divelab/Sys2Bench huggingface dataset:
Question: A worker is paid x rupees for painting every 10 metres of a wall and y rupees for painting every extra metre. During one week, he painted 10 metres on Monday, 13 metres on Tuesday, 12 metres on Wednesday, 11 metres on Thursday and 12 metres on Friday. What is his average daily earning rupees for the five day week?
Options:
A)x + (8/5)y
B)(5x + 9y)/5
C)10x + (8/5)y
D)None of these
E)Cannot be determined
Rationale: Explanation :
On Monday, the payment = Rs x
On Tuesday, the payment = Rs (x + 3y)
On Wednesday, the payment = Rs (x + 2y)
On Thursday, the payment = Rs (x + y)
On Friday, the payment = Rs (x + 2y)
Average daily earning during the five-day week = (5x + 8y)/5
= x + (8/5)y
Answer : A
Answer: A
Question: A train covers a distance of 12 km in 10 min. If it takes 6 sec to pass a telegraph post, then the length of the train is?
Options:
A)298 m
B)188 m
C)120 m
D)178 m
E)189 m
Rationale: Speed = (12/10 * 60) km/hr = (72 * 5/18) m/sec
= 20 m/sec.
Length of the train = 20 * 6
= 120 m.
Answer:C
Answer: C
Question: A distributor sells a product through an online store, which take a commission of 20% of the price set by the distributor. The distributor obtains the product from a producer at the price of $20 per item. What is the price that the buyer observers online if the distributor wants to maintain a 20% profit on the cost of the item?
Options:
A)$30
B)$31
C)$32
D)$33
E)$34
Rationale: Let x be the price that buyers see online.
The distributor wants to receive 1.2(original price) which should be 80% of x.
1.2(20) = 0.8x
x = 1.2(20) / 0.8 = 1.5(20) = $30
The answer is A.
Answer: A
Question: In a Basket there are 10 apples. Jack wants only 5 apples to eat. How many Possible Combinations that he have?
Options:
A)60
B)720
C)84
D)100
E)120
Rationale: Jack Wants only 5 apples. so the combination comes under 10C5.
10C5= 10!/(10-5)!*5!=10!/5!*5!
10!=10*9*8*7*6*5!
5!=5*4*3*2*1
10C5=10*9*8*7*6/5*4*3*2=84
There are 84 possible ways to choose apples in a basket
Answer: C
Question: A town consists of only one street in the form of a circle. The town authorities give out four licenses for a particular kind of business. The inhabitants of the town live in equal density along the circle and will always go to the closest business for what they need. Business A gets to choose a location first, then business B, then C, and finally D. Each business desires to carve out as much business for themselves as possible but each knows the others all have the same motive. Assume that if a business is indifferent between locating in two different sections of the circle it will choose a section at random. Also assume that the business that goes last will choose a location in the middle of the largest (or one of the largest) sections. Where should business B choose relative to the location of A?
Options:
A)more then 1/3
B)barely less then 1/3
C)less then 2/3
D)more then 2/3
E)more then 3/3
Rationale: The answer is just barely less then 1/3 of the circumference of the circle away from A.
This is only a quick overview of the solution, the details are left up to you. Let the circumference of the circle be 1 and that A chooses a location at point 0. Business D will choose a location in the middle of the largest section.
Business C will also choose the midpoint of the larger of the two gaps between the first two businesses.
If business B chooses a point before 1/3 then C will choose a point halfway between B and 1. Business D will choose randomly between the halfway point between A and C or B and C. If x is the location of business B then the area which B will carve out of the circle will be either (1+3x)/8 if D goes between B and C or (1+x)/4 if D goes between A and C. The average of these is (3+5x)/16. The same logic applies if B chooses a point after 2/3.
If business B chooses a point after 1/3 (but before 1/2) then C will choose a point halfway between A and B going the long way and D will choose the halfway point between A and B the short way. This will leave B exactly a 1/4 share of the business. The same logic applies if A chooses a point between 1/2 and just before 2/3.
If B should choose a location at exactly 1/3 then C would choose at 2/3 and D would be indifferent between 1/6, 1/2, and 5/6. B would have a 2/3 chance of having 1/4 of the business share and 1/3 chance of having 1/3, the average being 5/18 =~ 0.27778 .
