mathewhe/aqua_rat huggingface dataset:
Question: Two friends plan to walk along a 43-km trail, starting at opposite ends of the trail at the same time. If Friend P's rate is 15% faster than Friend Q's, how many kilometers will Friend P have walked when they pass each other?
(A) 21
(B) 21.5
(C) 22
(D) 22.5
(E) 23
Rationale: If Q complete x kilometers, then P completes 1.15x kilometers.
x + 1.15x = 43
2.15x=43
x = 43/2.15 = 20
Then P will have have walked 1.15*20=23 km.
The answer is E.
Answer: (E) 23
Question: In the coordinate plane, points (x, 1) and (5, y) are on line k. If line k passes through the origin and has slope 1/5, then what are the values of x and y respectively?
(A) 4 and 1
(B) 1 and 5
(C) 5 and 1
(D) 3 and 5
(E) 5 and 3
Rationale: Line k passes through the origin and has slope 1/5 means that its equation is y=1/5*x.
Thus: (x, 1)=(5, 1) and (5, y) = (5,1) -->x=5 and y=1
Answer: C
Answer: (C) 5 and 1
Question: For all numbers p and q, the operation @ is defined by p@q = p^2 - pq. If xy ≠ 0, then which of the following can be equal to zero?
I. x@y
II. (xy)@y
III. x@(x + y)
(A) II
(B) I and II
(C) I and III
(D) II and III
(E) All of the above
Rationale: p@q = p^2 - pq=p(p-q).... so p@q will be zero if p=q or p=0.. but a cannot be equal to 0.. as per Q, x and y can take any int value except 0...
now lets look at the choices..
when x=y, it will be 0... so ok...
when we put xy=y, it is possible when x=1 and y any integer... so ok again
when we put x=x+y.... only possibility when y=0 and it is given x and y cannot be 0....so not possible
only l and ll possible ans B....
Answer: (B) I and II
Question: Carl is facing very difficult financial times and can only pay the interest on a $10,000 loan he has taken. The bank charges him a quarterly compound rate of 4%. What is the approximate interest he pays annually?
(A) $1600
(B) $2000
(C) $2150
(D) $2500
(E) $12000
Rationale: Usually, you are given the annual rate of interest and it is mentioned that it is annual rate.
The bank charges him a quarterly compounded ANNUAL rate of 16%.
Here you find per quarter rate as (16/4)% = 4%
I have actually never seen a question with quarter rate given but since this question did not mentionannual rate of interestand since the options did not make sense with 4% annual rate of interest, it is apparent that the intent was a 4% quarterly rate. So the bank charges 4% every quarter and compounds it in the next quarter. Had it been a simple quarterly rate, we would have just found 4 * 4% of 10,000 = $1600 as our answer.
But since, the interest is compounded, it will be a bit more than $1600. Option (A) looks correct.
Answer: (A) $1600
Question: The speed at which a man can row a boat in still water is 25 kmph. If he rows downstream, where the speed of current is 11 kmph, what time will he take to cover 80 metres?
(A) 18 seconds
(B) 27 seconds
(C) 26 seconds
(D) 12 seconds
(E) 8 seconds
Rationale: Speed of the boat downstream = 25 +11
= 36 kmph
= 36 * 5/18 = 10 m/s
Hence time taken to cover 80 m = 80/10
= 8 seconds.
Answer:E
Answer: (E) 8 seconds
Question: There are k-2 members in a certain band, including Jim and Ellen. Two members are to be selected to attend the Grammy awards ceremony. If there are 6 possible combinations in which Jim and Ellen are not selected, what is the value of k?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12
Rationale: There are k-2 members in the band, and k-4 members without Jim and Ellen.
(k-4)C2 = 6
(k-4)(k-5)/2 = 6
(k-4)(k-5) = 12 = 4*3
k = 8
The answer is A.
Answer: (A) 8
Question: If (x^2 + 4x - 11)/5 ≤ x + 1, then x could be represented by which of the following?
