divelab/Sys2Bench huggingface dataset:
Question: If x and y are both prime numbers greater than 10, which of the following CANNOT be true?
I. xy is an even number.
II. The difference between x and y equals 117.
III. The sum of x and y is even.
Options:
A) I only
B) II and III only
C) I and III only
D) I and II only
E) I, II and III
Rationale: MANHATTAN GMATOFFICIAL SOLUTION:
Since x and y are both prime numbers greater than 10, they must both be odd. Therefore xy must be an odd number, so Statement I cannot be true. Similarly, if x and y are both odd, then x - y cannot equal 117 (an odd number). This difference must be even. Therefore, Statement II cannot be true. Finally, since x and y are both odd, x + y must be even, so Statement III will always be true. Since Statements I and II CANNOT be true, but Statement III IS true,the correct answer is (D).
Answer : D
Answer: D
Question: What is the difference between the compound interest on Rs.12000 at 20% p.a. for one year when compounded yearly and half yearly?
Options:
A)399
B)383
C)120
D)288
E)133
Rationale: When compounded annually, interest
= 12000[1 + 20/100]1 - 12000 = Rs.2400
When compounded semi-annually, interest
= 12000[1 + 10/100]2 - 12000 = Rs.2520
Required difference = 2520 - 2400 = Rs.120
Answer: C
Answer: C
Question: Two cars cover the same distance at the speed of 50 and 64 kmps respectively. Find the distance traveled by them if the slower car takes 1 hour more than the faster car.
Options:
A)212 km
B)214 km
C)224 km
D)216 km
E)204 km
Rationale: 50(x + 1) = 64x
X = 3.57
60 * 3.57 = 214 km
ANSWER:B
Answer: B
Question: 1 barrel of gas is enough to power the lawnmower to mow both you and your neighbor's lawn 15 times before refilling. You're son decided he's going take over mowing these lawns and also mow 6 more neighbor's lawns. How many times will he be able to mow your lawn and the neighbor's lawns before refilling the barrel of gas?
Options:
A)4.15
B)2.75
C)1.875
D)3.75
E)5
Rationale: First set up a ratio with all the relevant values and also the unknown variable:
2*15 = 8*x
Now solve by multiplying 2 by 15 and then dividing that product by 8:
2*15 = 30
30/8 = x
x = 3.75
The value of x equals how many times 8 lawns can be mowed on one 1 barrel of gas.
Answer: D
Answer: D
Question: In a public show 60 % of the seats were filled. If there were 600 seats in the hall, how many seats were vacant?
Options:
A)100
B)110
C)120
D)140
E)240
Rationale: 75 % of 600
= 60/100 × 600
= 360
Therefore, the number of vacant seats = 600 - 360 = 240.
Answer:E
Answer: E
Question: 5,9,18,30,50,__
Options:
A)79
B)80
C)81
D)82
E)83
Rationale: (9+5) +4 = 18
(16+9) +5 = 30
(28+16)+6 = 50
(48+28)+7 = 83
ANSWER:E
Answer: E
Question: How many words can be formed from the letters of the word ‘EXTRA’ so that the vowels are never together?
Options:
A)72
B)25
C)14
D)65
E)18
Rationale: Sol. The given word contains 5 different letters.
Taking the vowels EA together, we treat them as one letter.
Then, the letters to be arranged are XTR (EA).
These letters can be arranged in 4! = 24 ways.
The vowels EA may be arranged amongst themselves in 2! = 2 ways.
Number of words, each having vowels together = (24x2) = 48 ways.
Total number of words formed by using all the letters of the given words
= 5! = (5x4x3x2x1) = 120.
Number of words, each having vowels never together = (120-48) = 72.
Ans:A
Answer: A
Question: A cistern which could be filled in 9 hours takes one hour more to be filled owing to a leak in its bottom. If the cistern is full in what time will the leak empty it?
Options:
A)33
B)77
C)28
D)90
E)28
Rationale: 1/9 - 1/x = 1/10
=> 90 hrs
Answer:D
Answer: D
Question: Amy, Bob, Cindy, and Daniel all have the same birthday, but they are all different ages. Today is their birthday. Cindy's age is divisible by Bob's age, and Amy's age is divisible by Cindy's age. If A is Amy's age in years, B is Bob's age, and C is Cindy's age, which of the following CANNOT be Daniel's age?
