MU-NLPC/Calc-aqua_rat huggingface dataset:
How many positive integers less than 60 are equal to the product of a positive multiple of 5 and an even number?
Choose one:  A) 4  B) 5  C) 9  D) 10  E) 11

Solution: Number of positive multiples of 5 less than 60 = 11   5*1= 
5*1 = 5
5   5*2= 
5*2 = 10
10   5*3= 
5*3 = 15
15   5*4= 
5*4 = 20
20   5*5= 
5*5 = 25
25   5*6= 
5*6 = 30
30   5*7= 
5*7 = 35
35   5*8= 
5*8 = 40
40   5*9= 
5*9 = 45
45   5*10= 
5*10 = 50
50   5*11= 
5*11 = 55
55   Only 5 of the above are a product of a positive multiple of 5 and an even number - 10,20,30,40,50   Answer 5
Answer: B
In what time will a train 120 m long cross an electric pole, it its speed be 185 km/hr?  Choose the correct answer.
A) 2.3 sec  B) 2.9 sec  C) 2.7 sec  D) 8.7 sec  E) 8.5 sec

Solution: Speed = 185 * 5/18 = 51 m/sec   Time taken = 120/51 = 2.3 sec.   2.3 secnswer: 2.3 sec
Answer: A
How many positive three-digit integers are divisible by both 2 and 7?  Choose the correct choice from the following answers.
A) 40
B) 46
C) 52
D) 64
E) 70

Solution: A number to be divisible by both 2 and 7 should be divisible by the least common multiple of 2 and 7 so by 14.   Multiples of 14 between 100 and 999, inclusive is   (last-first)/multiple+1=(994-112)/14+1=63+1= 
63+1 = 64
64   Answer: 64.
Answer: D
A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year? Choose the correct choice from the following answers:
A) 1:5
B) 625:3125
C) 100:105
D) 100^4:105^4
E) 725:3225

Solution: The interest earned in the 1st year = $500   The interest earned in the 2nd year = $500*1.05   The interest earned in the 3rd year = $500*1.05^2   The interest earned in the 4th year = $500*1.05^3   The interest earned in the 5th year = $500*1.05^4   (500*1.05^3)/(500*1.05^4) = 1/1.05=100/105.   Answer: 100:105.
Answer: C
There are 300 employees in a room. 99% are managers. How many managers must leave the room to bring down the percentage of managers to 98%? Choices:
A) 200
B) 175
C) 150
D) 125
E) 100

Solution: There are 297 managers and 3 others. The 3 others would compose 2% of the total number of people if there were 150 people in the room. Thus 150 managers must leave.   The answer is 150.
Answer: C
The area of a square garden is Q square feet and the perimeter is P feet. If Q=P+21, what is the perimeter of the garden in feet? Answers.
A) 24 B) 28 C) 32 D) 36 E) 40

Solution: Let x be the length of one side of the square garden.   x^2 = 4x + 21   x^2 - 4x - 21 = 0   (x-7)(x+3) = 0   x = 7, -3   P = 4(7) = 28   The answer is 28.
Answer: B
In a forest 10 rabbits were caught, tagged with electronic markers, then released. A month later, 10 rabbits were captured in the same forest. Of these 10 rabbits, it was found that 2 had been tagged with the electronic markers. If the percentage of tagged rabbits in the second sample approximates the percentage of tagged rabbits in the forest, and if no rabbits had either left or entered the forest over the preceding month, what is the approximate number of rabbits in the forest?	Choose one
A) 150 B) 70 C) 50 D) 1,500 E) 500

Solution: Given that 10 rabbits were caught and tagged in the first sample, and 10 were caught in the second sample of which 2 had tags, then 2/10 approximates the fraction of the population of tagged rabbits in the forest.   Proportion: A/x = C/D   A = rabbits tagged first sample   C/D = the fraction of the population of tagged rabbits in the forest   x = the population of rabbits in the forest   Proportion: 10/x = 1/5   x = (1)(10)(5) = 50. There are approximately 50 rabbits in the forest.   Answer is 50
Answer: C
A and B started a business with initial investments in the ratio 14 : 15 and their annual profits were in the ratio 7 : 6. If A invested the money for 10 months, for how many months did B invest his money? Choose the correct choice: A) 2 B) 4 C) 6 D) 8 E) 9

