MU-NLPC/Calc-aqua_rat huggingface dataset:
Two cyclists are running on a circular track of 500m.They start from the same point A at the same time.If they meet for the first time at a point 300 m from A, which of the following could be the possible ratio of their speeds?	Choose the correct choice from the following options:
A) 1:3	B) 2:3	C) 3:2	D) 5:7	E) 7:11

Solution: well, the key aspect of this question is the distance mentioned, when they met for the first time, which is 300m away from the starting point. This distance between their first meeting can be less than 500 in only two circumstance.   1) if both are travelling in the same direction and one of them has been given a head start say by few meters.   2) or if both are travelling in the opposite direction.   now since they start their journey at the same time. hence option 1 is ruled out. hence they must be running in the opposite direction. hence answer should be 3:2   Why?? here is a solution   As here time is constant, therefore, distance traveled will be directly proportional to the the respective velocities/speed of the two runners.(d1/d2=v1/v2)   now one of them has covered 300m, therefore other must have covered 200m, therefore ratio of their speeds/velocities will be 3:2   3:2
Answer: C
There are 4 more women than there are men on a local co-ed softball team. If there are a total of 18 players on the team, what is the ratio of men to women?  Choose the correct answer.
A) 10/16	B) 6/16	C) 7/11	D) 6/10	E) 4/10

Solution: w = m+4   w+m = 18   m+4+m = 18   2m = 14   m = 7   w = 11   ratio:7:11   Ans: 7/11
Answer: C
What value should come in place of question mark (?) in the following equations ?
64% of 1775 + 36% of 550 = ? % of 6670  Choose the correct choice from the following:
A) 20
B) 24
C) 30
D) 40
E) 45

Solution: Explanation :   64x1775/100 + 36x550/100 = ?x6670/100   ∴ ? = 1136 + 198/6670 x 100 = 20%   20nswer : Option 20
Answer: A
Solution X is 10 percent alcohol by volume, and solution Y is 30 percent alcohol by volume. How many milliliters of solution Y must be added to 200 milliliters of solution X to create a solution that is 20 percent alcohol by volume? Answers
A) 200  B) 300  C) 400  D) 500  E) 600

Solution: 20% is an equal distance between 10% and 30%.   Thus there should be equal parts of both solutions.   We should add 200 ml of solution Y.   The answer is 200.
Answer: A
If each term in the sum a1+a2+a3+.....+an is either 7 or 77 and the sum equals 329, which of the following could be equal to n? Choices:	A) 37	B) 38	C) 39	D) 41	E) 42

Solution: Since, there is no 47 in the answer choices (329/7 = 47), we know there is at least one 77.   329 - 77= 
329 - 77 = 252
252   252/7= 
252/7 = 36
36   36+1= 
36+1 = 37
37.   If 37 wasn't there, I would have subtracted 77 from 252 and continued in a similar way.   Ans. 37
Answer: A
If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?
Pick:
A) 3 B) 6 C) 7 D) 14 E) 26

Solution: Basically the question is asking us what is the highest power of 18 that can give us a number that is a factor of 31!   So remember 18 after prime factorization is (3^2)(2). Now there are going to be less factors of 3 in 31!, than factors of 2.   Therefore, let's find how many factors of 3 in 31! We can use this quick method   31/3^1 = 10   31/3^2= 3   31/3^3=1   Sum = 14   Just ignore the remainders.   So we have that 3^14 must be the least. Now don't forget that 18 is 3^2k so k must be ONLY 7, because 2k will give us the 14.   Hence answer is (7)
Answer: C
The average weight of 16 students in a class is 50.25 kg and that of the remaining 8 students is 45.15 kg. Find the average weights of all the students in the class.	Choose the correct choice from the following answers
A) A)48.55  B) B)49  C) C)51  D) D)61  E) E)62