Options:
A)AC/B
B)(A+C)/B
C)(A+B)/C
D)AB/C
E)(A-C)/B
Rationale: Since today's is Daniel's birthday, his age must be an integer. As per the question, let A = 12, B = 2, and C = 6. Plugging these into the answer choices, only C does not yield an integer.
Answer: C
Question: Some of the 68% solution of acid was removed and this was replaced with an equal amount of 11% solution of acid. As a result, a 35% solution of acid was obtained. What fraction of the original solution was replaced?
Options:
A)7/11
B)7/17
C)8/17
D)8/19
E)11/19
Rationale: Let x be the fraction of the original solution that was replaced.
0.68*(1-x)+0.11(x)=0.35
0.57x=0.33
x=33/57=11/19
The answer is E.
Answer: E
Question: A and B go around a circular track of length 600 m on a cycle at speeds of 30 kmph and 60 kmph. After how much time will they meet for the first time at the starting point?
Options:
A)120 sec
B)176 sec
C)178 sec
D)72 sec
E)189 sec
Rationale: Time taken to meet for the first time at the starting point
= LCM { length of the track / speed of A , length of the track / speed of B}
= LCM { 600/ (30 * 5/18) , 600/ (60 * 5 /18) }
= 72 sec.
Answer: D
Answer: D
Question: J and M were each paid x dollars in advance to do a certain job together. J worked on the job for 10 hours and M worked for 4 hours less than J. If M gave J y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount, in terms of y, that J was paid in advance?
Options:
A)a)4y
B)b)5y
C)c)6y
D)d)8y
E)e)9y
Rationale: J and M were both paid x dollars
If M gave Y dollars from her X to J then at the end
J will have X+Y dollars
M will have X- Y dollars
Now since the hourly wages have to be same then
Hourly wage of J = (X+Y) / 10 (Since J worked for 10 hrs)
Hourly wage of M = (X-y) /6 (Since M worked 4 hrs less than J)
Equating both
(X+Y)/10=(X-Y)/6
on simplifying X=4y
Answer A
Answer: A
Question: In a certain sequence , the first term is 1, and each successive term is 1 more than the reciprocal of the term that immediately precedes it. What is the fifth term of the sequence?
Options:
A) 3/5
B) 5/8
C) 8/5
D) 5/3
E) 9/2
Rationale: First term = 1
Second term = (1/1)+ 1 = 2
Third Term = (1/2)+1 = 3/2
Forth Term = [1/(3/2)]+1 = 5/3
Fifth Term = [1/(5/3)]+1 = 8/5
Answer: Option C
Answer: C
Question: An error 2% in excess is made while measuring the side ofa square. The % of error in the calculated area of the square is?
Options:
A)4%
B)4.04%
C)4.12%
D)4.17%
E)4.18%
Rationale: 100 cm is read as 102 cm.
A1 = (100 x 100) cm2 and A2 (102 x 102) cm2.
(A2 - A1) = [(102)2 - (100)2]
= (102 + 100) x (102 - 100)
= 404 cm2.
Percentage error = 404 x 100 % = 4.04%
100 x 100
B
Answer: B
Question: A cistern has a leak which would empty the cistern in 20 minutes. A tap is turned on which admits 5 liters a minute into the cistern, and it is emptied in 24 minutes. How many liters does the cistern hold?
Options:
A)480
B)287
C)289
D)270
E)600
Rationale: 1/x - 1/20 = -1/24
x = 120
120 * 5 = 600
Answer: E
Answer: E
Question: Mayank, Mirza, Little and Jaspal bought a motorbike for $60. Mayank paid one-half of the sum of the amounts paid by the other boys. Mirza pad one-third of the sum of the amounts paid by the other boys. Little paid one-fourth of the sum of the amounts paid by the other boys. How much did Jaspal have to pay?
Options:
A)$15
B)$13
C)$17
D)None of these
E)Cannot be determined
Rationale: Explanation :
Mayank paid 1/2 of what others paid.
=> Mayank paid 1/3 rd of the total amount = $20.
Similarly, Mirza paid $15 and Little paid $12.
Remaining amount of $60 - $20 - $15 - $12 = $13 is paid by Jaspal.
Answer : B
Answer: B
Question: Fry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain).
The captain always proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go “Aye”, the loot is divided as proposed, as no pirate would be willing to take on the captain without superior force on their side.
