MU-NLPC/Calc-aqua_rat huggingface dataset:
Factor | Percent of Respondents |
User-friendly | 56% |
Fast response time | 48% |
Bargain prices | 42% |
The table gives three factors to be considered when choosing an Internet service provider and the percent of the 1,400 respondents to a survey who cited that factor as important. If 30 percent of the respondents cited both “user-friendly” and “fast response time,” what is the maximum possible number of respondents who cited “bargain prices,” but neither “user-friendly” nor “fast response time?”	Choose the correct choice from the following.
A) 312  B) 336  C) 364  D) 384  E) 420

Solution: The way I looked at is is as follows:   UserFriendly(UF) + FastResponse (FR) = 30%   UF leftover = 56 - 30= 
56 - 30 = 26
26%   FR leftover= 48 - 30= 
48 - 30 = 18
18%   Sum these= 74% (ie. 74% were either UF+FR, UF, FR)   26% leftover is the maximum number for Bargain Price   26/100 * 1400 = 26 * 14= 
26 * 14 = 364
364 (364).
Answer: C
A man is trying to climb over a frozen mountain that is sixty feet high. Every minute, he can climb 3 feet, but due to the slippery ice, he slips back 2 feet.
How long will it take him to reach the top of the mountain?	Choose the correct choice:	A) 57 minutes	B) 58 minutes	C) 59 minutes	D) 53 minutes	E) 54 minutes

Solution: Solution:   It will take the man 58 minutes to reach the top.   Since he is climbing 3 feet and falling 2 feet back every passing minute, he is actually climbing 1 feet every minute.   Therefore, he would have climbed 57 feet in 57 minutes. The remaining distance is 3 feet. He will take one more minute to climb it and reach the surface. Since he has reached the top surface, he won't slip back.   58 minutes   Answer 58 minutes
Answer: B
The decimal d is formed by writing in succession all the positive, odd integers in increasing order after the decimal point; that is d = 0.1357911131517...
What is the 100th digit of d to the right of decimal point?	Choose the correct choice.
A) 5  B) 1  C) 0  D) 9  E) 7

Solution: One-digit odd numbers: There are (1)(10/2) = 5 odd one-digit numbers.   0.13579   5 digits down, 95 remaining. The next 5 odd numbers will be 11, 13, 15, 17, and 19, ... so it follows that: (2)(5) = 10 digits.   Two-digit numbers: 90 digits, which brings us to the number 99.   Three-digit numbers: we only need 5 more digits. The next odd numbers are 101, and 103, thus 0 is the 100th digit.   Answer: 0
Answer: C
There is 60 lit of milk and water in which milk forms 84%.Howmuch water must be added to this solution to make it solution in which milk forms 68%
Choices:
A) 14.11  B) 19.75  C) 20.75  D) 21.75  E) 22.75

Solution: 60*84/100= 
60*84/100 = 252/5 = around 50.4
50.40 lit milk that is 9.60 lit water   let x lit water will be added then   (60+x)*68/100=50.40   so x=14.11   14.11NSWER:14.11
Answer: A
A sum of money lent out at S.I. amounts to a total of $520 after 2 years and to $820 after a further period of 5 years. What was the initial sum of money that was invested?
Select: A) $340 B) $360 C) $380 D) $400 E) $420

Solution: S.I for 5 years = $820 - $520 = $300   The S.I. is $60/year   S.I. for 2 years 820 -  520= 
820 -  520 = 300
$300   The S.I. is $60/year   S.I. for 2 years= $120   Principal = $520 - $120 = $400   The answer is $400.520 -  120= 
520 -  120 = 400
$400   The answer is $400.
Answer: D
A cricket bat is sold for $850, making a profit of $215. The profit percentage would be Choose one.
A) 24% B) 25% C) 34% D) 36% E) 40%

Solution: 215/(850 - 215) = 215/635 = 43/127 =34%.   Answer: 34%.
Answer: C
A and B starts a business with Rs.8000 each, and after 4 months, B withdraws half of his capital . How should they share the profits at the end of the 18 months?	Choose the most appropriate option:
A) 18:11 B) 18:16 C) 18:10 D) 18:14 E) 14:11