Solution: Required average   = (50.25 x 16 + 45.15 x 8)/(16 + 8)   = (804 + 361.20)/24   = 1165.20/24= 
1165.20/24 = 48.55
48.55kg   48.55nswer is 48.55
Answer: A
A cylinder of radius R and height H is reshaped. Which of the following changes results in the cylinder of greatest volume?	Select the correct option:	A) a 20% decrease in H and a 50% increase in R.	B) a 500% increase in H and a 50% decrease in R.	C) a 300% increase in H and a 30% decrease in R	D) a 70% decrease in H and a 100% increase in R	E) a 100% increase in H and a 10% decrease in R

Solution: Volume of a cylinder = Pi. R^2.H   Check for options as per given Q.   A. (1.5)^2 * (0.8) = 2.25 * 0.8= 
2.25 * 0.8 = 1.8
1.8 (Since 20% decrease in H and a 50% increase in R.)   B. (0.5)^2 * (6)= 0.25 * 6= 
0.25 * 6 = 1.5
1.5 (Since 500% increase in H and a 50% decrease in R)   C. (0.7)^2 * (4)= 0.49 * 4= 
0.49 * 4 = 1.96
1.96 (Since 300% increase in H and a 30% decrease in R )   D. (2)^2 * (0.3)= 4 * 0.3= 
4 * 0.3 = 1.2
1.2 (Since 70% decrease in H and a 100% increase in R. )   E. (0.9)^2 * (2)= 0.81 * 2= 
0.81 * 2 = 1.62
1.62 (Since 100% increase in H and a 10% decrease in R. )   Maximum value is 1.96   Option a 300% increase in H and a 30% decrease in R
Answer: C
A bucket full of nuts was discovered by the crow living in the basement. The crow eats a sixth of the total number of nuts in 4 hours. How many hours i total will it take the crow to finish a quarter of the nuts?	Options.
A) 6  B) 8  C) 10  D) 12  E) 14

Solution: In one hour, the crow eats 1/24 of the nuts.   (1/4) / (1/24)= 
(1/4) / (1/24) = 6
6 hours   The answer is 6.
Answer: A
A computer program produces a sequence a1, a2, a3, ... such that a1=a2=1 and ak=a(k-1)*3-1 for 2 < k < n for some n. If n=9, what is a6? Choose one.
A) 35	B) 37	C) 39	D) 41	E) 43

Solution: a1=1   a2=1   a3=1*3-1= 
1*3-1 = 2
2   a4=2*3-1= 
2*3-1 = 5
5   a5=5*3-1= 
5*3-1 = 14
14   a6=14*3-1=42-1= 
42-1 = 41
41   41
Answer: D
Every letter in the alphabet has a number value that is equal to its place in the alphabet. Thus, the letter A has a value of 1, the letter B has a value of 2, the letter C has a value of 3, etc... The number value of a word is obtained by adding up the value of the letters in the word and then multiplying that sum by the length of the word. What is the number value of the word "DOG"?  Select the correct option:  A) 72  B) 75  C) 78  D) 81  E) 84

Solution: "DOG" = (4+15+7)*3= 
(4+15+7)*3 = 78
78.   The answer is 78.
Answer: C
All of the students of Music High School are in the band, the orchestra, or both. 60 percent of the students are in only one group. There are 106 students in the band. If 10 percent of the students are in the band only, how many students are in the orchestra only?	Choose the correct choice from the following answers.
A) 120	B) 110	C) 116	D) 106	E) 112

Solution: % of students in one group only = 60%   % of students in both groups = 100 - 60= 
100 - 60 = 40
40%   % of students in the band only= 10%   Number of students who are in band = 106   40% + 10% = 50% 40  + 10= 
40  + 10 = 50
50%= 106 Students   Finding total students:   x = (106 x 100)/50   x = 212   Students in orchestra only = 212 - 106= 
212 - 106 = 106
106   Answer:106
Answer: D
There are 500 employees in a room. 99% are managers. How many managers must leave the room to bring down the percentage of managers to 98%? Choose the correct choice from the following answers:
A) 150	B) 200	C) 250	D) 350	E) 400