If the captain fails to obtain support of at least half his crew (which includes himself), he faces a mutiny, and all pirates will turn against him and make him walk the plank. The pirates start over again with the next senior pirate as captain.
What is the maximum number of coins the captain can keep without risking his life?
Options:
A)98
B)77
C)55
D)39
E)82
Rationale: A
98 Coins
The captain says he will take 98 coins, and will give one coin to the third most senior pirate and another coin to the most junior pirate. He then explains his decision in a manner like this…
If there were 2 pirates, pirate 2 being the most senior, he would just vote for himself and that would be 50% of the vote, so he’s obviously going to keep all the money for himself.
If there were 3 pirates, pirate 3 has to convince at least one other person to join in his plan. Pirate 3 would take 99 gold coins and give 1 coin to pirate 1. Pirate 1 knows if he does not vote for pirate 3, then he gets nothing, so obviously is going to vote for this plan.
If there were 4 pirates, pirate 4 would give 1 coin to pirate 2, and pirate 2 knows if he does not vote for pirate 4, then he gets nothing, so obviously is going to vote for this plan.
As there are 5 pirates, pirates 1 & 3 had obviously better vote for the captain, or they face choosing nothing or risking death.
Answer: A
Question: Sam purchased 20 dozens of toys at the rate of Rs. 375 per dozen. He sold each one of them at the rate of Rs. 33. What was his percentage profit?
Options:
A)3.5
B)4.5
C)5.6 percent
D)6.5
E)None
Rationale: Solution
C.P of 1 toy	= Rs.(375/12)
= Rs.31.25
S.P of 1 toy	= Rs.33.
Therefore,Profit	= (1.75/31.25×100)%
=(28/5)%
=5.6%.
Answer C
Answer: C
Question: 9log9(6) = ?
Options:
A)1
B)6
C)3
D)4
E)5
Rationale: Exponential and log functions are inverse of each other. Hence
aloga(x) = x , for all x real and positive.
and therefore
9log9(6) = 6
correct answer B
Answer: B
Question: What is the probability of rolling five six-sided dice, and getting a different number on each die?
Options:
A)1/12
B)1/3
C)4/9
D)5/54
E)7/18
Rationale: MAGOOSHOFFICIAL SOLUTION
For the first die, we can roll any one of six numbers. For the second die, we can roll any number save for the number we rolled on the first die, giving us 5 possibilities. For the third die, we can roll four different numbers (we can’t roll the number we rolled on the first or second die.
6 x 5 x 4x3x2 = 720 possibilities out of 216 total possibilities. (For total possibilities we get 6 x 6 x 6x6x6 = 6^5).
720/6^5 =5/54
Answer: D.
Answer: D
Question: Mary's income is 60% more Than Tim's income and Tim's income is 20% less than Juan's income.What % of Juan's income is Mary's income.
Options:
A)128%
B)B.120%
C)96%
D)80%
E)64%
Rationale: Even I got 96%
J = 100
T = 100*0.8 = 80
M = 80*1.6 = 128
If Mary's income is x percent of J
M = J*x/100
x = M*100/J = 128*100/100 = 128
Ans:A
Answer: A
Question: A bag of P peanuts can be divided into 9 smaller bags with 6 peanuts left over. Another bag of M peanuts can be divided into 12 smaller bags with 4 peanuts left over. Which of the following is the remainder when PM is divided by 18?
Options:
A)4
B)6
C)3
D)2
E)1
Rationale: P = 9x + 6
M = 12y + 4
PM = (9x + 6)*(12y + 4) = 108xy + 36x + 72y + 24
Remainder of nm/18 = (108xy + 36x + 72y + 24)/18
Observe that the first three terms are a multiple of 18
24 when divided by 18 leaves remainder 6
Hence mn/18 will leave remainder 6
Correct Option: B
Answer: B
Question: Martin bought 10 concert tickets, some at the full price of $2.00 per ticket, and some at a discounted price of $1.60 per ticket. If he spent a total of $18.40, how many discounted tickets did he buy?
Options:
A)3
B)4
C)5
D)6
E)7
Rationale: Let x be the number of tickets he bought at $2 per ticket.
then 2 x + (10-x)1.6 = 18.4
0.4x = 2.4 => x = 6
discounted tickets = 10 -x = 4
Ans : B
Answer: B
Question: A farm has only chickens and pigs. When the manager of the farm counted the heads of the animals on the farm, the number totaled up to 70. When the number of legs was counted, the number totaled up to 204. How many more chickens than pigs are there on the farm? Note: Each pig has 4 legs and each chicken had 2 legs.
