divelab/Sys2Bench huggingface dataset:
Question: Two trains leave the same train station at 6:00 am and 6:45 am and they travel at 100 kph and 120 kph respectively. How many kilometers from the train station will the two trains be together?
Options:
A)420
B)450
C)480
D)510
E)540
Rationale: At 6:45, the first train is 75 km ahead.
The second train gains 20 km each hour.
The time it takes the second train to catch the first train is 75 /20 = 3.75 hours.
In this time, the second train travels 3.75*120=450 km.
The answer is B.
Answer: B
Question: A small pool filled only with water will require an additional 700 gallons of water in order to be filled to 80% of its capacity. If pumping in these additional 700 gallons of water will increase the amount of water in the pool by 30%, what is the total capacity of the pool in gallons?
Options:
A)1000
B)1250
C)1300
D)1600
E)2125
Rationale: Since pumping in additional 700 gallons of water will increase the amount of water in the pool by 30%, then initially the pool is filled with 1,000 gallons of water.
So, we have that 1,000 + 700 = 0.8*{total} --> {total} = 2,125.
Answer: E.
Answer: E
Question: If b is a positive integer, which of the following could NOT be the square of b?
Options:
A)3165
B)9436
C)4726
D)6445
E)1432
Rationale: A number ending in 2 cannot be a perfect square.
1432 ends in a number 2, making that the correct answer
(E) 1432
Answer: E
Question: A girl scout was selling boxes of cookies. In a month, she sold both boxes of chocolate chip cookies ($1.25 each) and boxes of plain cookies ($0.75 each). Altogether, she sold 1,585 boxes for a combined value of $1,587.75. How many boxes of plain cookies did she sell?
Options:
A)0
B)233
C)500
D)695
E)787
Rationale: Let #plain cookies sold be X then #chocolate cookies = (total cookies-X)
Equating for X
(0.75)*X +(1.25)*(1585-X)=1587.75
=>x=787
E
Answer: E
Question: A work can be finished in 16 days by twenty women. The same work can be finished in fifteen days by sixteen men. The ratio between the capacity of a man and a woman is
Options:
A)1:3
B)4:3
C)2:3
D)2:1
E)3:1
Rationale: Explanation :
Work done by 20 women in 1 day = 1/16
Work done by 1 woman in 1 day = 1/(16×20)
Work done by 16 men in 1 day = 1/15
Work done by 1 man in 1 day = 1/(15×16)
Ratio of the capacity of a man and woman =1/(15×16) : 1/(16×20) = 1/15 : 1/20
= 1/3 :1/4 = 4:3
Answer : Option B
Answer: B
Question: The owner of a cell phone shop charges his customer 23% more than the cost price. If a customer paid Rs.6,040 for a cell phone, then what was the cost price of the cell phone?
Options:
A)Rs.5,845
B)Rs.6,750
C)Rs.5,700
D)Rs.4,910
E)None of these
Rationale: Explanation:
123% of CP =SP
Therefore, CP = Rs.6,040x100/123 = Rs.4910
Answer: Option D
Answer: D
Question: Find the invalid no.from the following series 13, 18, 25, 30, 37, 40
Options:
A)13
B)18
C)40
D)37
E)25
Rationale: The differences between two successive terms from the beginning are 7, 5, 7, 5, 7, 5. So, 40 is wrong.
C
Answer: C
Question: A bag contains 50p, 25p and 10p coins in the ratio 3 : 7 : 4, amounting to Rs. 109.50. Find the number of coins of each type.
Options:
A)80,250,140
B)75,220,135
C)90,210,120
D)80,210,120
E)None of these
Rationale: Solution:
Let the number of 50p, 25p and 10p coins be 3x,7x and 4x respectively.
Then, 3x/2 + 7x/4 + 4x/10 = Rs. 109.50.
=> (30x + 35x + 8x)/20 = 109.50. => 73x = 109.5 * 20 => 73x =2190.
=> x =2190/73 = 30.
.'. number of 50p coins = 3*30 =90; number of 25p coins = 7*30 =210;
number of 10p coins = 4*30 =120.
ANSWER IS C
Answer: C
Question: 30 square stone slabs of equal size were needed to cover a floor area of 58.8 sq.m. Find the length of each stone slab?
Options:
A)140cm
B)767cm
C)88cm
D)666cm
E)776cm
Rationale: Area of each slab
=58.8/30m2=1.96m2
Length of each slab √1.96=1.4m=140cm
Answer: A
Question: What is the greatest 5-digit number when divided by 9 and 10 leaves a remainder of 7 and 9 respectively?