(A) − 3 ≤ x ≤ 4
(B) − 4 ≤ x ≤ 3
(C) − 3 ≤ x ≤ 3
(D) − 4 ≤ x ≤ − 3
(E) 3 ≤ x ≤ 4
Rationale: IMO A is correct answer
solving through eqautions
x^2 +4x-11<= 5x+5
(x+3)(x-4)<=0
Answer: (A) − 3 ≤ x ≤ 4
Question: Find the smallest number of five digits exactly divisible by 22,33,66 and 44.
(A) 10101
(B) 11000
(C) 10110
(D) 10111
(E) 10100
Rationale: Smallest number of five digits is 10000.
Required number must be divisible by L.C.M. of 22,33,66,44 i.e 132,
On dividing 10000 by 132,we get 32 as remainder.
Therefore, Required number = 10000 +( 132 â€“ 32 ) = 10100.
Answer is E.
Answer: (E) 10100
Question: The entrance fee for a fair is $5 for persons under the age of 18, and 20% more for persons older. Each ride at the fair costs $0.50. If Joe goes with her 6 years old twin brothers, and they each took 3 rides in total. How much money does Joe end up spending at the fair?
(A) 16
(B) 20.5
(C) 17.5
(D) 20
(E) 4.5
Rationale: Total entrance fee is (2*$5) + (1.20*5)= $16
Total rides fee is (0.50*3)*3= $4.50
Total money spent is $20.50
Answer is B
Answer: (B) 20.5
Question: If X and Y are digits and 8XY is a 3-digit number that is divisible by 2, which of the following is a possible product of X and Y?
(A) 15
(B) 31
(C) 12
(D) 27
(E) 91
Rationale: Key to this question is to remember the fact that a number divisible by 2 must end with even OR 0 (i.e Y).
If Y had to be 0, product should also be 0 regardless of X.
Otherwise, product is a multiple of 2. Only one answer choice meets the requirement.
Ans C.
Answer: (C) 12
Question: If Tim had lunch at $50 and he gave 20% tip, how much did he spend?
(A) A)$60.00
(B) B)$35.42
(C) C)$60.60
(D) D)$21.56
(E) E)$78.45
Rationale: The tip is 20% of what he paid for lunch.
tip = 20% of 50.00 = (20/100)*50.00 = = $10.00
Total spent
50.00 + 10.00 = $60.00
correct answer is A)$60.00
Answer: (A) A)$60.00
Question: Rs. 825 becomes Rs. 956 in 3 years at a certain rate of simple interest.If the rate of interest is increased by 4% ,What amount will Rs. 825 become in 3 years ?
(A) Rs. 1020.80
(B) Rs. 1025
(C) Rs. 1055
(D) Data inadequate
(E) None of these
Rationale: Solution
S.I. = Rs.(956-825 )=Rs.131
Rate = (100x131/825x3) = 524/99%
New rate = (524/99 +4)% = 920/99%
New S.I. = Rs.(825 x 920/99 x 3/100) Rs. 230.
∴ New amount = Rs.(825+230)= Rs. 1055.
Answer C
Answer: (C) Rs. 1055
Question: q is a positive integer and multiple of 2; p = 4^q, what is the remainder when p is divided by 10?
(A) 10
(B) 6
(C) 4
(D) 0
(E) It Cannot Be Determined
Rationale: It is essential to recognize that the remainder when an integer is divided by 10 is simply the units digit of that integer. To help see this, consider the following examples:
4/10 is 0 with a remainder of 4
14/10 is 1 with a remainder of 4
5/10 is 0 with a remainder of 5
105/10 is 10 with a remainder of 5
It is also essential to remember that the q is a positive integer and multiple of 2. Any integer that is a multiple of 2 is an even number. So, q must be a positive even integer.
With these two observations, the question can be simplified to:what is the units digit of 4 raised to an even positive integer?
