MU-NLPC/Calc-aqua_rat huggingface dataset:
If 6 men and 8 boys can do a piece of work in 10 days while 26 men and 48 boys can do the same in 2 days, the time taken by 15 men and 20 boys in doing the same type of work will be: Choose the correct choice from the following options
A) 7 B) 9 C) 4 D) 13 E) None

Solution: Let 1 man's 1 day's work = x and 1 boy's 1 day's work = y   Then, 6x + 8y = 1/10 and 26x + 48y = 1/2   Solving these two equations, we get : x = 1/100 and y = 1/200   15 men + 20 boy)'s 1 day's work = (15/100 + 20/200) = 1/4   15 men and 20 boys can do the work in 4 days.   Answer: 4
Answer: C
Different breeds of dogs get older at different rates in “dog years.” Livonian wolfhounds age 6 times as fast as humans, whereas Khazarian terriers age 5 times as fast and Akkadian retrievers age 4 times as fast. If Dan bought a newborn Akkadian on January 1, 2010, a newborn Khazarian 1 year later, and a newborn Livonian 1 year after that, in what year will the sum of the dog-year ages of the Akkadian and the Khazarian first be exceeded by twice the age of the Livonian in dog years, rounding all ages down to the nearest integer?  Pick one:
A) 2017  B) 2014  C) 2015  D) 2016  E) 2013

Solution: chart starting at 2011: A=4   2012 = A=8 K=5 A+K =13   L=0.   Then I added 9 to a+k for each year and 12 to L for each year.   2013:2017K=22 L=12   2014:2017K=31 L=24   2015:2017K=40 L=36   2016 :2017K=49 L=48   2017: 2017K=59 L=60   2017
Answer: A
If the sides of a triangle are 196 cm, 81 cm and 277 cm, what is its area?
Choose the correct choice from the following answers
A) 5000 B) 5656 C) 7878 D) 7900 E) 7938

Solution: The triangle with sides 196 cm, 81 cm and 277 cm is right angled, where the hypotenuse is 277 cm.   Area of the triangle = 1/2 * 81 * 196= 
1/2 * 81 * 196 = 7_938
7938 cm2   Answer: Option 7938
Answer: E
Presently a mixture in a tub contains substances A and B in the ratio of 7:5. If 9 litres of this solution is drawn out and B is added to the mixture in certain quantity the resulting mixture becomes 7:9. How much litres of liquid A was kept in the tub initially? Choose the correct choice from the following choices:  A) 87  B) 265  C) 88  D) 21  E) 91

Solution: Explanation:   Currently the mixture is in ratio of A/B: 7:5   => This means fraction of A in the solution: 7/12   => Fraction of B in the solvent: 5/12   =>New concentration of A= 7-(7/12)9   =>New concentration of B= 5-(5/12)9+x   => (7-(7/12)9)/(5-(5/12)9+x) = 7/9   =>x=21   ANSWER: 21
Answer: D
For each hour of production, a certain factory requires 1 assembly line worker for every 25 units to be produced in that hour. The factory also requires 14 managers regardless of the number of units to be produced. Which of the following expressions represents the total number of assembly line workers and managers that this factory requires to produce 50N in one hour, where N is a positive integer? Choose the correct option:
A) 12 + 50N	B) 12 + 2N	C) 62N	D) 37N	E) 14 + 2N

Solution: 25 units = 1 worker;   50N units = 50N/25 = 2N workers.   So, the answer is 2N workers plus 14 managers.   Answer: 14 + 2N.50 /25= 
50 /25 = 2
2N workers.   So, the answer is 2N workers plus 14 managers.   Answer: 14 + 2N.
Answer: E
A and B can do a work in 25 days and 10 days respectively. A starts the work and B joins him after 2 days. In how many days can they complete the remaining work?  Choose the correct choice from the following answers.
A) 4.28 days
B) 4.38 days
C) 5.28 days
D) 6.28 days
E) 2.28 days

Solution: Work done by A in 2 days = 2/5   Remaining work = 3/5   Work done by both A and B in one day = 1/25 + 1/10 = 7/50   Remaining work = 3/5 * 50/7 = 4.28 days.   4.28 daysNSWER:4.28 days
Answer: A
A 20 litre mixture of milk and water contains milk and water in the ratio 3 : 2. 10 litres of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removal and replacement, what is the ratio Q of milk and water in the resultant mixture?	Choose the correct choice from the following answers.
A) 17 : 3  B) 9 : 1  C) 3 : 17  D) 5 : 3  E) 11: 2