Options:
A)56780
B)78910
C)97479
D)97918
E)97929
Rationale: When you divide a positive integer by 10, the remainder will just be the units digit. We know the remainder is 9 when we divide by 10, so E is the only possible answer.
Answer: E
Question: A water tank is two-fifth full. Pipe A can fill a tank in 10 minutes and pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?
Options:
A)6 min
B)8 min
C)7 min
D)9 min
E)1 min
Rationale: Clearly, pipe B is faster than pipe A and so, the tank will be emptied.
Part to be emptied = 2/5
Part emptied by (A + B) in 1 minute = (1/6 - 1/10) = 1/15
1/15 : 2/5 :: 1 : x
x = (2/5 * 1 * 15) = 6 min.
So, the tank will be emptied in 6 min.
Answer: A
Answer: A
Question: A certain stock exchange designates each stock with a two-letter or three-letter code, where each letter is selected from the 26 letters of the alphabet. If the letters may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes without using the vowels A, E, I, O, or U?
Options:
A)9372
B)9482
C)9592
D)9702
E)9812
Rationale: The number of two-letter codes is 21*21=441.
The number of three-letter codes is 21*21*21=9261.
The total number of codes is 441+9261=9702.
The answer is D.
Answer: D
Question: What is the least number of square tiles required to pave the floor of a room 10 m 17 cm long and 9m 9 cm broad?
Options:
A)724
B)804
C)11413
D)844
E)None
Rationale: Solution
Length of largest tile = H.C.F. of 1017cm & 909cm = 9cm.
Area of each tile = (9 x 9)cm2
∴ Required number of tiles	= [1017x909/9x9]
= 11413.
Answer C
Answer: C
Question: In a bag of small balls 1/4 are green, 1/8 are blue, 1/12 are yellow and the remaining 26 white. How many balls are blue?
Options:
A)5 balls.
B)9 balls.
C)10 balls.
D)7 balls.
E)6 balls.
Rationale: Let us first find the fraction of green, blue and yellow balls
1/4 + 1/8 + 1/12 = 6 / 24 + 3 / 24 + 2 / 24 , common denominator
= 11 / 24 , add numerators
The fraction of white balls is given by
24 / 24 - 11 / 24 = 13 / 24
So the fraction 13 / 24 corresponds to 26 balls. If x is the total number of balls then
(13 / 24) of x = 26 balls
or (13 / 24) ? x = 26
x = 26 ? (24 / 13) = 48 , total number of balls
The fraction of blue balls is 1 / 8 of x. The number of blue balls is given by
(1 / 8) of 48 = 6 balls.
correct answer E
Answer: E
Question: Dacid obtained 76, 65, 82, 67 and 85 marks (out of 100) in English, Mathematics, Physics, Chemistry and Biology. What are his average marks?
Options:
A)32
B)88
C)77
D)99
E)75
Rationale: Average = (76 + 65 + 82 + 67 + 85)/5 = 375/5 = 75.
Answer: E
Answer: E
Question: The average (arithmetic mean) of four numbers is 4x + 2. If one of the numbers is x, what is the average of the other three numbers?
Options:
A)x + 1
B)3x + 3
C)5x + 1
D)5x + 8/3
E)15x + 12
Rationale: If the average of four numbers is 4x + 3, then the sum of all four numbers if 4 * (4x + 2) = 16x + 8. If one of the numbers is x, then the average of the remaining three numbers would be: (16x + 8 - x)/3 = (15x + 8)/3 = 5x + 8/3.
Answer is D.
Answer: D
Question: 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?
Options:
A)33/01
B)33/91
C)33/90
D)33/99
E)83/98
Rationale: Explanation:
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: Option B
Answer: B
Question: By weight, liquid X makes up 0.8 percent of solution A and 1.8 percent of solution B. If 600 grams of solution A are mixed with 700 grams of solution B, then liquid X accounts for what percent of the weight of the resulting solution?
Options:
A)1.74%
B)1.94%
C)10%
D)15%
E)19%
Rationale: I think there is a typo in question.
