
w.k.t x = yn (i.e Q) + yk=60 (i.e remainder)
therefore :
1) y*0.18=60 -- we can't get y an integer.
2) y*0.17=60 -- we can't get y an integer.
3) y*0.16=60 -- we can get y an integer.
Therefore 3 are possible answers.
Ans C.
Answer: (C)  III only
Question: Steve traveled the first 2 hours of his journey at 40 mph and the last 3 hours of his journey at 90 mph. What is his average speed of travel for the entire journey?
(A) 60 mph
(B) 56.67 mph
(C) 53.33 mph
(D) 70 mph
(E) 66.67 mph
Rationale: Answer
Average speed of travel = Total distance travelled/Total time taken
Total distance traveled by Steve = Distance covered in the first 2 hours + distance covered in the next 3 hours.
Distance covered in the first 2 hours = speed * time = 40 * 2 = 80 miles.
Distance covered in the next 3 hours = speed * time = 90 * 3 = 270 miles.
Therefore, total distance covered = 80 + 270 = 350 miles.
Total time taken = 2 + 3 = 5 hours.
Hence, average speed = Total distance travelled/Total time taken=350/5 = 70 miles per hour.
Choice D
Answer: (D) 70 mph
Question: Integer X represents the product of all integers between 1 to 24 (inclusive). The smallest prime factor of (x+1) must be
(A) Between 1 to 10
(B) Between 11 to 15
(C) Between 15 to 20
(D) Between 20 to 25
(E) Greater than 24
Rationale: Answer = E = Greater than 24
This problem is asking smallest prime factor of (24!+1)
25! already have there prime factors 2,3,5,7,11,13.......... so on upto 23 (1 cannot be considered prime factor)
Just adding 1 to 24! will remove all the factors stated above;
so the smallest possible prime factor has to be greater than 24
Answer = E
Answer: (E) Greater than 24
Question: Three people (1 couple and 1 single) are seated at random in a row of 8 chairs. What is the probability that the couple does not sit together?
(A) 2/3
(B) 2/5
(C) 3/8
(D) 3/4
(E) 4/5
Rationale: The total number of arrangements of 3 people is 8*7*6 = 336
Let's think of the couple as a single unit.
The number of arrangements when the couple sits together is 7*6*2 = 84.
Note that we multiply by 2 because the couple can switch places with each other.
The number of ways the couple won't sit together is 336 - 84 = 252
P(couple won't sit together) = 252/336 = 3/4
The answer is D.
Answer: (D) 3/4
Question: There are two examinations rooms A and B. If 10 students are sent from A to B, then the number of students in each room is the same. If 20 candidates are sent from B to A, then the number of students in A is double the number of students in B. The number of students in room A is:
(A) 100
(B) 110
(C) 120
(D) 130
(E) 140
Rationale: Let the number of students in rooms A and B be x and y respectively.
Then, x - 10 = y + 10 x - y = 20 .... (i)
and x + 20 = 2(y - 20) x - 2y = -60 .... (ii)
Solving (i) and (ii) we get: x = 100 , y = 80.
The required answer A = 100.
Answer: (A) 100
Question: A student was asked to find 4/5 of a number. But the student divided the number by 4/5, thus the student got 18 more than the correct answer. Find the number.
(A) 30
(B) 35
(C) 40
(D) 45
(E) 50
Rationale: Let the number be x.
(5/4) * x = (4/5)*x + 18
25x = 16x + 360
9x = 360
x = 40
The answer is C.
Answer: (C) 40
Question: Rohit walked 25 m towards south. Then he turned to his left and walked 20 m. He then turned to his left and walked 25 m. He again turned to his right and walked 15 m. At what distance is he from the starting point and in which direction?
(A) 35 m East
(B) 35 m North
(C) 30 m West
(D) 45 m East
(E) 55 m East
Rationale: ! 20 !--->15
! !
!--->!
20+15=35
ANSWER:A
Answer: (A) 35 m East
Question: If the selling price is hexatrupled, the profit is septupled, what is the percentage of profit??
(A) 50%
(B) 100%
(C) 150%
(D) 200%
(E) 250%
Rationale: we can make two equations from the given info..
let selling price be S and cost price be C..
so profit=P=S-C...
if S is doubled profit is tripled..
so 7P=6S-C..
or S=6P...
therefore C=P..
% of profit=P/P=100%
B
Answer: (B) 100%
Question: What is the total number of P integers between 100 and 200 that are divisible by 3?