Solution: There are 495 managers and 5 others. The 5 others would compose 2% of the total number of people if there were 250 people in the room. Thus 250 managers must leave.   The answer is 250.
Answer: C
A shirt goes on sale for 60% of its original price. One week later, the sale price is marked down 10%. The final price is what percent of the original price?
Choose the correct choice from the following answers:
A) 72% B) 70% C) 52% D) 50% E) 28%

Solution: Just assume original price is 100.   Sale price = 60   Then it is marked down by 10% = 60-6=52.   Hence it is 52% od the original price. Hence answer is 52%.
Answer: C
The third proportional to 1 and 5 is? Choose the correct option: A) 8 B) 27 C) 24 D) 25 E) 3

Solution: (5*5)/1= 
(5*5)/1 = 25
25   Answer:25
Answer: D
A man covers a certain distance Q in a train. If the train moved 4km/hr faster, it would take 30 min less. If it moved 2km/hr slower, it would take 20 mins more. Find the distance?
Choose the correct choice:
A) 200 km B) 50 km C) 20 km D) 60 km E) 80 km

Solution: Not really. When you solve the 2 equation above, you get,   6T-4/3 = 5R/6 from simplifying equation 1   4T-2=R/2 from simplifying equation 2   You can now multiply equation 2 by 5 to get   5(4T-2=R/2 ) = 20T-10=5R/2 and then subtract this new equation from equation 1 to get T=3, followed by R=20 to give you distance Q = R*T=20*3= 
20*3 = 60
60 km.60 km
Answer: D
Sides of a rectangular park are in the ratio 3: 2 and its area is 3750 sq m, the cost of fencing it at 40 ps per meter is?
Select the correct option
A) 287	B) 369	C) 100	D) 279	E) 361

Solution: 3x * 2x = 3750 => x = 25   2(75 + 50) = 250 m   250 * 0.40   = Rs.100   Answer:100
Answer: C
Dana borrows 5000 pounds annually for her college education. If Dana gives her parents 4% of that amount back each month, how much will she still owe her parents after four years of college?	Choose one.
A) 9800 B) 10000 C) 10200 D) 10400 E) 10600

Solution: dana borrows 5000 and returns 4% per month.   so returns 200 per month. and 2400 per year.   so dana owes  5000-2400= 
 5000-2400 = 2_600
2600 per year.   after 4 years she will owe 10400.   Ans (10200)
Answer: C
Running at the same constant rate,6 identical machines can produce a total of 270 bottles per minute.At this rate,how many bottles could 12 such machines produce in 4 minutes ?
Pick one
A) 648
B) 1800
C) 2700
D) 2160
E) None of these

Solution: Solution   Let the required number of bottles be x.   More machines,More bottles (Direct Proportion)   More minutes,More bottles (Direct Proportion)   âˆ´ 6 Ã— 1 Ã— x = 12 Ã— 4 Ã— 270 â‡” x = 12x4x270/6= 2160 . Answer 2160
Answer: D
If 7125¸1.25= 5700< the value of 712.5÷12.5 is:
Pick one.
A) 5.7 B) 57 C) 570 D) 0.57 E) 0.057

Solution: Given 7125/1.25= 
7125/1.25 = 5_700
5700   712.5/12.5=71.25/1.25 = 7125*1/1.25*100 = 5700/100= 
5700/100 = 57
57   ANSWER:57
Answer: B
Find the percentage gain if a shop owner sells 10 mtr of fabric and gains Cost Price of 2 meters?  Options:
A) 10%
B) 12%
C) 19%
D) 25%
E) 20%

Solution: Let cost of each metre be Rs. 100.   Therefore, cost price of 2m cloth = 2*100= 
2*100 = 200
200   cost price of 10m cloth= 10*100= 
10*100 = 1_000
1000   Since the trader made the CP of 2m from the sale of 10m therefore   Profit on sale of 10m= cost price of 2 m cloth = 200   Selling price of 10m = 1000+200= 
1000+200 = 1_200
1200   Profit %= 20%   20%
Answer: E
A sum is invested at compounded interest payable annually. The interest in the first two successive years was Rs. 400 and Rs. 420. The sum is	Pick.
A) Rs. 8000
B) Rs. 8200
C) Rs.7500
D) Rs. 8500
E) Rs. 9000

