divelab/Sys2Bench huggingface dataset:
Question: Josh spends a total of $5.50 buying N items in the convenience store. If each of the items is either a 5 cents single bubblegum, or a 50 cents bubblegum pack, then N may be which of the following?
Options:
A)99
B)100
C)101
D)109
E)140
Rationale: Let x be the number of single bubblegums and y be the number of bubblegum pack.
then
5x+50y=550
or x+10y=110
or x=110-10y
Now,
let y=0, then x= 110 and N=x+y or N=0+110 =110.
let y=1, then x=100 and N=100+1= 101 (given in the options)
let y=2, then x= 90 and N=92
the value will further decrease and hence we do not need to calculate anymore values.
ANSWER:- C
Answer: C
Question: Alice deposited Rs 2 million in bank at 4 % simple interest per year. After 10 year how many Rs she will get from bank?
Options:
A)75000
B)90000
C)10000
D)80000
E)60000
Rationale: i = (200000*10*4)/100
i = 80000
Answer: D
Answer: D
Question: By selling 33 metres of cloth , one gains the selling price of 11 metres . Find the gain percent .
Options:
A)96%
B)48%
C)50%
D)36%
E)45%
Rationale: (SP of 33m)-(CP of 33m)=Gain=SP of 11m
SP of 22m = CP of 33m
Let CP of each metre be Re.1 , Then, CP of 22m= Rs.22,SP of 22m=Rs.33.
Gain%=[(11/22)*100]%=50%
Ans: C
Answer: C
Question: There are many numbers that can be expressed as the sum of three squares in three different ways.
Can you find out the smallest of such number?
Options:
A)55
B)54
C)49
D)50
E)51
Rationale: B
54
7^2 + 2^2 + 1^1
6^2 + 3^2 + 3^2
2^2 + 5^2 + 5^2
Answer: B
Question: The average temperature for Monday, Tuesday, Wednesday and Thursday was 48 degrees and for Tuesday, Wednesday, Thursday and Friday was 46 degrees. If the temperature on Monday was 42 degrees. Find the temperature on Friday?
Options:
A)22
B)679
C)62
D)34
E)12
Rationale: M + Tu + W + Th = 4 * 48 = 192
Tu + W + Th + F = 4 * 46 = 184
M = 42
Tu + W + Th = 192 -42 = 150
F = 184 – 150 = 34.Answer: D
Answer: D
Question: Working alone, A can complete a certain kind of job in 10 hours. A and D, working together at their respective rates, can complete one of these jobs in 5 hours. In how many hours can D, working alone, complete one of these jobs?
Options:
A) 9
B) 10
C) 11
D) 12
E) 13
Rationale: Let total time taken by D to complete the job = D
total time taken by A to complete the job = 10
Work done by A in an hour
1/A = 1/10
Working together A and D can complete the job in 5 hours
1/A + 1/D = 1/5
=>1/D = 1/5 - 1/A
= 1/5 - 1/10
= 1/10
=> D = 10 hours
Answer B
Answer: B
Question: Susan, Tim,and Kim need to be seated in 3 identical chairs in straight line so that Susan is seated always left to Tim. How many such arrangements are possible ?
Options:
A)A)6
B)B)120
C)C)80
D)D)240
E)E)60
Rationale: Total number of arrangements = 3! = 6
In exactly half, Susan will be to the left of Tim, which gives us 3 arrangements
Option (A)
Answer: A
Question: Simplify 682 * 9
Options:
A)2459
B)4236
C)6895
D)6138
E)1478
Rationale: Explanation:
Although it is a simple question, but the trick is to save time in solving this.
Rather than multiplying it we can do as follows:
682 * (10 - 1) = 6820 - 682 = 6138
Answer: Option D
Answer: D
Question: Two cyclist start from the same places in opposite directions. One is going towards north at 18kmph and the other is going towards south 20kmph. What time will they take to be 76km apart?