Options:
A)4
B)5
C)6
D)7
E)8
Rationale: Let x be the number of chickens.
Then 70-x is the number of pigs.
2x+4(70-x)=204
2x=76
x=38 (chickens)
70-x=32 (pigs)
There are 38-32=6 more chickens on the farm.
The answer is C.
Answer: C
Question: The average height of 50 pupils in a class is 155 cm. Five of them whose height is 146 cm, leave the class and five others whose average height is 156 cm, join. The new average height of the pupils of the class (in cm) is ?
Options:
A)156 cm
B)151 cm
C)276 cm
D)287 cm
E)297 cm
Rationale: Total height = 155 * 50 = 7750 cm.
New average = [7750 - 5 * 146 + 5 * 156 ] / 50
= 156 cm.
Answer:A
Answer: A
Question: There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability R that the both positions will be taken by members of just one party?
Options:
A)1/25
B)12/49
C)1/4
D)24/49
E)1/2
Rationale: R probability of one party having both spots:
(1/2) * (24/49) = 12/49
(1/2) or (25/50) because it does not matter which party or which person gets the first spot. (24/49) because after one person from a particular party is chosen, there are 24 members of the same party left out of 49 total candidates.
Since this result can happen for both parties, (12/49) + (12/49) = (24/49)
Answer: D
Answer: D
Question: A can do a work in 60 days. He works for 15 days and later B alone completes the work in 30 days. In what time will they together complete the work?
Options:
A)24 days
B)25 days
C)30 days
D)32 days
E)None of these
Rationale: let A can do a work in = 1/60 days
he does 15 days so work done by A in 15 days= (15) * (1/60) = 1/4
remaining work is = 1- 1/4= 3/4
B's one day work is =1/30
so 3/4 work is done by B in=(3/4) * (1/30) = 1/40 days
time require to complete the work is= 1/60 + 1/40 = 5/120
5/120 = 1/24
so 24 days require to complete the work together.
ANSWER:A
Answer: A
Question: If P represents the product of the first 14 positive integers, then P is not a
multiple of:
Options:
A)a) 99
B)b) 84
C)c) 51
D)d) 65
E)e) 57
Rationale: a) 99 = 9*11
b) 84 = 4*7*3
c) 51 =17*3
d) 65 = 5*13
e) 57 =19*3
since 17 is not there in first 15 positive numbers it is the only possibility
C
Answer: C
Question: A batsman makes a score of 90 runs in the 11th inning and thus increases his average by 5. Find his average after 11th inning.
Options:
A)40
B)50
C)62
D)45
E)56
Rationale: Let the average after 11th inning = x
Then, average after 10th inning = x-5
10(x-5)+90 = 11x
x = 90-50 = 40
Answer is A
Answer: A
Question: What is the remainder when 2990990900032 is divided by 32 ?
Options:
A)16
B)8
C)4
D)0
E)2
Rationale: Though i was unaware of the divisibility test for 32 but i guessed the pattern!!
divisibility rule for 4- last two digits must be divisible by 4
divisibility rule for 8- last three digits must be divisible by 8
similarly, divisibility rule for 32 - last five digits must be divisible by 32
Hence, Ans D
Answer: D
Question: A certain drink of type A is prepared by mixing 4 parts milk with 3 parts fruit juice. Another drink of type B is prepared by mixing 4 parts of fruit juice and 3 parts of milk. How many liters of fruit juice must be added to 84 liters of drink A to convert it to drink B?
Options:
A)24
B)28
C)32
D)36
E)40
Rationale: In 84 liters of drink A, there are 48 liters of milk and 36 liters of juice.
With 48 liters of milk, we need a total of 64 liters of juice to make drink B.
We need to add 28 liters of juice.
The answer is B.
Answer: B
Question: What is (16^7+16) / 16?
Options:
A)15467118
B)16777217
C)17827343
D)18047455
E)19357579
Rationale: (16^7+16) / 16 =
16*(16^6+1) / 16 =
16^6 + 1
Clearly this is a number which ends with a 7 in the units place.
The answer is B.
Answer: B
Question: If an integer n is to be selected at random from 1 to 100, inclusive, what is probability n(n+1) will be divisible by 16?
