MU-NLPC/Calc-aqua_rat huggingface dataset:
In how many different W ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?  Choose the correct choice from the following options: A) 70 B) 288 C) 576 D) 10,080 E) 20,160

Solution: Treat the 4 ladies as one object, now you have 5 objects to arrange around a table (m1,m2,m3,m4,women). This can be done in (5-1)! ways   And there are 4! ways to arrange ladies among themselves   Answer W= (4!)^2 = 576 or 576
Answer: C
In a certain appliance store, each model of television is uniquely designated by a code made up of a particular ordered pair of letters. If the store has 40 different models of televisions, what is the minimum number of letters that must be used to make the codes?  Choose the correct answer:
A) 6	B) 7	C) 7	D) 9	E) 10

Solution: If n is the number of distinct letters used to create the two lettered codes, then a total of n∗n=n^2 different codes can be created. We need n2≥40. The smallest n which fulfills this condition is n = 7.   Answer 7
Answer: C
Walking with 3/5 of my usual speed, I miss the bus by 5 minutes. What is my usual time?	Choose one.
A) 16 min
B) 26 min
C) 34 min
D) 20 min
E) 15 min

Solution: Speed Ratio = 1:3/5 = 5:3Time Ratio   = 3:5 1 -------- 5 3 --------- ? è 15   Answer: 15 min
Answer: E
A car covers a distance of 780 km in 6 ½ hours. Find its speed? Choose the most appropriate option:	A) 104 kmph	B) 187 kmph	C) 164 kmph	D) 130 kmph	E) 106 kmph

Solution: 780/6= 
780/6 = 130
130 kmph   Answer: 130 kmph
Answer: D
Statements: Prime school-going children in urban India have now become avid as well as more regular viewers of TV, even in households without a TV. As a result there has been an alarming decline inthe extent of readership of newspapers. Conclusions: 1) Method of increasing the readership of newspapers should be devised. 2) A team of experts should be sent to other countries to study the impact of TV. on the readership of newspapers. Choose the most appropriate option:
A) Only conclusion I follows B) Only conclusion II follows C) Either I or II follows D) Neither I nor II follows E) Both I and II follow

Solution: The statement concentrates on the increasing viewership of TV. and does not stress either on increasing the readership of newspapers or making studies regarding the same. So, neither I nor II follows.   Neither I nor II follows
Answer: D
Call a number Guardian if it is composite but not divisible by 2, 3, or 5. The three smallest guardian numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many Guardian numbers are there less than 1000?
Select:
A) 100	B) 102	C) 104	D) 106	E) 108

Solution: S1 = total number of numbers divisible by 2 from 1-1000 = 500   S2 = total number of numbers divisible by 3 from 1-1000 = 333   S3 = total number of numbers divisible by 5 from 1-1000 = 200   S4 = total number of numbers divisible by 2 and 3 = 166   S5 = total number of numbers divisible by 2 and 5 = 100   S6= total number of numbers divisible by 3 and 5 =66   S7 = total number of numbers divisible by 2,3 and 5 = 33   thus total number of numbers divisible by 2,3 or 5 are = 500 +333+200-166-100-66+33= 
500 +333+200-166-100-66+33 = 734
734   now since we have already included 2,3 and 5 while calculating their multiple in the above calculations, hence total number of prime numbers from 1-1000 becomes= 168-3= 
168-3 = 165
165   also, since 1 is neither composite nor prime, hence we will subtract 1 from the final answer.   thus total number of guardian numbers are 1000-734-1-165= 
1000-734-1-165 = 100
100   100NSWER:100
Answer: A
Rajan is sixth from the left end and vinay is tenth from the right end in a row of boys. If there are eight boys between Rajan and Vinay, how many boys are there in the row ? Pick:
A) 24
B) 66
C) 88
D) 27
E) 91

Solution: Explanation:   Number of boys in the row = (6 + 10 + 8)= 
(6 + 10 + 8) = 24
24   24nswer: 24) 24
Answer: A
I remember during the school days , the teacher asked the class "can you tell me the sum of the first 50 odd numbers ?".
I ran quickly to the teacher and told her "the answer is 2500". The teacher replied "lucky guess".
She then asked me "can you tell me the sum of first 74 odd numbers ?".
I wait for approx 10 seconds and replied with the correct answer.
How can i answer so quickly and whats the correct answer ?
Choose the correct answer.
A) 8715
B) 0152
C) 3581
D) 5476
E) 9126