Thus B should try to maximize (3+5x)/16 without choosing x equal or greater to 1/3. The optimal choice of location would be just a hair short of 1/3 (or just a hair after 2/3). At this point B will have a 50/50 chance at having either 1/4 or 1/3 of the market share for an average of 7/24=~ 0.29166667 of the market share.
correct answer B
Answer: B
Question: Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. Fast train completely passes a man sitting in the slower train in 5 sec. What is the length of the fast train?
Options:
A)27 7/7 m
B)27 8/9 m
C)29 7/9 m
D)27 7/9 m
E)27 7/2m
Rationale: Relative speed = (40 - 20) = 20 km/hr.
= 20 * 5/ 18 = 50/9 m/sec.
Length of faster train = 50/9 * 5
= 250/9
= 27 7/9 m.
Answer:D:
Answer: D
Question: |13 - 3| - |4 - 10| = ?
Options:
A)3
B)2
C)1
D)0
E)4
Rationale: |13 - 3| - |4 - 10| = | 10 | - | -6 | = 10 - 6 = 4
correct answer E
Answer: E
Question: If 3 people can do 3 times of a particular work in 3 days, then how many days would it take 4 people to do 4 times of that particular work?
Options:
A)1
B)2
C)3
D)4
E)7
Rationale: 3 people can do the work one time in one day.
1 person can do 1/3 of the work in one day.
4 people can do 4/3 of the work in one day.
4 people can do 4 times the work in 3 days.
The answer is C.
Answer: C
Question: There are 12 stamps from which two stamps will be selected. We have five stamps which are the same as five of the stamps which could be selected. What is the probability that the two stamps selected will be different from the stamps that we have?
Options:
A)3/14
B)5/19
C)7/22
D)9/23
E)11/28
Rationale: P(first stamp is different) = 7/12
P(second stamp is different) = 6/11
P(both stamps are different) = 7/12*6/11 = 7/22
The answer is C.
Answer: C
Question: Certain sum becomes 3 times it self at compound interest in 10 years. In how many years it becomes 9 times?
Options:
A)24
B)20
C)22
D)28
E)25
Rationale: P(1 + R/100)(power10) = 3P
=> P(1 + R/100)(power10) = 3
Let P(1 + R/100)(power n) = 9P
=> (1 + R/100)(power n) = 9
=> 3(power 2) = [(1 + R/100)(power10)](power2)
=> (1 + R/100)(power n) => (1 + R/100)(power 20)
=> n = 20 Years.
Answer is B.
Answer: B
Question: 5 years ago Kate was 5 times as old as her Son. 5 years hence her age will be 8 less than three times the corresponding age of her Son. Find the age of kate?
Options:
A)33
B)34
C)35
D)36
E)37
Rationale: let x be the mother's age and y be the son's age
x-5=5(y-5)
x+5=3(y+5)-8
by solving this we get x=35
ANSWER:C
Answer: C
Question: The last day of a century cannot be
Options:
A)Monday
B)Wednesday
C)Tuesday
D)Friday
E)3922
Rationale: Answer: C) Tuesday
Answer: C
Question: What is the ratio between perimeters of two squares one having 3 times the diagonal then the other?
Options:
A)32.5
B)32.4
C)32.9
D)32.3
E)32.1
Rationale: 36/7 r = 6.3
= 32.4
Answer:B
Answer: B
Question: If x is the interest on y and y is the interest on z, the rate and time is the same on both the cases. What is the relation between x, y and z?
Options:
A)Y2 = XY
B)Y2 = XZ
C)Y2 = XZ
D)X2 = XZ
E)Y2 = ZY
Rationale: X = (Y*NR)/100 Y = (Z*NR)/100
X/Y = NR/100 Y/Z = NR/100
X/Y = Y/Z
Y2 = XZ
Answer: C:
Answer: C
Question: A gas station sold 50 000 liters of petrol at $1.50 per liter, 40 000 liters of diesel at $2.20 per liter and 100 000 liters of bio diesel at $1.10 per liter. what % of its total revenue is from bio diesel?