Solution: Sol.   Suppose A invested Rs. 14x for 10 months and B invested Rs. 15x for y months. Then,   14x * 10 / 15x * y = 7/6 ⇒ y = 840 / 105= 
840 / 105 = 8
8   Hence, B invested the money for 8 months.   Answer 8
Answer: D
Sum of 3 consecutive even no.'s is 34 more than the 1st no. of the series. Find the middle no.?	Choices:
A) 8  B) 12  C) 14  D) 16  E) 18

Solution: Let the numbers be x, x+2 and x+4 then   x + x+2 + x+4 = x+34   3x + 6 = x + 34   2x = 28   \inline \therefore x = 14   \inline \therefore Middle number = x + 2 = 14+2= 
14+2 = 16
16   16
Answer: D
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.18000. A's share is?  Choose the correct answer:
A) 6028
B) 6020
C) 6000
D) 6098
E) 6008

Solution: x* 12 : 2x* 6: 3x* 4   1:1:1   1/3 * 18000= 
1/3 * 18000 = 6_000
6000   Answer: 6000
Answer: C
How many integers between 265,205and 758,805 have tens digit 1 and units digit 3?  Select
A) 4,629
B) 4,639
C) 3,946
D) 4,926
E) 4,936

Solution: There is one number in hundred with 1 in the tens digit and 3 in the units digit: 13, 113, 213, 313, ...   The difference between 265,205 and 758,805 is 758,805-265,205=493,600 - one number per each hundred gives 133,900/100=4,936 numbers.   Answer: 4,936.
Answer: E
A bottle contains a certain solution. In the bottled solution, the ratio of water to soap is 3:2, and the ratio of soap to salt is two times this ratio. The solution is poured into an open container, and after some time, the ratio of water to soap in the open container is halved by water evaporation. At that time, what is the ratio of water to salt in the solution?	Pick:
A) 1:1
B) 2:3
C) 3:2
D) 9:4
E) 18:8

Solution: Water:soap = 3:2   Soap:Salt=6:2   => For 6 soap, salt = 2   => For 2 Soap, salt = (2/6)*2 = 2/3   So, water:soap:salt = 3:2:2/3 = 18:12:4   After open container, water:soap:salt = 9:18:4   So, water:salt = 9:4 = 18:8   18:8
Answer: E
At a certain supplier, a machine of type A costs $20,000 and a machine of type B costs $55,000. Each machine can be purchased by making a 20 percent down payment and repaying the remainder of the cost and the finance charges over a period of time. If the finance charges are equal to 40 percent of the remainder of the cost, how much less would 2 machines of type A cost than 1 machine of type B under this arrangement? Choose the most appropriate option:  A) $10,000  B) $11,200  C) $12,000  D) $12,800  E) $19,800

Solution: Total Cost of 2 Machines of Type A   = 20% of (Cost of 2 machine A) + remainder + 40 % remainder   = 20% of 40000 + (40000 - 20% of 40000) + 40% of (40000 - 20% of 40000)   = 52800   Total Cost of 1 Machine of Type B   = 20% of (Cost of 1 machine B) + remainder + 40 % remainder   = 20% of 55000 + (55000 - 20% of 55000) + 40% of (50000 - 20% of 55000)   = 72600   Diff = 72600 - 52800= 
72600 - 52800 = 19_800
19800   Hence, $19,800.
Answer: E
If A, B, and C invested some amount in the ratio 3:2:5 in a buisiness. If the total profit incurrec after a year was Rs.12100, find the share of B.
Choose the correct choice from the following:
A) Rs.6050
B) Rs.3630
C) Rs.2420
D) Rs.3360
E) Rs.6100

Solution: Total profit after a year =12100 Ratio of profit share =3x+2x+5x=10x 10x 3 +2 +5= 
3 +2 +5 = 10
10x 10x= 12100. Therefore x=1210 B's share = 2*1210 =Rs.2420 Answer : Rs.2420
Answer: C
If an amount of Rs 13,640 is divided equally among
62 students, approximately how much amount will
each student get?
Choose one:
A) Rs 206	B) Rs 210	C) Rs 220	D) Rs 218	E) None

Solution: Amount received by each student = 13640/62 â‰ˆ Rs   220   Answer Rs 220
Answer: C
Three friends sai, vani and ram divide $1020 amongs them in such a way that if $10, $20 and $15 are removed from the sums that sai, vani and ram received respectively, then the share of the sums that they got will be in the ratio of 8 : 4 : 2. How much did ram receive?
Pick one
A) $147
B) $142
C) $242
D) $122
E) $143