Solution: Explanation:   his means that, simple Interest on Rs.400 for 1 year = 420 - 400= 
420 - 400 = 20
20   Rate= (100 Ã—SI)/PT = (100 Ã— 20)/(400Ã—1) = 5%   Rs.400 is the interest on the sum for 1st year   Hence, sum = (100 Ã— SI)/RT = (100 Ã— 400)/(5 Ã— 1) = Rs. 8000   Rs. 8000NSWER IS Rs. 8000
Answer: A
A chocolate has 12 equal piece. Monju gave 1/4th of it to Anju, 1/3rd of it to Sujata and 1/6th of it to Fiza. The number of piece of chocolate left with Manju is? Answers.
A) 1	B) 2	C) 3	D) 4	E) 5

Solution: Answer   The number of pieces of chocolate left with Manju   = 1- [(1/4) + (1/3) + (1/6)   = 1 - (3 + 4 + 2)/12   = 1 - 9/12 = (12 - 9) / 12   = 3/12   Option: 3
Answer: C
A certain manufacturer sells its products	to stores in 110 different	regions worldwide, with an average (arithmetic mean) of 181 stores per region. If last year these stores sold an	average of 51,752 units of the manufacturer’s product per store, which of the following is closest to the total number of units of manufacturer’s product	sold	worldwide last	year? Choose the most appropriate option
A) 10^5  B) 10^6  C) 10^7  D) 10^8  E) 10^9

Solution: 51752*181*110 or   51000*180*110 or   50000*200*100   =10^9   10^9
Answer: E
The total of company C's assets in 1994 was 100% greater than the total in 1993, which in turn was 400% greater than the total in 1992. If the total of company C's assets in in 1992 was N dollars, which one of the following represents company C's assets in 1994: Choose the correct answer
A) 7N  B) 8N  C) 9N  D) 10  E) 20

Solution: Let's use actual numbers. Starting in 1992, let's say company C had $100 in assets.   In 1993, the total assets were 400% GREATER, which means 1992 PLUS 400% of 1992:   $100 + 4x$100 = $100 + $400 =$500   In 1994, the total assets were 100% GREATER than they were in 1993, which means 1993 PLUS 100% of 1993:   $500 + 1x$500 100 +  400= 
100 +  400 = 500
$500   In 1994, the total assets were 100% GREATER than they were in 1993, which means 1993 PLUS 100% of 1993:   $500 + 1x$500= $500 + $500 =$1000   This is 10 times the 1992 number, so the correct answer is 10N.   10500 +  500= 
500 +  500 = 1_000
$1000   This is 10 times the 1992 number, so the correct answer is 10N.   10
Answer: D
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?	Choose the correct choice from the following choices:
A) 10  B) 6  C) 4  D) 0  E) It Cannot Be Determined

Solution: Given: z is an even Integer   The cyclicity of Unit Digit of 4 is 2 i.e. Unit digit of powers of 4 repeat after every two powers in the pattern {4, 6, 4, 6, 4, ....}   4^1=4   4^2=16   4^3=64   4^4=256   i.e. Every Even Power of 4 gives the Unit digit = 6 = Remainder   Answer: Option 6
Answer: B
How many positive integers, from 2 to 100, inclusive, are not divisible by odd integers greater than 1?
Answers:
A) 5
B) 6
C) 8
D) 10
E) 50

Solution: The no will be of the form 2^n to achieve this. Since any other form will have odd no in the prime factorization.   Hence we need to find solution of n for equation - 1< 2^n < 101.   2^6 = 64, 2^7 - 128   Hence n can take values from 1 to 6.   Hence Answer : 6
Answer: B
If q and r are both odd numbers C , which of the following must also be odd?
Choose the correct choice from the following answers.
A) q – r
B) (q + r)^2
C) q(q + r)
D) (qr)^2
E) q/r

