MU-NLPC/Calc-aqua_rat huggingface dataset:
A bike agent buys 30 bicycles, of which 8 are first grade and the rest are second grade, for Rs 3,150, Find at what price he must sell the first grade bikes so that if he sells the second grade bikes at three quarters of the price, he may make a profit of 40% on his outlay.	Choose the correct answer:
A) Rs 200 B) Rs 240 C) Rs 180 D) Rs 210 E) Rs 230

Solution: Suppose the price of first grade bike = Rs X   And the price of second grade bike = Rs Y   Therefore, 8X + 22Y = 3150   Suppose he sells the first grade bike at the rate of Rs Z per cycle   Therefore, 8Z + 22 x 3/4Z = 3150 + 40% of 3150   98Z = 17640 => Z = 180   S.P. of the first grade bike = Rs 180   S.P. of the second grade bike = Rs 135   ANSWER:Rs 180
Answer: C
A drawer holds 5 red hats and 5 blue hats. What is the probability of getting exactly three red hats or exactly three blue hats when taking out 8 hats randomly out of the drawer and returning each hat before taking out the next one?	Select the correct option:
A) 3/4
B) 5/8
C) 7/16
D) 15/32
E) 31/64

Solution: Let's find the probability of getting exactly 3 red hats.   One possible case is red-red-red-blue-blue-blue-blue-blue.   The probability of this is 1/2*1/2*1/2*1/2*1/2*1/2*1/2*1/2 = 1/256   There are 8C3 = 56 such cases so we need to multiply this by 56.   P(exactly 3 red hats) = 56*1/256 = 7/32   Clearly the probability of getting exactly 3 blue hats will also be 7/32.   The sum of the probabilities is 7/32+7/32 = 14/32 = 7/16   The answer is 7/16.
Answer: C
If d is the smallest positive integer such that 3,150 multiplied by d is the square of an integer, then d must be
Choose the correct choice from the following options:
A) 2	B) 5	C) 6	D) 7	E) 14

Solution: 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 d is the square of an integer. We can write this as:   2^1 x 3^2 x 5^2 x 7^1 x d = 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, d = 7 x 2 = 14   Answer is 14.
Answer: E
John takes 6 hrs to print 32 pages. Peter takes 5hrs to print 40 pages. In how many hours they will print 110 pages together ?	Pick
A) 8 hours  B) 8 hours 15 minutes  C) 9 hours 15 minutes  D) 11 hours  E) 12 hours

Solution: Ram can type 32/6 pages per hour   Shaam can type 8 pages per hourThey can type 32/6 + 8 = 40/3 pages per hour   So they will take 110/40 × 3 = 33/4 hours or 8 hours 15 minutes   8 hours 15 minutes)
Answer: B
70,78, 80,85,90,105,105,130,130,130
The list consists of the times in seconds that it took each of the 10 school children to run a distance of 400 mts . If the standard deviation of the 10 running times is 22.4, rounded to the nearest tenth of a second, how many of the 10 running times are more than 1 standard deviation below the mean of the 10 running times.	Choose the correct choice:
A) 1
B) 2
C) 3
D) 4
E) 5

Solution: How many of the 10 running times are more than one SD below the meanmeans how many data points from given 10 are less thanmean-1SD.   We are given that SD=22.4, so we should find mean --> mean=100 --> there is only 1 data points below  100-22.4= 
 100-22.4 = 77.6
77.6, namely 70   1nswer: 1.
Answer: A
The Shopkeeper increased the price of a product by 25% so that customer finds it difficult to purchase the required amount. But somehow the customer managed to purchase only 70% of the required amount. What is the net difference in the expenditure on that product?	Choose the correct choice: A) 11% B) 12% C) 12.5% D) 13% E) 14%

Solution: Quantity X Rate = Price   1 x 1 = 1   0.7 x 1.25 = 0.875   Decrease in price = (0.125/1) × 100 = 12.5%   12.5%
Answer: C
The greatest number of four digits which is divisible by 25, 40, 54 and 75 is:	Answers: A) 5000 B) 5400 C) 5600 D) 5800 E) 5200

Solution: Greatest number of 4-digits is 9999.   L.C.M. of 25, 40, 54 and 75 is 600.   On dividing 9999 by 5400, the remainder is 4599.   Required number  (9999 - 4599)= 
 (9999 - 4599) = 5_400
5400.   Answer: Option 5400
Answer: B
The length of a rectangle is increased to 2 times its original size and its width is increased to 3 times its original size. If the area of the new rectangle is equal to 1800 square meters, what is the area of the original rectangle?	Pick:
A) 100 square meters B) 200 square meters C) 300 square meters D) 400 square meters E) 500 square meters