Solution: A invests Rs.8000 for 18 months, but B invests Rs.8000 for the first 4 months and then withdraws Rs.4000. So, the investment of B for remaining 14 months is Rs.4000 only.   18:11 : B   8000*18 : (8000*4) + (4000*14)   14400 : 88000   18:11:B = 18:11   18:11nswer: 18:11
Answer: A
You are gifted a coin that is dated 378 B.C. How much worth the coin is as per your assumption?
Choose the correct choice from the following choices
A) 1 B) 0 C) 2 D) 5 E) 6

Solution: Solution:   The worth of that coin is zero. This is because the B.C. and A.D. dating system was not empowered till the year 525. So the coin is probably fake.   Answer 0
Answer: B
Dustin's Pizza Shop sells 3 sizes of pizza: Small, Medium, and Large. A Small pizza contains 4 slices, A Medium Pizza contains 6 slices and a Large pizza contains 10 slices. Lisa, a customer having a birthday party, visits the shop with her 2 daughters and 9 friends. To ensure each of the visitors receive an equal amount of pizza with none leftover, which of the following combinations are possible? Choose the correct choice from the following answers:  A) 1 Small and 1 Large  B) 4 Small and 1 Medium  C) 2 Small and 4 Medium  D) 4 Medium and 1 Large  E) 1 Medium and 3 Large

Solution: There are 12 total customers (Lisa, 2 daughters, and 9 friends) The correct combination must be divisible by 12 with no remainder.   A= 14 Slices (1 slice per guest with 2 slices leftover)   B= 22 Slices (1 Medium and 3 Largeach kid would have 2 slices, but Lisa would have none)   C= 32 Slices (8 visitors would have 3 slices, but 4 visitors would only have 2)   D= 34 Slices (10 Visitors would have 3 slices, but 2 visitors would only have 2)   1 Medium and 3 Large= 36 Slices (1 Medium and 3 Largeach visitor would have 3 slices, with nothing leftover) *Correct Answer*   The Correct Answer is 1 Medium and 3 Large
Answer: E
If s,u, and v are positive integers and 2s=2u+2v, which of the following must be true?
i. s=0
ii. u is not equal to v
iii.u > v	Choose one:
A) None
B) I only
C) II only
D) III only
E) II and III

Solution: Notice two things: 1. we are asked to find out which of the following MUST be true, not COULD be true and 2. s, u, and v are positive integers.   Given: 2s=2u+2v --> s=u+v. Now, since s, u, and v are positive integers then s is more than either u or v, so I is never true and III is always true. As for II: it's not necessarily true, for example 4=2+2. .   Nonenswer: None
Answer: A
Which of the following equations describes a line that is perpendicular to y = x/2 + 6?	Choose the correct answer.
A) y = x − 6  B) y = −1 − 2x  C) y = -3x  D) y = 6x-1  E) y = 1/x + 6

Solution: Two perpendicular lines have slopes that are inverse reciprocals of each other.   The slope of y = x/2 + 6 is 1/2 so the slope of a perpendicular line will be -2.   The answer is y = −1 − 2x.
Answer: B
Three friends had dinner at a restaurant. When the bill was received, Amita paid 2/3 as much as Veena paid and Veena paid 1/2 as much as Tanya paid. What faction of the bill did Veena pay ? Choose the correct choice from the following:
A) 1/3 B) 3/11 C) 12/13 D) 5/8 E) 9/8

Solution: let veena paid x,   so anita paid 2/3 x,   and tanya paid 2x,   so total bill paid is given by   x+(2/3)x +2x = 1   i.e. x=3/11   ANSWER:3/11
Answer: B
40% of Ram's marks is equal to 20% of Rahim's marks which percent is equal to 30% of Robert's marks. If Robert's marks is 80, then find the average marks of Ram and Rahim?  Choose the correct choice from the following options:
A) 60
B) 70
C) 80
D) 90
E) 100

Solution: Given, 40% of Ram's marks = 20% of Rahim's marks = 30% of Robert's marks.   Given, marks of Robert = 80   30% of 80 = 30/100 * 8 = 24   Given, 40% of Ram's marks = 24.   => Ram's marks = (24 * 100)/40= 
(24 * 100)/40 = 60
60   Also, 20% of Rahim's marks= 24   => Rahim's marks = (24 * 100)/20= 
(24 * 100)/20 = 120
120   Average marks of Ram and Rahim= (60 + 120)/2= 
(60 + 120)/2 = 90
90.   ANSWER:90
Answer: D
An Olympic diver received the following scores: 7, 4, 8, 10, and 6. The standard deviation of her scores is in which of the following ranges?
Choose the correct choice from the following options.
A) 0-1.9 B) 2-3.9 C) 4 – 6.9 D) 7 – 7.9 E) 8 – 9.9