It should have been "By weight Liquid 'X ' makes up..... "
Weight of Liquid X = 0.8 % of Weight of A + 1.8% of Weight of B
When 600gms of A and 700gms of B is mixed :
Weight of Liquid X = (0.8 * 600)/100 + (1.8* 700)/100 = 17.4gms
% of Liquid X in resultant mixture = (17.4/1000) *100 = 1.74%
A
Answer: A
Question: The average age of 15 students of a class is 15 years. Out of these, the average age of 8 students is 14 years and that of the other 6 students is 16 years, The age of the 15th student is
Options:
A)17
B)12
C)13
D)14
E)15
Rationale: Explanation:
Age of the 15th student = [15 * 15 - (14 * 8 + 16 * 6)] = 17 years. Answer: A
Answer: A
Question: Working together at their respective constant rates, Machine A and Machine B can produce 1,600 units in 8 hours. Working alone, Machine B would complete that same output in 50% more time. If Machine A were to work on its own for an 8-hour shift, what percent of the 1,600 unit total would it produce?
Options:
A)25
B)33
C)50
D)67
E)75
Rationale: 1/A+1/B= 1/T
1/A +1/12= 1/8 (50% more of 8 is 12)
1/A= 1/24
Machine A can produce 1600 units in 24 hrs, so it can produce 1600*8/24= 533.33 units is 8 hrs.
533.33 is 33.33% of 1600.
B is the answer
Answer: B
Question: Three bells chime at intervals of 18 min., 24 min. and 32 min. respectively. At a certain time they begin together. What length D of time will elapse before they chime together again?
Options:
A)2 hr. and 24 min.
B)4 hr. and 48 min.
C)1 hr. and 36 min.
D)5 hr.
E)2 hr.
Rationale: It should be simple. All the three bells will chime again together whenver their time intervals intersect eachother.
So the LCM of the three time intervals (18, 24,32) would be the answer.
LCM (18, 24, 32) D= 288 => 4 hours 48 mins.B
Answer: B
Question: A circular path of 13 m radius has marginal walk 2 m wide all round it. Find the cost of leveling the walk at 25p per m2?
Options:
A)Rs.18
B)Rs.47
C)Rs.44
D)Rs.28
E)Rs.28
Rationale: π (152 - 132) = 176
176 * 1/4 = Rs.44
Answer: C
Answer: C
Question: Peter takes a loan of $100,000 with 12% annual interest: the interest is paid once, at the end of the year. Martha takes a loan of $100,000 with 12% annual interest, compounding monthly at the end of each month. At the end of one full year, compared to Peter's loan interest, approximately how much more does Martha have to repay?
Options:
A)$68.25
B)Zero
C)$6825.00
D)$6.82
E)$682.50
Rationale: Peters interest= $100,000*0.12 = $12,000 or $1,000 each month.
Martha’s interest, 12%/12 = 1% each month:
For the 1st month = $100,000*0.01 = $1,000;
For the 2nd month = $1,000 + 1% of 1,000 = $1,010, so we would have interest earned on interest (very small amount);
For the 3rd month = $1,010 + 1% of 1,010 = ~$1,020;
For the 4th month = $1,020 + 1% of 1,020 = ~$1,030;
...
For the 12th month = $1,100 + 1% of 1,100 = ~$1,110.
The difference between Peters interest and Martha’s interest = ~(10 + 20 + ... + 110) = $660.
Answer: E.
Answer: E
Question: What is the remainder when 1041*1047*1050*1053 is divided by 33?
Options:
A)3
B)27
C)30
D)21
E)18
Rationale: take the remainder from each of 1041/33, 1047/33 and so on..
1044/33 gives remainder = 18
1047/33 gives remainder = 24
1050/33 gives remainder = 27
1053/33 gives remainder = 30
the net remainder is the product of above individual remainders. i.e = 18*24*27*30
break them into pairs 18*24/33 gives remainder 3
and 27*30/33 gives remainder 18
so 3*18/33 gives remainder 21.
D
Answer: D
Question: Find the number of ways you can fill a 3 x 3 grid (with 4 corners defined as a, b, c, d), if you
have 3 white marbles and 6 black marbles
Options:
A)9c3
B)9c2
C)9c4
D)8c3
E)7c3
Rationale: The area of the grid is = 9.
Now, No. of ways we can arrange 3 white balls in 9 places = 9c3.
and 6 black balls in (9-3) = 6 places = 6c6 = 1.
So total = 9c3 * 1 = 9c3.
In another way,
No ways we can arrange 6 black balls in 9 places = 9c6 = 9c3
and no ways 3 white balls in (9-6) places = 3c3 = 1
No. of ways we can arrange 3 white balls and 9 black balls in 3x3 grid is = 9c3*1 = 9c3
ANSWER:A
Answer: A
Question: On dividing 14698 by a certain number, we get 89 as quotient and 14 as remainder. What is the divisor?