Solution: Explanatory Answer   Let the sums of money received by A, B and C be x, y and z respectively.   Then x - 10 : y - 20 : z -15 is 8a : 4a : 2a   When $10, $20 and $15 are removed, we are removing a total of $45 from $1020.   Therefore, 8a + 4a + 2a = 1020 - 45= 
1020 - 45 = 975
975   i.e., 14a= 975   or a = 70.   We know that z - 2 = 70a = 2 * 70= 
2 * 70 = 140
140   Therefore, z= 140 + 2= 
140 + 2 = 142
$142   The correct choice is ($142)
Answer: B
A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half (1/2). The number of pupils in the class is:
Choose the correct answer
A) 38  B) 39  C) 40  D) 46  E) 49

Solution: Let there be x pupils in the class.   Total increase in marks = (x x 1/2) = x/2   x /2 = (83 - 63)   x /2= 20   x= 40   Answer is 40
Answer: C
There are two circles of different radii. The are of a square is 784 sq cm and its side is twice the radius of the larger circle. The radius of the larger circle is seven - third that of the smaller circle. Find the circumference of the smaller circle? Answers.
A) 16	B) 15	C) 12	D) 87	E) 16

Solution: Let the radii of the larger and the smaller circles be l cm and s cm respectively. Let the side of the square be a cm.   a2 = 784 = (4)(196) = (22).(142)   a = (2)(14) = 28   a = 2l, l = a/2 = 14   l = (7/3)s   Therefore s = (3/7)(l) = 6 Circumference of the smaller circle = 2∏s   = 12∏ cm.   Answer:12
Answer: C
A train running at the speed of 90 km/hr crosses a pole in 6 seconds. What is the length of the train?  Pick one
A) 150  B) 2766  C) 187  D) 267  E) 191

Solution: Speed=(60 * 5/18) m/sec = (50/3) m/sec Length of the train = (Speed x Time) = (50/3 * 9)= 
(50/3 * 9) = 150
150 m.   150nswer: 150
Answer: A
A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year?  Choose the correct option
A) 32:47  B) 32:49  C) 32:45  D) 32:46  E) 32:41

Solution: A : B   (8000*4)+(4000*8) : (9000*6)+(6000*6)   64000 : 90000   32 : 45   Answer: 32:45
Answer: C
In a graduate physics course, 60 percent of the students are male and 30 percent of the students are married. If two-sevenths of the male students are married, what fraction of the female students is single? Choose the correct choice from the following choices.
A) 2/7
B) 17/30
C) 1/2
D) 27/40
E) 5/7

Solution: let assume there are 100 students of which 60 are male and 40 are females   if 30 are married then 70 will be single.   now its given that two-sevenths of the male students are married that means 2/7 of 60 = 17 males are married   if 30 is the total number of students who are married and out of that 17 are males then the remaining 13 will be females who are married.   total females = 40   married females = 13   then single females = 40-13= 
40-13 = 27
27   we need to find the fraction of female students who are single i.e single female students / total female student= 27/40 [27/40]
Answer: D
In how many ways, a committee of 5 members can be selected from 7 men and 5 ladies, consisting of 3 men and 2 ladies?	Options.
A) 75	B) 152	C) 175	D) 350	E) 412

Solution: (3 men out 7) and (2 ladies out of 5) are to be chosen   Required number of ways = (7C3 * 5C2) = 350   Answer is 350
Answer: D
Compound X contains elements A and B at an approximate ratio, by weight, of 2:10. Approximately how many grams of element B are there in 300 grams of compound X?  Select the correct option: A) 54 B) 162 C) 250 D) 270 E) 322

Solution: total number of fractions =2+10= 
2+10 = 12
12   element B constitutes=10 out of 12 parts of X   So in 300 gms of X have 300*10/12= 
300*10/12 = 250
250 gms of B   and  300-250= 
 300-250 = 50
50 gms of A.   cross check:- A/B= 50/250 =2/10 (as given)   Ans 250
Answer: C
The sum of the present ages of two persons A and B is 60. If the age of A is twice that of B, find the sum of their ages 7 years hence? Choose the correct choice from the following answers.
A) 58
B) 74
C) 62
D) 70
E) 74

Solution: Explanation:   A + B = 60, A = 2B   2B + B = 60 => B = 20 then A = 40.   7 years, their ages will be 47 and 27.   Sum of their ages = 47 + 27= 
47 + 27 = 74
74.   74)
Answer: B
A trader bought a car at 20% discount on its original price. He sold it at a 40% increase on the price he bought it. What percent of profit did he make on the original price?	Choose the correct choice from the following:
A) 10%  B) 11%  C) 12%  D) 15%  E) 13%