Solution: If L and W be the original length and width of the rectangle and its area is given by   L ? W   After increase the length becomes 2 L and the width becomes 3 W. The area is then given by (2 L) ? (3 W) and is known. Hence   (2 L) ? (3 W) = 1800   Solve the above equation to find L ? W   6 L ? W = 1800   L ? W = 1800 / 6= 
1800 / 6 = 300
300 square meters , area of original rectangle   correct answer 300 square meters
Answer: C
A man invests some money partly in 12% stock at 132 and partly in 10% stock at 80.To obtain equal dividends from both,he must invest the money in the ratio :	Select.
A) 3 : 4 B) 3 : 5 C) 11:8 D) 16 : 15 E) None

Solution: Solution   For an income of Rs. 1 in 12% stock at 132, investment = Rs. (132/12)= Rs. 11 .   For an income of Rs. 1 in 10% stock at 80, investment = Rs.(80/10) = Rs. 8.   ∴ Ratio of investments = 11:8 =11:8   Answer 11:8
Answer: C
Working alone, Jerry finishes cleaning half the house in a third of the time it takes Nick to clean the entire house alone. Jerry alone cleans the entire house in 6 hours. How many hours will it take Nick and Jerry to clean the entire house if they work together?	Select the correct option
A) 1.5 B) 2 C) 2.4 D) 3 E) 3.6

Solution: Answer is 3.6 hours.   Jerry does the complete house in 6 hours while Nick does it in 9 hours. 1/ (1/6+1/9)= 3.6   answer is 3.6
Answer: E
.008/?= .01
Choose one:
A) .8
B) .09
C) .009
D) .0009
E) None of them

Solution: Let .008/x = .01;   Then x = .008/.01= 
.008/.01 = 0.8
.8/1= 
.8/1 = 0.8
.8   .8nswer is .8
Answer: A
I. a3 - 988 = 343,
II. b2 - 72 = 49 to solve both the equations to find the values of a and b? Choose one
A) a > b	B) a ≥ b	C) a < b	D) a ≤ b	E) a = b or the relationship between a and b cannot be established.

Solution: Explanation:   a3 = 1331 => a = 11   b2 = 121 => b = ± 11   a ≥ b   a ≥ b)
Answer: B
Tanks Q and B are each in the shape of a right circular cylinder. The interior of tank Q has a height of 10 meters and a circumference of 8 meters, and the interior of tank B has a height of 8 meters and a circumference of 10 meters. The capacity of tank Q is what percent of the capacity of tank B?
Choose the correct choice from the following choices.
A) 75%  B) 80%  C) 100%  D) 120%  E) 125%

Solution: B.   For Q, r=8/2pi. Its capacity = (4pi)^2 * 10= 160pi   For B, r=10/pi. Its capacity = (5pi)^2 *8 = 200pi   Q/80% = 160pi/200pi = 0.8160  /200= 
160  /200 = 4/5 = around 0.8
0.8
Answer: B
4 dice are thrown simultaneously on the board. Find the probability show the same face.	Choose the correct choice from the following answers.
A) 1/115  B) 2/113  C) 1/216  D) 2/121  E) 3/114

Solution: The total number of elementary events associated to the random experiments of throwing four dice simultaneously is:   =6×6×6×6=64=6×6×6×6=64   n(S)=64n(S)=64   Let XX be the event that all dice show the same face.   X={(1,1,1,1,),(2,2,2,2),(3,3,3,3),(4,4,4,4),(5,5,5,5),(6,6,6,6)}X={(1,1,1,1,),(2,2,2,2),(3,3,3,3),(4,4,4,4),(5,5,5,5),(6,6,6,6)}   n(X)=6n(X)=6   Hence required probability,   =n(X)n(S)=664=n(X)n(S)=664   =1/216   1/216
Answer: C
10 men, working 4 hours a day can complete a work in 18 days. How many hours a day must 15 men work to complete the work in 12 days ?
Choose the correct choice from the following choices: A) 4 hours a day B) 5 hours a day C) 6 hours a day D) 7 hours a day E) 8 hours a day