Options:
A)743
B)154
C)852
D)741
E)165
Rationale: Divisor *quotient+remainder=dividend
divisor=(dividend)-(remainder)/quotient
(14698-14)/89=165
Answer(E)
Answer: E
Question: The binary equivalent of 5.375 is
Options:
A)101.101110111
B)101.011
C)101011
D)None of above
E)Cannot be determined
Rationale: Explanation :
The process to convert a floating point number to its binary equivalent :
1. Separate the number, based on the decimal point.
Ex. : 5.375 => 5 and .375
3. Convert the whole number part to its binary equivalent. In this case it is 5, so 101.
The final result will be 101.(dot)(something) .
4. Now we have to convert .375 in to its decimal equivalent.
Step 1: Begin with the decimal fraction and multiply by 2. The whole number part of the result is the first binary digit to the right of the point.
.375*2 = 0.75
So the binary result till now is : 101.0
Step 2: Discard the whole number part and again multiply the decimal fraction by 2 and again append the whole number part to the right of the point. Follow this process until we get a zero as our decimal part or until we recognize an infinite repeating pattern.
0.75*2 = 1.50
So the Binary result till now is 101.01
.50 * 2 = 1.00
So, the Final result will 101.011 i.e. option B
Answer : B
Answer: B
Question: In how many ways 4 boys and 3 girls can be seated in a row so that they are alternative?
Options:
A)120
B)144
C)160
D)210
E)180
Rationale: 4 boys can be seated in 4!
three girls can be seated in 3!
Required number = 4!3! = 144
Answer is B
Answer: B
Question: Find the quadratic equations whose roots are the reciprocals of the roots of 2x2 + 5x + 3 = 0?
Options:
A)+ 5x - 2 = 0
B)+ 5x + 2 = 0
C)- 5x + 2 = 0
D)- 5x - 2 = 0
E)of these
Rationale: The quadratic equation whose roots are reciprocal of 2x2 + 5x + 3 = 0 can be obtained by replacing x by 1/x.
Hence, 2(1/x)2 + 5(1/x) + 3 = 0
=> 3x2 + 5x + 2 = 0
ANSWER:B
Answer: B
Question: To mail a package, the rate is 10 cents for the first pound and 5 cents for each additional pound. Two packages weighing 3 pounds and 5 pounds, respectively can be mailed seperately or combined as one package. Which method is cheaper and how much money is saved?
Options:
A)Combined, with a saving of 8 cents
B)Combined, with a saving of 5 cents
C)Separately, with a saving of 2 cents
D)Separately, with a saving of 4 cents
E)Combined, with a saving of 10 cents
Rationale: 3 pounds = 1*10+2*5 = 20
5 pounds = 1*10+4*5 =30
total = 50
8 pound = 1*10+7*5 = 45
dif. 5
Answer : B
Answer: B
Question: Laura has a deck of standard playing cards with 11 of the 55 cards designated as aheart.If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart?
Options:
A)1/4
B)1/5
C)5/26
D)12/42
E)13/42
Rationale: No replacement there, 10 cards are dealt and we are asked to find the probability that 10th card is a heart.
If there is no replacement then how is the (P) that the 10th card is 11/55 ??
there are many cases here to be considered here if there is no replacement such as:
H- Denotes heart X-may be any diamond, spade or club.
1. HXXXXXXXXH
2. HHXXXXXXXH
3. HHHXXXXXXH
.
.
.
.
.
9. HHHHHHHHHH
10. XXXXXXXXXH
All cases from 1 to 10 will have different probabilities for heart to be at the 10th place and it will take hell lot of time to calculate all of them.
For according to me the above solution by Ian is only valid if cards are replaced (Every card has a 11/55 = 1/5 chance of being a heart; it doesn't matter if it's the top card in the deck or the tenth card in the deck.)If that's the case that brings back me to my original question ----how do we determine that the cards are replaced or not?? based on the question given ....
B
Answer: B
Question: Out of 5 fruits in a basket, 2 are rotten. If two fruits are drawn at random from the basket, the probability of both being rotten is
Options:
A)1/10
B)10/21
C)20/21
D)11/21
E)2/21
Rationale: The number of exhaustive events = 5C2 = 10.
Let E be event of the 2 fruits being rotten. The number of favourable cases are
2C2 = 1 way.
Required probability = 1/10.
