MU-NLPC/Calc-aqua_rat huggingface dataset:
To be considered for “movie of the year,” a film must appear in at least 1/4 of the top-10-movies lists submitted by the Cinematic Academy’s 775 members. What is the smallest number of top-10 lists a film can appear on and still be considered for “movie of the year”?  Choices:	A) 191	B) 194	C) 193	D) 212	E) 213

Solution: Total movies submitted are 775.   As per question we need to take 1/4 of 775 to be considered for top 10 movies = 193.75   approximate the value we 194.   IMO option 194 is the correct answer...
Answer: B
A train 800 m long is running at a speed of 78 km/hr. If it crosses a tunnel in 1 min, then the length of the tunnel is?	Choose the correct choice.
A) 288 m  B) 256 m  C) 500 m  D) 278 m  E) 287 m

Solution: Speed = 78 * 5/18 = 65/3 m/sec.   Time = 1 min = 60 sec.   Let the length of the train be x meters.   Then, (800 + x)/60   = 65/3   x = 500 m.   Answer:500 m
Answer: C
The cost of a Personal computer was $2,000 in 1990. In 2000, the price had decreased to $500. What was the percent decrease in the cost of the Personal computer?
Choose the correct option.
A) 25%  B) -30%  C) -75%  D) 40%  E) 90%

Solution: Decrease = 2000-500= 
2000-500 = 1_500
1500   % decrease= 1500*100/2000 = -75%   Answer: option -75%
Answer: C
A man rows his boat 105 km downstream and 45 km upstream, taking 2 1/2 hours each time. Find the speed of the stream?
Choose the correct choice from the following choices: A) 5 kmph B) 7 kmph C) 12 kmph D) 8 kmph E) 1 kmph

Solution: Speed downstream = d/t = 105/(2 1/2) = 42 kmph   Speed upstream = d/t = 45/(2 1/2) = 18 kmph   The speed of the stream = (42 - 18)/2= 
(42 - 18)/2 = 12
12 kmph   Answer:12 kmph
Answer: C
The first train leaves the station at 6:00 am with a speed of 60 km/hr. The second train leaves at 7:15 am, in the same direction from the same station, with a speed of 75 km/hr. What time will the second train catch the first train? Choose the correct choice from the following answers:	A) 11:15 am	B) 11:45 am	C) 12:15 pm	D) 12:45 pm	E) 1:15 pm

Solution: In seventy-five minutes, the first train travels 75 km.   The second train catches the first train with a relative speed of 15 km/h.   The second train will catch the first train in 5 hours.   The second train will catch the first train at 12:15 pm.   The answer is 12:15 pm.
Answer: C
How many such letter-pairs are there in the word SERVANT having the same no. of letters left between them in the word as they have in the series?  Choose the most appropriate option:
A) 2
B) 3
C) 4
D) 5
E) 6

Solution: 1 pair is S & V   1 pair is N &S   total 2 pair   2NSWER: 2
Answer: A
At 1:00 pm, there were 10.0 grams of bacteria. The bacteria increased to x grams at 4:00 pm, and 22.5 grams at 7:00 pm. If the amount of bacteria present increased by the same fraction during each of the 3-hour periods, how many grams of bacteria were present at 4:00 pm? Choose the correct choice from the following answers:
A) 14.4 B) 14.7 C) 15.0 D) 15.3 E) 15.6

Solution: Let x be the factor by which the bacteria increases every three hours.   At 4:00 pm, the amount of bacteria was 10x and at 7:00 pm it was 10x^2.   10x^2 = 22.5   x^2=2.25   x=1.5   At 4:00 pm, the amount of bacteria was 10(1.5)=15 grams.   The answer is 15.0.
Answer: C
In a division sum, the remainder is 6 and the divisor is 5 times the quotient and is obtained by adding 27 to the thrice of the remainder. The dividend is
Choose one
A) 74  B) 78  C) 186  D) 92  E) 98

Solution: Divisor = (6 * 3) + 27= 
(6 * 3) + 27 = 45
45   5 * Quotient= 45   Quotient = 9.   Dividend = (Divisor * Quotient) + Remainder   Dividend = (20 * 9) + 6= 
(20 * 9) + 6 = 186
186.   186)
Answer: C
A 300 m long train crosses a platform in 39 sec while it crosses a signal pole in 18 sec. What is the length of the platform? Answers:	A) 300	B) 350	C) 360	D) 370	E) 380