Options:
A)1 1/4 hours
B)2 1/3 hours
C)4hours
D)3 3/4 hours
E)2 hours
Rationale: To be (18+20) km apart, they take 1 hour
To be 76 km apart, they take 1/38 * 76 = 2 hrs
Answer is E
Answer: E
Question: Calculate the L.C.M of 4/9, 5/ 7, 9/13, 7/15 is :
Options:
A)1230
B)1290
C)1260
D)1240
E)2260
Rationale: Required L.C.M =L.C.M. of 4, 5, 9, 7/H.C.F. of 9, 7, 13, 15 =1260/1=1260
Answer is C
Answer: C
Question: If x=2y=4z, what is y-z, in terms of x?
Options:
A)2x
B)x/2
C)x/4
D)3x/5
E)3x/7
Rationale: x=2y=4z
x/4 = z
and y = x/2
y-z = x/2 - x/4 = x/4
Answer is C
Answer: C
Question: In a garden, 26 trees are planted at equal distances along a yard 700 metres long, one tree being at each end of the yard. What is the distance between two consecutive trees?
Options:
A)10
B)28
C)12
D)14
E)16
Rationale: 26 trees have 25 gaps between them.
Length of each gap = 700/25 = 28
i.e., distance between two consecutive trees = 28
Answer is B.
Answer: B
Question: In a tree, 2/5 of the birds are robins while the rest are bluejays. If 1/3 of the robins are female and 2/3 of the bluejays are female, what fraction of the birds in the tree are male?
Options:
A)3/5
B)4/15
C)7/15
D)8/15
E)17/30
Rationale: The fraction of birds that are male robins is (2/3)(2/5)=4/15.
The fraction of birds that are male bluejays is (1/3)(3/5)=1/5.
The total fraction of male birds is 4/15 + 1/5 = 7/15.
The answer is C.
Answer: C
Question: The time taken by a man to row his boat upstream is twice the time taken by him to row the same distance downstream. If the speed of the boat in still water is 42 kmph, find the speed of the stream?
Options:
A)19 kmph
B)18 kmph
C)14 kmph
D)13 kmph
E)17 kmph
Rationale: The ratio of the times taken is 2:1.
The ratio of the speed of the boat in still water to the speed of the stream = (2+1)/(2-1) = 3/1 = 3:1
Speed of the stream = 42/3 = 14 kmph.
Answer: C
Answer: C
Question: For each month of a given year except December, a worker earned the same monthly salary and donated one-tenth of that salary to charity. In December, the worker earned N times his usual monthly salary and donated one-fifth of his earnings to charity. If the worker's charitable contributions totaled one-nineth of his earnings for the entire year, what is the value of N?
Options:
A)8/5
B)5/2
C)3
D)11/3
E)22
Rationale: Let monthly salary for each of the 11 months except December was x, then
11x*1/10 + Nx*1/5 = 1/9(11x + Nx);
11/10 + N/5 = 1/9(11 + N)=>
11+2N/10 = 11+N/9=>66+12N=110+10N =>2N=44
N = 22
Answer: E
Answer: E
Question: Zoe is interested in dividing her high school trophies into boxes so they can be put away in storage. Unfortunately, they do not divide equally. It turns out that in order for the boxes to each have an equal number of trophies, Zoe won't need one of the boxes and will have to add two of her college trophies to the last box. Which of the following can describe the number of initial high school trophies and the initial number of boxes (in the order trophies, boxes)?
Options:
A)26; 8.
B)50; 7.
C)27; 4.
D)25; 3.
E)45; 10.
Rationale: Option A.) Initially 26 trophies, increased to 28 trophies; and initially 8 boxes decreased to 7 boxes:
28 / 7 = 4 trophies per box.
Answer: A
Question: Two trains are traveling on a collision course. If train A is traveling at a speed of 350 mph and train B is traveling 28% slower, how much time will it take the trains to collide if the initial distance between the two is 2408 miles?
Options:
A)Two hours and 30 minutes.
B)One hour and 10 minutes.