Solution: he 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, there will be 12 litres of milk in the mixture and 8 litres of water in the mixture.   Step 1.   When 10 litres of the mixture is removed, 6 litres of milk is removed and 4 litres of water is removed. Therefore, there will be 6 litres of milk and 4 litres of water left in the container. It is then replaced with pure milk of 10 litres. Now the container will have 16 litres of milk and 4 litres of water.   Step 2.   When 10 litres of the new mixture is removed, 8 litres of milk and 2 litres of water is removed. The container will have 8 litres of milk and 2 litres of water in it. Now 10 litres of pure milk is added. Therefore, the container will have 18 litres of milk and 2 litres of water in it at the end of the second step. Therefore, the ratio of milk and water is 18 : 2 or 9 : 1.   Shortcut.   We are essentially replacing water in the mixture with pure milk.   Let W_o be the amount of water in the mixture originally = 8 litres.   Let W_r be the amount of water in the mixture after the replacements have taken place.   Then,{W_r}/{W_o}= (1-R/M)^n   where R is the amount of the mixture replaced by milk in each of the steps, M is the total volume of the mixture and n is the number of times the cycle is repeated.   Hence, {W_r}/{W_o} =(1/2)^2  =1/4   Therefore Q,W_r ={W_o}/4= 8/4= 
8/4 = 2
2 litres.9 : 1
Answer: B
The ratio of the ages of Anil and his son at present is 7:3. Six years hence, the ratio of the ages of the Anil's wife and the son will be 2:1. Find the ratio of the present ages of Anil and his wife?
Choices.
A) 2:2  B) 2:9  C) 2:4  D) Cannot be determined  E) None of these

Solution: Let the present ages of Anil and his son be 7x and 3x years respectively.   Let the present age of the son's mother be m years.   (m + 6)/(3x + 6) = 2/1   m = 6x + 6   Ratio of the present ages of Anil and his son's mother = 7x/(6x + 6).   This cannot be found uniquely.   Answer:Cannot be determined
Answer: D
Twenty six years ago,a father was three times as old as his son.Now the father is only twice as old as his son. Then the sum of the present ages of the son and the father is :
Select:
A) 120
B) 156
C) 108
D) 105
E) 86

Solution: Let the present ages of the father and son be 2x and x years respectively.   then,(2x-26)=3 (x-26)   x=52.   Required sum =(2x+x)=3x=156 years.   Answer is 156
Answer: B
55% of respondents to a survey consisted of those who used both M and N and those who used neither M nor N. 47% of respondents did not use M. 40% used N. What percent of respondents used only N?	Choose the correct choice:
A) 16%
B) 21.5%
C) 20%
D) 18.5%
E) 17.5%

Solution: From the question, we can summarize the information as follows:   N Not N   M X   Not M 16%ns y 47   40 100   Then we can set a system of 3 equations 3 unknowns described below:   X + Y = 55   X + 16%ns = 40   Y + 16%ns = 47   Sum up all three equations to get 2X + 2Y + 216%ns = 142   Divide both sides by 2 to get X + Y + 16%ns = 71   Replace X + Y with 55 to get   55 + 16= 
  55 + 16 = 71
71   Substract 55 from both sides to get 16%ns= 71 - 55= 
71 - 55 = 16
16   The answer must be 16%.
Answer: A
A palindrome is a word or a number that reads the same forward and backward. For example, 2442 and 111 are palindromes. If 5-digit palindromes are formed using one or more of the digits 1, 2, 3, 4, 5, and 6, how many palindromes are possible?  Choices.
A) 36
B) 96
C) 132
D) 216
E) 256

Solution: There are 6 choices for each of the first three digits.   The number of possible palindromes is 6^3=216.   The answer is 216.
Answer: D
A train 360 m long is running at a speed of 45 km/hr. In what time will it pass a bridge 140 m long? Pick one
A) 40 sec	B) 32 sec	C) 12 sec	D) 85 sec	E) 56 sec