Solution: Speed = 300/18 = 50/3 m/sec.   Let the length of the platform be x meters.   Then, (x + 300)/39 = 50/3   3x + 900 = 1950 => x = 350 m.   Answer: Option 350
Answer: B
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after a time of 30 mins and 12 seconds. If the second blast occurred exactly 30 mins after the first, how many meters was he from the site when he heard the second blast? ( speed of sound = 330 m/s) Select
A) 3120 B) 3330 C) 3540 D) 3750 E) 3960

Solution: The distance the sound traveled to the man is 12*330= 
12*330 = 3_960
3960 meters   The answer is 3960.
Answer: E
A man sells two articles for Rs.3300 each and he gains 21% on the first and loses 21% on the next. Find his total gain or loss? Choose the correct choice.
A) 9%loss B) 2%loss C) 7%loss D) 8%loss E) 4%loss

Solution: 4%lossxplanation:   (21*21)/100   = 4%loss   Answer:4%loss
Answer: E
Marisa is taking a cash deposit to the bank. She has $10 bills and $5 bills in a deposit
pouch. Altogether, she has 130 bills with a total value of $890. How many ten dollar bills are in the pouch?
Choose the correct choice from the following options.
A) 28
B) 58
C) 48
D) 67
E) 76

Solution: Let x represent the number of $10 bills in the bag. Next, use the same variable to represent the   number of $5 bills. If there are 130 bills in all, then 130 x represents the number of $5 bills.   Now you need to establish a relationship between the number of bills and their value.   Using the expression above, the total value of the $10 bills is 10x, and the total value of   the $5 bills is 5(130 x).   Write an equation and solve. 10x 5(130 x) 890 Multiply both terms in parentheses by 5.   10x 650 5x 890 Combine like terms.   5x 650 890 Subtract 650 from both sides.   5x 240 Divide both sides by 5 to get the   x 48 number of $10 bills.   130 x 82 Subtract to find the number of $5 bills.   Answer: There are 48 ten-dollar bills and 82 five-dollar bills.   Check: (48 $10) (82 $5) $480 $410 $890. The answer is correct.   correct answer 48
Answer: C
The ratio of the ages of Anil and his son at present is 7:3. Six years hence, the ratio of the ages of the Anil's wife and the son will be 2:1. Find the ratio of the present ages of Anil and his wife?  Options:  A) 7  B) 8  C) 9  D) 6  E) 5

Solution: Let the present ages of Anil and his son be 7x and 3x years respectively.   Let the present age of the son's mother be m years.   (m + 6)/(3x + 6) = 2/1   m = 6x + 6   Ratio of the present ages of Anil and his son's mother = 7x/(6x + 6).   This cannot be found uniquely.   Answer:6
Answer: D
There are 7 pairs of socks and 2 socks are worn from that such that the pair of socks worn are not of the same pair. what is the number of pair that can be formed.  Choices:	A) 40	B) 49	C) 41	D) 56	E) 42

Solution: First of all you should remember that there is a difference in left and right sock.   now no. of way to select any of the sock = 7   and for second = 6   so total methods = 7*6= 
7*6 = 42
42   ANSW42R:42
Answer: E
There are 60 supermarkets in the FGH chain. All of them are either in the US or Canada. If there are 22 more FGH supermarkets in the US than in Canada, how many FGH supermarkets are there in the US?	Choose the correct choice from the following choices: A) 20 B) 31 C) 41 D) 53 E) 64

Solution: x + (x - 22) = 60 --> x =41.   Answer: 41.
Answer: C
A company consists of 10 senior and 5 junior staff officers. If a committee is created with 3 senior and 2 junior staff officers, in how many ways can the committee be formed?
Pick one: A) 120 B) 300 C) 450 D) 800 E) 1200

Solution: Choose 3 senior from 10 senior and choose 2 junior from 5 junior:   10C3 * 5C2 = 120 * 10= 
120 * 10 = 1_200
1200   Ans: 1200
Answer: E
A certain animal in the zoo has consumed 39 pounds of food in six days. If it continues to eat at the same rate, in how many more days will its total consumption be 143 pounds?	Pick one
A) 8  B) 7  C) 12  D) 16  E) none of the above

Solution: Ans is 16:   39 pounds --> 6 days   143 pounds --> x days   x = 143*6/39= 
143*6/39 = 22
22   the animal has already consumed food in 6 days so the the number of days   for it's total consumption be 143 pounds is   22-6= 
  22-6 = 16
16
Answer: D
Mr. Kramer, the losing candidate in a two-candidate election, received 942,568 votes, which was exactly 30 percent of all votes cast. Approximately what percent of the remaining votes would he need to have received in order to have won at least 50 percent of all the votes cast?
Choose the most appropriate option:
A) 10%
B) 12%
C) 15%
D) 17%
E) 28%