C)Two hours and 25 minutes.
D)Three hours and 15 minutes.
E)Four hours and 0 minutes.
Rationale: Answer is E. Four hours and 0 minutes.
Since they are racing towards each other, the speed is accumulative, ie. 350 mph + 350*0.72 = 602 (0.72 because the second train is 28% slower)
So time before collision is total distance / total speed = 2408/602 = 4 hours and 0 mins
Answer: E
Question: Running 3/4th of his usual rate, a man is 15min late. Find his usual time in hours ?
Options:
A)2/3 hrs
B)3/4 hrs
C)1/3 hrs
D)1/4 hrs
E)1 hr
Rationale: Explanation:
Walking at 3/4th of usual rate implies that time taken would be 4/3th of the usual time. In other words, the time taken is 1/3rd more than his usual time
so 1/3rd of the usual time = 15min
or usual time = 3 x 15 = 45min = 45/60 hrs = 3/4 hrs.
ANSWER IS B
Answer: B
Question: The ages of two persons differ by 32 years. If 5 years ago, the elder one be 5 times as old as the younger one, their present ages (in years) are respectively
Options:
A)20,20
B)46,14
C)25,15
D)30,10
E)None of these
Rationale: Explanation:
Let their ages be x and (x + 32) years.
5 (x - 5) = (x + 32 - 5) or 4x = 52 or x = 14.
Their present ages are 46 years and 14 years
Option B
Answer: B
Question: A saleswoman makes a 5% commission on the selling price for each products she sells. If she sells 250 products and makes $750 in commission, what was the selling price of each product?
Options:
A)30
B)60
C)90
D)120
E)300
Rationale: Commission per one product: 750/250=3
3 = 5% of the selling price
3 = 0.05x
x = 60
Answer is B.
Answer: B
Question: What is the least number which when divided by 5, 6, 9 and 18 leaves remainder 4 in each care?
Options:
A)30
B)44
C)36
D)56
E)94
Rationale: Explanation:
LCM of 5, 6, 9 and 18 is 90
Required number = 90 + 4 = 94
Answer: Option E
Answer: E
Question: If it takes 30 construction workers 70 days to build a skyscraper, how long would it take 50 workers?
Options:
A)52 days
B)47 days
C)42 days
D)37 days
E)32 days
Rationale: 30 * 70 = 50 * x
x = 42
Answer: C
Answer: C
Question: Sides of a rectangular park are in the ratio 3: 2 and its area is 5766 sq m, the cost of fencing it at 50 ps per meter is?
Options:
A)287
B)1287
C)125
D)155
E)271
Rationale: 3x * 2x = 5766 => x = 31
2(93 + 62) = 310 m
310 * 1/2 = Rs.155
Answer: D
Answer: D
Question: If toaster X costs 10 percent more than toaster Y costs, which of the following COULD be true?
I. Toaster Y costs $10 less than toaster X costs
II. Toaster X costs $20 more than toaster Y costs
III. Toaster Y costs 10 percent less than toaster X costs
Options:
A)I only
B)I and II only
C)I and III only
D)II and III only
E)I, II and III
Rationale: 1. In case p=100 , Toaster X = 110
Toaster Y = 100
1 could be true .
2. In case p=200 ,
Toaster X = 220
Toaster Y = 200
2 could be true .
3. Never true .
Toaster Y will be (1/11)x100 percent lesser i.e 9.09 percent
Answer B.
Answer: B
Question: If x < 12, then it must be true that...
Options:
A)-x < -12
B)-x - 2 < 14
C)-x + 2 < -10
D)x + 2 < 10
E)x - 4 < 15
Rationale: A. -x < -12 --> x>12. Not true.
B. -x - 2 < 14 --> x>-16. Not true, consider x=-20.
C. -x + 2 < -10 --> x>12. Not true.
D. x + 2 < 10 --> x<8. Not true, consider x=10.
E. x - 4 < 15 --> x<19. Since given that x<12, then x is for sure less than 13. Thus this statement must be true.
Answer: E.