Solution: Explanation:   More Men, Less Hours {Indirect Proportion}   Less Days, More Hours {Indirect Proportion}   [men 15 10   days 12 18] ::4:x   =>xâˆ—15âˆ—12=10âˆ—18âˆ—4   =>x=10âˆ—18âˆ—4/ 15âˆ—12   =>x=4   Option 4 hours a day
Answer: A
If (x + 10)^2 = 144, which of the following could be the value of 2x?  Choose the correct choice from the following choices: A) –26 B) –22 C) –44 D) 66 E) 13

Solution: You can also get to the answer by backsolving using answer choices   Starting with x= -22   (x + 10)^2 = 144   (-22 + 10)^2 = 144 =====> (-12)^2 = 144 Bingo, –44learly, among the answer choices only -22 will result in 144 in the equation so x must be -22,   Then 2x = 2 * -22 = -44   Answer: –44
Answer: C
Let f(x,y) be defined as the remainder when (x–y)! is divided by x. If x=36, what is the maximum value of y for which f(x,y)=0?  Choose the correct choice.
A) 9  B) 12  C) 18  D) 20  E) 30

Solution: The question is finding y such that (36-y)! is a multiple of 36. That means we need to have 2^2*3^2 in (36-y)!   6! is the smallest factorial number with 2^2*3^2 as a factor.   36-y = 6   y = 30   The answer is 30.
Answer: E
What percent is 6 gm of 1 kg?
Choose one
A) 0.06%
B) 0.6%
C) 0.006%
D) 6%
E) None of these

Solution: Explanation:   1 kg = 1000 gm   6/1000 * 100 = 600/1000   =3/5= 
3/5 = 3/5 = around 0.6
0.6 %   Answer: Option 0.6%
Answer: B
Ten years ago A was half of B in age. If the ratio of their present ages is 3 : 4, what will be the total of their present ages	Choose the correct choice from the following: A) 30 B) 35 C) 37 D) 41 E) 42

Solution: Explanation:   Let A's age 10 years ago = x years.   Then, B's age 10 years ago = 2x years.   (x + 10) / (2x+ lO) = 3/4   => x = 5.   So, the total of their present ages =(x + 10 + 2x + 10)   = (3x + 20) = 35 years   Option 35
Answer: B
The least number, which when divided by 12, 15, 20 and 54 leaves in each case a remainder of 5 is: Choose one:
A) 545
B) 488
C) 542
D) 548
E) 560

Solution: Required number = (L.C.M. of 12, 15, 20, 54) + 5   = 540 + 5= 
540 + 5 = 545
545.   answer :545
Answer: A
A can do a piece of work in 12 days. He worked for 15 days and then B completed the remaining work in 10 days. Both of them together will finish it in.
Choose the correct choice:  A) 12 1/2 days  B) 12 1/9 days  C) 12 8/2 days  D) 17 1/2 days  E) 32 1/2 days

Solution: 15/25 + 10/x = 1 => x = 25   1/25 + 1/25 = 2/25   25/2 = 12 1/2 days   12 1/2 daysnswer: 12 1/2 days
Answer: A
The average age of students of a class is 15.8 years. The average age of boys in the class is 16.4 years and that of the girls is 15.4 years. The ration of the number of boys to the number of girls in the class is?	Choose the correct choice from the following answers.
A) 2:6
B) 2:3
C) 2:5
D) 2:1
E) 2:4

Solution: Let the ratio be k : 1. Then,   k * 16.4 + 1 * 15.4 = (k + 1) * 15.8   = (16.4 - 15.8)k = (15.8 - 15.4)   = k = 0.4/0.6 = 2/3   Required ratio = 2/3 : 1 = 2:3.   Answer:2:3
Answer: B
If 7875 / 5.25 = 1500, then 787.5 / 52.5 is equal to ?	Select.
A) 15 B) 19 C) 12 D) 14 E) 16

Solution: 15nswer   Given expression 787.5 / 52.5 = 7875 / 525   = 7875 / (525 x 100)   = 1500 / 100   15   Correct Option: 15
Answer: A
The ratio of number of boys and girls in a school is 2:5. If there are 490 students in the school, find the number of girls in the school?
Choose the correct choice from the following choices:
A) 150 B) 350 C) 300 D) 370 E) 280

Solution: Let the number of boys and girls be 2x and 5x   Total students = 490   Number of girls in the school = 5*490/7= 
5*490/7 = 350
350   Answer is 350
Answer: B
What percent is 320 gm of 1 kg?
Choose the correct choice from the following choices:
A) 25%	B) 40%	C) 10%	D) 8%	E) 32%