Solution: Lets assume that candidate got 30% votes and total votes is 100.   Candidate won = 30   Remaining = 70   To get 50%, candidate requires 20 votes from 100 which is 20% and 20 votes from 70.   20/70 = 28.57%   Which is approx 28%. Hence the answer is 28%
Answer: E
8 men can do a piece of work in 12 days. 4 women can do it in 48 days and 10 children can do it in 24 days. In how many days can 10 men, 4 women and 18 children together complete the piece of work? Options
A) 5 days B) 15 days C) 28 days D) 6 days E) 7 days

Solution: Explanation:   1 man’s 1 day’s work = 1/8 × 12 = 1/96   10 men’s 1 day’s work = 1 × 10/96 = 5/48   1 woman’s 1 day’s work = 1/192   4 women’s 1 day’s work = 1/192 × 4 = 1/48   1 child’s 1 day’s work = 1/240   18 children’s 1 day’s work = 3/40   Therefore, (10 men + 4 women + 18 children)’s 1 day’s work = 5/48 + 1/48 + 3/40   = 1/5   The required No. of days = 5 days   5 daysnswer: Option 5 days
Answer: A
In a colony of 70 residents, the ratio of the number of men and women is 4 : 2. Among the women, the ratio of the educated to the uneducated is 1 : 4. If the ratio of the number of educated to uneducated persons is 8 : 27, then find the ratio of the number of educated to uneducated men in the colony?  Choose the correct answer:
A) A)1:6 B) B)1:1 C) C)1:8 D) D)1:9 E) E)1:3

Solution: Number of men in the colony = 4/7 * 70= 
4/7 * 70 = 40
40.   Number of women in the colony= 2/7 * 70= 
2/7 * 70 = 20
20.   Number educated women in the colony= 1/5 * 30= 
1/5 * 30 = 6
6.   Number of uneducated women in the colony= 4/5 * 50 = 24.   Number of educated persons in the colony = 8 /35 * 70= 
8 /35 * 70 = 16
16.   As 6 females are educated, remaining 10 educated persons must be men.   Number of uneducated men in the colony= 40 - 10= 
40 - 10 = 30
30.   Number of educated men and uneducated men are in the ratio 10 : 30 i.e., 1:3.   Answer:1:3
Answer: E
Two employees X and Y are paid a total of Rs. 880 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week?
Options.
A) s.150
B) s.200
C) s.250
D) s.350
E) s.400

Solution: Let the amount paid to X per week = x   and the amount paid to Y per week = y   Then x + y = 880   But x = 120% of y = 120y/100 = 12y/10   âˆ´12y/10 + y = 880   â‡’ y[12/10 + 1] = 880   â‡’ 22y/10 = 880   â‡’ 22y = 8800   â‡’ y = 8800/22 = 800/2 = Rs.400   s.400)
Answer: E
Last year a certain bond yielded 10 percent of its face value in interest. If that interest was approximately 4 percent of the bonds selling price of $7,500, what is the bonds face value?
Choose the correct choice from the following answers:
A) $3,000 B) $6,750 C) $7,425 D) $7,500 E) $9,375

Solution: Interest = 0.10*face value = 0.04*7,500 --> face value = 0.04*7,500/0.1 = 3,000.   $3,000nswer: $3,000.
Answer: A
An order was placed for a carpet whose length and width were in the ratio of 3 : 2. Subsequently, the dimensions of the carpet were altered such that its length and width were in the ratio 3 : 1 but were was no change in its perimeter. What is the ratio of the areas of the carpets?	Options:
A) 12:7 B) 15:11 C) 25:16 D) 30:23 E) 32:25

Solution: Let the length and width of one carpet be 3x and 2x.   Let the length and width of the other carpet be 3y and y.   2(3x + 2x) = 2(3y + y)   5x = 4y   (5/4) * x = y   The ratio of the areas of the carpet in both cases:   = 3x * 2x : 3y * y   = 6x^2 : 3y^2   = 6x^2 : 3*(25/16) * x^2   = 96 : 75   = 32 : 25   The answer is 32:25.
Answer: E
A motorist knows seven different routes from Bristol to Birmingham. From Birmingham to Sheffield he knows three different routes and from Sheffield to Carlisle he knows two different routes. How many routes does he know from Bristol to Carlisle ? Select the correct option
A) 4	B) 8	C) 12	D) 24	E) 42