Solution: We're told that q and r are both ODD numbers C. We're asked which of the following must also be odd.   IF...   q = 1   r = 3   Answer A) q – r = 1-3 = -2 NOT odd   Answer B) (q + r)^2 = (1+3)^2 = 16 NOT odd   Answer C) q(q + r) = (1)(4) = 4 NOT odd   Answer (qr)^2) (qr)^2 = (3)^2 = 9 This IS O(qr)^2(qr)^2   Answer E) q/r = 1/3 NOT odd   Final Answer:   (qr)^2
Answer: D
23, 25, 53, 163, 657, 3291, ?  Choose the correct option.
A) 19853
B) 18753
C) 17753
D) 19753
E) 17532

Solution: 23*1+2= 
23*1+2 = 25
25   25*2+3= 
25*2+3 = 53
53   53*3+4= 
53*3+4 = 163
163   163*4+5= 
163*4+5 = 657
657   657*5+6= 
657*5+6 = 3_291
3291   3291*6+7= 
3291*6+7 = 19_753
19753   ANSWER:19753
Answer: D
If you have 5 New Year greeting cards and you want to send them to 3 of your friends, in how many ways can this be done?  Choose the correct choice from the following
A) 60	B) 720	C) 20	D) 70	E) None

Solution: Solution   We have to find number of permutations of 3 objects out of 5 objects. This number is 5P3=5(5-1)(5-2)=5*4*3= 
5*4*3 = 60
60   60nswer 60
Answer: A
If Rs.7500 are borrowed at C.I at the rate of 4% per annum, then after 6 years the amount to be paid is? Select the correct option: A) 3377 B) 2688 C) 2688 D) 9490 E) 1268

Solution: A = 7500(26/25)^6 = 9490   Answer: 9490
Answer: D
A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour? Select the correct option.
A) 7.1  B) 7.2  C) 7.5  D) 7.4  E) 7.3

Solution: distance = 600 meter   time = 5 minutes = 5 x 60 seconds = 300 seconds   Speed = distance/time=600/300= 
600/300 = 2
2m/s=2×18/5 km/hr=36/5 km/hr=7.2 km/hr   Answer is 7.2.
Answer: B
If a man can cover 14 metres in one second, how many kilometres can he cover in 3 hours 45 minutes?
Choose the correct choice from the following answers
A) 189km	B) 195km	C) 197km	D) 200km	E) 205km

Solution: 14 m/s = 14 * 18/5 kmph   3 hours 45 minutes = 3 3/4 hours = 15/4 hours   Distance = speed * time = 14 * 18/5 * 15/4= 
14 * 18/5 * 15/4 = 189
189 km.   189kmnswer:189km
Answer: A
For 2consecutive yrs, my incomes are in the ratio of 4:7 and expenses in the ratio of 3:5. If my income in the 2nd yr is Rs. 42000 & my expenses in the first yr in Rs.21000, my total savings for the two-year is  Choose the correct choice from the following options:
A) Rs. 7000 B) Rs. 8000 C) Rs. 9000 D) Rs. 9900 E) Rs. 10000

Solution: Sol. Income in first year = * x 42000 = Rs. 24000   Expenses in second year = \ x 21000 = Rs. 35000 Total savings = Total income - Total expenses = (42000 + 24000)- (21000 + 35000)   = 66000 - S6000 = Rs. 10000   Rs. 10000
Answer: E
1!+2!+3!+4!+5!+.......100! is divided by 24. Find the remainder? Choices:
A) 5	B) 7	C) 9	D) 11	E) 14