Solution: D   5476   n^1   74 * 74= 
74 * 74 = 5_476
5476 (sum of first 74 odd numbers).   50 * 50= 
50 * 50 = 2_500
2500 (sum of first 50 odd numbers).
Answer: D
The average (arithmetic mean) of 4 different integers is 68. If the largest integer is 90, what is the least possible value of the smallest integer? Pick:
A) 5  B) 19  C) 29  D) 30  E) 33

Solution: Total of Integers=68*4= 
68*4 = 272
272   Lowest of the least possible integer is when the middle 2 intergers are at the maximum or equal to the highest possible integer.   But all integers are distinct. So if the largest integer is 90, then the middle 2 will be 88 and 89   Lowest of least possible integer= 272-(90+89+88)=272-267= 
272-267 = 5
5   5nswer: 5
Answer: A
In an examination, it is required to score 32% of the aggregate marks for a student to pass the examination.Raju has scored 250 marks and was declared failed by 22 marks.Find the aggregate marks?	Choose the correct choice from the following options
A) 720
B) 750
C) 800
D) 850
E) 900

Solution: 32% of x =272.   x=850   ANSWER:850
Answer: D
The monthly incomes of A and B are in the ratio 5 : 2. B's monthly income is 12% more than C's monthly income. If C's monthly income is Rs. 13000, then find the annual income of A? Choose the correct choice from the following answers:
A) Rs. 420000
B) Rs. 180000
C) Rs. 436800
D) Rs. 504000
E) None of these

Solution: B's monthly income = 13000 * 112/100 = Rs. 14560   B's monthly income = 2 parts ----> Rs. 14560   A's monthly income = 5 parts = 5/2 * 14560 = Rs. 36400   A's annual income = Rs. 36400 * 12 = Rs. 436800   ANSWER:Rs. 436800
Answer: C
How many paying stones, each measuring 2 1/2 m * 2 m are required to pave a rectangular court yard 30 m long and 16 1/2 m board? Choose the correct choice from the following answers
A) 99 B) 86 C) 21 D) 26 E) 11

Solution: 30 * 33/2 = 5/2 * 2 * x => x   = 99   99
Answer: A
A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With help of C, they did the job in 4 days only. Then, C alone can do the job in? Answers
A) 9
B) 9 3/5
C) 10
D) 11
E) None of these

Solution: (A + B + C)'s 1 day's work =	1	,   4   A's 1 day's work =	1	,   16   B's 1 day's work =	1	.   12   Therefore C's 1 day's work =	1	-	(	1	+	1	)	=	(	1	-	7	)	=	5	.   4	16	12	4	48	48   So, C alone can do the work in	48	= 9	3	days.   5	5   Answer is 9 3/5
Answer: B
A ladder was rested along a wall of height 5m. If ladder slides 2m away from the wall then ladder touches the foot of the wall .What is the height of the ladder. Answers.
A) 6.25 mtrs  B) 7.25 mtrs  C) 8.25 mtrs  D) 9.25 mtrs  E) 5.25 mtrs

Solution: suppose , earlier foot of ladder was x mtres away from wall, then   length of ladder = x+2 mtrs   hence   (x+2)^2 = x^2+5^2   x= 21/4= 
21/4 = 21/4 = around 5.25
5.25 mtrs   length of ladder= 5.25+2= 
5.25+2 = 7.25
7.25 mtrs   ANSWER:7.25 mtrs
Answer: B
From January 1, 2015, to January 1, 2017, the number of people enrolled in health maintenance organizations increased by 12 percent. The enrollment on January 1, 2017, was 45 million. How many million people, to the nearest million, were enrolled in health maintenance organizations on January 1, 2015 ?	Choose the correct choice from the following choices:
A) 38  B) 39  C) 40  D) 41  E) 42

Solution: Soln:-   12x=45 --> 28/25*x=45 --> x=45*25/28=1125/28=~40   Answer: 40.
Answer: C
When x is divided by y, the quotient is q with remainder r. Which of the following is y in terms of r, x, and q? (Assume both y and q are nonzero.)  Choose the correct choice from the following answers:
A) x – r – q
B) (x+r)/q
C) x/q−r
D) (x−r)/q
E) xr – q

Solution: Given that when x is divided y, the quotient is q and remainder is r. This implies x-r is divisible by y and yq=(x-r) or y = (x-r)/q.   Answer (x−r)/q.
Answer: D
A person purchased a TV set for Rs. 16000 and a DVD player for Rs. 6250. He sold both the items together for Rs. 32150. What percentage of profit did he make?	Options:
A) 22  B) 44.5  C) 40  D) 26  E) 11