Solution: Explanation:   Total number of routes from Bristol to Carlisle = (7 x 3 x 2) = 42.   Answer: 42
Answer: E
The simple interest on a sum of money will be Rs.700 after 10 years. If the principal is trebled after 5 years what will be the total interest at the end of the tenth year?	Choices:
A) 2261  B) 2888  C) 1400  D) 2699  E) 2771

Solution: P --- 10 ---- 700   P --- 5 ----- 350   3P --- 5 ----- 1050   ------   => 1400   Answer: 1400
Answer: C
The speed of a boat in upstream is 40 kmph and the speed of the boat downstream is 80 kmph. Find the speed of the boat in still water and the speed of the stream?	Choose the correct choice from the following answers
A) 10 kmph  B) 67 kmph  C) 22 kmph  D) 20 kmph  E) 12 kmph

Solution: Speed of the boat in still water   = (40+80)/2= 
(40+80)/2 = 60
60 kmph. Speed of the stream= (80-40)/2= 
(80-40)/2 = 20
20 kmph.   Answer:20 kmph
Answer: D
In a factory, there are 60% technicians and 100% non-technicians. If the 60% of the technicians and 100% of non-technicians are permanent employees, then the percentage of workers who are temporary is? Options.
A) 62% B) 57% C) 52% D) 85% E) 42%

Solution: Total = 160   T= 60 NT= 100   60*(100/100)=37.5 100*(60/100)=37.5   37.5 + 37.5= 
37.5 + 37.5 = 75
75=> 160 - 75= 
160 - 75 = 85
85%   Answer:85%
Answer: D
4 buses runs between Chennai and Mysore. If a man goes from Chennai to Mysore by a bus and comes back to Mysore by another bus, then the total possible ways are	Answers
A) 4 B) 8 C) 12 D) 16 E) 18

Solution: Explanation :   Since the man can go in 4 ways and can back in 3 ways. Therefore total number of ways are 4 x3 = 12ways.   Answer : 12
Answer: C
If BASKET is coded as 116,then COLOUR is coded as,	Select:
A) 158	B) 168	C) 178	D) 188	E) 198

Solution: NUM168ER THE ALPHA168ETS   168=2, A=1, S=19, K=11, E=5, T=20   168ASKET = 116   2(168+A+S+K+E+T) ==>2(2+1+19+11+5+20) = 2(58) = 116   Similarly C=3, O=15, L=12, O=15, U=21, R=18   2(C+O+L+O+U+R) = 2(3+15+12+15+21+18) = 2(84) = 168   ANSWER:168
Answer: B
An article is bought for Rs.685 and sold for Rs.900, find the gain percent?
Choose the most appropriate option.
A) 30 1/3%	B) 33 1/3%	C) 31 53/137%	D) 35 1/3%	E) 29 1/3%

Solution: 31 53/137%   31 53/137%   685 ---- 215   100 ---- ? => 31 53/137%
Answer: C
For the set {2, 2, 3, 3, 4, 4, 5, 5,6,6 x}, which of the following values of x will most increase the standard deviation?  Choose the correct choice from the following options
A) 1
B) 2
C) (-3)^2
D) 4
E) 5

Solution: Standard Deviation Step 1, as pointed out by others, is to find out the mean = 4   Step 2, For each number: subtract the Mean and square the result =   (1-4)^2=(-3)^2   (2-4)^2=(-2)^2   (3-4)^2 =(-1)^2   (4-4)^2=(0)^2   (5-4)^2=(1)^2   Clearly (1-3.5)^2=(-2.5)^2 will give you the greatest value among all the other options.   Hence (-3)^2
Answer: C
If the number of boys in a class are 8 times the number of girls, which value can be the total number of students? Choose the correct choice from the following options:
A) A - 42
B) B - 45
C) C - 41
D) D - 40
E) E - 39

Solution: Let the number of girls = x and the number of boys = 8x Then, total number of students = x + 8x = 9x i.e., the total number of students must be a multiple of 9 From the given choices, only 45 cannot be a multiple of 9. Hence, 45 is the total number of students. Answer is B - 45.
Answer: B
What sum of money will produce Rs.70 as simple interest in 4 years at 4 1/2 percent?	Choose the most appropriate option:	A) 263	B) 500	C) 388.9	D) 368	E) 339

