MU-NLPC/Calc-aqua_rat huggingface dataset:
32 = a + 2b
|a| > 2
If ‘a’ and ‘b’ are both integers, then what is the smallest possible values of ‘a’ that can be used to solve the above equation.
Select the correct option
A) 2
B) 3
C) 4
D) 5
E) 6

Solution: Let us understand the meaning of |a| > 2   Mod is very easy concept if you solve mod question by considering as a distance. when a mod is written as |x-(a)| = b, this means the distance from point 'a' (both side left and right of 'a' on number line) is b. |x-(a)| < b means the distance is between the two extreme distance(left and right side of 'a' on number line, considering the max distance is 'b' from 'a' - as per this scenario.....hence the value of 'a' must be between these two extremes. |x-(a)| > b means the distance is greater than the distance of 'b'..i.e the value of a could be anywhere more than 'b' (i.e. current case).   Now come to the question. First its given |a| > 2 ==> a > 2 i.e. a can have any value bigger than 2 (i.e. 3, 4, 5…).   Now, lets move to equation a + 2b = 32 ==> b = (32 – a)/2 ==> b = 16 – (a/2). According to question, b is an integer, hence to make ‘b’ integer a must be divisible by 2. Minimum possible value of ‘a’ that is divisible by 2 is 4 (a is greater than 2 so the next number that is divisible by 2 is 4). So the answer is 4 (value 4).
Answer: C
A and B enter into partnership with capital as 7:9. At the end of 8 months, A withdraws. If they receive the profits in the ratio of 8:9 find how long B's capital was used?
Select the correct option.
A) 4  B) 5  C) 34  D) 7  E) 9

Solution: 7 * 8 : 9 * x = 8:9 => x= 7   Answer:7
Answer: D
What is the dividend. divisor 20, the quotient is 8 and the remainder is 6  Choose the correct answer
A) A)145	B) B)148	C) C)150	D) D)153	E) E)166

Solution: D = d * Q + R   D = 20 * 8 + 6   D = 160 + 6= 
160 + 6 = 166
166   Answer : 166
Answer: E
What is the greatest positive integer x such that 3^x is a factor of 9^9?	Choose the correct choice from the following choices
A) 5	B) 9	C) 10	D) 18	E) 30

Solution: What is the greatest positive integer x such that 3^x is a factor of 9^9?   9^9 = (3^2)^9 = 3^18   D. 18
Answer: D
each man in a gang of 100 investors has investments ineither equities or securities or both. Exactly 2525 of the investors in equities haveinvestments in securities, exactly 4040 of the investorsin securities have investments inequities. How many have investmentsin equities?	Choose the correct choice from the following options:
A) 70	B) 80	C) 90	D) 100	E) 120

Solution: The investors can be categorized into three groups:   (1) Those who have investments in equities only.   (2) Those who have investments in securities only.   (3) Those who have investments in both equities and securities.   Let xx, yy, and zz denote the number of people in the respective categories. Since the total number of investors is 110, we have   x+y+z=110x+y+z=110 ------------- (1)   Also,   The number of people with investments in equities is x+zx+z and   The number of people with investments in securities is y+zy+z.   Since exactly 25%25% of the investors in equities have investments in securities, we have the equation   25100×(x+z)=z25100×(x+z)=z   25100×x=25100×z25100×x=25100×z   x=3zx=3z ------------------- (2)   Since exactly 40%40% of the investors in securities have investments in equities, we have the equation   40100×(y+z)=z40100×(y+z)=z   (y+z)=5z2(y+z)=5z2   y=3z2y=3z2   Substituting equations (2) and (3) into equation (1) yields   3z+3z2+z=1103z+3z2+z=110   11z2=11011z2=110   z=110×211=20z=110×211=20   Hence, the number of people with investments in equities is:   x+z=3z+z=3×20+20=60+20x+z=3z+z=3×20+20=60+20= 
60+20 = 80
80   80
Answer: B
An alloy is to contain copper and zinc in the ratio 5:3. The zinc required to be melted with 40kg of copper is? Choose the correct choice from the following options
A) 15kg B) 20kg C) 8kg D) 24kg E) 32kg

Solution: Let the required quantity of copper be x kg   5:3::40:x   5x = 40*3   x = 24kg   Answer is 24kg
Answer: D
Pipe A can fill a tank in 9 hours, pipe B in 18 hours and pipe C in 54 hours. If all the pipes are open,in how many hours will the tank be filled ?	Select.
A) 5.1
B) 5.4
C) 5
D) 5.2
E) 5.5