Answer: E
Question: Two trains are running in opposite directions with the same speed. If the length of each train is 420 m and they cross each other in 12 sec, then the speed of each train is?
Options:
A)11
B)26
C)126
D)99
E)27
Rationale: Let the speed of each train be x m/sec.
Then, relative speed of the two trains = 2x m/sec.
So, 2x = (420 + 420)/12 => x = 35
Speed of each train = 10 m/sec.
= 35 * 18/5 =126 km/hr.
Answer: C
Answer: C
Question: Six high school boys gather at the gym for a modified game of basketball involving three teams. Three teams of 2 people each will be created. How many ways are there to create these 3 teams?
Options:
A)15
B)30
C)42
D)90
E)108
Rationale: To put it in a different way:
Say 6 boys are a,b,c,d,e,f and we need to form 3 teams say t1,t2,t3
With a : a,b a,c a,d a,e a,f --->5
With b : b,c b,d b,e b,f --->4
With c : c,d c,e c,f --->3
With d : d,e d,f --->2
With e : e,f --->1
With f : --- --->0
Hence Total is 5+4+3+2+1+0 = 15.
ANSWER:A
Answer: A
Question: A number when divided by a divisor leaves a remainder of 24.
When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?
Options:
A)13
B)59
C)35
D)37
E)38
Rationale: Let the original number be 'a'
Let the divisor be 'd'
Let the quotient of the division of aa by dd be 'x'
Therefore, we can write the relation as a/d=xa/d=x and the remainder is 24.
i.e., a=dx+24
When twice the original number is divided by d,2a is divided by d.
We know that a=dx+24. Therefore, 2a=2dx+48
The problem states that (2dx+48)/dleaves a remainder of 11.
2dx is perfectly divisible by dd and will therefore, not leave a remainder.
The remainder of 11 was obtained by dividing 48 by d.
When 48 is divided by 37, the remainder that one will obtain is 11.
Hence, the divisor is 37.
Option (D)
Answer: D
Question: The average age of 20 men in the class is 15.6 years. Five new men join and the new average becomes 15.56 years. What was the average age of five new men?
Options:
A)15.5
B)15.4
C)15.25
D)15.3
E)15.6
Rationale: Total age of 20 men = 15.6 x 20 = 312
Now, total age of 25 men = 389.
Total age of five men added later = 389 - 312 = 77.
Hence, the total average of five men = 77/5 = 15.4
ANSWER:B
Answer: B
Question: A train 605 meters long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
Options:
A)51
B)64
C)11
D)22
E)33
Rationale: Speed of train relative to man = (60 + 6) km/hr = 66 km/hr
[66 * 5/18] m/sec = [55/3] m/sec.
Time taken to pass the man = [605 * 3/55] sec = 33 sec
Answer: E
Answer: E
Question: Tough and Tricky questions: Exponents.
If a and b are positive integers and x = 16^a and y = 9^b, which of the following is a possible units digit of xy?
Options:
A)1
B)6
C)5
D)7
E)8
Rationale: Only Units digit of X and Y contribute to the Units digit of XY.
Finding all values ofXfor positive values ofa
When,
a=1; X=16
a=2; X=256
a=3; X=4096........ Units digit repeat in a cycle of 6,6,6,6......
Similarly, Finding all values ofYfor positive values ofb
When,
b=1; Y=9
b=2; Y= 81
b=3; Y= 729....... Units digit repeat in cycle of 9,1,9,1....
So, possible value of Units digit of XY can be found by considering all possibilities of Units digit of X and Y
When X=6 and Y=9: Units digit of XY = 4
When X=6 and Y=1: Units digit of XY = 6
When X=6 and Y=9: Units digit of XY = 4
When X=6 and Y=1: Units digit of XY = 6
Only two possibilities for Units digit of XY i.e. 4 and 6.