Solution: 40 secnswer: Option 40 sec   Speed = 45 * 5/18 = 25/2 m/sec   Total distance covered = 360 + 140= 
360 + 140 = 500
500 m   Required time= 500 * 2/25= 
500 * 2/25 = 40
40 sec
Answer: A
A and B walk around a circular track. They start at 8 a.m. from the same point in the opposite directions. A and B walk at a speed of 2 rounds per hour and 3 rounds per hour respectively. How many times shall they cross each other before 9.30 a.m. ?  Pick: A) 9.38 a.m B) 9.36 a.m C) 9.30 a.m D) 9.10 a.m E) 9.40 a.m

Solution: Relative speed = 2 + 3= 
2 + 3 = 5
5 rounds per hour.   So, they cross each other 5 times in an hour and 2 times in half an hour.   Hence, they cross each other 7 times before 9.30 a.m.   Answer:9.30 a.m
Answer: C
Find the principle on a certain sum of money at 5% per annum for 2 2/5 years if the amount being Rs.1120?  Answers:
A) 1000
B) 289
C) 209
D) 280
E) 892

Solution: 1120 = P [1 + (5*12/5)/100]   P= 1000   1000nswer: 1000
Answer: A
One train is traveling 45 kmph and other is at 10 meters a second. Ratio of the speed of the two trains is?	Pick:
A) 5:4
B) 2:8
C) 5:1
D) 5:2
E) 5:9

Solution: 45 * 5/18 = 10   25:20 => 5:4   5:4nswer:5:4
Answer: A
The sum of the ages of a father and son is 45 years. Five years ago, the product of their ages was four times the fathers age at that time. The present age of father and son  Choose the correct option:
A) 34,11 B) 35,10 C) 36,9 D) 40,5 E) 41,5

Solution: Explanation:   Let sons age = x years. Then fathers age = (45 - x)years.   (x—5)(45—x—5) = 4(45- x - 5) hence (x—5) = 4 so x = 9   Their ages are 36 years and 9 years.   Answer: Option 36,9
Answer: C
A train traveling at 100 kmph overtakes a motorbike traveling at 64 kmph in 18 seconds. What is the length of the train in meters? Pick:
A) 180 meters  B) 1111 meters  C) 1777 meters  D) 1822 meters  E) None of these

Solution: Train overtakes a bike means that we are talking about total length of the train. (train's head is close to bike when it started and its tail crosses the bike when it overtakes the bike)   Relative speed= 100-64= 
100-64 = 36
36Km/h= 36000 m/h   time= 18 seconds   Distance= speed*time    36000* 18/3600= 
 36000* 18/3600 = 180
180 meters.   180 meters is the answer.
Answer: A
A, B and C can do a work in 7 , 14 and 21 days respectively. They completed the work and got Rs.242. What is the share of C?
Choose the correct choice from the following choices
A) 33
B) 44
C) 55
D) 11
E) 31

Solution: The ratio of their working rates = 1/7 : 1/14 : 1/21 = 6 : 3 : 2.   Since, they work together, the share of C   = 2/11 * 242 = Rs.44   Answer: 44
Answer: B
How many times the keys of a writer have to be pressed in order to write first 50 counting no's?	Options.
A) 90 B) 91 C) 92 D) 93 E) 94

Solution: 1 to 9 = 9 * 1= 
9 * 1 = 9
9   10 to 50= 41 * 2 = 84   -----------   93   93
Answer: D
An electric motor makes 3,000 revolutions per minutes. How many degrees does it rotate in one second?	Choose the correct choice from the following: A) 19,000 degrees / second B) 17,000 degrees / second C) 16,000 degrees / second D) 18,000 degrees / second E) 28,000 degrees / second

Solution: 3000 revolutions / minute   = 3000*360 degrees / 60 seconds   = 18,000 degrees / second.   Answer is 18,000 degrees / second.
Answer: D
Two pipes A and B can fill a tank in 10 hrs and 40 hrs respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?	Answers:
A) 8 hours	B) 6 hours	C) 4 hours	D) 2 hours	E) None of these

Solution: Explanation :   Pipe 8 hours can fill 1⁄10 of the tank in 1 hr   Pipe B can fill 1⁄40 of the tank in 1 hr   Pipe 8 hours and B together can fill 1⁄10 + 1⁄40 = 1⁄8 of the tank in 1 hr   i.e., Pipe 8 hours and B together can fill the tank in 8 hours .8 hoursnswer : Option 8 hours
Answer: A
If 6th March, 2021 is Thursday, what was the day of the week on 6th March, 2020?	Choose the most appropriate option: A) Sunday B) Saturday C) Tuesday D) Wednesday E) None of these