Solution: 70 = (P*4*9/2)/100   P = 388.9   Answer: 388.9
Answer: C
Three sets of 80, 50 and 60 students appeared for an examination and the pass percentage was 100, 90 and 50 respectively. the pass percentage of the whole set is	Choose the correct choice from the following options
A) 86.57%  B) 89.57%  C) 82.57%  D) 81.97%  E) 81.57%

Solution: 81.57%xplanation:   total passed student =80+45+30= 
80+45+30 = 155
155   155 is 81.57% of 190   ANSW81.57%R:81.57%
Answer: E
A driver took a trip consisting of three segments at three different speeds. The driver drove a distance of (5D) at a speed of (2V), then a distance of (10D) at a speed of (3V), then a distance of D at a speed of (6V). In terms of D and V, what was the total time of the trip? Pick one: A) 3D/V B) 4D/V C) 5D/V D) 6D/V E) 7D/V

Solution: The total time of the trip was:   56D/V/2V + 106D/V/3V + 6D/V/6V =   306D/V/12V + 406D/V/12V + 26D/V/12V =   726D/V/12V = 66D/V/V   The answer is 6D/V.
Answer: D
The average height of 20 students in a class was calculated as 175 cm. It has later found that the height of one of the students in the class was incorrectly written as 151 cm whereas the actual height was 111 cm. What was the actual average height of the students in the class?
Choose the most appropriate option
A) 174 cm	B) 173 cm	C) 172 cm	D) 171 cm	E) 170 cm

Solution: The total height was 40 cm too much.   The average height should be reduced by 40 cm/20 = 2 cm   The answer is 173 cm.
Answer: B
A Man can row upstream at 85kmph and downward stream at 47 kmph. Find Man's rate of the current  Choose the correct choice.
A) 19km/hr
B) 20km/hr
C) 65km/hr
D) 2.8km/hr
E) 4.4km/hr

Solution: Rate of the current=1/2(85-47)km/hr   =19km/hr   correct option is :19km/hr
Answer: A
The ratio of three numbers is 1:2:3 and the sum of their squares is 2744. The sum of the numbers is?
Choose the correct choice from the following answers
A) 10 B) 12 C) 15 D) 14 E) 9

Solution: Let the numbers be x,2x,3x   then, x^2+4x^2+9x^2 = 2744   14x^2 = 2744   x^2 = 196   x=14   Answer is 14
Answer: D
Reena is twice as old as Sunita. Three years ago, she was three times as old as Sunita. How old is Reena now ?  Answers:
A) 14 years  B) 6 years  C) 12 years  D) 18 years  E) 16 years

Solution: Let Sunita's present age = x years   Then Reena present age = 2x years   Three years ago   (2x-3) = 3(x-3)   2x - 3 = 3x - 9   or x = 6   Reena age = 2x = 2*6= 
2*6 = 12
12 year.   Answer: Option 12 years
Answer: C
There are 10 fictions and 6 non-fictions. How many cases are there such that 2 fictions and 2 non-fictions are selected from them?
Choose the correct choice from the following options:
A) 90	B) 120	C) 675	D) 180	E) 200

Solution: Number of ways of selecting 2 fiction books = 106752   Number of ways of selecting 2 non fiction books = 66752   106752 * 66752 = 45 * 15= 
45 * 15 = 675
675   Answer: 675
Answer: C
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.	Choose the correct choice from the following options:
A) 2	B) 3	C) 4	D) 5	E) 6

Solution: Required number = H.C.F. of (91 - 43), (183 - 91) and (183 - 43)   = H.C.F. of 48, 92 and 140 = 4.   answer :4
Answer: C
Two bottles A and B contain diluted ammonium nitric acid. In bottle A, the amount of water is double the amount of acid while in bottle B, the amount of acid is 3 times that of water. How much mixture Should be taken from each bottle in order to prepare 5 litres of diluted sulphuric acid containing equal amount of acid and water?  Choose the correct choice from the following options
A) 1, 4  B) Required ratio= 3, 2  C) 2, 3  D) 4, 1  E) 1,5

Solution: A B   Acid water Acid water   1 : 2 3 : 1   Acid=1/3 Acid=3/4   Mixture=Acid:Water=1:1 Required acid=1/2   Now,Required ratio=(1/4): (1/6)=3:2   So the required quantity is 3 and 2 litres respectively.   ANSWER:Required ratio= 3, 2
Answer: B
A man purchases a cow for Rs.3000 and sold it on the same day allowing the buyer a credit of 2 years. If the buyer will pay Rs.3600 after 2 year at a rate of simple interest 50/7 % per annum, then the man has a gain of
Choose the correct choice from the following options.
A) Rs 181.43
B) Rs 171.43
C) Rs 191.43
D) Rs 161.43
E) Rs 151.43