The units digit of 4 raised to an integer follows a specific repeating pattern:
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^(odd number) --> units digit of 4
4^(even number) --> units digit of 6
There is a clear pattern regarding the units digit. 4 raised to any odd integer has a units digit of 4 while 4 raised to any even integer has a units digit of 6.
Since q must be an even integer, the units digit of p=4^q will always be 6. Consequently, the remainder when p=4^q is divided by 10 will always be 6.
In case this is too theoretical, consider the following examples:
q=2 --> p=4^q=16 --> p/10 = 1 with a remainder of 6
q=4 --> p=4^q=256 --> p/10 = 25 with a remainder of 6
q=6 --> p=4^q=4096 --> p/10 = 409 with a remainder of 6
q=8 --> p=4^q=65536 --> p/10 = 6553 with a remainder of 6
Answer: B.
Answer: (B) 6
Question: If q is the square of a positive integer, which of the following must be equal to the square of the next positive integer?
(A) √n + 1
(B) n + 1
(C) n^2 + 1
(D) q + 2√q + 1
(E) n^2 + 2n + 1
Rationale: If q is the square of a positive integer, which of the following must be equal to the square of the next positive integer?
q = (x)^2 where x is a positive integer
To calculate -
(x+1)^2 = x^2 + 2x + 1
root(q) = x
Ans - q + 2 root(q) + 1
This should be D
Answer: (D) q + 2√q + 1
Question: Rs. 5600 is divided into three parts A, B and C. How much A is more than C if their ratio is 1/7:1/7:1/14?
(A) 300
(B) 992
(C) 1120
(D) 552
(E) 312
Rationale: 1/7:1/7:1/14 = 2:2:1
1/5*5600 = 1120
2240-1120 = 1120
Answer: C
Answer: (C) 1120
Question: If a/b=3/4 and 8a+5b=22,then find the value of a.
(A) 1/2
(B) 3/2
(C) 5/2
(D) 4/2
(E) 7/2
Rationale: (a/b)=3/4 b=(4/3) a.
Therefore, 8a+5b=22 = 8a+5*(4/3) a=22  8a+(20/3) a=22
=44a = 66 = a=(66/44)=3/2
Answer is B.
Answer: (B) 3/2
Question: Given that k/l < 1, and both k and l are positive integers, which one of the following must be greater than 1?
(A) k/l^2
(B) k^2/l
(C) k^2/l^2
(D) l/k
(E) √(k/l)
Rationale: Since k/l is a fraction l must always be > 1
Given -
Which one of the following must be greater than 1
We can get the result one only when the denominator in k/l ( Which is less than 1 ) becomes numerator..
Among the given options only (D) has the required characteristic we are looking for...
Hence answer will be (D)
Answer: (D) l/k
Question: Mike took 5 mock tests before appearing for the GMAT. In each mock test he scored 10 points more than the previous mock test. If he scored 760 on the GMAT and his average score for the mocks and the GMAT was 716.67, what was the difference in the score of his last mock and his GMAT score?
(A) 20
(B) 32
(C) 40
(D) 50
(E) 60
Rationale: One way to do this would be weighted average method..
1) let the average of 5 mocks be x...
so take it as a mix of5 quantitites of xand1 quantity of 760resulting in an average of 716.67...
By alligation/weighted average..
the difference in 760 and 716.67 is 5/6 of difference of 760 and x..
760-716.67 = 5/6 * (760-x)
760-x= 43.33*6/5=52...
so x = 708..
the last of mock test will be 708+10+10=728...
so ans = 760-728=32
B
Answer: (B) 32
Question: John found that the average of 15 numbers is 40. If 10 is added to each number then the mean of number is?
(A) 50
(B) 45
(C) 65
(D) 78
(E) 64
Rationale: (x+x1+...x14)/15 = 40
50
Option A
Answer: (A) 50
Question: A person is traveling at 20 km/hr and reached his destiny in 2.5 hr then find the distance?
