MU-NLPC/Calc-aqua_rat huggingface dataset:
If a , b, and c are integers and a∗b2/ca∗b2/cis a positive even integer, which of the following must
be true?
I. ab is even
II. ab > 0
III. c is even Choose the correct option:
A) l
B) ll only
C) l and ll
D) l and lll
E) l,ll,and lll

Solution: Given: a∗b2c=even>0a∗b2c=even>0 --> ab2=c∗even=evenab2=c∗even=even --> either aa is even or bb or both.   I. abab is even --> according to the above this must be true;   II. ab>0ab>0 --> not necessarily true, bb could be positive as well as negative (for example a=1a=1, c=1c=1 and b=−2b=−2);   III. cc is even --> not necessarily true, see above example.   lnswer: l (I only).
Answer: A
Of the 40 electronics components that a factory must manufacture, 30 percent would be most e¢ ciently manufactured by Machine A and the remaining 70 percent would be most efficiently manufactured by Machine B, though either machine could manufacture any of the 40 components. If 70 percent of the components were manufactured by Machine A and the remainder were manufactured by Machine B, what is thehighest possible number of components that were manufactured by the machine that would manufacture them the most efficiently?	Choose the correct choice: A) 30 B) 32 C) 60 D) 35 E) 36

Solution: A can manufacture 30 % of 40 efficiently, which is 12.   B can manufacure 70 % of 40 efficiently, which is 28.   Now 80 % of 40 are produced by A. This means 32 parts. Since A can produce 12 efficiently, all 32 are produced efficiently. Rest (8) is produced by B, which can produce max. 28 parts efficiently.   Hence parts that were produced efficiently = 32+28= 
32+28 = 60
60. (60).
Answer: C
What is the 36th digit to the right of the decimal point in the decimal expansion of 1/37?	Choose one:	A) 0	B) 2	C) 4	D) 7	E) 9

Solution: 1/37=0.027027... So, we have a repeating cycle of 027. Every third digit (3rd, 6th, 9th, ...) to the right of the decimal point is 7, thus 36th digit to the right of the decimal point is also 7.   Answer: 7.
Answer: D
The simple interest and the true discount on a certain sum for a given time and at a given rate are Rs. 85 and Rs. 78 respectively. The sum is:
Choices: A) 1360 B) 1450 C) 1600 D) 947 E) None

Solution: Sol.   Sum = S.I. * T.D. / (S.I)-(T.D.) = 85 * 78 / (85 - 78) = Rs.947.   Answer 947
Answer: D
A person distributed 20% of his income to his 3 children each. He deposited 30% of his income to his wife's account. He donated 5% of remaining amount to an orphan house.Finally he has $60000. Find his total income?	Choose the correct choice from the following options.
A) A)1200000
B) B)562000
C) C)800000
D) D)500000
E) E)652000

Solution: 3 children got = 3*20= 
3*20 = 60
60%   wife got= 30%   orphan house = 5%   total = 60+30+5= 
60+30+5 = 95
95%   remaining= 100-95= 
100-95 = 5
5%   5%= 60000   100% = 60000*100/5= 
60000*100/5 = 1_200_000
$1200000   1200000nswer is 1200000
Answer: A
In particular old age home, 60% are suffering from diabetic and at the same time 80% are suffering from blood pressure.If suppose 75% are suffering from diabetic,what percentage of old people suffering from blood pressure at the same old age home?  Options:
A) 23
B) 16
C) 26
D) 36
E) 20

Solution: given : % of people suffering from diabetics = 60%; % of people suffering from blood pressure = 80% (ie ) 0.8   total =75*100; Those who are suffering from diabetics= 75 - 60= 
75 - 60 = 15
15; Those who are suffering from blood pressure=   100 - 80= 
100 - 80 = 20
20; so those who are suffering from blood pressure=0.8*20= 
0.8*20 = 16
16; Total number of people suffering from blood pressure= 20+16= 
20+16 = 36
36. Answer: 36
Answer: D
If P/6 = Q/7 = R/8 then find P:R:Q Choices:
A) 1:2:3 B) 6:8:7 C) 2:4:9 D) 4:9:7 E) 3:5:7

Solution: Let P/6 = Q/7 = R/8 = K   P = 6K   Q = 7K   R = 8K   P:R:Q = 6:8:7   Answer is 6:8:7
Answer: B
ABCD is a square. E and F are the midpoints of sides CD and BC, respectively. What is the ratio of the shaded region area to the unshaded region?
Answers.
A) 1:1
B) 2:1
C) 3:1
D) 5:3
E) 8:3

Solution: Let's join the midpoints as shown in the figure then we'll get four small squares...   AODX be 1 and similarly all squares..and we can split the unit 1 into half and half as per the mid points...   Then shaded will be 1+1/2+1/2+1/2 = 5/2   and unshaded is 1/2+1/2+1/2 = 3/2..   shaded : unshaded is 5:3..   ANS option 5:3.
Answer: D
The average of five consecutive odd numbers is 51. What is the difference between the highest and lowest number?  Choose the correct choice from the following answers:  A) 3  B) 7  C) 8  D) 11  E) None of these