Solution: 1/9+1/18+1/54=10/54=1/5.4. so 5.4 hrs   ANSWER:5.4
Answer: B
A student scored an average of 80 marks in 3 subjects: Physics, Chemistry and Mathematics. If the average marks in Physics and Mathematics is 90 and that in Physics and Chemistry is 70, what are the marks in Physics? Choose the correct option:
A) 87
B) 37
C) 80
D) 36
E) 13

Solution: Given M + P + C = 80 * 3= 
80 * 3 = 240
240 --- (1)   M + P= 90 * 2= 
90 * 2 = 180
180 --- (2)   P + C= 70 * 2= 
70 * 2 = 140
140 --- (3)   Where M, P and C are marks obtained by the student in Mathematics, Physics and Chemistry.   P= (2) + (3) - (1) = 180 + 140 - 240= 
180 + 140 - 240 = 80
80   Answer: 80
Answer: C
12 welders work at a constant rate they complete an order in 3 days. If after the first day, 9 welders start to work on the other project, how many more days the remaining welders will need to complete the rest of the order?  Choose the correct choice from the following options.
A) 5	B) 2	C) 8	D) 4	E) 6

Solution: 1. We need to find out the time taken by 3 workers after day 1.   2. total no.of wokers * total time taken = time taken by 1 worker   3. Time taken by 1 worker = 12*3= 
12*3 = 36
36 days   4. But on day 1 twelve workers had already worked finishing 1/3 of the job. So 3 workers have to finish only 2/3 of the job.   5. Total time taken by 3 workers can be got from formula used at (2). i.e., 3* total time taken= 36. Total time taken by 3 workers to finish the complete job is 36/ 3= 
36/ 3 = 12
12 days.   6. Time taken by 6 workers to finish 2/3 of the job is 2/3 * 12= 
2/3 * 12 = 8
8 days.   The answer is choice 8
Answer: C
Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:  Select:  A) 175.5  B) 170  C) 169.5  D) 180  E) 190

Solution: Since first and second varieties are mixed in equal proportions.   average price = 126+135/2 = 130.50   o, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say, Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x.   x - 153 = 22.50   x = 175.50   175.5NSWER 175.5
Answer: A
What approximate value should come in place of the question mark (?) in the following equation ?
5⁄7 of 1596 + 3015 = ? – 2150  Choose the correct choice from the following answers
A) 7200	B) 48000	C) 5300	D) 58000	E) 6300

Solution: ? ≈ 5 × 230 + 3000 + 2150   = 1150 + 3000 + 2150= 
1150 + 3000 + 2150 = 6_300
6300   Answer 6300
Answer: E
A 12% stock yielding 8% is quoted at?
Answers:
A) 165	B) 170	C) 175	D) 150	E) 185

Solution: Assume that face value = Rs.100 as it is not given   To earn Rs.8, money invested = Rs.100   To earn Rs.12, money invested = 100×12/8 = Rs.150   ie, market value of the stock = Rs.150   Answer is 150.
Answer: D
3x + 2y = 19 , and x + 3y = 1. Find the value of 4x + 5y
Choose the correct choice
A) 10
B) 18
C) 11
D) 20
E) 5

Solution: Add these two equations   4x + 5y = 20   Divide by 2 ( to Get 4x + 5y)   Answer will be D. 20
Answer: D
In the coordinate plane, the points F (-1,1), G (1,4), and H (4,1) lie on a circle with center P. What are the coordinates of point P ?	Choose the correct choice from the following answers:
A) (0,0)	B) (1,1)	C) (1.5,1.5)	D) (1,-2)	E) (2.5, -2.5)

Solution: I thought I need to find the 4th point that would lie on the circle   Hence my answer was (1,-2) (1,-1)   But the the center of the circle will lie on (1.5,1.5)   (1,-2)
Answer: D
George's drawer has 10 loose black socks, 13 loose blue socks, and 8 loose white socks. If George takes socks out of the drawer at random, how many would he need to take out to be sure that he had at least two matching pairs of socks in two different colors?
Choose the correct choice from the following options.
A) 7	B) 9	C) 11	D) 16	E) 20