Solution: By applying rule 2, we divide the terms of the above expression individually, and add them to get the final remainder. But from 4! onwards all the terms leave a remainder 0 when divided by 24.   So the remainder = 1 + 2 + 6 + 0 + 0....... = 9   9
Answer: C
Joe’s average (arithmetic mean) test score across 4 equally weighted tests was 50. He was allowed to drop his lowest score. After doing so, his average test score improved to 55. What is the lowest test score that was dropped? Choose the correct answer
A) 20	B) 35	C) 55	D) 65	E) 80

Solution: The arithmetic mean of 4 equally weighted tests was 50. So what we can assume is that we have 4 test scores, each 50.   He dropped his lowest score and the avg went to 55. This means that the lowest score was not 50 and other three scores had given the lowest score 5 each to make it up to 50 too. When the lowest score was removed, the other 3 scores got their 5 back. So the lowest score was  3 *5= 
 3 *5 = 15
15 less than 50.   So the lowest score=50 - 15= 
50 - 15 = 35
35   Answer (35)
Answer: B
A 18% stock yielding 8% is quoted at?
Choose the correct option
A) 165  B) 170  C) 175  D) 180  E) 225

Solution: Assume that face value = Rs.100 as it is not given   To earn Rs.8, money invested = Rs.100   To earn Rs.18, money invested = 100×18/8 = Rs.225   ie, market value of the stock = Rs.175   Answer is 225.
Answer: E
What is the sum of all consecutive multiples of 8 between 400 and 600, inclusive? Options:
A) 8000  B) 10,400  C) 12,500  D) 13,000  E) 15,000

Solution: 400 is 8*50 and 600 is 8*75...   Average of 400 and 600= 500, extreme multiples of 8..   # of multiples of 8 =75-50+1=26..   Sum =#*avg= 26*500= 
26*500 = 13_000
13000   ANSWER:13,000
Answer: D
P and Q can complete a work in 40 days and 24 days respectively. P alone started the work and Q joined him after 8 days till the completion of the work. How long did the work last?  Choose the most appropriate option:
A) 5 days  B) 10 days  C) 14 days  D) 20 days  E) 26 days

Solution: Explanation :   Work done by P in 1 day = 1/40   Work done by Q in 1 day = 1/24   Work done by P in 8 days = 8 Ã— (1/40) = 1/5   Remaining work = 1 â€“ 1/5 = 4/5   Work done by P and Q in 1 day = 1/40 + 1/24 = 1/15   Number of days P and Q take to complete the remaining work = (4/5) / (1/15)= 
(4/5) / (1/15) = 12
12   Total days= 8 + 12= 
8 + 12 = 20
20   Answer : Option 20 days
Answer: D
A certain quantity is measured on two different scales, the B-scale and the S-scale, that are related linearly. Measurements on the B-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the B-scale corresponds to a measurement of 100 on the S-scale?  Answers
A) 20
B) 36
C) 48
D) 60
E) 84

Solution: First, we have to understand what linearly means. It's not a straight ratio (since 6:30 does NOT equal 24:60). We need to look at the increases in each measurement to see what the scalar actually is.   From 6 to 24 we have an increase of 18. From 30 to 60 we have an increase of 30. Therefore, the increase ratio is 18:30 or 3:5. In other words, for every 3 that B increases, S increases by 5.   We know that S is 100. To get from 60 to 100, we went up by 40, or 8 jumps of 5; therefore, B will go up by 8 jumps of 3.   24 + 8(3) = 24 + 24= 
24 + 24 = 48
48=48
Answer: C
Which one of the following is not a prime number?	Choices:
A) 31  B) 61  C) 71  D) 91  E) 93

Solution: 91 is divisible by 7. So, it is not a prime number.   91)
Answer: D
What sum of money will produce Rs.70 as simple interest in 4 years at 2 1/2 percent?	Choose the correct choice from the following options:
A) 337 B) 500 C) 700 D) 288 E) 211

Solution: 70 = (P*4*5/2)/100   P = 700   Answer: 700
Answer: C
There are 7 executives, including the CEO and CFO, that are asked to form a small team of 4 members. However, the CEO and CFO may not both be assigned to the team. Given this constraint, how many ways are there to form the team?	Choose the correct choice from the following answers:
A) 25
B) 28
C) 30
D) 32
E) 35