Solution: 7+4+8+10+6/5 =7   | 7-7= 
 7-7 = 0
0 0^2= 0   |4-7| = 3^2 = 9   | 8-7= 
 8-7 = 1
1^2=1   | 10-7= 
 10-7 = 3
3^2= 9   |6-7| = 1^2 = 1   SD = sqrt 20/5 = 2   Answer : 2-3.9
Answer: B
A 10-by-7 inch piece of paper is used to form the lateral surface of a cylinder. If the entire piece of paper is used to make the lateral surface, which of the following must be true of the two possible cylinders that can be formed?
Choose the correct choice.
A) The volume of the cylinder with height 10 is 70/pi cubic inches greater than the volume of the cylinder with height 6.  B) The volume of the cylinder with height 7 is 70/pi cubic inches greater than the volume of the cylinder with height 10.  C) The volume of the cylinder with height 10 is 70pi cubic inches greater than the volume of the cylinder with height 6.  D) The volume of the cylinder with height 7 is 70pi cubic inches greater than the volume of the cylinder with height 10.  E) The volume of the cylinder with height 6 is 240/pi cubic inches greater than the volume of the cylinder with height 10.

Solution: Pretty straightforward. Plug in the values to compute the volume of the cylinder. Keep in mind 7 and 10 are the circumference of the base and not the radius. The volume of the cylinder with height 7 is 70/pi cubic inches greater than the volume of the cylinder with height 10. is the right answer.
Answer: B
In how many different ways can 7 girls and 7 boys form a circle such that the boys and the girls alternate?	Choose the correct choice from the following:  A) 2880211  B) 3628800  C) 120000  D) 36286200  E) None of these

Solution: Explanation :   In a circle, 7 boys can be arranged in 6! ways   Given that the boys and the girls alternate.   Hence there are 7 places for girls which can be arranged in 7! ways   Total number of ways = 6! x 7! = 720 x 5040 = 3628800. Answer : Option 3628800
Answer: B
On 8th march,2005,Wednesday falls. What day of the week was it on 8th march,2004? Choices.
A) 28 B) 00 C) 27 D) 01 E) 8

Solution: Answer:01
Answer: D
Two employees X and Y are paid a total of Rs. 638 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week?
Pick: A) s.250 B) s.280 C) s.290 D) s.299 E) s.300

Solution: Let the amount paid to X per week = x   and the amount paid to Y per week = y   Then x + y = 638   But x = 120% of y = 120y/100 = 12y/10   ∴12y/10 + y = 638   ⇒ y[12/10 + 1] = 638   ⇒ 22y/10 = 638   ⇒ 22y = 6380   ⇒ y = 6380/22 = 580/2 = Rs.290   s.290
Answer: C
If the number is decreased by 5 and divided by 7 the result is 7. What would be the result if 2 is subtracted and divided by 13? Choose the correct choice.
A) 4	B) 7	C) 8	D) 5	E) 3

Solution: Explanation:   Let the number be x. Then,   (x - 5)/7 = 7 => x - 5 = 49   x = 54   .: (x - 2)/13 = (54 - 2)/13= 
(54 - 2)/13 = 4
4   4nswer: Option 4
Answer: A
An outlet pipe empties a tank which is full in 5 hours. If the inlet pipe is kept open, which lets water in at the rate of 8 litres/min then outlet pipe would take 3 hours longer. Find the capacity of the tank.
Choose the correct choice from the following choices:
A) 8600 litres
B) 200 litres
C) 12800 litres
D) 6400 litres
E) 13200 litres

Solution: Let the rate of outlet pipe be x liters/hour;   Rate of inlet pipe is 8 litres/min, or  8*60= 
 8*60 = 480
480 liters/hour;   Net outflow rate when both pipes operate would be x-480 liters/hour.   Capacity of the tank= x*5 hours = (x-480)*(5+3) hours   5x=(x-480)*8 --> x=1280 --> capacity = 5x = 6400 liters.   Answer: 6400 litres
Answer: D
If the simple interest on a sum of money for 3 years at 7% per annum is Rs. 570, what is the compound interest on the same sum at the rate and for the same time? Pick one:	A) Rs. 610.83	B) Rs. 810.83	C) Rs. 617	D) Rs. 618	E) Rs. 610.83