Solution: The total CP = Rs. 16000 + Rs. 6250 = Rs. 22250 and SP = Rs. 32150   Profit(%) = (32150 - 22250)/22250 * 100 = 44.5%.Answer:44.5
Answer: B
Two persons start running simultaneously around a circular track of length 380 m from the same point at speeds of 24 kmph and 28 kmph. When will they meet for the first time any where on the track if they
are moving in the opposite direction ? Choose the correct choice
A) 144 B) 36 C) 124 D) 26 E) 38

Solution: Time taken to meet the first time = length of track/relative speed   = 380/ (24 +28) (5/18)   = 380/52 *(18/5) = 26 sec.   Answer : 26
Answer: D
The number 341 is equal to the sum of the cubes of two integers. What is the product of those integers? Choose the correct choice from the following answers.
A) 8  B) 15  C) 21  D) 30  E) 39

Solution: 5^3+6^3= 341 number is   5*6= 
  5*6 = 30
30   30
Answer: D
50 is divided into two parts such that the sum of their reciprocals is 1/12. Find the two parts. Select the correct option: A) 40,10 B) 22,28 C) 25,25 D) 35,15 E) 30,20

Solution: Let the two parts be x and (50 - x).   Then, 1 / x + 1 / (50 – x) = 1 / 12 => (50 – x + x) / x ( 50 – x) = 1/12   => x^2 – 50x + 600 = 0 => (x – 30)( x – 20)=0 => x = 30 or x = 20.   So, the parts are 30 and 20.   Answer 30,20)30,20.
Answer: E
Angelo and Isabella are both salespersons. In any given week, Angelo makes $550 in base salary plus 8 percent of the portion of his sales above $4,000 for that week. Isabella makes 10 percent of her total sales for any given week. For what amount of weekly sales would Angelo and Isabella earn the same amount of money?  Choose the correct choice from the following answers:	A) 11,500	B) 24,500	C) 25,500	D) 26,500	E) 27,500

Solution: OFFICIAL SOLUTION:   The problem asks for the amount of weekly sales it takes for Angelo and Isabella to earn the same amount of money. You can write an equation that sets Angelo’s and Isabella’s weekly earnings equal to each other, with x representing weekly sales. Weekly earnings for each salesperson equal base salary plus commission. So Angelo’s earnings are 550 + (0.08)(x – 4,000), and Isabella’s are 0.10x. Set up the equation and solve:   550 + (0.08)(x – 4,000) = 0.10x   Distribute the 0.08:   550 + 0.08x – 320 = 0.10x   Combine terms and subtract 0.08x from both sides:   230 = 0.02x   Divide both sides by 0.02:   11,500 = x   Your answer is 11,500.
Answer: A
What is the 28th digit to the right of the decimal point in the decimal form of 4/11?  Choose one: A) 3 B) 4 C) 5 D) 6 E) 7

Solution: 4/11 = 0.363636...   The even numbered positions in the decimal expansion are all 6.   The answer is 6.
Answer: D
What is the value of(a,y) in (13^-20-a*13^y)=168*13^-22? Options.
A) (1,2)
B) (0,1)
C) (-1,22)
D) (1,-22)
E) (1,-20)

Solution: (13^-20-a*13^y)=168*13^-22   Divide the LHS and RHS by 13^-22   13^2-a*13^(y+22) =168   168 can be expressed as 13^2 - 1   13^2-a*13^(y+22) =13^2 - 1   a*13^(y+22) =1   Now look at options and substitute .. Its easy for y=-22 . 13^0=1 a also = 1   ANSWER:(1,-22)
Answer: D
If the sum of the interior angles of a regular polygon measures up to 1080 degrees, then how many sides does the polygon have? Choose the correct option
A) 10
B) 16
C) 8
D) 11
E) 14

Solution: one of the interior angle = 180 - 360/n   sum of the interior angles is = n (180 - 360/n)   given   n(180 - 360/n)=1080   180n - 360 = 1080   180n = 1440   n = 1440/180= 
1440/180 = 8
8   ANSWER:8
Answer: C
In an isoscele right angled triangle, the perimeter is 20 metre. Find its area. Choose the correct choice:  A) 9,320 m2  B) 8,750 m2  C) 7,980 m2  D) 6,890 m2  E) None of these