Solution: 1 kg = 1000 gm   320/1000 Ã— 100 = 32000/1000= 
32000/1000 = 32
32 %   Answer is 32%
Answer: E
A train moves 100m in 2 sec and then find the speed travelled?	Choose the correct choice from the following:
A) 28
B) 900
C) 50
D) 180
E) none

Solution: Length = Speed * time   Speed = L/T   S = 100m/2sec   S = 50 M/Sec   Speed100 /2= 
100 /2 = 50
50 M/Sec   Speed= 50*18/5 (To convert M/Sec in to Kmph multiply by 18/5)   Speed = 180Kmph.Answer:180
Answer: D
A selection committee is to be chosen consisting of 5 ex-technicians. Now there are 12 representatives from four zones. It has further been decided that if Mr. X is selected, Y and Z will not be selected and vice-versa. In how many ways it can be done?	Choose the correct choice from the following answers.
A) 572	B) 672	C) 472	D) 372	E) None

Solution: Solution:   10C5: when both are not included.   10C4: when one of them is included.   Number of ways = 10C5 +10C4+ 10C4 = 672   Answer: Option 672
Answer: B
7 men and 2 boys working together can do four times as much work as a man and a boy. Working capacity of man and boy is in the ratio
Pick one.
A) 1:2	B) 1:3	C) 2:1	D) 2:3	E) None of these

Solution: Explanation:   Let 1 man 1 day work = x   1 boy 1 day work = y   then 7x + 2y = 4(x+y)   => 3x = 2y   => x/y = 2/3   => x:y = 2:3   Option 2:3
Answer: D
A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If four marbles are picked at random, what is the probability that none is blue?  Answers:
A) 33/16	B) 33/91	C) 33/29	D) 33/18	E) 33/11

Solution: Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles. When four marbles are picked at random, then the probability that none is blue is   = ¹²C₄/¹⁵C₄   = (12 * 11 * 10 * 9)/(15 * 14 * 13 * 12)   = 33/91   Answer: 33/91
Answer: B
An empty pool being filled with water at a constant rate takes 8hours to fill to 4/5 of its capacity.how much more time will it take to finish filling the pool? Choose the correct choice:
A) 5hr 30min  B) 5hr 20min  C) 4hr 48min  D) 3 hr 12min  E) 5hr 40 min

Solution: As pool is filled to 3/5 of its capacity then 2/5 of its capacity is left to fill.   To fill 3/5 of the pool took 8 hours --> to fill 2/5 of the pool will take 8/(3/5)*2/5=16/3 hours = 5 hours 20 minutes (because if t is the time needed to fill the pool then t*3/5=8 --> t=8*5/3 hours --> to fill 2/5 of the pool 8*5/3*2/5=16/3 hours will be needed).   Or plug values: take the capacity of the pool to be 5 liters --> 3/5 of the pool or 3 liters is filled in 8 hours, which gives the rate of 3/8 liters per hour --> remaining 2 liters will require: time = job/rate = 2/(3/8) = 16/3 hours = 5 hours 40 minutes.   Answer: 5hr 40 min.
Answer: E
The radii of two cones are in ratio 2:1, their volumes are equal. Find the ratio of their heights. Choose the correct option
A) 1:4
B) 1:3
C) 1:2
D) 1:5
E) None of these

Solution: Explanation:   Let their radii be 2x, x and their heights be h and H resp.   Then,   Volume of cone =1/3πr2h   1/3∗π∗(2x)2∗h /1/3∗π∗x2∗H   =>hH=14   =>h:H=1:4   Option 1:4
Answer: A
A sum of Rs.312 was divided among 100 boys and girls in such a way that the boy gets Rs.3.60 and each girl Rs. 2.40 the number of girls is  Choose one
A) 35
B) 40
C) 45
D) 50
E) 55

Solution: Explanation:   Step (i): Let x be the number of boys and y be the number of girls.   Given total number of boys and girls = 100   x+y=100 -------------- (i)   Step (ii): A boy gets Rs. 3.60 and a girl gets Rs. 2.40   The amount given to 100 boys and girls = Rs. 312   3.60x + 2.40y = 312 -------------- (ii)   Step (iii):   Solving (i) and (ii)   3.60x + 3.60y = 360 --------- Multiply (i) by 3.60   3.60x + 2.40y = 312 --------- (ii)   1.20y = 48   y = 48 / 1.20= 
48 / 1.20 = 40
40   Number of girls= 40   Answer: 40
Answer: B
A and B together can do a piece of work in 6 days and A alone can do it in 13 days. In how many days can B alone can do it?  Choose the correct choice:
A) 12 days  B) 15 days  C) 78/7 days  D) 21 days  E) 22/7 days