ANSWER:A
Answer: A
Question: A dishonest person wants to make a profit on the selling of milk. He would like to mix water (costing nothing) with milk costing 33 $ per litre so as to make a profit of 40% on cost when he sells the resulting milk and water mixture for 36$. In what ratio should he mix the water and milk?
Options:
A)1:20
B)1:10
C)17:8
D)17:4
E)17:60
Rationale: First of all, let's consider 1 liter of the stuff he is going to sell --- naive customers think it's pure milk, but we know it's some milk-water mixture. He is going to sell this liter of milk-water for $36.
This $36 should be a 40% increase over cost. Here, we need to think about percentage increases as multipliers.
Using multipliers
(cost)*1.40 = $36
cost = 36/1.4 = 360/12 = $25.71
If he wants a 40% increase over cost on the sale of one liter of his milk-water, the cost has to be $25.71.
Well, a liter of milk costs $33, so if he is going to use just $30 of milk in his mixture, that's 25.71/33 = 60/77 of a liter. If milk is 60/77 of the liter, then water is 17/77 of the liter, and the ratio of water to milk is 17:60. Answer choice(E)
Answer: E
Question: A can do a piece of work in 10 days; B can do the same in 20 days. How many days will it take the work to get finished if A and B work together?
Options:
A)4.21
B)5.32
C)6.66
D)7.52
E)8.02
Rationale: A completes 1/10th of the work in one day and B completes 1/20th.
(1/10)+(1/20)=(1/x)
3/20=1/x => x=20/3=6.66 days
Answer: C
Answer: C
Question: A bullet train 220 m long is running with a speed of 59 kmph. In what time will it pass a man who is running at 7 kmph in the direction opposite to that in which the bullet train is going?
Options:
A)23 sec
B)15 sec
C)12 sec
D)11 sec
E)16 sec
Rationale: C
12 sec
Speed of the bullet train relative to man = (59 + 7) kmph
= 66 *5/18 m/sec = 55/3 m/sec.
Time taken by the bullet train to cross the man = Time taken by it to cover 220 m at (55/3) m / sec
= (220 *3/55) sec = 12 sec
Answer: C
Question: In an entrance exam, 3 marks is awarded for every correct answer and (-1) for every wrong answer. If a student gets 38 marks after attempting all questions, find the number of questions answered correctly if the total questions were 70.
Options:
A)27
B)88
C)26
D)29
E)71
Rationale: Explanation:
Let x be the number of questions answered correctly. Hence, number of questions whose answer were wrong = (70 – x)
3*x + (70 – x)*(-1) = 38
4x = 38 + 70 = 108
x = 27
ANSWER:A
Answer: A
Question: The mean of 50 observations is 200. But later he found that there is decrements of 15 from each observations. what is the the updated mean is ?
Options:
A)165
B)185
C)190
D)198
E)199
Rationale: 185
Answer is B
Answer: B
Question: A bag contains an equal number of one rupee, 50 paise and 25 paise coins respectively. If the total value is 35, how many coins of each type are there?
Options:
A)20 coin
B)30 coins
C)28 coins
D)25 coins
E)None of these
Rationale: Let number of each type of coin = x. Then,
1 × x + .50 × x + .25x = 35
⇒ 1.75x = 35 ⇒ x = 20 coin
Answer A
Answer: A
Question: A basket of 2 apple is to be chosen from four apples. What is the probability ways of apples that can be chosen?
Options:
A)6
B)8
C)10
D)4
E)2
Rationale: The question is out of 4 apples 2 apples should be chosen.
Required probability is 4C2=4!/2!*2!=6
The option A is correct.
Answer: A
Question: AB
+BA
-----------
AAC
In the correctly worked addition problem shown here, where the sum of the two-digit positive integers AB and BA is the three-digit integer AAC, and A, B, and C are different digits, what is the units digit of the integer AAC?
Options:
A)9
B)6
C)3
D)0
E)2
Rationale: AB+BA = AAC -> EQ 1
EQ 1 IMPLIES: B+A = 10+C - EQ 2 (DIFFICULT TO INTERPRET. I AGREE. JUST SUM UP THE UNITS DIGIT IN EQ 1)
NOW SPLIT UP EQ 1
10A+B+10B+A = 100A+10A+C - EQ 3
EQ 3 IMPLIES: 11(A+B) = 110A+C
SUBSTITUTE EQ 2 (B+A = 10+C) IN ABOVE
11(10+C) = 110A+C => 110+11C = 110A + C
IMPLIES A = 1C = 0
D
Answer: D
Question: Triangle ABC will be constructed in a xy-plane according to the following conditions: Angle ABC is 90 degrees, and AB is parallel to the y-axis; for each of points A, B and C, both the x-coordinate and the y-coordinate must be integers; the range of possible x-coordinates is 0=<x=<5 and the range of possible y-coordinates is -4=<y=<6. Any two triangles with non-identical vertices are considered different. Given these constraints, how many different triangles could be constructed?