(A) 53 km
(B) 55 km
(C) 52 km
(D) 60 km
(E) 50 km
Rationale: T = 2.5 hrs = 5/2 hrs
D= T*S = 20*5/2 = 50 km
Answer is E
Answer: (E) 50 km
Question: The first five numbers in a regular sequence are 4, 10, X, 46, and 94. What is x ?
(A) 28
(B) 26
(C) 30
(D) 22
(E) 24
Rationale: it is a good Q to learn to pick up number properties in the given set, even if we do not get a Q on these line in actuals..
when we see the sequence 4, 10, x, 46, and 94, we see that each succeeding number is some value more than twice the previous number..
10=2*4+2..
x=2*10+2 = 22..
and so on..
so next number= 22
D
Answer: (D) 22
Question: Tim has 350 pounds of cement in 100, 50, and 25 pound bags. He has an equal number of each size bag. How many bags of cement does Tim have?
(A) 2
(B) 4
(C) 6
(D) 8
(E) 10
Rationale: Let the number of each sized bag = c100c + 50c + 25c = 350175c = 350c = 2Therefore, Tim has 3c = 6 bags of cementAnswer: C
Answer: (C) 6
Question: What is the least value of x, So that 2x5475 is divisible by 9
(A) 7
(B) 8
(C) 4
(D) 3
(E) 2
Rationale: Explanation:
The sum of the digits of the number is divisible by 9.
Then the number is divisible by 9.
2 + x + 5 + 4 + 7 + 5 = 23 + x
Least value of x may be '4',
So that the total 23 + 4 = 27
is divisible by 9.
Answer: Option C
Answer: (C) 4
Question: Square P is inscribed in circle Q. If the perimeter of P is 40, what is the circumference of Q?
(A) 11√ 2π
(B) 10√ 2π
(C) 9√ 2π
(D) 8√ 2π
(E) 7√ 2π
Rationale: square forms two right angled triangles.
Any time we have a right angle triangle inside a circle, the hypotenuse is the diameter.
hypotenuse here = diagonal of the square = 10 sqrt(2) = diameter
=> radius = 5 sqrt(2)
Circumference of the circle = 2pi r = 10 pi sqrt(2)
Answer is B.
Answer: (B) 10√ 2π
Question: A debtor reached an agreement with his creditor to repay a loan by making a monthly payment which is double of the amount she paid the previous month. If the debtors first payment was $200, and he is expected to pay back 51200, how many months will it take to fully repay the loan?
(A) 9
(B) 10
(C) 9
(D) 12
(E) 10
Rationale: First payment--$200
Total amount to be repaid--$25600
First month payment--$200
Second month payment( double the previous month payment)--$200*2=$400
Third month payment( double the previous month payment)--$400*2=$800
Fourth month payment( double the previous month payment)--$800*2=$1600
Fifth month payment( double the previous month payment)--$1600*2=$3200
Sixth month payment( double the previous month payment)--$3200*2=$6400
Seventh month payment( double the previous month payment)--$6400*2=$12800
Eight month payment( double the previous month payment)--$12800*2=$25600
Ninth month payment( double the previous month payment)--$25600*2=$51200
Answer is A
Answer: (A) 9
Question: As a bicycle salesperson, James earns a fixed salary of $30 per week plus $6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true?
I. y<2x
II. y>x
III. y>3
(A) I,II only
(B) I only
(C) II,III only
(D) II only
(E) III only
Rationale: II. y>x --> since this week, James earned more than he did last week and the total salary is in direct relationship with the # of bicycle sold, then y (# of bicycle sold this week) must be more than x (# of bicycle sold last week);
III. y>3 --> if James sold 3 bicycles this week then this week he earned 30+3*6=$48, which cannot be more than twice as much as he earned the last week, since the minimum salary is fixed to $30. So y must be more than 3;
I. y<2x --> is not always true.
Answer: C
Answer: (C) II,III only
Question: A school currently maintains a fixed number of students per class. If the ratio of students per class were to be increased by 1, 10 fewer classes would be run for a total of 120 students. What is the current ratio Q of students per class?