Solution: Solution   The year 2020 is a leap year. So, it has 2 odd days.   But, Feb 2020 not included because we are calculating from March 2020 to March 2021. So it has 1 odd day only.   The day on 6th March, 2021 will be 1 day beyond the day on 6th March, 2020. Given that, 6th March, 2021 is Thursday.   6th March, 2020 is Wednesday (1 day before to 6th March, 2021)   Answer Wednesday
Answer: D
Simran started a software business by investing Rs. 50,000. After six months, Nanda joined her with a capital of Rs. 80,000. After 3 years, they earned a profit of Rs. 24,500. What was Simran's share in the profit?	Select.
A) Rs.10528
B) Rs.10521
C) Rs.10511
D) Rs.10411
E) Rs.10500

Solution: Simran : Nanda = (50000 * 36) : (80000 * 30) = 3:4   Simran's share = 24500 * 3/7   = Rs.10500   Answer: Rs.10500
Answer: E
How many distinct four-digit numbers can be formed by the digits {1, 2, 3, 4, 5, 5, 6, 6}? Options.
A) 280	B) 360	C) 486	D) 560	E) 606

Solution: Option 1: each group of 4 different digits will produce 4!=24 four-digit numbers   - Option 2: each group of 4 digits that contains two 5's will produce 4!/2=12 four-digit numbers   - Option 3: same with option 2 -> 4!/2=12 four-digit numbers   - Option 4: 4!/4=6 four-digit numbers   In total, there are: 15*24 + 10*12 + 10*12 + 1*6= 
15*24 + 10*12 + 10*12 + 1*6 = 606
606 four-digit numbers.   ANSW606R:606
Answer: E
A train 360 m long is running at a speed of 72 km/hr. In what time will it pass a bridge 140 m long?
Pick one
A) 40 sec B) 29 sec C) 25 sec D) 27 sec E) 34 sec

Solution: Speed = 72 * 5/18= 
72 * 5/18 = 20
20 m/sec   Total distance covered= 360 + 140= 
360 + 140 = 500
500 m   Required time= 500 * 1/20= 
500 * 1/20 = 25
25 sec   Answer: 25 sec
Answer: C
The length of a rectangle is reduced by 22%. By what % would the width have to be increased to maintainthe original area?
Answers:  A) 15%  B) 20%  C) 28%  D) 35%  E) 40%

Solution: Sol. Required change = (22*100)/(100-22)=28%   28%
Answer: C
The profit earned by selling an article for Rs. 832 is equal to the loss incurred when the same article is sold for Rs. 448. What should be the sale price for making 50% profit? Choose the correct answer:  A) 260.  B) 960.  C) 968.  D) 9690.  E) 260.

Solution: Let C.P. = Rs. x.   Then, 832 - x = x - 448   2x = 1280 => x = 640   Required S.P. = 150% of Rs. 640 = 150/100 * 640 = Rs. 960.   Answer:960.
Answer: B
The sum of the ages 4 members of a family 5years ago was 94.Today,when the daughter has been married off and replaced by a daughter-in-law,the sum of their ages is 100.Assuming that there has been no other change in the family structure and all members are alive,what is the difference in the ages of daughter and daughter-in-law ?  Answers:
A) 22 B) 11 C) 25 D) 14 E) 16

Solution: Solution:sum of ages of 4 members 5 years ago=94   =>sum of present ages of 4 members=94+ 4*5= 
94+ 4*5 = 114
114   14ifference in the sum of the ages=14ifference in the ages of daughter and daughter-in-law   14ifference in the sum of the ages=114-100= 
114-100 = 14
14=>14ifference in the ages of daughter and daughter-in-law=14   Answer 14
Answer: D
In a Lake, there is a patch of lily pads. Every day, the patch doubles in size. It takes 50 days for the patch to cover the entire lake, how many days would it take the patch to cover half of the lake?
Choose the correct choice from the following options:  A) 36  B) 2^4*3  C) 24  D) 38  E) 49

Solution: So 49 days   Answer 49=49
Answer: E
In a garden, there are three blue flowers, three blue flowers, three yellow flowers, and three grey flowers. What is the probability that a girl will choose three flowers of the same color when randomly picking three flowers? Choices: A) 4/55 B) 3/55 C) 7/55 D) 4/33 E) 1/55