(A)  33
(B)  32
(C)  31
(D)  30
(E)  29
Rationale: yes there is a different way of arriving at that answer ....
u can also use airthmetic progression to get the answer
since the first term to be divisble by 3 is 102 ..take that as A .. the starting no
and since 198 is the last digit to be divisible by 3 take that as N ...
since the difference is 3 take that as D
no u have to find what term is 198 take that as nth term
the formula for that is N = A + (n-1) * d
198 = 102 +(n-1) * 3
from this u get n =33
Answer: (A)  33
Question: There are two examinations rooms A and B. If 10 students are sent from A to B, then the number of students in each room is the same. If 20 candidates are sent from B to A, then the number of students in A is double the number of students in B. The number of students in room A is:
(A) 20
(B) 80
(C) 100
(D) 200
(E) 120
Rationale: Let the number of students in rooms A and B be x and y respectively.
Then, x - 10 = y + 10
x - y = 20 .... (i)
and x + 20 = 2(y - 20)
x - 2y = -60 .... (ii)
Solving (i) and (ii) we get: x = 100 , y = 80.
The required answer A = 100.
ANSWER:C
Answer: (C) 100
Question: Which one of the following numbers is the greatest positive integer x such that 4^x is a factor of 64^5?
(A) 5
(B) 8
(C) 10
(D) 15
(E) 19
Rationale: 64 = 4*4*4= 4^3
64^5 =(4^3)^5
=4^15
we have to find the greatest positive integer value for x
when x=15, it divides exactly once and that is the least positive integer value of the quotient
therefore, the correct option is D
Answer: (D) 15
Question: The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm, 12 m 95 cm is:
(A) 15 cm
(B) 25 cm
(C) 35 cm
(D) 42 cm
(E) 45 cm
Rationale: Required length = H.C.F. of 700 cm, 385 cm and 1295 cm = 35 cm.
Answer: Option C
Answer: (C) 35 cm
Question: In a friendship gang Kala has 11 gang, in how many ways can he invite one or more of the gang to his house?
(A) 1244
(B) 2047
(C) 3698
(D) 5789
(E) 78895
Rationale: Kala can select one or more than one of his 8 gang.
=> Required number of ways = 2^11 – 1= 2047.
B
Answer: (B) 2047
Question: A train moves past a house and a river bridge 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train ?
(A) 79.2 km/hr
(B) 60.5 km/hr
(C) 65 km/hr
(D) 62.5 km/hr
(E) 75.2 km/hr
Rationale: length of the train be p metres and its speed - q m/sec.
p/q = 8 => p = 8q
p+264/20 = q
8q+264 = 20q
q = 22
speed = 22 m/s = (22*18/5) = 79.2 km/hr
ANSWER A
Answer: (A) 79.2 km/hr
Question: If s is the smallest positive integer such that 3,150 multiplied by s is the square of an integer, then s must be
(A)  2
(B)  5
(C)  6
(D)  7
(E)  14
Rationale: Solution:
This problem is testing us on the rule that when we express a perfect square by its unique prime factors, every prime factor's exponent is an even number.
Let’s start by prime factorizing 3,150.
3,150 = 315 x 10 = 5 x 63 x 10 = 5 x 7 x 3 x 3 x 5 x 2
3,150 = 2^1 x 3^2 x 5^2 x 7^1
(Notice that the exponents of both 2 and 7 are not even numbers. This tells us that 3,150 itself is not a perfect square.)
We also are given that 3,150 multiplied by s is the square of an integer. We can write this as:
2^1 x 3^2 x 5^2 x 7^1 x s = square of an integer
According to our rule, we need all unique prime factors' exponents to be even numbers. Thus, we need one more 2 and one more 7. Therefore, s = 7 x 2 = 14
Answer is E.
Answer: (E)  14
Question: The last day of a century cannot be
(A) Monday
(B) Wednesday
(C) Tuesday
(D) Friday
(E) None of these
Rationale: Explanation :
We know that number of odd days in 100 years = 5
Hence last day of the first century is Friday
Number of odd days in 200 years = 5 x 2 = 10 = 3 (As we can reduce multiples of 7
from odd days which will not change anything)
Hence last day of the 2nd century is Wednesday
Number of odd days in 300 years = 5 x 3 = 15 = 1 (As we can reduce multiples of 7
from odd days which will not change anything)
Hence last day of the 3rd century is Monday
We know that umber of odd days in 400 years = 0. (? 5 x 4 + 1 = 21 = 0)
Hence last day of the 4th century is Sunday
Now this cycle will be repeated.
Hence last day of a century will not be Tuesday or Thursday or Saturday
Answer : Option C
Answer: (C) Tuesday
Question: What is the product of all positive factors of 21?
(A) 39
(B) 324
(C) 441
(D) 451
(E) 551
Rationale: Positive factors of 21 are 1, 3, 7, 21.
Positive factors of 21 is: 21*7*3 =441
Answer: C
Answer: (C) 441