Solution: Explanation:   A and B can do work 1/6 in 1 day   A alone can do 1/13 work in 1 day   B alone can do (1/6 -1/13) =7/78 work in 1 day   => complete work can be done in 78/7 days by B   Answer: Option 78/7 days
Answer: C
From a Group of 8 People, Including Gary and Nina, 3 people are to be selected at random to work on a certain project. What is the probability that 3 people selected will include Gary but not Nina	Pick:  A) 5/56  B) 9/56  C) 15/56  D) 21/56  E) 25/56

Solution: Number of ways of selecting 3 people out of 8 people = 8C3   In the three members Gary will always be there in the team. At this step we have vacancy for 2 more members and 7 members are available. Nina cannot be there in the team. So 2 members have to be selected and the number of available members = 7 - Nina = 6   Number of ways to form a 3 member team that includes Gary and excludes Nina = 615/562   Probability = 615/562/815/563 = 15/56   Answer: 15/56
Answer: C
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
Answers:  A) 28  B) 32  C) 48  D) 60  E) 120

Solution: Sisters can sit separately in two ways:   1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case:  2*12= 
 2*12 = 24
24   Or   2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case:  2*4= 
 2*4 = 8
8.   Total=24+8= 
24+8 = 32
32. ANS:32
Answer: B
How many bricks each measuring 20cm x 10cm x 8cm, will be needed to build a wall 10m x 8m x 24.5m  Pick:
A) 12250  B) 13400  C) 12500  D) 14340  E) None of these

Solution: Explanation:   no. of bricks= volume of the wall/volume of 1 brick = (1000 x 800 x 24.5) / (20 x 10 x 8) = 12250   12250nswer: 12250
Answer: A
In Richard's office one half of the workers are Indians,quarter are from America,one-fifth are from Canada and the other 100 are China. How many workers total are there in the company? Pick one.
A) 250	B) 500	C) 750	D) 1000	E) 2000

Solution: 1/2x + 1/4x +1/5x = 19/20x   Remaining 1/20 x= 100   x= 100*20= 
100*20 = 2_000
2000   2000 is the answer.
Answer: E
Rahim bought 32 books for Rs.1500 from one shop and 60 books for Rs.340 from another. What is the average price he paid per book ? Options
A) s.17
B) s.18
C) s.12
D) s.11
E) s.20

Solution: Average price per book   = (1500+ 340) / (32 + 60) = 1840/ 92   = Rs.20   Answer:s.20
Answer: E
Each day a man meets his wife at the train station after work, and then she drives him home. She always arrives exactly on time to pick him up. One day he catches an earlier train and arrives at the station an hour early. He immediately begins walking home along the same route the wife drives. Eventually his wife sees him on her way to the station and drives him the rest of the way home. When they arrive home the man notices that they arrived 30 minutes earlier than usual. How much time did the man spend walking?	Choose the correct answer:
A) 45 minutes	B) 50 minutes	C) 40 minutes	D) 55 minutes	E) 35 minutes

Solution: As they arrived 30 minutes earlier than usual, they saved 30 minutes on round trip from home to station (home-station-home) --> 15 minutes in each direction (home-station) --> wife meets husband 15 minutes earlier the usual meeting time --> husband arrived an hour earlier the usual meeting time, so he must have spent waking the rest of the time before their meeting, which is hour-15 minutes= 45 minutes.   45 minutesnswer: 45 minutes
Answer: A
How much time will a train of length 200 m moving at a speed of 72 kmph take to cross another train of length 300 m, moving at 36 kmph in the same direction?
Select
A) sec  B) sec  C) sec  D) sec  E) sec

Solution: The distance to be covered = Sum of their lengths = 200 + 300= 
200 + 300 = 500
500 m.   Relative speed= 72 -36= 
72 -36 = 36
36 kmph= 36 * 5/18= 
36 * 5/18 = 10
10 mps.   Time required= d/s = 500/10= 
500/10 = 50
50 sec.   secnswer: sec
Answer: A
How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall of 8 m x 6 m x 22.5 cm? Select the correct option
A) 5600	B) 6000	C) 6400	D) 7200	E) 8600