(A) Q=3
(B) Q=4
(C) 6
(D) 8
(E) 12
Rationale: Another way to look at the problem...
Since the total is 120, RATIO * CLASSES = R*C = 120.....(i)
we are looking where ratio increases by 1 and # of classes decreases by 10 = (R+1)(C-10) = RC+C-10R-10=120....(ii)
(ii)-(i)....
C=10R+10 = 10(R+1).......
so # of classes has to be multiple of 10
AS RC=120.... 10(R+1)*R = 120...................R(R+1) = 12..
so 12 is a multiple of consecutive numbers ONLY 3 *4 fits in..... and R=3
A
Answer: (A) Q=3
Question: A survey reveals that the average income of a company’s customers is $45,000 per year. If 50 customers respond to the survey and the average income of the wealthiest 10 of those customers is $95,000, what is the average income of the other 40 customers?
Is there a way to solve this using weighted average concept instead of doing tedious calculations?
(A)  $32,500
(B)  $35,000
(C)  $37,500
(D)  $42,500
(E)  $50,000
Rationale: let x be the average of 40 customers
40*x + 10* 95000 = 50*45000
solving this we have x= 32500
Answer is A.
Answer: (A)  $32,500
Question: The slant height of a cone is 35 cm and radius of the base is 14cm, find the curved surface of the cone.
(A) 4150
(B) 1780
(C) 1540
(D) 1500
(E) 6100
Rationale: π * 14 * 35= 1540
Answer:C
Answer: (C) 1540
Question: A cheese factory sells its cheese in rectangular blocks. A normal block has a volume of 8 cubic feet. If a small block has half the width, half the depth, and half the length of a normal block, what is the volume of cheese in a small block in cubic feet?
(A) 1
(B) 5
(C) 4
(D) 6
(E) 2
Rationale: Volume of cube=lbh=8
New cube l ,b, h are decreases of .5l, .5b,.5h
New volume of cube =.5l*.5b*.5h=.125*lbh
=.125*8
=1
Answer: A
Answer: (A) 1
Question: In a new housing development, trees are to be planted along the sidewalk of a certain street. Each tree takes up one square foot of sidewalk space, and there are to be 20 feet between each tree. How many trees can be planted if the road is 148 feet long?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 16
Rationale: Let T be the number of trees. Then the length required for trees on the sidewalk will be 1*T= T
To maximize the number of trees, the number of 20 feet spaces between trees should be 1 less than total number of trees.
For example, If there are 3 trees, then there should be 2 spaces between them.
So the number of 20 feet spaces will be T-1. Then, the length of sidewalk required for 20 feet spaces will be 20*(T-1)
It is given that total length of sidewalk is 148 feet.
or 20(T-1)+T = 148
or 20T-20+T = 148
or 21T = 168
or T=8
Answer:-A
Answer: (A) 8
Question: What will be the difference in simple and compound interest on 2000 after three years at the rate of 10 percent per annum?
(A) 160
(B) 42
(C) 62
(D) 20
(E) None of these
Rationale: For 3 years:
Diff.=Sum×(rate)2(300+rate)/(100)3
= 2000×10×10×310/100×100×100 = 62
Answer C
Answer: (C) 62
Question: A $500 investment and a $1,500 investment have a combined yearly return of 19 percent of the total of the two investments. If the $500 investment has a yearly return of 7 percent, what percent yearly return does the $1,500 investment have?
(A) 9%
(B) 10%
(C) 23%
(D) 21%
(E) 22%
Rationale: The equation we can form the question :
Return on Total Investment = Sum of individual Investments
(500+1500)(19)=(500âˆ—7)+(1500x), where x is the return on investment of 1500.