Options:
A)45.3%
B)40.3%
C)41.3%
D)42.3%
E)46.3%
Rationale: Correct Answer:B
The company's total revenue from petrol=50000*$1.50=$75000
The company's total revenue from diesel=40000*$2.20=$88000
The company's total revenue from bio diesel =100000*$1.10=$110000
The combined total of all sales=$273000
Therefore % of bio diesel sales=110000/273000*100= 40.3%
Answer: B
Question: How many multiples of 7 are there between 10 and 100 (both are inclusive)?
Options:
A)11
B)13
C)15
D)17
E)21
Rationale: 7 multiples are 14,21,28,---------,98
It should be mentioned whether 10 and 100 are inclusive.
the answer is (98-14)/7+1=13
Answer is B
Answer: B
Question: if a men rows at the rate of 5 kmph in still water and his rate against the current is 3.5 kmph, then the man’s rate along the current is :
Options:
A)4.25 kmhr
B)6 kmph
C)6.5 kmph
D)8.5 kmph
E)None of these
Rationale: Solution
Let the rate along the current be x kmph.
Then,1/2(x +3.5)= 5 or x= 6.5 kmph. Answer C
Answer: C
Question: between 100 to 200 how many no are divisible by both 3 and 2 and 100,200 are inclusive?
Options:
A)77
B)78
C)79
D)80
E)81
Rationale: Prime numbers are 21
So 100-21=79
ANSWER:C
Answer: C
Question: Suppose that another fresh fruit contains 40% water and dry fruit contains 25% water. How much dry fruit can be obtained from 100kg of fresh fruit?
Options:
A)A)32kg
B)B)40kg
C)C)80kg
D)D)85kg
E)E)100kg
Rationale: Quantity of pulp in 100kg of fresh fruits = (100-40)% of 100kg = 60kg
Let the quantity of dry fruit obtained be x kg
then, (100-25)% of x = 60
(75/100)*x = 60
x = 80
correct option is C
Answer: C
Question: How many seconds will a 500 meter long train take to cross a man walking with a speed of 3 km/hr in the direction of the moving train if the speed of the train is 63 km/hr?
Options:
A)25 sec
B)40 sec
C)30 sec
D)60 sec
E)50 sec
Rationale: Let length of tunnel is x meter
Distance = 800+x meter
Time = 1 minute = 60 seconds
Speed = 78 km/hr = 78*5/18 m/s = 65/3 m/s
Distance = Speed*Time
800+x = (65/3) * 60
800+x = 20 * 65 = 1300
x = 1300 - 800 = 500 meters
ANSWER C
Answer: C
Question: Cheese, bologna, and peanut butter sandwiches were made for a picnic in a ratio of 5 to 7 to 8. If a total of 360 sandwiches were made, how many Cheese sandwiches
were made?
Options:
A)15
B)30
C)90
D)42
E)48
Rationale: For deciding such task we should calculate all parts 5 + 7 + 8 = 20 parts
and we should calculate how many sandwiches holds 1 part:
360 / 20 = 18 sandwiches in one part
For cheese we have 5 parts so:
5 * 18 = 90
Answer is C
Answer: C
Question: If |20x-10| = 130, then find the product of the values of x?
Options:
A)-45
B)50
C)-42
D)35
E)-30
Rationale: |20x-10| = 130
20x-10 = 130 or 20x-10 = -130
20x = 140 or 20x = -120
x = 7 or x = -6
product = -6*7 = -42
Answer is C
Answer: C
Question: One side of a rectangular field is 4 m and its length along diagonal is 5 m. What is the area of the field?
Options:
A)3
B)4
C)6
D)8
E)9
Rationale: M = 4
S = 2
US = 4 - 2 = 2
D = 6
T = 6/2 = 3
Answer: A
Answer: A
Question: Together, 15 type A machines and 7 type B machines can complete a certain job in 4 hours. Together 8 type B machines and 15 type C machines can complete the same job in 11 hours. How many hours E would it take one type A machine, one type B machine, and one type C machine working together to complete the job (assuming constant rates for each machine)?
Options:
A) 22 hours
B) 30 hours
C) 44 hours
D) 60 hours
E) It cannot be determined from the information above.