Options:
A)50
B)66
C)2500
D)3300
E)4356
Rationale: Since give that 0=<x=<5 and -4=<y=<6 then we have a rectangle with dimensions 6*11 (6 horizontal and 11 vertical dots). AB is parallel to y-axis, BC is parallel to x-axis and the right angle is at B.
Choose the (x,y) coordinates for vertex B: 6C1*11C1;
Choose the x coordinate for vertex C (as y coordinate is fixed by B): 5C1, (6-1=5 as 1 horizontal dot is already occupied by B);
Choose the y coordinate for vertex A (as x coordinate is fixed by B): 10C1, (11-1=10 as 1 vertical dot is already occupied by B).
6C1*11C1*5C1*10C1=3,300.
Answer: D
Answer: D
Question: Two trains of equal lengths take 10 sec and 15 sec respectively to cross a telegraph post. If the length of each train be 120 m, in what time will they cross other travelling in opposite direction?
Options:
A)22
B)12
C)77
D)99
E)21
Rationale: Speed of the first train = 120/10 = 12 m/sec.
Speed of the second train = 120/5 = 8 m/sec.
Relative speed = 12 + 8 = 20 m/sec.
Required time = (120 + 120)/20 = 12 sec.
Answer: B
Answer: B
Question: In a box, there are 10 red, 6 blue and 7 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?
Options:
A)4/51
B)10/23
C)7/12
D)11/31
E)13/17
Rationale: Total number of balls = 10+6+7 = 23
E = event that the ball drawn is neither red nor green
= event that the ball drawn is red
n(E) = 10
P(E) = 10/23
Answer is B
Answer: B
Question: A bicyclist travels uphill from town A to town B for 2 hours at an average speed of 4 miles per hour and returns along the same road at an average speed of 5 miles per hour. What is the bicyclist’s average speed for the round trip, in miles per hour?
Options:
A)(a) 24/5
B)(b) 5
C)(c) 40/9
D)(d) 27/5
E)(e) 28/5
Rationale: from the uphill we can find out the distance = 4*2 = 8
average speed = total distance/total time
= 16/ (2+ (8/5)) = 40/9
Answer is C.
Answer: C
Question: The milk level in a rectangular box measuring 62 feet by 25 feet is to be lowered by 6 inches. How many gallons of milk must be removed? (1 cu ft = 7.5 gallons)
Options:
A)100
B)250
C)750
D)5812.5
E)5835.5
Rationale: 6 inches = 1/2 feet (there are 12 inches in a foot.), so 62*25*1/2=775 feet^3 of milk must be removed, which equals to 775*7.5=5812.5 gallons.
Answer: D.
Answer: D
Question: Goldman Limited has 5 Senior Managing Directors and 6 Managing Directors. Each investment banking deal requires the guidance of four managing directors, at least one of which must be a Senior Managing Director. How many possible investment banking deals can Morgan Limited possibly provide guidance for?
Options:
A) 116
B) 236
C) 315
D) 514
E) 614
Rationale: P(at least 1 S D) = P(Total) - P(0 Senior Director)
=11C4 - 6C4
= 315
Answer: C
Question: What is the smallest positive integer m such that the following equation
holds for some polynomials u(x) and v(x) with integer coefficients?
(x + 2)(x + 5)(x + 7) u(x) − (x − 2)(x − 5)(x − 7) v(x) = m
Options:
A)70
B)1260
C)1680
D)3780
E)7560
Rationale: Substituting x = 5 into the given equation, we deduce 7 · 10 · 12 · u(5) = m. Substituting x = 7
into the given equation, we deduce 9 · 12 · 14 · u(7) = m. It follows that m must be divisible by
each of 7 · 10 · 12 and 9 · 12 · 14 which implies that 23 · 33 · 5 · 7 = 7560 divides m. Of the choices,
we see that only (e) is possible here. Verifying that such a u(x) and v(x) exist requires a little more
work, but we note that one can take u(x) = x2 − 14x + 54 and v(x) = x2 + 14x + 54.
correct answer E
Answer: E