Solution: Explanation:   Let the numbers be x, x + 2, x + 4, x + 6 and x + 8.   Then, [x + (x + 2) + (x + 4) + (x + 6) + (x + 8)] / 5 = 51   5x + 20 = 255   x = 47   So, required difference = (47 + 8) – 47 = 8   ANSWER 8
Answer: C
In a city, the streets are aligned in a grid, where the east-west roads are called 1st Rd, 2nd Rd, 3rd Rd, etc, increasing in number as one moves northward. The north-south roads are called 1st Ave, 2nd Ave, 3rd Ave, etc, increasing in number as one moves eastward. If Bill needs to walk from the corner of 2nd Rd and 3rd Ave to the corner of 10th Rd and 5th Ave in the shortest possible time, how many different routes could he take?  Choose the correct option:  A) 45  B) 60  C) 75  D) 90  E) 105

Solution: Bills needs to walk 8 blocks north and 2 blocks east.   One possible solution is NNNNNNNNEE.   There are 10C8 = 45 possible solutions.   The answer is 45.
Answer: A
If two of the four expressions x+y, x+5y, x-y, and 5x-y are chosen at random, what is the probability W that their product will be of the form of x^2-(by)^2, where b is an integer?  Choices
A) 1/2 B) 1/3 C) 1/4 D) 1/5 E) W=1/6

Solution: Only (X+y)(x-y) pair will give the form x^2-(by)^2   The probability of selecting these two pairs are 1/4 * 1/3 = 1/12, assuming x+y is picked first, then x-y, but x-y can be picked first followed by x+y. So the probability W= 1/12 * 2 = 1/6   Ans W=1/6
Answer: E
How many numbers between 11 and 90 are divisible by 7? Answers:
A) 7
B) 9
C) 11
D) 8
E) 10

Solution: The required numbers are 14, 21, 28, 35, .... 77, 84.   This is an A.P. with a = 14 and d = (21 - 14)= 
(21 - 14) = 7
7.   Let it contain n terms.   Then, T^n= 84 => a + (n - 1) d = 84   => 14 + (n - 1) x 7 = 84 or n = 11.   Answer is 11.
Answer: C
HCF of 2472,1284 and a third number 'n'is 12.If their LCM is 8*9*5*103*107.then the number 'n'is.	Choose one:	A) 160	B) 170	C) 180	D) 190	E) 200

Solution: 2472=8*3*103   1284=4*3*107   so n = 4*3*x   dividing by hcf   2472/12=2*103   1284/12= 
1284/12 = 107
107   lcm/12=2*3*5*103*107   x=lcm/first*second   x=2*3*5*103*107/2*103*107=3*5= 
3*5 = 15
15   number is= 15*12= 
15*12 = 180
180   ANSWER:180
Answer: C
Arya starts cycling along the boundaries of the squares. She starts from a point A and after 90 minutes she reached to point C diagonally opposite to A. If she is travelling with 20km/hr, then find the area of square field.
Pick one.
A) 125  B) 225  C) 325  D) 425  E) None of these

Solution: Explanation :   D = 20*3/2= 
20*3/2 = 30
30 km. So side of square is 15km, so area – 225km^2   Answer – 225
Answer: B
The length of a rectangle is increased by 25% and its breadth is decreased by 20%. What is the effect on its area?
Pick.
A) 10000
B) 10008
C) 10005
D) 10004
E) 10001

Solution: 100 * 100= 
100 * 100 = 10_000
10000   125 * 80= 
125 * 80 = 10_000
10000   No change   10000nswer:10000
Answer: A
The H.C.F. and L.C.M. of two numbers are 15 and 5040 respectively If one of the numbers is 144, find the other number
Choose the most appropriate option.
A) 400
B) 256
C) 120
D) 420
E) 525

Solution: Explanation:   Solve this question by using below formula.   Product of 2 numbers = product of their HCF and LCM   144 * x = 15 * 5040   x = (15*5040)/144= 
(15*5040)/144 = 525
525   Option 525
Answer: E
A certain junior class has 1000 students and a certain senior class has 800 students. Among these students, there are 60 siblings pairs each consisting of 1 junior and 1 senior. If 1 student is to be selected at random from each class, what is the probability that the 2 students selected will be a sibling pair? Choose the correct choice:
A) 3/40000  B) 1/3600  C) 9/2000  D) 1/60  E) 1/15