Solution: Sum = (570 * 100) / (7 * 3) = Rs. 2,714.29   Amount = [2,714.29 * (1 + 7/100)3] = Rs. 3,325.12   C.I. = (3,325.12 - 2,714.29) = Rs. 610.83.   ANSWRs. 610.83R:Rs. 610.83
Answer: E
While working alone at their constant rates, computer X can process 240 files in 12 hours, and computer Y can process 240 files in 4 hours. If all files processed by these computers are the same size, how many hours would it take the two computers, working at the same time at their respective constant rates, to process a total of 240 files?  Choices:
A) 2
B) 2.4
C) 2.7
D) 3
E) 3.5

Solution: Both computers together process files at a rate of 240/12 + 240/4 = 20+60= 
20+60 = 80
80 files per hour.   The time required to process 240 files is  240/80= 
 240/80 = 3
3 hours   The answer is 3.
Answer: D
A train crosses a platform of 150 m in 15 sec, same train crosses another platform of length 250 m in 20 sec. then find the length of the train?  Choose the most appropriate option:
A) 150m
B) 188m
C) 267m
D) 268m
E) 287m

Solution: Length of the train be ‘X’   X + 150/15 = X + 250/20   4X + 600 = 3X + 750   X = 150m   150mnswer: 150m
Answer: A
A group of 10 representatives is to be selected out of 15 seniors and 10 juniors. In how many different ways can the group be selected, if it should have 5 seniors and 5 juniors?	Choose the most appropriate option
A) 15C₅ * 10  B) 15C₇ * 10  C) 15C10 * ¹⁰C₅  D) 12 * ¹⁰C₅  E) None of these

Solution: Here, five seniors out of 15 seniors can be selected in 15C₅ ways. Also, five juniors out of ten juniors can be selected ¹⁰C₅ ways. Hence the total number of different ways of selection = 1515C10 * ¹⁰C₅₅ * ¹⁰15C10 * ¹⁰C₅₅ =1515C10 * ¹⁰C₅10 * ¹⁰15C10 * ¹⁰C₅₅   since 1515C10 * ¹⁰C₅₅ = 1515C10 * ¹⁰C₅10   ANSWER:15C10 * ¹⁰C₅
Answer: C
Michelle deposited a certain sum of money in a savings account on July 1st, 2007. She earns an 10% interest compounded semiannually. The sum of money in the account on December 31st, 2009 is approximately what percent of the initial deposit? Pick one
A) 127%  B) 120%  C) 121%  D) 135%  E) 140%

Solution: Since Michelle earns 10% interest compounded semiannually, then she earns 5% interest every 6 months.   Now, the simple interest earned in 5 periods (30 months = 5*6 months) would be 5%*5=25%. But, since the interest is compounded every 6 months, then there would be interest earned on interest (very small amount) thus the actual interest earned would be a little bit more than 25%, only answer choice A fits.   127%nswer: 127% 5 *5= 
 5 *5 = 25
25%. But, since the interest is compounded every 6 months, then there would be interest earned on interest (very small amount) thus the actual interest earned would be a little bit more than 25%, only answer choice A fits.   127%nswer: 127%
Answer: A
4.036 divided by 0.04 gives	Choose the correct choice from the following:
A) 1.009 B) 100.09 C) 10.9 D) None of these E) cannot be determined

Solution: Solution   4.036 / 0.04	= 403.6 / 4= 
403.6 / 4 = 100.9
100.9   1.009nswer 1.009
Answer: A
What will come in place of the x in the following Number series? 18, 22, 30, 46 ,X, 142	Choose the correct choice from the following answers: A) 35 B) 56 C) 78 D) 87 E) 98

Solution: (78)   The pattern is +4, +8, +16, +32, +64   So the missing term is = 78
Answer: C
One thief entered a cave where he found a pile of diamonds. At first he thought of taking all. But listening to his conscience he thought of taking only half of them. But while returning he picked up two more from the pile. After that Second thief entered the cave and bagged half of remaining diamonds and picked up two more while returning. Later third thief entered and did the same. Even the fourth thief did the same. When the fifth thief entered he found only one diamond in the cave. How many diamonds are there in the cave at first.
Choose the correct answer: A) 86 B) 94 C) 80 D) 96 E) 55