Only 6 is given in answer choices, therefore answer is B
Answer: B
Question: The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:
Options:
A)101
B)107
C)111
D)185
E)196
Rationale: Let the numbers be 37a and 37b.
Then, 37a x 37b = 4107
ab = 3.
Now, co-primes with product 3 are (1, 3).
So, the required numbers are (37 x 1, 37 x 3) i.e., (37, 111).
Greater number = 111.
Answer: Option C
Answer: C
Question: A man takes 3 hours 45 minutes to row a boat 20 km downstream of a river and 2 hours 30 minutes to cover a distance of 3 km upstream. Find the speed of the current.
Options:
A)1 km/hr
B)2 km/hr
C)3 km/hr
D)4 km/hr
E)None of these
Rationale: Explanation:
First of all, we know that
speed of current = 1/2(speed downstream - speed upstream) [important]
So we need to calculate speed downstream and speed upstream first.
Speed = Distance / Time [important]
Speed upstream =(20/3 3/4)km/hr
=16/3km/hr
Speed Downstream = (3/2 1/2)km/hr
=6/5km/hr
So speed of current = 1/2(16/3−6/5)
=2km/hr
Option B
Answer: B
Question: A sells a bicycle to B at a profit of 60% and B sells it to C at a loss of 40%. Find the resultant profit or loss.
Options:
A)-4%
B)5%
C)-5%
D)6%
E)-7%
Rationale: The resultant profit or loss = 60 - 40 - (60*40)/100 = -4%
loss = 4%
Answer is A
Answer: A
Question: For integers a and b, 16a = 32^b. Which of the following correctly expresses a in terms of b?
Options:
A)a = 2^b
B)a = 4^b
C)a = 2^(5b − 4)
D)a = 4^(5b − 4)
E)a = 2^(5b)
Rationale: I tried this one with an example, although I am not quite sure if this is correct. I would appreciate if there would be a common rule for this.
I tried it out with b = 2 and only c correctly expresses the following:
16a = 32^b b = 2
16a = 1024
a = 64
choice c --> a = 2^6 = 64
However, this is a quite time consuming process. Is there something faster?
Answer C
Answer: C
Question: A, B and C invested Rs.6300, Rs.4200 and Rs.10500 respectively, in a partnership business. Find the share of A in profit of Rs.12100 after a year?
Options:
A)3630
B)2979
C)2789
D)2662
E)2123
Rationale: 6300:4200:10500
3:2:5
3/10 * 12100
= 3630
Answer:A
Answer: A
Question: A man has Rs. 412 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 ?
Options:
A)45
B)60
C)77.25
D)90
E)105
Rationale: Let number of notes of each denomination be x.
Then x + 5x + 10x = 412
16x = 412
x = 25.75.
Hence, total number of notes = 3x = 77.25.
Answer = C
Answer: C
Question: The average (arithmetic mean) of 8 numbers is Z, and one of the numbers is 14. If 14 is replaced with 28, then what is the new average (arithmetic mean) in terms of Z?
Options:
A)Z + 1/2
B)Z + 7/4
C)Z + 2
D)Z + 4
E)2Z + 1
Rationale: (8z−14+28)/8
Or, (8z+14)/8
Or, (4z+7)/4
Or, z+7/4
So, correct answer will be (B) z+7/4
Answer: B
Question: Find the odd man out. 1050, 510, 242, 107, 46, 16, 3
Options:
A)107
B)106
C)510
D)1050
E)None of these
Rationale: Explanation :
1050
(1050 - 30)/2 = 510
(510 - 26)/2 = 242
(242 - 22)/2 = 110
(110 - 18)/2 = 46
(46- 14)/2 = 16
(16- 10)/2 = 3
Hence, 110 should have come in place of 107. Answer : Option A
Answer: A
Question: A plane flying north at 500 kmph passes over a city at 12 noon. A plane flying east at the same attitude passes over the same city at 12.30 pm. The plane is flying east at 400 kmph. To the nearest hundred km, how far apart are the two planes at 2 pm?