Solution: P(all the same color) = P(1st flower is ANY color AND 2nd flower is same as first AND 3rd flower is also the same color)   = P(1st flower is ANY color)xP(2nd flower is same as 1st)ANDP(3rd flower is the same color)   = 1x2/11x1/10   = 1/55   Answer: 1/55
Answer: E
The radius of a cylinder is 2 r units and height is 3 r units. Find the curved surface? Choose one:
A) 27	B) 16	C) 12	D) 18	E) 11

Solution: 2 * π * 2r * 3r = 12 πr2   Answer:12
Answer: C
j, b, and c are positive integers. If j, b, and c are assembled into the six-digit number jbcjbc, which one of the following must be a factor of jbcjbc?  Choices:
A) 16	B) 13	C) 5	D) 3	E) none of the above

Solution: Plug in some values and check -   jbcjbc = 123123   Not divisible by 16 and 5   let jbcjbc = 125125   Not divisible by 3   Only option (B) and (E) is left in both the cases...   Check once more to marke (13) as correct answer   let jbcjbc = 135135   Again divisible by 13   So, mark answer as (13) 13
Answer: B
A car traveling at a certain constant speed takes 2 seconds longer to travel 1 kilometer than it would take to travel 1 kilometer at 90 kilometers per hour. At what speed, in kilometers per hour, is the car traveling?  Choose the correct choice from the following answers
A) 71.5  B) 72  C) 72.5  D) 73  E) 85.7

Solution: Many approaches are possible, one of them:   Let the distance be 1 kilometer.   Time to cover this distance at 90 kilometers per hour is 1/90 hours = 3,600/90 seconds = 40 seconds;   Time to cover this distance at regular speed is 40+2= 
40+2 = 42
42 seconds= 42/3,600 hours = 7/600 hours;   So, we get that to cover 1 kilometer 1/(7/600) hours is needed --> regular speed 85.7 kilometers per hour (rate is a reciprocal of time or rate=distance/time).   Answer: 85.7.
Answer: E
If Rs.540/- are divided among A,B and C in such a way that A’s share is ½nd of B share and B’s share is 1/3rd of C’s share. The share of A is?  Choose the correct choice from the following:  A) Rs.40/-  B) Rs.40/-  C) Rs.60/-  D) Rs.70/-  E) Rs.90/-

Solution: A:B:Rs.60/- = 1:2:6   Total parts = 9   A's share is = 1 parts   9 -----> Rs.540/-   1 -----> Rs.60/-   Answer: Option 'Rs.60/-'
Answer: C
Trains A and B start simultaneously from stations 200 miles apart, and travel the same route toward each other on adjacent parallel tracks. If Train A and Train B travel at a constant rate of 20 miles per hour and 20 miles per hour, respectively, how many miles will Train A have traveled when the trains pass each other, to the nearest mile?
Pick one: A) 100 B) 133 C) 150 D) 167 E) 188

Solution: Since we know the distance (200) and the combined rate (40), we plug it into the formula:   Distance = Rate * Time   200 = 40 * Time   We can solve for the time they will meet cause we added the rate of Train A and Train B together.   So the time will be 200/40 from dividing 40 on both sides to isolate Time in the equation above.   Time will be 5 hours so now you can plug that in for Train A’s distance.   Distance = Rate * Time   Distance = 20 * 5= 
20 * 5 = 100
100 according to answer choice 100.
Answer: A
Caleb spends $66.50 on 50 hamburgers for the marching band. If single burgers cost $1.00 each and double burgers cost $1.50 each, how many double burgers did he buy? Select:
A) 5  B) 33  C) 20  D) 40  E) 45

Solution: Solution -   Lets say, Single hamburgersxand Double hamburgersy   Given that,   x+y=50 and 1x+1.5y=66.50. By solving the equations y=33. ANS 33.
Answer: B
If a train, travelling at a speed of 90 kmph, crosses a pole in 5 sec, then the length of train is?  Pick:
A) 281 m	B) 112 m	C) 117 m	D) 125 m	E) 123 m

Solution: 125 m   125 m = 90 * 5/18 * 5= 
90 * 5/18 * 5 = 125
125 m
Answer: D
A certain ball team has an equal number of right- and left-handed players. On a certain day, two-thirds of the players were absent from practice. Of the players at practice that day, two-third were right handed. What is the ratio of the number of right-handed players who were not at practice that day to the number of left handed players who were not at practice?	Choose the correct choice from the following:
A) 1/3 B) 2/3 C) 5/7 D) 7/5 E) 3/2