Solution: In an isoscele right angled triangle,   9,320 m2rea = 23.3 × perimeter2   = 23.3 × 20(2) = 9320 m2   9,320 m2nswer 9,320 m2
Answer: A
In the expression 51840/x^4, for which of the following values of x will the expression NOT be an integer?	Choose the most appropriate option
A) 1/2 B) 1 C) 2 D) 3 E) 5

Solution: We can straightaway eliminate A & B, since the expression will always yields an integer for those values   Out of C, D, and E. Let us consider option E. 51840 has only one trailing zero and tens digit is not 5, so It has just one 5 in it. But for the expression to be an integer numerator should have 5^4.   ANSW5R:5
Answer: E
HCF and LCM two numbers are 12 and 396 respectively. If one of the numbers is 36, then the other number is?	Choose the correct choice
A) 36  B) 66  C) 132  D) 264  E) 364

Solution: 12 * 396 = 36 * x   x = 132   ANSWER:132
Answer: C
Find the number of ways in which 4 letters may be selected from the word "Examination"?
Choose the correct choice from the following.
A) 66
B) 70
C) 136
D) 330
E) 4264

Solution: EXAMINATION has 11 letters, and in which 'A', 'I' and 'N', all occur twice. 11C4, would have been fine if all letters were distinct.   So, we have E, X, M, T, O, (AA), (II), (NN). 8 distinct letters.   1. 4 letters selected, which are all distinct: 8C4 = 70   2. 2 letters alike, and 2 distinct (eg: AAEX) = 3C1 x 7C2 = 63   3. 2 letters alike, and 2 letters alike (eg: AAII) = 3C2 = 3   So answer is, 70 + 63 + 3 = 136.   ANSWER:136
Answer: C
By selling a book for 250, 20% profit was earned. What is the cost price of the book?  Choices:	A) A)215	B) B)208	C) C)230	D) D)235	E) E)240

Solution: SP = 120% of CP; :. CP = 250 × 100/120 = 208   Option '208'
Answer: B
Calculate the ratio between x and y if 90% of x equal to 60% of y?	Choose the correct choice: A) 1 : 2 B) 3 : 5 C) 3 : 2 D) 3 : 4 E) 5 : 2

Solution: Explanation:   90 x = 60 y   x: y = 90: 60 = 3 : 2   Answer:3 : 2
Answer: C
There are 8 students. 4 of them are men and 4 of them are women. If 4 students are selected from the 8 students. What is the probability R that the number of men is equal to that of women?
Choose the most appropriate option
A) R=18/35 B) 16/35 C) 14/35 D) 13/35 E) 12/35

Solution: METHOD-1:   Favorable outcomes (i.e. No. of Men= No. of Women =2) = 4C2 * 4C2 = 6*6= 
6*6 = 36
36   Total ways of selecting 4 out of 8 Students= 8C4 = 8! / (4! * 4!) = 70   Probability = 36/70 = 18/35   A   METHOD-2:Also see the mistake done by the person who posted this question   Probability of First selected person being Man = 4/8   Probability of Second selected person being Man = 3/7   Probability of First selected person being Woman = 4/6   Probability of Second selected person being Woman = 3/5   i.e. Probability = (4/8)*(3/7)*(4/6)*(3/5) *[4!/(2!*2!)]= 18/35   The important part for the readers is to understand the reason of multiplying[4!/(2!*2!)]here   When we take the probability of each case like we have have taken in this method then it always include R=18/35RRR=18/35NGEMENTS as well and so we have to take every arrangement of the events as well   The arrangement of these 4 events can be done in 4! ways but since the second man can't be selected before 1st so we have to exclude their arrangement by dividing by 2! and similarly since the second Woman can't be selected before 1st so we have to exclude their arrangement by dividing by 2!   R=18/35
Answer: A
Farm Tax is levied on the 60% of the cultivated Land. The tax department collected total $ 4000 through the farm Tax from the village of Mr. willam. Mr. willam paid only $ 500 as farm tax. The percentage of total land of Mr. willam over the total taxable land of the village is :  Choices:  A) 15%  B) 25%  C) 12.5%  D) 20.83%  E) none