Options:
A)600 km
B)1000 km
C)1100 km
D)1200 km
E)1300 km
Rationale: The plane flying north would have traveled for 2 hours at 2 PM.
It travels at 500 kmph.
Hence, the plane flying north would have covered 1000 kms from the city at 2 PM.
The plane flying east would have traveled for 1.5 hours at 2 PM.
It travels at 400 kmph.
Hence, the plane flying east would have covered 1.5*400 = 600 kms from the city at 2 PM.
The two planes are flying at right angles to each other.
The shortest distance between the two planes is the length of the hypotenuse of the right triangle whose legs are the distances covered by the two planes.
Hence, distance = 1000square + 200 square = 1166 km
The nearest 100 km distance between the two planes = 1200 kms.
Answer D
Answer: D
Question: John started driving on a highway at a constant speed of R miles per hour at 13:00. Then, 2 hours later, Tom started driving on the same highway at a constant speed of 2R miles per hour at 15:00. If both drivers maintained their speed, how many miles had John driven on the highway when Tom caught up with John?
Options:
A)2R
B)3R
C)4R
D)5R
E)6R
Rationale: In 2 hours, John drove 2R miles.
Tom can catch John at a rate of 2R - R = R miles per hour.
The time it takes Tom to catch John is 2R / R = 2 hours.
In 2 hours, John can drive another 2R miles for a total of 4R miles.
The answer is C.
Answer: C
Question: If n=5p where p is a prime number greater than 1, how many different positive even divisors does n have, including n?
Options:
A)1
B)2
C)32
D)4
E)3
Rationale: N = 5P and that P is a prime number greater than 1. Let's TEST P = 2; so N = 10
The question now asks how many DIFFERENT positive EVEN divisors does 10 have, including 10?
10:
1,10
5,2
How many of these divisors are EVEN? 2, 10…..2 even divisors.
ANSWER:B
Answer: B
Question: Before January, the price of a dress was D and the price of a matching pair of shoes was H. In January, the price of the dress increased by 40% and the price of the shoes increased by 50%. In March, Roberta bought both items with a 30% discount. If D = 5H, which of the following represents the amount that Roberta paid?
Options:
A)D + 40
B)D + H – 1
C)D + 2H
D)5.95H
E)1.21D
Rationale: Before Discount price was: 1.4D+1.5H
After Discount: 1.4D +1.5H-.42D-.45H
substitute D=5H
final value: 5.95H
Ans Is D : 5.95H
Answer: D
Question: In march Mrs Bil's earnings were 80 percent of the Bil family's total income. In april Mrs Bil earned 10 percent more than in May. If the rest of the family's income was the same both months, then, in June, Mrs Bil's earnings were approximately what percent of the Bil family's total income ?
Options:
A)83
B)56
C)34
D)12
E)26
Rationale: Lets say the Family Income is 100
In May, lee earned 80Family income is 20
In june, Lee earned 20% more than May, so it is (80+20*80/100 = 96)
Family Income is same 20
In June Lee's Income percent is 96*100/116 ~ 83
Ans is A
Answer: A
Question: A started a business with Rs. 18000. After 4 months B joins with Rs. 24000. After 2 more months C joins with Rs. 30000. At the end of 10 months C received Rs. 1850 as his share. Find the total profit.
Options:
A)Rs. 7955
B)Rs. 7030
C)Rs. 8510
D)Rs. 6845
E)None of these
Rationale: Explanation:
Ratio of capitals of A, B and C
= (18000 × 10) : (24000 × 6) : (30000 × 4)
= 15 : 12 : 10
Let the total profit be Rs. x.
C's share = Rs. 10x/37
=> 10x/37 = 1850
=> x (1850 × 37)/10 = 6845
Hence, the total profit = Rs. 6845
Answer: Option D
Answer: D
Question: A beggar on the street can make one cigarette out of every 6 cigarette butts he finds. After one whole day of searching and checking public ashtrays the beggar finds a total of 72 cigarette butts. How many cigarettes can he make and smoke from the butts he found?