Options:
A)1/4
B)1/8
C)1/2
D)2/3
E)3/4
Rationale: because n(n+1) is always an even product of even*odd or odd*even factors,
there is a probability of 1 that that it will be divisible by 2,
and, thus, a probability of 1/2 that it will be divisible by 4
and, thus, a probability of 1/4 that it will be divisible by 8
and, thus, a probability of 1/8 that it will be divisible by 16
1*1/8=1/8
Answer : B
Answer: B
Question: If A=x^4 + y^4 = 100, then the greatest possible value of x is between
Options:
A)0 and 3
B)3 and 6
C)6 and 9
D)9 and 12
E)12 and 15
Rationale: My attempt:
If A=x^4+y^4=100, then the greatest possible value of x would be when y is minimum.
Let y^4 be 0. Now x^4 = 100. x should be definitely greater than 3 but less than 4. The only option that fits this range is B
Hence answer is --
b) 3 and 6.
Answer: B
Question: What is the 4 digit number in which the 1st digit is 1/3 of the second, the 3rd is the sum of the 1st and 2nd, and the last is three times the second?
Options:
A)1243
B)1345
C)1349
D)1456
E)2467
Rationale: First digit is 1/3 second digit => The numbers can be 1 & 3, 2& 6, 3 & 9.
First + second = third => we can eliminate 3 & 9 since 3 + 9 = 12.
Last is 3 times the second => we can eliminate option 2 & 6 since 3 * 6 = 18.
Hence the number is 1349
C
Answer: C
Question: What is the ratio between perimeters of two squares one having 3 times the diagonal then the other?
Options:
A)3:6
B)3:9
C)3:2
D)3:1
E)3:5
Rationale: d = 3d d = d
a√2 = 3d a√2 = d
a = 3d/√2 a = d/√2
=> 3: 1
Answer:D
Answer: D
Question: A can do a work in 6 days. B can do the same work in 12 days. If both A & B are working together in how many days they will finish the work?
Options:
A)3
B)5
C)4
D)2
E)6
Rationale: A rate = 1/6
B rate = 1/12
(A+B) rate = (1/6)+(1/12) = 1/4
A & B finish the work in 4days
correct option is C
Answer: C
Question: If 10 ounces of a strong vinegar solution are diluted with 50 ounces of water to form a five-percent vinegar solution, what was the concentration of the original solution?
Options:
A)25%
B)28%
C)30%
D)32%
E)36%
Rationale: Let x be the concentration of the original solution.
10x = 0.05(60)
x = 0.3
The answer is C.
Answer: C
Question: If y is 40% greater than x, than x is what % less than y?
Options:
A)24.4%
B)28.6%
C)32.3%
D)36.5%
E)40.9%
Rationale: y = 1.4x
x = y/1.4 = 10y/14 = 5y/7
x is 2/7 less which is 28.6% less than y.
The answer is B.
Answer: B
Question: Lionel left his house and walked towards Walt's house, 48 miles away. Four hours later, Walt left his house and ran towards Lionel's house. If Lionel's speed was 2 miles per hour and Walt's 6 miles per hour, how many miles had Lionel walked when he met Walt?
Options:
A)12
B)18
C)20
D)24
E)28
Rationale: In the first 4 hours Lionel at the rate of 2 miles per hour covered distance=rate*time=4*2=8 miles. So, the distance between him and Walt was 48-8=40 miles when Walt left his house. Now, their combined rate to cover this distance was 2+6=8 miles per hour, hence they will meet (they will cover that distance) in time=distance/rate=40/8=5 hours. Total time that Lionel was walking is 4+5=9 hours, which means that he covered in that time interval distance=rate*time=2*9=18 miles.
Answer: B.
Answer: B
Question: At what price must an article costing Rs.38 be marked in order that after deducting 5% from the list price. It may be sold at a profit of 25% on the cost price?
Options:
A)52.5
B)50
C)62.3
D)62.1
E)62.4
Rationale: CP = 38
SP = 38*(125/100) = 38
MP*(95/100) = 47.5
MP = 50
Answer:B
Answer: B
Question: If x = 3k, where k is a prime number greater than 2, how many different positive even divisors does x have, including x?
Options:
A)0
B)1
C)2
D)3
E)It cannot be determined from the information given
Rationale: k is prime greater than 2 thus k is odd
3k is odd
Thus no even factors
Ans:A
Answer: A
Question: Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?