Rationale: Say the rates of machines A, B and C are a, b, and c, respectively.
Together 15 type A machines and 7 type B machines can complete a certain job in 4 hours --> 15a + 7b = 1/4;
Together 8 type B machines and 15 type C machines can complete the same job in 11 hours --> 8b + 15c = 1/11.
Sum the above: 15a + 15b + 15c = 1/4 + 1/11 = 15/44 --> reduce by 15: a + b + c = 1/44 --> so, the combined rate of the three machines is 1/44 job/hour --> time is reciprocal of the rate, thus machines A, B and C can do the job E in 44 hours.
Answer: C.
Answer: C
Question: What percent is 10 gm of 1 kg?
Options:
A)0.5 %
B)1%
C)1.5%
D)2%
E)3%
Rationale: 1 kg = 1000 gm
10/1000 × 100 = 1000/1000
=1 %
B)
Answer: B
Question: Jerry’s average (arithmetic mean) score on the first 3 of 4 tests is 94. If Jerry wants to raise his average by 2 points, what score must he earn on the fourth test?
Options:
A)87
B)89
C)90
D)93
E)102
Rationale: Total score on 3 tests = 94*3 = 282
Jerry wants the average to be = 96
Hence total score on 4 tests should be = 96*4 = 384
Score required on the fourth test = 384 - 282 = 102
Option E
Answer: E
Question: At a certain food stand, the price of each apple is 40¢ and the price of each orange is 60¢. Mary selects a total of 10 apples and oranges from the food stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 52¢. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 44¢?
Options:
A)1
B)2
C)3
D)4
E)5
Rationale: If the average price is 52, then Mary has 6 oranges and 4 apples (a ratio of 3:2).
For the average price to be 44, the ratio should be 1:4.
Mary must put back 5 oranges.
The answer is E.
Answer: E
Question: The total price of a basic computer and printer are $2,500. If the same printer had been purchased with an enhanced computer whose price was $500 more than the price of the basic computer, then the price of the printer would have been 1/3 of that total. What was the price of the basic computer?
Options:
A)1500
B)1600
C)1750
D)1900
E)2000
Rationale: Let the price of basic computer be C and the price of the printer be P: C+P=$2,500.
The price of the enhanced computer will be C+500 and total price for that computer and the printer will be 2,500+500=$3,000. Now, we are told that the price of the printer is 1/3 of that new total price: P=1/3*$3,000=$1000
Plug this value in the first equation: C+1000=$2,500 --> C=$1,500.
Answer: A.
Answer: A
Question: The number of girls in a class is 5 times the number of boys. Which one of the following numbers cannot represent the total number of children in the class
Options:
A)27`
B)28
C)277
D)198
E)171
Rationale: Answer:B
Answer: B
Question: The number of girls in a class is 5 times the number of boys. Which one of the following numbers cannot represent the total number of children in the class.
Options:
A)36
B)41
C)48
D)72
E)11
Rationale: Answer:A
Answer: A
Question: The first term in a sequence is 1 and the second term is 15. From the third term on each term is the average (arithmetic mean) of all preceding terms. What is the 35th term in the sequence?
Options:
A)8
B)15
C)35
D)70
E)135
Rationale: The third term is 8 since this is the average of 1 and 15.
Then the next terms will all be 8 since the average stays at 8.
The answer is A.
Answer: A
Question: Anne traveled from City A to City B in 4 hours, and her speed was between or equal to 25 miles per hour and 30 miles per hour. John traveled from City A to City B along the same route in 2 hours, and his speed was between or equal to 25 miles per hour and 55 miles per hour. Which of the following could be the distance, in miles, from City A to City B?
Options:
A)95
B)108
C)125
D)160
E)180
Rationale: In four hours Anne must travel 4 times her speed
so Anne's possible distances =
4(25 -30) or 100-120 miles
in two hours John travels 2 times his speed or
2(25-55) or 50-110 miles
the statement is only true if both distances are possible,
so you can tell its between or equal to
100-110
B is the answer.
Answer: B
Question: 1, 4, 27, 16, ?, 36, 343
Options:
A)125
B)50
C)78
D)68
E)58
Rationale: The series consists of cubes of odd numbers and square of even numbers.