Options:
A)4
B)10
C)14
D)16
E)8
Rationale: If the begger can make a whole cigarette from 6 butts then he can make 12 cigarettes from the 72 he found. Once he smokes those, he then will have another 12 butts, which gives him enough to make another 2 cigarettes 12+2=14. A total of 14.
Answer: C
Question: Which of the following numbers is between 2⁄3 and 1⁄2?
Options:
A).45
B).35
C).29
D).22
E).55
Rationale: Isn't it E?
2/3 = .66
1/2 = .50
The only answer between that is E, .55.
Answer: E
Question: What is the dividend. divisor 17, the quotient is 9 and the remainder is 5?
Options:
A)130
B)134
C)148
D)158
E)160
Rationale: D = d * Q + R
D = 17 * 9 + 5
D = 153 + 5
D = 158
Answer: D
Question: Bruce and Anne can clean their house in 4 hours working together at their respective constant rates. If Anne’s speed were doubled, they could clean their house in 3 hours working at their respective rates. How many W hours does it currently take Anne to clean the house on her own?
Options:
A)6
B)7
C)8
D)12
E)14
Rationale: Lets suppose Anne and Bruce take A and B hrs working separately
So in 1 hour they can together finish 1/A + 1/B portion of the work which equals 1/4 (As the work is completed in 4 hours)
After anne doubles her rate of work the portion completed by the both is 1/A +2/B which is W equal to 1/3 (As the work is completed in W=3 hours)
Solving these 2 equations we can find B as 12
So, D
Answer: D
Question: Find (7x + 4y ) / (x-2y) if x/2y = 5/2 ?
Options:
A)13
B)25
C)26
D)27
E)29
Rationale: x/2y=3/2
=>x=10y/2=5y
=>(7x+4y)/(x-2y)=((7*(5y))+4y)/(5y-2y)
=>39y/3y=13
ANSWER:A
Answer: A
Question: If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people who are married to each other, how many B such committees are possible?
Options:
A)B=20
B)40
C)50
D)B=80
E)120
Rationale: Each couple can send only onerepresentativeto the committee. Let's see in how many ways we can choose 3 couples (as there should be 3 members) each to send onerepresentativeto the committee: 5C3=10.
But these 3 chosen couples can send two persons (either husband or wife): 2*2*2=2^3=8.
Total # of ways: 5C3*2^3=80.
Answer: D.
Answer: D
Question: There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. Let A, B and C be three distinct points on the perimeter of the outer circle such that AB and AC are tangents to the inner circle. If the area of the outer circle is 12 square centimetres then the area (in square centimetres) of the triangle ABC would be
Options:
A)π√​12
B)9/π
C)9√3/π square centimetres
D)6√3/π
E)None of these
Rationale: Explanation :
Outer circle:
Since πR² = 12, R = √(12/π) = 2√(3/π).
Inner circle:
Since the area is 1/4 the area of the outer circle, πr² = 3 and r = √(3/π).
AO is the Radius of the outer circle and OD is the radius of the inner circle.
A line tangent to a circle and a radius drawn to the point of tangency form a right angle.
Thus, ∆AOD is a right triangle.
Hypotenuse AO is twice the length of leg OD.
A right triangle whose hypotenuse is twice the length of a leg is a 30-60-90 triangle.
Thus, ∆AOD is a 30-60-90 triangle.
In a 30-60-90 triangle, the sides are proportioned x : x√3 : 2x.
Thus, AD = √3 * √(3/π) * = 3/√π.
Using the same reasoning for ∆OCD, we can deduce that CD = 3/√π.
Thus, AC = 2 * 3/√π = 6/√π.
The formula for the area of an equilateral triangle = (s²√3)/4.
Since AC = 6/√π, the area of ∆ABC = (6/√π)² * √3/4 = 36/π * √3/4 = (9√3)/π.
Answer : C
Answer: C