Solution: Only trick n this question is to ignore 60% information as farm tax is levied uniformly in the village and that includes Mr William's land.   What percentage of tax Mr William paid ? This will be equal to the percentage of total cultivated land he holds over the total cultivated land in the village. That leads to (500/4000)x100 =12.5%in percentage terms.   But the question asks ratio between his total land to total cultivated land. Hence the answer is 12.5%x (100/60) = 20.833% and the answer is not there in the options.   The correct answer is (20.83%).
Answer: D
A train running at the speed of 60 km/hr crosses a pole in 8 seconds. Find the length of the train.
Choose the correct choice from the following choices:	A) 133.33	B) 882	C) 772	D) 252	E) 121

Solution: Speed = 60*(5/18) m/sec = 50/3 m/sec   Length of Train (Distance) = Speed * Time   (50/3) * 8 = 133.33 meter.133.33nswer: 133.33
Answer: A
If an article is sold at 18% profit instead of 9% profit, then the profit would be $54 more. What is the cost price?  Choose the correct choice from the following answers.
A) $500 B) $600 C) $700 D) $800 E) $900

Solution: 9% * cost price = $54   1% * cost price = $54 / 9= 
54 / 9 = 6
$6   The cost price is $600.   The answer is $600.
Answer: B
A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?  Choose the correct choice from the following choices.
A) 1000
B) 3000
C) 2000
D) 4000
E) 5000

Solution: Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.   Then, 4x - 3x = 1000   x = 1000.   B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.   Answer is 2000.
Answer: C
2, 6, 12, 20, 30, 42, 56, ?	Choose one:
A) 63  B) 67  C) 69  D) 72  E) 75

Solution: The pattern is 1 × 2, 2 × 3, 3 × 4, 4 × 5, 6 × 7, 7 × 8,....   Answer : 72.
Answer: D
The following sets each have a mean of 10 and the standard deviations are given in variable form.
Set I = {10, 10, 10, 10, 10, 10}, standard deviation = P
Set II = {6, 6, 6, 14, 14, 14}, standard deviation = Q
Set III = {7, 8, 9, 11, 12, 13}, standard deviation = R
Rank these three standard deviations from least to greatest.	Choices: A) P, Q, R B) P, R, Q C) Q, P, R D) Q, R, P E) R, Q, P

Solution: SD is minimum when the terms are closer to the mean.   In set 1, we have all terms equal to mean. hence, its SD is 0.(The Lowest)   Comparing Set 2 and Set 3, Set 3 has terms more closer to mean than set 2. hence, Set 2 Sd is the highest.   So, Answer is P, R, Q
Answer: B
In a graduating class of 238 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?	Choose the most appropriate option.
A) 144  B) 119  C) 113  D) 88  E) 31

Solution: Greatest Possible Number taken both should be 144 (As it is maximum for One)   Smallest Possible Number taken both should be given by   Total - Neither = A + B - Both   Both = A+B+Neither - Total (Neither must be 0 to minimize the both)   So 144 + 119 + 0 - 238= 
144 + 119 + 0 - 238 = 25
25   Greatest - Smallest is  144-25= 
 144-25 = 119
119   So answer must be B. 119
Answer: B
A sum was put a simple interest at a certain rate for 2 years. Had it been put at 4% higher rate, it would have fetched Rs. 60 more. The sum is:
Choices.
A) Rs.750  B) Rs.700  C) Rs.820  D) Rs.940  E) Rs.900

Solution: Explanation :   Let the sub be Rs.x and the initial rate be R%.Then   x × (R+4) × 2/100−x×R×2/100=60   ⇒x × 4×2/100=60   ⇒x × 2 /100=15   ⇒2x=1500   ⇒x=750   Rs.750nswer : Option Rs.750
Answer: A
A and B invests Rs.3000 and Rs.4000 respectively in a business. If A doubles his capital after 6 months. In what ratio should A and B divide that year's profit?
Select the correct option.
A) 9:5	B) 9:8	C) 9:6	D) 9:1	E) 9:2

Solution: (3*6 + 6*6): (4*12)   54:48 => 9:8   Answer: 9:8
Answer: B
The regular price per can of a certain brand of soda is $0.15. If the regular price per can is discounted 10 percent when the soda is purchased in 24-can cases, what is the price of 75 cans of this brand of soda purchased in 24-can cases?  Answers.
A) $16.32
B) $18.00
C) $10.125
D) $24.48
E) $28.80

Solution: The discounted price of one can of soda is (0.9)($0.15), or $0.135   Therefore, the price of 75 cans of soda at the discounted price would be (75)($0.135) = 10.125   Answer: $10.125.
Answer: C
Line E has the equation 3x + y = 7. Which of the following lines is perpendicular to Line E?
Choose the most appropriate option
A) y = 3x + 4	B) y = –3x – 6	C) y = (1/3)x – 1	D) y = (–1/3)x + 2	E) y = (–7/3)x – 5