Solution: value of Rs3000 after 2 years at 50/7 % S.I will be = 3000 + (3000*2*50/7)/100 = 3000 + 428.57= 
3000 + 428.57 = 3_428.57
3428.57   therefore C.P of article after 2 years will be Rs 3428.57   but the buyer pays Rs 3600   therefore gain= 3600-3428.57= 
3600-3428.57 = 171.43
Rs 171.43   ANSWER:Rs 171.43
Answer: B
In the simple subtraction problem below, some single digits (not necessarily distinct) are replaced by letters. Find the value of 2*A + 2*B + 4*C*D.
A9C5
-	2B79
--------------
633D  Choose one:
A) 50	B) 52	C) 66	D) 48	E) 46

Solution: A9C5   -2B79   ----------   6336 since,15-9=6. from there,D=6 and solving all,we get A=8,B=5,C=1.   So 2*8+2*5+4*1*6= 
2*8+2*5+4*1*6 = 50
50   50NSWER:50
Answer: A
A number is said to be a “digifac” if each of its digits is a factor of the number itself. What is the sum B of the missing digits of the following five-digit digifac: 9, 5, 3 _ _ ?	Choose the correct choice from the following answers:
A) 5  B) 7  C) 9  D) 10  E) 14

Solution: Here, the term “digifac” should look intimidating. You probably haven’t studied digifacs before, so how should you approach this problem? Well, keep in mind that digifacs aren’t being tested; in fact, the author of this question just made that term up, and then defined it for you. What makes this question hard is that the non-challenge-seeker (I think I just made that term up, too…) will see the unfamiliar term “digifac” and lose faith immediately. “I don’t know what that is!” She who finds the challenge in the GMAT fun, however, will read the definition and think “got it – I need to find the two digits that ensure that 9, 5, and 3 are both factors of the overall number, and that the remaining two digits are also factors”. And work from there. The number must be divisible by 5, so the only units digits that work are 0 or 5. And the number must be divisible by 9 (and also 3), so we need the sum B of all digits to be a multiple of 9. 9 + 5 + 3 = 17, so our only options are to get the sum to 18 (by adding 1) or to 27 (by adding 10). A quick glance at the answer choices shows that 0 1 isn’t an option. Why not? That would require 0 to be one of the digits…and 0 isn’t a factor of anything. So the units digit must be 5, making the tens digit 5, and we have 95,355. That number is a multiple of 5, 3, and 9, so it works:the correct answer is 10, and more importantly this fun challenge required no “trivial” information about digifacs…that term only existed to obscure the link between the given information and the path to the answer.10
Answer: D
Right triangle RST can be constructed in the xy-plane such that RS is perpendicular to the y-axis and the right angle is at R. The x and y-coordinates of R, S, and T are to be nonzero integers that satisfy the inequalities −3 ≤ x ≤ 4 and −7 ≤ y ≤ 3. Given these restrictions , how many different triangles can be constructed?  Choose the correct choice.
A) 3780  B) 4200  C) 4900  D) 6160  E) 7744

Solution: −3 ≤ x ≤ 4 and −7 ≤ y ≤ 3 gives a rectangle with 8*11 dimensions (8 horizontal and 11 vertical dots). We are given that RS is parallel to x-axis, RT is parallel to y-axis and the right angle is at R.   Choose the (x,y) coordinates for vertex R: 7C1*10C1 (we are told that coordinates of R, S, and T must be nonzero integers, so we are choosing from 7 and 10 instead of 8 and 11 because we should exclude 0);   Choose the x coordinate for vertex S (as y coordinate is fixed by R): 6C1, (7-1=6 as 1 horizontal dot is already occupied by R);   Choose the y coordinate for vertex T (as x coordinate is fixed by R): 9C1, (10-1=9 as 1 vertical dot is already occupied by B).   7C1*10C1*6C1*9C1=3,780.   3780nswer: 3780.
Answer: A
What is the smallest no. that should be added to 236 to make it exactly divisible by 9?
Choose the correct choice.
A) 1
B) 3
C) 5
D) 7
E) 9

Solution: If a number is divisible by 9, the sum of its digits must be a multiple of 9.   Here, 2+3+6=11, the next multiple of 9 is 18.   7 must be added to 236 to make it divisible by 9   7
Answer: D