Solution: Original price = 100   12%P = 80   S = 80*(140/100)= 
80*(140/100) = 112
112   100 - 112= 12%   ANSWER:12%
Answer: C
Two hoses are pouring water into an empty pool. Hose 1 alone would fill up the pool in 6 hours. Hose 2 alone would fill up the pool in 3 hours. How long would it take for both hoses to fill up two-thirds of the pool? Choices:
A) 5/12 hours	B) 5/8 hours	C) 4/3 hours	D) 12/7 hours	E) 12/5 hours

Solution: Hose 1 would fill the pool alone in 6 hours   i.e. 1 Hour work of Hose 1 = 1/6   Hose 2 would fill the pool alone in 3 hours   i.e. 1 Hour work of Hose 2 = 1/3   i.e. 1 Hour work of Hose1 and Hose2 together = (1/6)+(1/3) = (1/2)   i.e. (1/2) work can be finished by Hose1 and Hose2 together in 1 hour   i.e. (2/3) work can be finished by Hose1 and Hose2 together in 1*(2)*(2/3) = 4/3 hour   Answer: Option 4/3 hours
Answer: C
10^185 - 10^9 = Which of the following best approximates the value of the expression above?
Choices:
A) 10^170	B) 10^180	C) 10^(185)	D) 10^160	E) 10^179

Solution: 10^185 - 10^9 =   Which of the following best approximates the value of the expression above?   Yes, answer 10^185 is correct. Note that we need approximate value of the given expression and as 10^(185) is much larger number than 10^(9) then 10^(9) is pretty much negligible in this case: 10^(185)-10^(9)=~10^(185).   Answer : 10^(185)
Answer: C
45 pupil,out of them 12 in debate only and 22 in singing only.then how many in both? Choices:
A) 9  B) 10  C) 11  D) 12  E) 13

Solution: The intersection for two = 45 - 12 - 22= 
45 - 12 - 22 = 11
11 play both games.   ANSWER:11
Answer: C
In the first 10 overs of a cricket game, the run rate was only 4.8. What should be the run rate in the remaining 40 overs to reach the target of 282 runs?  Options
A) 4.25	B) 5.85	C) 6.25	D) 7	E) 7.5

Solution: Required run rate = 282 - (4.8 x 10) =	234   234/40= 
234/40 = 117/20 = around 5.85
5.85   5.85)
Answer: B
A and B can do a work in 6 days, B and C in 2 days and C and A in 8 days. In how many days will the work be completed, if all three of them work together? Choose the correct choice from the following answers:
A) 12/24
B) 48/63
C) 45/56
D) 48/19
E) 28/12

Solution: One day work of A and B = 1/6   One day work of B and C = 1/2   One day work of C and A = 1/8   2(A + B + C) = 1/6 + 1/2 + 1/8   2(A + B + C) = 19/24   (A + B + C) = 19/48   Number of days required = 48/19 days.   Answer: 48/19
Answer: D
A series of book was published at seven years intervals when the seventh book was published the total sum of publication year was 13, 1524. first book was published in? Choose the most appropriate option:
A) 1911  B) 1912  C) 1913  D) 1914  E) 1915

Solution: First of all total sum of publication years is 13524 not 131524.   Let the first year is x.   so the years are x, x+7, x+14, x+21, x+28, x+35 and x+42.   Now Sum of all years = (n/2)(2a+(n-1)d) [n=number of years, a=first year, d=difference of two years]   =(7/2)(2x+(7-1)7)   =7x+147   1911gain Sum of all years = 13524   So, 7x+147=13524   => x=1911   1911NSWER:1911
Answer: A
Every student of a certain school must play an instrument. In last year, 1/2 of the students picked a woodwind instrument, 2/5 of the students picked a brass instrument, and all of the other students took percussion. In this year, 1/2 of the students who play woodwind and 1/4 of the students who play brass instruments left school, other students did not leave, and no fresh student come in. What fraction of all students play either a woodwind or brass instrument?  Answers:
A) 7/13  B) 4/5  C) 9/13  D) 11/13  E) 7/10

Solution: Lets pick smart numbers. Total number of students: 20   Woodwind (1/2): 10   Brass (2/5): 8   Percussion (1/10): 2   After leaving school   Woodwind: 5   Brass: 6   Percussion: 2   New Total number of students: 13   Woodwind and Brass: 11   Answer 11/13 or 11/13
Answer: D
Two trains 111 meters and 165 meters in length respectively are running in opposite directions, one at the rate of 80 km and the other at the rate of 65 kmph. In what time will they be completely clear of each other from the moment they meet? Options:
A) 6.85
B) 7.18
C) 7.16
D) 7.15
E) 7.12