Solution: Say the total number of players is 18, 9 right-handed and 9 left-handed.   On a certain day, two-thirds of the players were absent from practice --> 12 absent and 6 present.   Of the players at practice that day, one-third were right-handed --> 6*2/3=4 were right-handed and 2 left-handed.   The number of right-handed players who were not at practice that day is 9-4 --  6*2/3= 
 --  6*2/3 = 4
4 were right-handed and 2 left-handed.   The number of right-handed players who were not at practice that day is  9-4= 
 9-4 = 5
5.   The number of left-handed players who were not at practice that days is   9-2= 
  9-2 = 7
7.   The ratio= 5/7.   Answer: 5/7
Answer: C
A certain gallery is hanging a series of 6 paintings. All the paintings will be exhibited in a row along a single wall. Exactly 2 of the paintings are on panel, the remainder are on canvas. In how many ways can the paintings be exhibited if the works on panel must be the second and sixth in the row?
OE	Pick:
A) 240  B) 200  C) 122  D) 80  E) 36

Solution: We need to consider them separately because the question specifies:   1. Panel paintings have to be placed only at 2nd, 3nd and 6th place.   2. Implies that Canvas has to be placed 1,4,5th place.   So we have two separatebucketsto consider.   So panels can be ordered in 3!   Canvas can be ordered in 3!   since relative position of panels to canvas is fixed (Panels 2nd and 6th) the answer is 3! x 3! = 36   36
Answer: E
In how many ways can five boys and three girls sit in a row such that all boys sit together?	Choose the correct choice from the following answers.
A) A)2888	B) B)2885	C) C)2880	D) D)2876	E) E)2224

Solution: Treat all boys as one unit. Now there are four students and they can be arranged in 4! ways. Again five boys can be arranged among themselves in 5! ways.   Required number of arrangements = 4! * 5! = 24 * 120= 
24 * 120 = 2_880
2880.   Answer:2880
Answer: C
If n is a positive integer, what is the maximum possible number of prime numbers in the following sequences: n + 1, n + 2, n + 3, n + 4, n + 5, and n + 6? Options: A) 2 B) 3 C) 4 D) 5 E) 6

Solution: for n=1,   the numbers are 2,3,4,5,6,7   out if these 2,3,5,7 these 4 are prime numbers.   now for any value of n >1 all six consecutive numbers will be 3 odd and 3 even. so at max we will get 3 prime numbers(assuming all the odd numbers are prime)   so the ans is 4   ANSWER:4
Answer: C
Rahul's housing colony have One tap, working alone at its maximum constant rate, can fill a certain tank in 230 seconds. A second tap, working alone at its maximum constant rate, can fill the same tank in 234 seconds. If both taps work together at their respective maximum constant rates, the time it will take to fill the tank is closest to
Choose one.
A) 116 seconds
B) 150 seconds
C) 177 seconds
D) 200 seconds
E) 270 seconds

Solution: Tap 1 : 230 secs   Tap 2 : 234 secs   Considering the average of these 2 taps : 232 secs.   So 1 tap can fill the tank in 232 secs, so 2 taps can fill the tank in 232/2= 
232/2 = 116
116 secs.   Closest answer is 116 seconds.
Answer: A
When the no.7y86038 is exactly divisible by 11, then the smallest whole no.in place of y?
Choose one.
A) 3  B) 4  C) 6  D) 7  E) 9

Solution: The given number =7y86038   Sum of the odd places =8+0+8+7= 
8+0+8+7 = 23
23   Sum of the even places= 3+6+y   (Sum of the odd places)- (Sum of even places) = Number (exactly divisible by 11)   23-(9+y) = divisible by 11   14 � y = divisible by 11.   Y must be 3, to make given number divisible by 11.   3
Answer: A
In the Land of Oz only one or two-letter words are used. The local language has 66 different letters. The parliament decided to forbid the use of the seventh letter. How many words have the people of Oz lost because of the prohibition?  Pick one:
A) 65 B) 66 C) 67 D) 131 E) 132

Solution: The seventh letter (Let it be @) can be used with other 65 in two type of arrangents   one when seventh letter is in 1st slot @_ (65 ways)   another, when seventh letter is in 2nd slot _@ (65 ways)   seventh letter (@) itself   Lost words = 1 + 65 + 65= 
1 + 65 + 65 = 131
131   ANSWER:131
Answer: D