Solution: Number of bricks =Volume of the wall/Volume of 1 brick =	(800 x 600 x 22.5)/(25 x 11.25 x 6) = 6400.   Answer: Option 6400
Answer: C
A can do a work in 14 days. B is 40% more efficient than A. How many days does B alone take to do the same job?
Choices
A) 9  B) 10  C) 11  D) 12  E) 8

Solution: Ratio of times taken by A and 10 = 140:100 = 7:5   suppose 10 alone to take x days to do the work   then, 7:5::14:x   7x= 5*14   x=10 days   correct option is 10
Answer: B
Each light bulb at a hotel is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and eighty percent of the fluorescent bulbs are switched on. If 72 percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent? Choose the correct choice from the following answers
A) 24%
B) 16%
C) 11%
D) 7%
E) 5%

Solution: Let I be the number of incandescent bulbs.   Let F be the number of fluorescent bulbs.   0.4I + 0.8F = 0.72(I+F)   0.08F = 0.32I   F = 4I   This means that for every 1 incandescent bulb, there are 4 fluorescent bulbs.   The percent of bulbs that are switched on which are incandescent is:   0.4I / (0.4I + 0.8F) = 0.4I / (0.4I + 0.8*4I) = 0.4I / 3.6I = 1/9 which is about 11%.   The answer is 11%.
Answer: C
A crate of mangoes contains one bruised mango for every thirty mango in the crate. If three out of every four bruised mango are considerably unsaleable and there are 12 unsaleable mangoes in the crate then how many mango are there in the crate? Options:	A) 321	B) 430	C) 530	D) 172	E) 480

Solution: Let the total no of mangoes in the crate be x   Then the no of bruised mango = 1/30 x   Let the no of unsalable mangoes =3/4 (1/30 x)   1/40 x =12   x=480   Option 480
Answer: E
If -1 < x < 1, which of the following must be true?
I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2
Pick one:
A) I only
B) I and II only
C) II and III only
D) I and III only
E) I, II and III

Solution: I. x^3 < x^2 --> from -1 < x < 1 x must be a decimal so multiplying it by itself makes it a smaller value positive or negative. x^2 is always positive so x^3 being a small value always makes x^3<x^2 --="" 1="" <="" ii.="" is="" so="" statement="" this="" true.="" x="" x^5="" –=""> x(x^4+1) < 1 --> since 1(1+1)</x^2>=2 > 1, so this statement is false.   III. x^4 < x^2 --> reduce by x^2 (we can safely do that since from -1 < x < 1 is follows that x^2>0): x^2 < 1. Again, as -1 < x < 1, then x^2 must be less than 1. Hence, this statement is also true.   Answer: I and III only.
Answer: D
If # is an operation which results in adding the digits of integer until a single digit is left, what is the probability that a number picked up in first 90 positive integers will have the result of # as an odd digit ?
( example 99 = 9+9 = 18=1+8=9, so #=9)
Select:
A) 4/10
B) 4/9
C) 1/2
D) 6/10
E) 5/9

Solution: First of all let us find out what is the maximum sum we will get by adding digits of numbers from 1 to 90.   That will be 17 (8+9+17) Why did I calculate this? I will come to that in a moment.   So to get an odd digit by carrying out the operation #, the sum of the digits of the integer should be 1, 3, 5, 7, 9, 10, 12, 14, and 16 (Why not 18? Because we know the maximum sum can be 17)   Number of integers, whose digits add to 1: 1 and 10 = 2   Number of integers, whose digits add to 3: 3, 12, 21, 30 = 4   Number of integers, whose digits add to 5: 5, 14, 23, 32, 41, 50 = 6   Number of integers, whose digits add to 7: 7, 16, 25, 34, 43, 52, 61, 70 = 8   Number of integers, whose digits add to 9: 10 multiples of 9 up to 90 = 10   Number of integers, whose digits add to 10: 19, 28, 37, 46, 55, 64, 73, 82 = 8   Number of integers, whose digits add to 12: 39, 48, 57, 66, 75, 84 = 6   Number of integers, whose digits add to 14: 59, 68, 77, 86 = 4   Number of integers, whose digits add to 16: 79, 88 = 2   So in total there are  2+4+6+8+10+8+6+4+2= 
 2+4+6+8+10+8+6+4+2 = 50
50 such numbers   Probability= 50/90 = 5/9   Correct answer 5/9
Answer: E