Options:
A)4.2
B)4.4
C)4.7
D)5.1
E)None
Rationale: Explanation: In this simple addition series, each number increases by 0.8.
Answer: Option C
Answer: C
Question: How many words, with or without meaning, can be formed using all letters of the word RED using each letter exactly once?
Options:
A)6
B)7
C)8
D)9
E)10
Rationale: The Word RED has exactly 3 letters which are all different.
Therefore the number of words that can be formed = Number of permutations of 3 letters taken all at a time.
= P (3, 3) = 3!
= 3 x 2 × 1
= 6
ANSWER:A
Answer: A
Question: Mary’s annual income is $15,000 and John’s annual income is $18,000. By how much must Mary’s annual income increase so that it constitutes 50% of Mary and John’s combined income?
Options:
A) $3,000
B) $4,000
C) $7,000
D) $11,000
E) $25,000
Rationale: Let Mary's income increase by x, then the equation will be
15,000 + x= (50 /100)* (15,000 + x + 18000 )
15,000 + x = (1/2) * (33,000 + x)
30,000 + 2x = x+ 33,000
x =3,000
So, Answer will be A
Answer: A
Question: The price of an item is discounted 10 percent on day 1 of a sale. On day 2, the item is discounted another 20 percent, and on day 3, it is discounted an additional 40 percent. The price of the item on day 3 is what percentage of the sale price on day 1?
Options:
A)28%
B)48%
C)64.8%
D)70%
E)72%
Rationale: Let initial price be 1000
Price in day 1 after 10% discount = 900
Price in day 2 after 20% discount = 720
Price in day 3 after 40% discount = 432
So, Price in day 3 as percentage of the sale price on day 1 will be = 432/900*100 => 48%
Answer will definitely be (B)
Answer: B
Question: At a certain conference, 150 students spoke one or more of the following languages: Spanish, English, and French. If 43 students spoke Spanish, 90 spoke English, 65 spoke French, and 32 spoke only 2 languages, how many students spoke three different languages?
Options:
A)7
B)8
C)12
D)15
E)18
Rationale: 150=43+90+65-32-2x
x=8
B. 8
Answer: B
Question: A man can row upstream at 10kmph and downstream at 18 kmph. The speed of the stream is
Options:
A)2.5 km/hr
B)4 km/hr
C)5 km/hr
D)10.5 km/hr
E)None
Rationale: Sol.
Speed of stream = 1/2(18 -10) kmph = 4 kmph.
Answer B
Answer: B
Question: If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 45x and 20y?
Options:
A)5
B)5(x-y)
C)20x
D)20y
E)35x
Rationale: We are looking for a choice that CANNOT be the greatest common divisor of 45x and 20y ...which means 45x and 20y when divided by the answer choice the quotient should not be a integer.
lets check
a. 5 45x/5 = 9x and 20y/5 = 4y both are integers so eliminate
b. 5(x-y) when x = 2 and y = 1 it could be be the greatest common divisor ..so eliminate
c. 20x when x = 1 its 20 and 20 cannot be the greatest common divisor of 435x and 20y ...
or 45x/20x = 9/4 which is not a integer.
so answer is C.
Answer: C
Question: Of the two-digit positive integers that have no digits equal to zero, how many have two digits that are equal to each other and the remaining digit different from the other two?
Options:
A)24
B)9
C)72
D)144
E)216
Rationale: Of the two-digit positive integers that have no digits equal to zero, how many have two digits that are equal to each other and the remaining digit different from the other two?
A. 24
B. 36
C. 72
D. 144
E. 216
Choosing the digit for X - 9 ways;
# of permutations of 3 digits in XX -2!/2!
Total: 9*2!/2!=9.
Answer: B.
Answer: B
Question: A certain company retirement plan has arule of 70provision that allows an employee to retire when the employee's age plus years of employment with the company total at least 70. In what year could a female employee hired in 1986 on her 30nd birthday first be eligible to retire under this provision?
Options:
A)2003
B)2004
C)2005
D)2006
E)2007
Rationale: She must gain at least 70 points, now she has 30 and every year gives her two more points: one for age and one for additional year of employment, so 30 + 2*(# of years) = 70 --> (# of years) = 20 --> 1986+20=2006.
Answer: D.
Answer: D