Solution: Explanation :   When 5th thief entered-1 diamond   When 4th thief entered -(1+2)*2=6 diamonds   When 3rd thief entered -(6+2)*2=16 diamonds   When 2nd thief entered -(16+2)*2=36 diamonds   When 1st thief entered -(36+2)*2=76 diamonds   Hence(D) is the correct answer.   Answer : 96
Answer: D
A train overtakes 2 girls who are walking inthe opposite direction in which the train is going at the rate of 3 km/hour & 6km/hour and passes them completely in 36sec & 30 sec respectively. Find the length of the train is?	Select:
A) 90 m
B) 120 m
C) 150 m
D) 190 m
E) 210 m

Solution: Let the length of the train e x meter, and let the speed of train be y km/h, then   \inline x=\left ( y+3 \right )\frac{5}{18}\times 36 ........(1)   and \inline x=\left ( y+6\right )\frac{5}{18}\times 30 ........(2)   From eq (1) and (2), we get   \inline (y+3)\times 36 = ( y+6)\times 30   y = 12 km/h   \inline \therefore \inline x=(y+3)\times \frac{5}{18}\times 36   x= 150 m   150 m
Answer: C
In a 500 m race, the ratio of the speeds of two contestants A and B is 2 : 4. A has a start of 300 m. Then, A wins by:  Choose the correct choice from the following options:	A) 60 m	B) 20m	C) 43m	D) 20m	E) 100 m

Solution: To reach the winning post A will have to cover a distance of (500 - 300)m, i.e., 200 m.   While A covers 2 m, B covers 4 m.   While A covers 200 m, B covers 4 x 200/2	m	= 400 m.   Thus, when A reaches the winning post, B covers 400 m and therefore remains 100 m behind.   A wins by 100 m.   Answer:	100 m
Answer: E
The sum of the present ages of a son and his father is 60 years. Six years ago, father's age was five times the age of the son. After 6 years, what will be son's age?  Choose the correct answer:
A) 20
B) 21
C) 22
D) 25
E) 18

Solution: Let the present age of the son =x   Then, present age of the father =(60−x)   Six years ago father's age was 5 times the age of the son   ⇒(60−x)−6=5(x−6)⇒84=6x⇒x=846=14   Son's age after 6 years   =x+6=14+6= 
14+6 = 20
20   20nswer is 20.
Answer: A
A person incures loss for by selling a watch for rs1140.at what price should the watch be sold to earn a 5% profit ? Choose the correct choice from the following options.
A) Rs.1160	B) Rs.1260	C) Rs.1200	D) Rs.1250	E) Rs.1280

Solution: let the new sp be Rs x.then   (100-loss%) : (1st sp)=(100+gain%) (2nd sp)   ={(100-5)/1400}={(100+5)/x}=> x={(105*1140)/95} =Rs.1260.   Answer is Rs.1260.
Answer: B
In a apartment, 20% of the people speak English, 60% speak Hindi and 60% speak both. If a people is selected at random, what is the probability that he has speak English or Hindi?  Pick.
A) 1/5
B) 2/3
C) 2/5
D) 2/7
E) 2/9

Solution: P (E) = 20 / 100 = 1 / 5 , P (H) = 60 / 100 = 3 / 5 and P (E âˆ© H) = 60 / 100 = 3 / 5   P (E or H) = P (E U H)   = P (E) + P (H) - P (E âˆ© H)   = (1 / 5) + (3 / 5) - (3 / 5) = 4 / 10 = 1/5   1/5
Answer: A
The difference between the length and breadth of a rectangle is 30 m. If its perimeter is 240 m, then its area is:
Options
A) 1520 m2	B) 2420 m2	C) 3375 m2	D) 2520 m2	E) None

Solution: EXPLANATION   We have: (l â€“ b) = 30 and 2(l + b) = 240 or (l + b) = 120.   Solving the two equations, we get: l = 75 and b = 45.   Area = (l x b) = (75x 45) m2 = 3375 m2.   Answer 3375 m2
Answer: C
A straight line in the xy- plane has slope 4 . On this line the x-coordinate of the point is 50 and y-coordinate is 300 then what is the y intercept of the plane?  Choose the correct choice from the following answers
A) 150 B) 110 C) 100 D) 200 E) 50

Solution: Eq of line = Y = MX +C   M= 4   X=50   Y=50*4 + C,   Substitute Y by 300 as given in question.   300=200+100 , 100 =100   correct option is 100
Answer: C
If a truck is travelling at a speed of 54 kmph, crosses a pole in 1 sec, then the length of train is?	Choose the correct choice from the following options: A) 13 m B) 14 m C) 15 m D) 16 m E) 17 m