Solution: The worst case scenario is choosing 13 blue socks. Then he would need 3 more socks to be sure of a matching pair of black socks or white socks.   The answer is 16.
Answer: D
If 50 apprentices can finish a job in 4 hours, and 30 journeymen can finish the same job in 8 hours, how much of the job should be completed by 10 apprentices and 15 journeymen in one hour?  Choose the correct choice from the following choices.
A) 1/9	B) 29/180	C) 9/80	D) 1/5	E) 39/121

Solution: 50 apprentices can finish the job in 4 hours,thus:   10 apprentices can finish the job in   4*5= 
  4*5 = 20
20 hours;   In 1 hour 10 apprentices can finish 1/20 of the job.   30 journeymen can finish the same job in 4,5 hours,thus:   15 journeymen can finish the job in  8*2= 
 8*2 = 16
16 hours;   In 1 hour 15 journeymen can finish 1/16 of the job.   Therefore, in 1 hour 10 apprentices and 15 journeymen can finish 1/20+1/16=9/80 of the job.   Answer: 9/80
Answer: C
A tank holds x gallons of a saltwater solution that is 20% salt by volume. One Fourth of the water is evaporated, leaving all of the salt. When 6 Gallons of water and 12 gallons of salt are added, the resulting mixture is 33 1/3 % salt by volume. What is the value of x? Pick one.
A) 37.5  B) 75  C) 100  D) 150  E) 90

Solution: Nope, 150. I can only get it by following PR's backsolving explanation. I hate that.   Original mixture has 20% salt and 80% water.   Total = x   Out of which Salt = 0.2x and water = 0.8x   Now, 1/4 water evaporates and all salt remains.   So what remains is 0.2x salt and 0.6x water.   Now 12 gallons salt is added and 6 gallons of water is added.   So salt now becomes -> (0.2x + 12) and water --> (0.6x+6)   Amount of salt is 33.33% of total. So amount of water is 66.66%. So salt is half of the volume of water.   So (0.2x+12) = (0.6x+6)/2 => 0.4x+24=0.6x+6 ->0.2x=18   Solving, x = 90   Answer : 90
Answer: E
A group of students was interviewed for that if it was asked whether or not they speak French and / or English. Among those who speak French, 10 speak English well, while 40 of them do not speak English. If 75% of students do not speak French, how many students were surveyed? Choose the correct option: A) 210 B) 225 C) 200 D) 250 E) 300

Solution: Number of students who speak French are 40 + 10= 
40 + 10 = 50
50   Of total students, the percentage of students who do not speak French was 75% --> percentage of who do is 25%   50-------25%   x ------- 100%   x= 50*100/25= 
50*100/25 = 200
200= number of all students   Answer is 200
Answer: C
Townville has 100 residents, and 40 of them are females. If half of Townville’s male residents are smokers, and 1/4 of the female residents are smokers, which of the following represents the total number of Townville’s residents who are NOT smokers?
Choose one: A) 54 B) 58 C) 62 D) 60 E) 64

Solution: Number of people =100   Females = 40   men = 100-40= 
100-40 = 60
60   Half of the men are smoker  60/2= 
  60/2 = 30
30, that means the remaining men 60-30= 
60-30 = 30
30 are non smokers.   1/4 females are smoker. i.e 1/4*40= 10.   40-10= 
40-10 = 30
30 females are non smokers   So, total number of non smokers in the town are 30+30= 
30+30 = 60
60   Answer : 60
Answer: D
A cricket player whose bowling average was 21.5 runs per wicket, takes 5 wicket for 52 runs in a match. Due to this his average decreases by 0.5. What will be the number of wickets taken by him till the last match?  Choose the correct choice from the following options:
A) 64  B) 72  C) 111  D) 721  E) 108

Solution: Average = Total Runs / Total wickets   Total runs after last match = 21.5w + 52   Total wickets after last match = w + 5   (21.5w + 52) / (w + 5) = 21.5 - 0.5= 
21.5 - 0.5 = 21
21   w= 106   so total wickets aftr last match = w+5 = 111   ANSWER:111
Answer: C
Three pieces of timber 54 m, 63 m and 81 m long have to be divided into planks of the same length, What is the greatest possible length of each plank ?
Choose the correct answer:
A) 9 m
B) 4 m
C) 4 m
D) 6 m
E) None of these