Solution: The total number of ways to form a team of 4 is 7C4=35.   We need to subtract the number of teams that have both the CEO and the CFO.   The number of teams with both the CEO and CFO is 5C2=10.   The number of ways to form an acceptable team is  35-10= 
 35-10 = 25
25.   The answer is 25.
Answer: A
The third proportional to 0.24 and 0.36 is :	Choose the correct option
A) 48  B) 64  C) 54  D) 58  E) 66

Solution: Explanation :   Solution: Let the third proportion to 0.24 and 0.36 be x.   Then, 0.24 : 0.36 : : 0.36 : x   => x = 0.36*0.36/0.24= 
0.36*0.36/0.24 = 0.54
0.54   Answer : 54
Answer: C
What is the number of different committees of 9 people that can be selected from a group of 10 people?	Pick.
A) a) 20 B) b) 40 C) c) 80 D) d) 10 E) e)1

Solution: 10c9 = 10   Answer is d) 10.
Answer: D
What are the last two digits of 35*37*82*71*41?
Choose the correct answer:
A) 10  B) 30  C) 50  D) 70  E) 80

Solution: We know that to find the last two digits, we need to find the remainder we get when we divide the product by 100.   Remainder of (63*35*37*82*71*41)/ 100   Note that we can simplify this expression by canceling out the 5 and 2 in the numerator and denominator. But before we do that, here is an important note:   Note: We cannot just cancel off the common terms in the numerator and denominator to get the remainder. But, if we want to cancel off to simplify the question, we can do it, provided we remember to multiply it back again.   So say, we want to find the remainder when 14 is divided by 10 i.e. 14/10 (remainder 4). But we cancel off the common 2 to get 7/5. The remainder here will be 2 which is not the same as the remainder obtained by dividing 14 by 10. But if we multiply 2 back by 2 (the number we canceled off), the remainder will become 2*2= 
2*2 = 4
4 which is correct.   Take another example to reinforce this – what is the remainder when 85 is divided by 20? It is 5.   We might rephrase it as – what is the remainder when 17 is divided by 4 (cancel off 5 from the numerator and the denominator). The remainder in this case is 1. We multiply the 5 back to 1 to get the remainder as 5 which is correct.   So keeping this very important point in mind, let’s go ahead and cancel the common 5 and 2.   We need the   Remainder of (63*7*37*41*71*41*5*2)/10*5*2   Remainder of (63*7*37*41*71*41)/10   Now using concept 2, let’s write the numbers in form of multiples of 10   Remainder of (60+3)*7*(30+7)*(40+1)*(70+1)*(40+1)/10   Remainder of 3*7*7*1*1*1/10   Remainder of 147/10= 7   Now remember, we had canceled off 10 so to get the actual remainder so we need to multiply by 10: 7*10 = 70.   When 63*35*37*82*71*41 is divided by 100, the remainder is 70. So the last two digits of 63*35*37*82*71*41 must be 50.   Answer (50)
Answer: C
A and B began business with Rs.3000 and Rs.4000 after 8 months, A withdraws Rs.1000 and B advances Rs.1000 more. At the end of the year, their profits amounted to Rs.630 find the share of A.	Choose the correct answer.
A) 240	B) 288	C) 277	D) 877	E) 361

Solution: (3*8 + 2*4):(4*8 + 5*4)   8:13   8/21 * 630= 
8/21 * 630 = 240
240   240nswer: 240
Answer: A
If k is an integer and 2 < k < 8, for how many different values of k is there a triangle with sides of lengths 3, 6, and k? Select:
A) 0 B) 1 C) 2 D) 3 E) 4

Solution: (6-3) < k < (6+3)   3 < k < 9   Thus k = 4, 5, 6, 7, 8   Since 2 < k < 8, then k can be 4, 5, 6, or 7.   The answer is 4.
Answer: E