Solving the equation, we get x = 23% ( Option C )	ANSWER:C
Answer: (C) 23%
Question: Find the constant k so that : -x2 - (k + 11)x - 8 = -(x - 2)(x - 4)
(A) 11
(B) 12
(C) 17
(D) 14
(E) 15
Rationale: -x2 - (k + 11)x - 8 = -(x - 2)(x - 4) : given
-x2 - (k + 11)x - 8 = -x2 + 6x - 8
-(k + 11) = 6 : two polynomials are equal if their corresponding coefficients are equal.
k = -17 : solve the above for k
correct answer C
Answer: (C) 17
Question: A positive whole number has factors of 3 and 5. The number MUST be divisible by: I. 15 II. 30 III. 60
(A) I & II
(B) III
(C) I & III
(D) II & III
(E) II
Rationale: 15 , 30 is not divisible by 60.But 60 is divisible by 3,5,15,30
So answer is III
Answer : B
Answer: (B) III
Question: Krishan and Nandan jointly started a business. Krishan invested six times as Nandan did and invested his money for double time as compared to Nandan. Nandan earned Rs. 6000. If the gain is proportional to the money invested and the time for which the money is invested then the total gain was?
(A) Rs.78000
(B) Rs.48000
(C) Rs.6000
(D) Rs.82000
(E) Rs.32000
Rationale: 6:1
2:1
------
12:1
1 ----- 6000
13 ----- ? => Rs.78,000
Answer: A
Answer: (A) Rs.78000
Question: There are 8 players in a chess group, and each player plays each of the others once. Given that each game is played by two players, how many total games will be played?
(A) 10
(B) 30
(C) 28
(D) 60
(E) 90
Rationale: 10 players are there.
two players play one game with one another.
so 8C2=8*7/2
=28
SO OPTION C is correct
Answer: (C) 28
Question: A pipe takes a hours to fill the tank. But because of a leakage it took 2 times of its original time. Find the time taken by the leakage to empty the tank
(A) 50 min
(B) 60 min
(C) 90 min
(D) 80 min
(E) 120 min
Rationale: pipe a can do a work 60 min.
lets leakage time is x;
then
1/60 -1/x=1/120
x=120 min
ANSWER:E
Answer: (E) 120 min
Question: Today is 22 days before Mariah's graduation. If she graduates on a Friday, what day of the week is it today?
(A) Monday
(B) Friday
(C) Tuesday
(D) Thursday
(E) Sunday
Rationale: Each weekday is repeated every 7 days. 22 divided by 7 is 3, with a remainder of 1. Therefore, 22 days before the graduation occurs 1 day before the weekday of the graduation. Answer: D
Answer: (D) Thursday
Question: Suppose a, b, and c are positive integers with a < b < c such that
1/a
+
1/b
+
1/c
= 1. What is
a + b + c?
(A) 1
(B) 4
(C) 9
(D) 11
(E) no such integers exist
Rationale: First note that we must have 1/a < 1, so a > 1. Since 1/a > 1/b > 1/c, we
must also have 1/a > 1/3; so a < 3. Thus, a/ 2. Now 1/b + 1/c = 1/2 where 2 < b < c.
Similar to before, 1/b > 1/4, so b < 4. Thus, b = 3. With a = 2 and b = 3 we have
1/2+1/3+1/c = 1, which is satisfied when c = 6. To conclude, a+b+c = 2+3+6 = 11.
correct answer D
Answer: (D) 11
Question: The age of man is three times the sum of the ages of his two sons.Four years hence,his age will be double of the sum of the ages of his sons.The fatherâ€™s present age is :
(A) 36 years
(B) 45 years
(C) 50 years
(D) 55 years
(E) 75 years
Rationale: Solution
Let the sum of present ages of the two sons be x years.
Then,father's present age = 3x years.
âˆ´ (3x + 4)=2 (x +8) â‡” 3x + 4 = 2x + 16 â‡” x =12.
Hence,father's present age = 36 years. Answer A
Answer: (A) 36 years
Question: A bottle of coke contains 200gm in place of 1kg of fluid. Find the actual % difference, given a 10% gain on initial fluid?