Solution: Explanation:   D = 54 * 5/18 * 1= 
54 * 5/18 * 1 = 15
15 m   Answer: Option 15 m
Answer: C
What is the sum of all the prime factors of 220 and 330?
Pick one:
A) 15
B) 21
C) 35
D) 41
E) 64

Solution: Prime factorization of both the numbers is as follows   220 = 2 * 2 * 5 * 11   sum of the prime factors of 220 = 20   330 = 2 *3 * 5 * 11   sum of prime factors of 330 = 21   addition of both = 20 + 21= 
20 + 21 = 41
41   ANSWER 41
Answer: D
Two twins sisters Sita and Geeta were standing back to back and suddenly they started running in opposite direction for 4 miles and then turn to left and run for another 3 miles.
what is the distance between the the two twins when they stop ?
Pick.
A) 10 miles  B) 11 miles  C) 18 miles  D) 16 miles  E) 13 miles

Solution: 10 miles   10 miles
Answer: A
If x and y are positive integers, then xy - y is Choose the correct choice from the following answers.
A) Always odd B) Odd whenever y is odd C) Odd whenever x+y is even D) Even whenever y is odd E) It cannot be determined

Solution: x and y are positive integers, then xy - y..   Let's take even as 4 and odd as 5.   IMO option A is correct answer..   If you consider your fourth scenario, that turns out to be even.   Hence option Odd whenever y is odd should be correct.
Answer: B
If x, y, and z are all factors of 21, which of the following must be true?	Choose the correct option:
A) The product xyz > 21
B) Either x = y, y = z, or x = z
C) x – y – z = even
D) The product xyz is odd
E) yz/x is an integer

Solution: All the factors of 21 are odd numbers.   The product of odd numbers is odd.   The answer is The product xyz is odd.
Answer: D
Ratio of the ages of Mahesh and Nilesh is 5 : x. Mahesh is 18 years younger to Ramesh. After nine years Ramesh will be 47 years old. If the difference between the ages of Mahesh and Nilesh is same as the age of Ramesh, what is the value of x?
Choose the correct option.
A) 72.9 B) 62.2 C) 14.2 D) 13.7 E) 14.5

Solution: Let the present ages of Mahesh, Nilesh and Ramesh be m, n and r respectively.   m/n = 5/x ------ (1)   m = r - 18 ------ (2)   r + 9 = 47 ------ (3)   m - n = r ----- (4)   (3) => r = 47 - 9= 
47 - 9 = 38
38 years   (2)=> m = 38 -18= 
38 -18 = 20
20 years   (1)=> 20/n = 5/x => n = 4x   (4) => 4x - 20 = 38   => 4x = 58 => x = 14.5   Answer:14.5
Answer: E
Which one of the following can't be the square of natural number ?  Choices:
A) 128881  B) 128242  C) 131044  D) 130321  E) 130322

Solution: Square of a natural number cannot end with 2 .   Hence 128242 cannot be the square of natural number   Answer is 128242
Answer: B
Every digit of a number written in binary is either 0 or 1. To translate a number from binary, multiply the nth digit (reading from right to left) by 2^(n-1)
What is the largest prime number (written in binary) that is a factor of both 10110000 and 101100000 ? Choose the correct choice from the following answers:
A) 10	B) 11	C) 101	D) 1011	E) 10001

Solution: Binary Divison can provide a quick answer if you are comfortable with it.   as option E is the biggest binary number we try with it first :   100010000/ 1011 =10000   1000100000/ 1011 =100000   so answer is option is 1011
Answer: D
If x = - |x| , then which one of the following statements could be true?
I. x <1
II. x < 0
III. x > 0	Choose the most appropriate option:
A) None
B) I only
C) II only
D) I and II
E) II and III

Solution: Answer copied from NOVA guide   Statement I could be false because 1/2 = −(1/2) = −1/2 .   Statement II could be true because the right side of the equation is always negative [ ± x = –(a positive number) = a negative number].   Now, if one side of an equation is always negative, then the other side must always be negative,   otherwise the opposite sides of the equation would not be equal.   Since Statement III is the opposite of Statement II, it must be false. But let’s show this explicitly:   Suppose x were positive. Then x = x, and the equation x = ± x becomes x = –x. Dividing both sides of this equation by x yields 1 = –1. This is contradiction.   Hence, x cannot be positive. The answer is II only).
Answer: C