Solution: 9 mnswer   Greatest possible length of each plank = H.C.F of 54, 63, 81 = 9 m   Correct Option: 9 m
Answer: A
Pipe A fills a tank in 30 minutes. Pipe B can fill the same tank 5 times as fast as pipe A. If both the pipes were kept open when the tank is empty, how much time will it take for the tank to overflow? Choose the most appropriate option
A) 4 mins	B) 6 mins	C) 5 mins	D) 3 mins	E) 7 mins

Solution: Let the total capacity of tank be 90 liters.   Capacity of tank filled in 1 minute by A = 3 liters.   Capacity of tank filled in 1 minute by B = 15 liters.   Therefore, capacity of the tank filled by both A and B in 1 minute = 18 liters.   Hence, time taken by both the pipes to overflow the tank = 90/18= 
90/18 = 5
5 minutes.   ANSWER:5 mins
Answer: C
Each month a retailer sells 100 identical items. On each item he makes a profit of $60 that constitutes 10% of the item's price to the retailer. If the retailer contemplates giving a 5% discount on the items he sells, what is the least number of items he will have to sell each month to justify the policy of the discount?
Choose the correct choice from the following answers
A) 191	B) 213	C) 221	D) 223	E) 226

Solution: For this question, we'll need the following formula:   Sell Price = Cost + Profit   We're told that the profit on 1 item is $20 and that this represents 10% of the cost:   Sell Price = Cost + $60   Sell Price = $600 + $60   Thus, the Sell Price is $660 for each item. Selling all 100 items gives the retailer...   100($60) = $2,000 of profit   If the retailer offers a 5% discount on the sell price, then the equation changes...   5%(660) = $33 discount   $627 = $600 + $27   Now, the retailer makes a profit of just $27 per item sold.   To earn $2,000 in profit, the retailer must sell....   $27(X) = $2,000   X = 2,000/27   X = 222.222222 items   You'll notice that this is not among the answer choices.... 221 and 223 are.   Selling 221 items would get us 9(221) = $1989 which is NOT enough money. To get back to AT LEAST $2,000, we need to sell 223 items.   Final Answer:   223
Answer: D
If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor S of 2x and 3y? Choose the correct choice from the following answers
A) 1  B) S=Cannot be determined  C) S=2  D) 5  E) 6

Solution: My explanation: from question stem we know that nothing is common between X and Y , X and Y are two prime numbers eg: X=2, Y=3 and their GCD(2,3) =1 and so 2X and 3Y will have a GCD(2X,3Y) = 1 . what if either X or Y was 1, eg: X=1,Y=4 then GCD(1,4) =1 , but GCD(2,12) = 2.   and hence answer S=Cannot be determined
Answer: B
0.0023*0.13/0.236=my age
0.022*0.12/0.220=my wife age
then what is difference b/w ages? Pick:
A) 0.01073  B) 0.02073  C) 0.03076  D) 0.04078  E) 0.0578

Solution: My age = 0.0023*0.13/0.236 = 0.000299/0.236 = 0.001266949   My wife age = 0.022*0.12/0.220 = 0.00264/0.220= 
0.00264/0.220 = 0.012
0.012   difference b/w ages= 0.012 - 0.001266949 = 0.01073   0.01073NSWER:0.01073
Answer: A
The ratio of cats to dogs to birds to fish is 5:6:7:8. There are no other types of animals in the pet store.which of the followingcannotbe the total number of animals in the store? Choose the correct option:	A) 52	B) 78	C) 104	D) 130	E) 223

Solution: The parts add up to 5 + 6 + 7 + 8= 
5 + 6 + 7 + 8 = 26
26, so we know that any possible number of total answers must be a multiple of 26. All five answer choices are multiples of 26:   (A) 52= 26*2   (B) 78= 26*3   (C) 104= 26*4   (D) 130= 26*5   (223) 223 = not a multiple of 26
Answer: E
Ashok has Rs.1000, He invests a part of it at 3% per annum and remainder at 8% per annum, He earns Rs.100 in 2 years. Find the sum invested at different rates of interest?  Options:
A) Rs.5000, Rs.500
B) Rs.600, Rs.400
C) Rs.700, Rs.300
D) Rs.300, Rs.700
E) Rs.300, Rs.800

Solution: (600÷100)×3×2=34   (400÷100)×8×2=64   Total=100   ANSWER:Rs.600, Rs.400
Answer: B
If q = 35! + 1, which of the following cannot be a prime factor of q?
I. 11
II. 19
III. 31	Choose the correct choice:
A) I​ only B) III only C) II and III D) I and II E) I​, II, and III