Solution: I first rewrote the equation in the standard y=mx+b form. Therefore, Line E as presented, 3x + y = 7, can be rewritten as follows: y=-3x+7.   Thought process next is what line would be perpendicular to Line E? Any line with a reciprocal of the slope but in the opposite direction. The reciprocal of any fraction/integer is 1 over that number/integer. Therefore the reciprocal of -3 is -1/3 - need to drop the negative sign because the line would kinda run parallel and we want perpendicular. Scan the answers choices and notice y = (1/3)x – 1 as the only one.
Answer: C
If the simple interest on a sum of money for 2 years at 5% per annum is Rs. 50, what is the compound interest on the same sum at the rate and for the same time? Pick:
A) Rs.51.25	B) Rs.51.26	C) Rs.51.22	D) Rs.51.98	E) Rs.51.11

Solution: Sum = (50 * 100) / (2 * 5)   = Rs. 500   Amount = [500 * (1 + 5/100)2]   = Rs. 551.25   C.I. = (551.25 - 500)   = Rs. 51.25.   Rs.51.25nswer: Rs.51.25
Answer: A
A train travels 125 km in 2.5 hours and 270 km in 3 hours.find the average speed of train. Choose the correct choice from the following answers:
A) 80 kmph B) 60 kmph C) 72 kmph D) 90 kmph E) None of these

Solution: As we know that Speed = Distance / Time   for average speed = Total Distance / Total Time Taken   Thus, Total Distance = 125 + 270= 
125 + 270 = 395
395 km   Thus, Total Speed= 5.5 hrs   or,Average Speed = 5=395/5.5   or, 72kmph.   ANSWER:72 kmph
Answer: C
A car travelling with 2/3 km of its actual speed covers 12km in 2hr 14 min 28 sec find the actual speed of the car?
Pick one.
A) 8.9kmph	B) 2.96kmph	C) 1.09kmph	D) 45.9kmph	E) 4.8kmph

Solution: time taken =2hr 14 min 28 sec=823/50hrs   let the actual speed be x kmph   then 2/3 x*823/50=12   x==1.09kmph   Answer (1.09kmph)
Answer: C
If m = 3^n, what is the greatest value of n for which m is a factor of 19!
Choose the correct choice from the following options:
A) 8  B) 10  C) 12  D) 14  E) 16

Solution: Solution-   Consider multiples of 25!=> 3,6,9,12,15,18   Count no. of 3 in each multiple.   3=3x1->1   6=3x2->1   9=3x3->2   12=3x4->1   15=3x5->1   18=3x3x2->2   ---- count 3's =8 so answer is 8   8nswer : 8
Answer: A
Sn=1^8+2^8+3^8+----------n^8
find the unit digit of S29 Select the correct option.
A) 6 B) 5 C) 7 D) 8 E) 9

Solution: unit digit of-   1^8=1   2^8=6   3^8=1   4^8=6   5^8=5   6^8=6   7^8=1   8^8=6   9^8=1   so 1+6+1+6+5+6+1+6+1= 
1+6+1+6+5+6+1+6+1 = 33
33 so from here we get 3 as unit digit for 1^8+2^8+.....+9^8   now for 11 to 19 same way unit digit is 3   then from 21 to 29 unit digit is 3   so 3+3+3= 
3+3+3 = 9
9   so unit digit is 9   ANSW9R:9
Answer: E
Two alloys A and B are composed of two basic elements. The ratios of the compositions of the two basic elements in the two alloys are 5 : 3 and 1 : 1, respectively. A new alloy X is formed by mixing the two alloys A and B in the ratio 4 : 3. What is the ratio of the composition of the two basic elements in alloy X ?
Select the correct option
A) 1 : 1
B) 2 : 3
C) 5 : 2
D) 4 : 3
E) 7 : 9

Solution: Mixture A has a total of 5 + 3= 
5 + 3 = 8
8 parts. If in the final mixture this represents 4 parts, then the total number of parts in mixture B should be  (8/4)*3= 
 (8/4)*3 = 6
6.   So, we should take of mixture B a quantity with 3 and 3 parts, respectively.   This will give us in the final mixture (5 + 3) : (3 + 3), which means 4:3   Answer 4 : 3.
Answer: D