Solution: T = (111 + 165)/ (80 + 65) * 18/5   T = 6.85   6.85nswer: 6.85
Answer: A
The market value of a 10.5% stock, in which an income of Rs. 756 is derived by investing Rs. 6000, brokerage being 1/4%, is: Choose one:
A) 83.08  B) 114  C) 114.75  D) 124  E) 124.75

Solution: Face Value = Rs. 6000.   Dividend = 10.5%.   Annual Income = Rs. 756.   Brokerage per Rs. 100 = Rs. 0.25.   Dividend is always paid on the face value of a share.   Face Value * Dividend / (Market Value + Brokerage per Rs. 100) = Annual income.   = 6000 * 10.5 / 756 = Market Value of Rs. 100 stock + Brokerage per Rs. 100.   = Market Value of Rs. 100 stock + Brokerage per Rs. 100 = Rs. 83.33.   = Market Value of Rs. 100 stock = Rs. 83.33 - Re. 0.25.   = Market Value of Rs. 100 stock = Rs. 83.08   answer : 83.08
Answer: A
A, B and C start a business each investing 20,000. After 5 months A withdrew 10000, B withdrew 15000 and C invests 5000 more. At the end of the year, a total profit of 58000 was recorded. Find the share of B.	Choose the correct choice from the following answers
A) 20,000	B) 21,200	C) 28,200	D) 13,500	E) None of these

Solution: Ratio of the capitals of A, B and C   = 20000 Ã— 5 + 10000 Ã— 7 : 20000 Ã— 5 + 5000 Ã— 7 : 20000 Ã— 5 + 25000 Ã— 7   =170000 : 135000 : 275000 = 170 : 135 : 275.   Bâ€™s share = (58000 Ã— 135â„580) = 13500   Answer 13,500
Answer: D
A train of 24 carriages, each of 60 meters length, when an engine also of 60 meters length is running at a speed of 60 kmph. In what time will the train cross a bridge 1.5 km long?  Choose the correct option
A) 1 Â½ mins  B) 3 mins  C) 3 Â½ mins  D) 7 mins  E) 5 mins

Solution: Explanation:   D = 25 * 60 + 1500= 
25 * 60 + 1500 = 3_000
3000 m   T= 3000/60 * 18/5= 
3000/60 * 18/5 = 180
180 sec= 3 mins   ANSWER IS 3 mins
Answer: B
A publisher has a shelf that contains biographies which normally sell for $20 each and mysteries that normally sell for $12 each. During a sale, the biographies and mysteries are discounted at different rates so that a customer saves a total of $19 from the normal price by buying 5 discounted biographies and 3 discounted mysteries. If the sum of the discount rates for the two types of books is 35 percent, what is the discount rate on mysteries book?  Answers.
A) 10%  B) 40%  C) 35%  D) 30%  E) 25%

Solution: Let B be the discount on biographies and M be the discount on mysteries   So., B + M = 0.35 -----(1)   And   (20*5 + 12*3) - (20*5*(1-B) + 12*3*(1-M)) = 19   --> 100(1-(1-B)) + 36(1-(1-M) = 19   100B + 36M = 19 ------(2)   Solving 12., we get M = 0.25 = 25%   25%
Answer: E
In a certain country ½ of 5 = 3. Assuming the same proportion, what would be the value of 1/3 of 10 ?
Select
A) 3  B) 4  C) 8  D) 7  E) 9

Solution: 4   4
Answer: B
Which of the following Equation is NOT equivalent to 25x^2 = y^2 - 49?Which of the following Equation is NOT equivalent to 25x^2 = y^2 - 4?  Choose one
A) 25x^2 + 4 = y^2
B) 75x^2 = 3y^2 - 12
C) 25x^2 = (y + 2)(y - 2)
D) 5x = y - 7
E) x^2 = (y^2 - 4)/25