(A) 32.5%
(B) 112.5%
(C) 35%
(D) 40%
(E) 50%
Rationale: fluid price of 200gm = 100+10 = 110
Difference = 110-20 = 90
% of difference = 90*100/80 =112.5 %
Answer is B
Answer: (B) 112.5%
Question: A person can walk at a constant rate of 8mph and can bike at a rate of 16mph. If he wants to travel 64 miles in 8 hours using bike and walking at their constant rates, how much distance would he require to walk?
(A) 20
(B) 30
(C) 48
(D) 64
(E) 72
Rationale: Total distance = 64
Distance = Speed * Time
Walking speed = s1 = 8
Walking time = t1
Bike speed = s2 = 16
Time traveled in bike = t2
d1 + d2 = 64
s1t1 + s2t2 = 64
8*t1 + 16*t2 = 64
t1 + 2*t2 = 8 ----- (1)
Given: t1 + t2 = 8 ----- (2)
(1) - (2) --> t2 = 0 and t1 = 8 - 0 = 8
Walking distance = s1*t1 = 8*8 = 64
Answer: D
Answer: (D) 64
Question: 5358 x 51 = ?
(A) 273258
(B) 273268
(C) 273348
(D) 273358
(E) None of these
Rationale: Explanation:
5358 x 51	= 5358 x (50 + 1)
= 5358 x 50 + 5358 x 1
= 267900 + 5358
= 273258.
ANSWER IS A
Answer: (A) 273258
Question: 3 persons (1 couple and 1 single) are seated at random in a row of 5 chairs. What is the probability that the couple does not sit together?
(A) 5/7
(B) 4/5
(C) 2/5
(D) 3/5
(E) 11/8
Rationale: Let's find the probability that a couple sits together (right next to each other) and subtract that value from 1.
Total # of ways 3 persons C1C1, C2C2 and SS to be seated in a row of 5 seats is 5!2!=605!2!=60. Consider this, we are interested in arrangement of C1, C2, S, E, EC1, C2, S, E, E, so in arrangement of 5 letters out of which 2 E's are identical (E denotes an empty seat);
# of ways for a couple to sit together is 4!2!∗2=244!2!∗2=24. Consider a couple as a single unit: {C1,C2}, S, E, E{C1,C2}, S, E, E, so we have total of 4 units out of which 2 E's are identical, # of arrangement of these units is 4!2!4!2!, but C1C1, C2C2 within their unit can be arranged in 2 ways ({C1,C2}{C1,C2} or {C2,C1}{C2,C1}), so total # of arrangement for this case is 4!2!∗2=244!2!∗2=24;
P=1−2460=35P=1−2460=35.
Answer: D.
Answer: (D) 3/5
Question: In a recent survey at a local deli, it was observed that 3 out of 5 customers bought a bagel and 5 out of 7 customers bought a coffee. Some customers bought both. If 8 customers are selected, what are the chances that at least 1 customer bought a coffee and a bagel?
(A) 27/343
(B) 3/7
(C) 27/125
(D) 199/245
(E) 9/125
Rationale: Let us take 7*5=35 as the total number of customers. So 7*3=21 customers bought a bagel and 5*5=25 customers bought a coffee.
chances that at least 1 customer bought a coffee and a bagel = 1 - chances that no customer bought a coffee and a bagel
chances that no customer bought a coffee and a bagel=24/35*23/34*22/33*21/32*20/31*19/30*18/29*17/28=46/245
chances that at least 1 customer bought a coffee and a bagel= 1 - 46/245 = 199/245
Answer D.
Answer: (D) 199/245
Question: |x+3| – |4-x| = |8+x| How many S solutions will this equation have?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Rationale: |x| = x when x >= 0 (x is either positive or 0)
|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0,
|0| = 0 = -0 (all are the same)
So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.A
Answer: (A) 0