Solution: The numbers 11, 19, and 31 are all factors of 35!   Therefore they can not be factors of 35! + 1.   The answer is I​, II, and III.
Answer: E
what is the sum of all the 4 digit numbers that can be formed using all of the digits 2,3,5 and 7?  Choose one
A) 113323  B) 113322  C) 114322  D) 113321  E) 112322

Solution: Actually there is the direct formula for this kind of problems.   1. Sum of all the numbers which can be formed by using the n digits without repetition is: (n-1)!*(sum of the digits)*(111…..n times).   2. Sum of all the numbers which can be formed by using the n digits (repetition being allowed) is: n^{n-1}*(sum of the digits)*(111…..n times).   so as per question:(4-1)!*(2+3+5+7)*1111=113322   ANSWER:113322
Answer: B
Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. Fast train completely passes a man sitting in the slower train in 5 sec. What is the length of the fast train?
Pick one:
A) 27 7/8
B) 27 7/6
C) 27 7/4
D) 27 7/9
E) 27 7/1

Solution: Relative speed = (40 - 20)= 
(40 - 20) = 20
20 km/hr.= 20 * 5/ 18 = 50/9 m/sec.   Length of faster train = 50/9 * 5 = 250/9 = 27 7/9 m.   Answer: 27 7/9
Answer: D
If 25% of x is 30 less than 20% of 1000, then x is?
Pick
A) 188
B) 216
C) 156
D) 680
E) 121

Solution: 25% of x = x/4 ; 20% of 1000 = 20/100 * 1000= 
20/100 * 1000 = 200
200   Given that, x/4= 200 - 30 => x/4 = 170 => x = 680.   Answer: 680
Answer: D
A sum of money is put out at compound interest for 2 years at 20%. It would fetch Rs.482 more if the interest were payable half-yearly, then it were pay able yearly. Find the sum?	Choose the correct choice from the following:
A) 198
B) 279
C) 267
D) 200
E) 269

Solution: P(11/10)4 - P(6/5)2   = 482   P = 200   Answer: 200
Answer: D
Rose and Anna are partners in a business. Rose runs the business and receives 12% of the profit for managing the business, the rest is divided in proportion of their capitals. Rose invested Rs.12000 and Anna invested Rs.20000. the rest is divided in proportion of their capitals. Out of a total profit of Rs.9000, how much did Rose receive?	Choose the correct choice.
A) 5872  B) 5839  C) 5832  D) 5838  E) 5832

Solution: ratio of their investments Rs.12000:20000=3:5   12% of the profit for managing the business =12/100*9000= 
12/100*9000 = 1_080
1080   remaining profit=9000-1080= 
9000-1080 = 7_920
7920   share profit in ratio of investment 7920/5= 
7920/5 = 1_584
1584   Moses's ratio share=3*1584= 
3*1584 = 4_752
4752   total money Moses collects=4752+1080= 
4752+1080 = 5_832
5832   Answer: 5832
Answer: C
Find the value of x from this equation? 7(x - 1) = 21
Pick one:  A) 2  B) 4  C) 6  D) 8  E) 10

Solution: 1. Divide both sides by 7:   2. Simplify both sides:   x - 1 = 3   3. Add 1 to both sides:   x - 1 + 1 = 3 + 1   4. Simplify both sides:   x = 4   4
Answer: B
A batsman in his 17th innings makes a score of 85 and their by increasing his average by 3. What is his average after the 17thinnings?	Choices.
A) 11	B) 7	C) 76	D) 37	E) 9

Solution: 16x + 85 = 17(x + 3)   x = 34 + 3= 
34 + 3 = 37
37   Answer: 37
Answer: D
Obra drove 150π meters along a circular track. If the area enclosed by the circular track on which she drove is 57,600π square meters, what percentage of the circular track did Obra drive?
Choices
A) 6.67%	B) 12.5%	C) 18.75%	D) 25%	E) 31.25%

Solution: area enclosed by the circular track on which she drove is 57,600π square meters   So, π(r^2) = 57,600π ---> (r^2)=57,600 ---> r=240   Circumference of the circular track = 2πr = 480π   Therefore, part of circumference covered = 150π/480π =31.25%   Hence, answer is 31.25%.
Answer: E
There are four different models that are to appear in a fashion show. Two are from Europe, two are from South America, and two are from North America. If all the models from the same continent are to stand next to each other, how many ways can the fashion show organizer arrange the models?	Options:	A) 72	B) 48	C) 64	D) 24	E) 8