Solution: 1) Take the original equation and try to manipulate it so that it becomes 4 of the 5 answers   2) Take each answer choice and see if you can manipulate it to make it into the original equation.   With Answer D, we have...   5X = Y - 7   If we square both sides, we end up with....   (5X)^2 = (Y-7)^2   (5X)(5X) = (Y-7)(Y-7)   25X^2 = Y^2 - 14Y + 49   This is NOT a match for the original equation, so it's the answer that is NOT equivalent.5x = y - 7
Answer: D
A certain bus driver is paid a regular rate of $14 per hour for any number of hours that does not exceed 40 hours per week. For any overtime hours worked in excess of 40 hours per week, the bus driver is paid a rate that is 75% higher than his regular rate. If last week the bus driver earned $976 in total compensation, how many total hours did he work that week?
Choose one.
A) 54
B) 51
C) 59
D) 55
E) 52

Solution: For 40 hrs = 40*14= 
40*14 = 560
560   Excess= 976-560= 
976-560 = 416
416   For extra hours=.75(14)=10.5+14= 
10.5+14 = 24.5
24.5   Number of extra hrs=416/24.5=18.82=19 approx.   Total hrs =40+19= 
40+19 = 59
59   Answer 59 59
Answer: C
24 oz of juice P and 25 oz of juice F are mixed to make smothies X and Y . The ratio of p to F in smothie X is 4 is to 1 and that in Y is 1is to 5.How many ounces of juice P are contained in the smothie X? Choose the correct choice
A) 5  B) 10  C) 15  D) 20  E) 25

Solution: Easy way to solve this question is start from the answer and then conform the information provided in the question.   we can start from option D i.e 20 ... as a quantity of juice P in X because it is the only one option that gets divided by 4 is 20 ... since in the X the juice P to F ratio is 4:1   this gives us that quantity of juice P in X = 20 therefore quantity of Juice F will be 5 ... hence ratio = 4:1   This will lead to quantity of juice P in X = 4 and quantity of Juice F = 20 ... hence ratio 1:5   if we calculate total Juice P = 24 and total of juice V = 25   it fits because totals are same as what mentioned in the question ...   thus ans is 20
Answer: D
Harkamal purchased 3 kg of grapes at the rate of 70 per kg and 9 kg of mangoes at the rate of 55 per kg. How much amount did he pay to the shopkeeper?	Choose the correct choice from the following answers: A) A)700 B) B)1065 C) C)1070 D) D)1075 E) E)1080

Solution: Cost of 3 kg grapes = 70 × 3 = 210.   Cost of 9 kg of mangoes = 55 × 9 = 490.   Total cost he has to pay = 210 + 490= 
210 + 490 = 700
700   700
Answer: A
There are 76 lights which are functional and each is controlled by a separate On/Off switch. Two children A and B start playing with the switches. A starts by pressing every third switch till he reaches the end. B, thereafter, presses every fifth switch till he too reaches the end. If all switches were in Off position at the beggining, How many lights are switched On by the end of this operation? Choose the correct choice from the following
A) 28  B) 30  C) 32  D) 34  E) 36

Solution: Editing my solution:   Number of switches = 76   Number of switches turned on by A: 3, 6, ... 75 = 25   Number of switches turned on by B: 5, 10, ....75 = 15   Few switches are turned on by A and later turned off by B: LCM(3,5) = 15x = 15, 30,....90 = 6.   Subtract the above 6 switches from both A and B as they are turned off.   Number of switches that are turned on = (25 - 6) + (15 - 6)= 
(25 - 6) + (15 - 6) = 28
28   28nswer: 28
Answer: A
For positive integers n, the integer part of the nth term of sequence B equals n, while the infinite decimal part of the nth term is constructed in order out of the consecutive positive multiples of n, beginning with 2n. For instance, B_1 = 1.2345678…, while B_2 = 2.4681012… The sum of the first seven terms of sequence B is between:	Choose the most appropriate option.
A) 28 and 29	B) 29 and 30	C) 30 and 31	D) 31 and 32	E) 32 and 33

Solution: First, construct the first seven terms, though only out to a few decimal places, following the given pattern.   B_1 = 1.23…   B_2 = 2.46…   B_3 = 3.69…   B_4 = 4.812…   B_5 = 5.10…   B_6 = 6.12…   B_7 = 7.14…   Now, to add up the first seven terms, you should be strategic about how many decimal places to keep. You can drop the hundredths place and get a good approximation with the tenths place—and if you find the sum too close to a boundary between choices, then you can refine your answer if necessary.   1.2 + 2.4 + 3.6 + 4.8 + 5.1 + 6.1 + 7.1= 
1.2 + 2.4 + 3.6 + 4.8 + 5.1 + 6.1 + 7.1 = 30.3
30.3   Including more decimal places would only add a couple of tenths to the sum—not enough to tip the sum over 31.   The correct answer is 30 and 31.
Answer: C