Answer : A.
Answer: A
Question: In the game of Dubblefud, red balls,blue balls and green balls are each worth 2, 4 and 5 points respectively. In a certain selection of balls,the product of the point values of the balls is 16,000. If the number of blue balls in this selection equals the number of green balls, how many red balls are in the selection?
Options:
A)1
B)2
C)3
D)4
E)5
Rationale: this is equivalent to :-
2x * 4y * 5z = 16000
y = z (given)
2x * 4y * 5y = 16000
2x * y^2 = 16000/20
2x * y^2 = 800
now from options given we will figure out which number will divide 800 and gives us a perfect square :-
which gives us x = 4 as
2* 4 * y^2 =800
y^2 = 100
y =10
Number of red balls = 4 hence D
Answer: D
Question: Three dice are thrown together, Find the probability of getting a total of 6?
Options:
A)1/12
B)3/31
C)2/9
D)1/9
E)5/108
Rationale: Three different dice are thrown at the same time.
Therefore, total number of possible outcomes will be 63 = (6 × 6 × 6) = 216.
Number of events of getting a total of 6 = 10
i.e. (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2).
Therefore, probability of getting a total of 6
P(E) = Number of favorable outcomes/Total number of possible outcome
= 10/216
= 5/108
Answer: E
Question: A polling company reports that there is a 30% chance that a certain candidate will win the next election. If the candidate wins, there is a 60% chance that she will sign Bill X and no other bills. If she decides not to sign Bill X, she will sign either Bill Y or Bill Z, chosen randomly. What is the chance that the candidate will sign Bill Z?
Options:
A)10
B)6
C)8
D)4
E)5
Rationale: 30% - Candidate Elected
100%-60% = 40% - Candidate doesnotsigh Bill X
50% - Candidate randomly chooses between two bills.
These are multiplicative:
30% x 40% x 50%
0.3 x 0.4 x 0.5 = 0.06 = 6%
Answer (B)
Answer: B
Question: Bill has a small deck of 10 playing cards made up of only 2 suits of 5 cards each. Each of the 5 cards within a suit has a different value from 1 to 5; thus, there are 2 cards in the deck that have the same value. Bill likes to play a game in which he shuffles the deck, turns over 3 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?
Options:
A)1/2
B)1/3
C)1/4
D)1/5
E)1/6
Rationale: P(no pairs) = 8/9*6/8 = 2/3
P(at least one pair) = 1 - 2/3 = 1/3
The answer is B.
Answer: B
Question: A shopkeeper buys mangoes at the rate of 6 a rupee and sells them at 4 a rupee. Find his net profit or loss percent?
Options:
A)33 1/8 %
B)33 7/3 %
C)50 %
D)33 5/8 %
E)34 1/3 %
Rationale: The total number of mangoes bought by the shopkeeper be 24.
If he buys 6 a rupee, his CP = 4
He selling at 4 a rupee, his SP = 6
Profit = SP - CP = 6 - 4 = 2
Profit Percent = 2/4 * 100 = 50 %
Answer: C
Answer: C
Question: The average runs of a cricket player of 13 innings was 22. How many runs must he make in his next innings so as to increase his average of runs by 5?
Options:
A)92
B)42
C)64
D)76
E)84
Rationale: Explanation:
Average after 14 innings = 27
Required number of runs = (27 * 14) – (22 * 13)
= 378 – 286 = 92
ANSWER A
Answer: A
Question: If a light flashes every 30 seconds, how many times will it flash in 1/2 of an hour?
Options:
A)30
B)42
C)61
D)59
E)45
Rationale: There are 60 minutes in an hour.
In 1/2 of an hour there are (60 * 1/2) minutes = 30minutes.
In 1/2 of an hour there are (60 * 30) seconds = 1800 seconds.
Light flashed for every 30 seconds.
In 1800 seconds 1800/30 = 60 times.
The count start after the first flash, the light will flashes 61 times in 1/2 of an hour.
Answer:C
Answer: C
Question: A man can row a boat at 40 kmph in still water. If the speed of the stream is 6 kmph, what is the time taken to row a distance of 60 km downstream?