Solution: Since we have 2 continental pairs (EU, SA, NA), these 2 pairs have 2*1= 
2*1 = 2
2 Combinations. Within each pair, you have however 2 different ways to put them together for each of the pair (2*2= 4). So we have 2*4= 
2*4 = 8
8.   Please correct me if im wrong.   Answer 8.
Answer: E
If the numerator of a fraction E is tripled, and the denominator of a fraction is doubled, the resulting fraction will reflect an increase of what percent?
Options:  A) 16 1⁄6%  B) 25%  C) 33 1⁄3%  D) 50%  E) 66 2⁄3%

Solution: This question can be dealt with conceptually or by TESTing VALUES.   We're told that the numerator of a fraction E is tripled and the denominator of that same fraction is doubled. We're asked for the resulting increase, in percentage terms, of that fraction.   Let's TEST 2/1   If we triple the numerator and double the denominator, we end up with....   (2x3)/(1x2) = 6/2= 
6/2 = 3
3   Since we're increasing 2 to 3, we're increasing that fraction by (3-2)/2= 1/2 of itself = 50%   50%
Answer: D
If X boxes of mango juice cost 112 cents, how much will Y boxes cost in dollars?
Answers:
A) 2Y*1.12/x
B) 3Y*1.12/x
C) Y*12/x
D) Y*0.12/x
E) Y*1.12/x

Solution: X boxes of mango juice cost 112 cents or 1.12 dollars.   cost of mango juice = 1.12/x   Cost of Y mango juices = Y*1.12/x   Hence Answer : Y*1.12/x
Answer: E
Two persons start running simultaneously around a circular track of length 300 m from the same point at speeds of 13 km/hr and 23 km/hr. When will they meet for the first time any where on the track if they are moving in opposite directions?	Choose the correct choice from the following.
A) 28 sec  B) 30 sec  C) 17 sec  D) 23 sec  E) 27 sec

Solution: Time taken to meet for the first time anywhere on the track   = length of the track / relative speed   = 300 / (13 + 23)5/18 = 300* 18 / 36 * 5 = 30 seconds.   Answer : 30 sec
Answer: B
You are given three coins: one has heads on both faces, the second has tails on both faces,
and the third has a head on one face and a tail on the other. You choose a coin at random and
toss it, and it comes up heads. The probability that the other face is tails is	Choose the most appropriate option
A) 1/4
B) 1/3
C) 1/2
D) 2/3
E) 4/3

Solution: the three coins are HH LL HT   total possiblity are 6   1st head probablity are 2/6,   2 nd head probablity are 0,   third head probablity are 1/6,   The total head probablity is 2/6+0+1/6=1/2,   Then tail probablity is 1-1/2=1/2   ANSWER:1/2
Answer: C
One hour after Yolanda started walking from X to Y, a distance of 31 miles, Bob started walking along the same road from Y to X. If Yolandaâs walking rate was 1 miles per hour and Bobâs was 2 miles per hour, how many miles had Bob walked when they met?  Choose the correct answer
A) 19	B) 20	C) 22	D) 21	E) 19.5

Solution: Let t be the number of hours that Bob had walked when he met Yolanda. Then, when they met, Bob had walked 4t miles and Yolanda had walked (t + 1) miles. These distances must sum to 31 miles, so 2t + (t + 1) = 31, which may be solved for t as follows   2t + (t + 1) = 31   2t + t + 1 = 31   3t = 30   T = 10 (hours)   Therefore, Bob had walked 2t = 2(10) = 20 miles when they met. The best answer is 20.
Answer: B
In a graduating class of 236 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?
Select the correct option
A) 144	B) 119	C) 113	D) 117	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 - 236= 
144 + 119 + 0 - 236 = 27
27   Greatest - Smallest is  144-27= 
 144-27 = 117
117   So answer must be D. 117
Answer: D
if a number cube is rolled once and a coin is tossed once. what is the probability that coin will show tails and composite number on on the cube
Pick:
A) 1/6
B) 2/6
C) 3/6
D) 4/6
E) 5/6

Solution: 1/2 * 2/6 = 1/6   1/6NSWER:1/6
Answer: A