Options:
A)30/83 hours
B)80/13 hours
C)30/13 hours
D)30/23 hours
E)32/13 hours
Rationale: Speed downstream = 40 + 6 = 46 kmph.
Time required to cover 60 km downstream
= d/s = 60/46
= 30/23 hours.
Answer:D
Answer: D
Question: A train covers a distance in 50 min, if it runs at a speed of 48kmph on an average. The speed at which the train must run to reduce the time of journey to 40min will be.
Options:
A)60 km/h
B)55 km/h
C)40 km/h
D)70 km/h
E)65 km/h
Rationale: Time=50/60 hr=5/6hr
Speed=48mph
distance=S*T=48*5/6=40km
time=40/60hr=2/3hr
New speed = 40* 3/2 kmph= 60kmph
Answer : A
Answer: A
Question: 357 : 73 ::?
Options:
A)429 : 94
B)201 : 21
C)138 : 38
D)93 : 39
E)95 : 39
Rationale: 357 : 73
to get 2nd number,middle digit is removed, 1st & last interchanged their position of 1st no.
only option a) match this.
ANSWER:A
Answer: A
Question: The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is
Options:
A)10.5
B)11.5
C)12.5
D)13.5
E)None of these
Rationale: Explanation:
Middle numbers = [(10.5 x 6 + 11.4 x 6) - 10.9 x 11] = (131.4 - 119-9) = 11.5 Answer: B
Answer: B
Question: How many total number of ways in which n distinct objects can be put into two different boxes is
Options:
A)n2
B)2n
C)2n
D)3n
E)None of these
Rationale: Let the two boxes be B1 and B2. There are two choices for each of the n objects. So, the total number of ways is
2 × 2 ×.....× 2 (n times ) = 2n
Answer B
Answer: B
Question: A, B and C enter into partnership. A invests some money at the beginning, B invests double the amount after 6 months, and C invests thrice the amount after 8 months. If the annual gain be Rs.15000. A's share is?
Options:
A)s.7500
B)s.7200
C)s.6000
D)s.5000
E)s.6750
Rationale: x* 12 : 2x* 6: 3x* 4
1:1:1
1/3 * 15000 = 5000
ANSWER:D
Answer: D
Question: Tom opened a shop investing Rs.3000. Jose joined him 2 months later, investing Rs.4500. They earned a profit of Rs.5400 after completion of one year. What will be Jose's share of profit?
Options:
A)3000
B)2200
C)2309
D)2400
E)3200
Rationale: SoL = ~s- So Anju’s share = [5/9] x 5400 = 3000
A
Answer: A
Question: A train running at 25 km/hr takes 40 seconds to pass a platform. Next, it takes 12 seconds to pass a man walking at 5 km/hr in the same direction. Find the length of the train and that of the platform.
Options:
A)a) 200/3m175/3m
B)b) 45m160m
C)c) 50m45m
D)d) 100/3 m200/3
E)e) none of these
Rationale: Hi Math gurus correct me if I am wrong. I wondered i reached the answer mentioned here i.e 100m and 25m
lets solve it
To find out length of train the train, which is must cover the static point of the man as well as the distance covered the man during 12 sec.
The man covered the distance during 12sec is 5*5/40*12=15/2m so the train covered a distance in 12 sec= 25*5/40*12=75/2
so the length of the train(Lt) is 15/2+75/2=45m
Now crossing the platform means length of train+length of platform(Lt+p)= 40*25*5/40=125m Now the length of platform (Lp) is= 125-45=80m
So answer is 45m and 80m Must beBeven though its mentioned 45m and 160m thinking so typo mistake...........
Answer: B
Question: At a garage sale, all of the items were sold at different prices. If the price of a radio sold at the garage sale was both the 17th highest price and the 24th lowest price among the prices of the items sold, how many items were sold at the garage sale?
Options:
A)38
B)39
C)40
D)41
E)42
Rationale: There were 16 items sold at a higher price than the radio and 23 items sold at a lower price than the radio. Including the radio, there were 16 + 23 + 1 = 40 items sold.
The answer is C.
Answer: C
Question: For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?
Options:
A)5
B)6
C)15
D)16
E)18
Rationale: Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.
Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.
Answer: D.
Answer: D