Solution: There are 60 siblings in junior class and 60 their pair siblings in the senior class. We want to determine probability of choosing one sibling from junior class and its pair from senior.   What is the probability of choosing ANY sibling from junior class? 60/1000 (as there are 60 of them).   What is the probability of choosing PAIR OF CHOSEN SIBLING in senior class? As in senior class there is only one pair of chosen sibling it would be 1/800 (as there is only one sibling pair of chosen one).   So the probability of that the 2 students selected will be a sibling pair is: 60/1000∗1/800=3/40000   3/40000nswer: 3/40000.
Answer: A
If r ≠ 1 and if sr/(s - r) = 1, what is the value of s in terms of r? Choose the correct choice from the following choices:	A) (r+1)/r	B) (r - 1)/r	C) r/(r + 1)	D) r/(1-r)	E) (1 - r)/r

Solution: This question comes with a particular 'quirk' (one that you probably won't see on Test Day). The GMAT won't test you on the concept of undefined numbers (re: numbers divided by 0), so any time that this concept is a possibility, the question writers have to add a restriction that removes the option that a 0 could occur in the denominator.   Here, we're told that B CANNOT = 1, which is a bit strange because that restriction doesn't seem to impact the original equation much. In fact, it impacts JUST ONE of the answer choices - so you have to ask WHY that restriction is even there. It's actually because that one answer is the correct one.   Final Answer:   r/(1-r)
Answer: D
Worker W produces n units in 5 hours. Workers V and W work independently but at the same time, produce n units in 2 hours. How long would it take V alone to produce n units?	Options:
A) 8/3 hours	B) 10/3 hours	C) 11/3 hours	D) 12/3 hours	E) 18/3 hours

Solution: w's 1 hours production=n/5   (w+v)'s 1 hours production=n/2   v's 1 hours production   n/5+V=n/2   v's 1 hours production=n/2-n/5   =3n/10   n/(10/3)   ans=10/3 hours   ANSWER:10/3 hours
Answer: B
A computer manufacturer produces a certain electronic component at a cost of $50 per component. Shipping costs for delivering the components are $8 per unit. Further, the manufacturer has costs of $16,000 a month related to the electronic component regardless of how many it produces. If the manufacturer produces and sells 200 components a month, what is the lowest price it can sell them for such that the costs don't exceed the revenues?  Choose the most appropriate option: A) 135 B) 129 C) 131 D) 138 E) 141

Solution: $16000 is a fixed cost   Each component is $58 ($50 to produce, $8 to ship)   Manufacturer will be producing and selling 200 components   so therefore the equation to find price would be   200*p = 16000+(200*50)+(200*8)   p = (16000+10000+1600)/200= 
(16000+10000+1600)/200 = 138
138   Answer: 138
Answer: D
A fruit drink is made of orange, watermelon, and grape juice, where 35 percent of the drink is orange juice and 35 percent is watermelon juice. If the drink is made with 45 ounces of grape juice, how many ounces is the drink total?  Pick:
A) 120  B) 140  C) 150  D) 180  E) 200

Solution: Let the total number of ounces in the drink be x.   % of orange = 35%   % of watermelon = 35%   % of grape = 100% - 70% = 30%   0.30x 100  - 70= 
100  - 70 = 30
30%   0.30x= 45   x = 150   Therefore there are a total of 150 ounces in the drink.   The answer is 150.
Answer: C
Find 44 % of 240 Options
A) 96
B) 105.6
C) 86
D) 74
E) 110

Solution: We know that R % of m is equal to R/100 × m.   So, we have 44 % of 240   44/100 × 240   =105.6   Answer:105.6
Answer: B
Suresh and Ramesh started a business investing Rs. 24,000 and Rs. 40,000 respectively. Out of total profit of Rs. 15,000, what is Ramesh's share?
Select.
A) Rs. 9375 B) Rs. 9379 C) Rs. 9376 D) Rs. 9372 E) Rs. 9371

Solution: Explanation:   Ratio of Suresh and Ramesh's share = 24,000 : 40,000 = 3 : 5   Ramesh's share = Rs. (15000 x 5 / 8) = Rs. 9375   Rs. 9375NSWER: Rs. 9375
Answer: A
?% of 360 = 108.0
Choose the most appropriate option:
A) 277
B) 36
C) 64
D) 72
E) 30

Solution: ?% of 360 = 108.0   or, ? = 108.0×100/360=30   Answer 30
Answer: E
what is the least number by which 46,800 must be divided to get a number perfect square Choices:
A) 21  B) 52  C) 24  D) 40  E) 45

Solution: 46,800/52=900   900=(30)^2   Ans-52   ANSWER:52
Answer: B
What is 15 percent of 44?
Choose the correct answer
A) 6.6	B) 7.8	C) 8.9	D) 9.4	E) 10.2

Solution: (15/100) * 44= 
(15/100) * 44 = 33/5 = around 6.6
6.6   The answer is 6.6.
Answer: A
Rohit and Rohan can complete a work in 12 days and 6 days respectively. How much time will they take when working together?  Choices: A) 4 B) 3 C) 5 D) 2 E) 1

Solution: Time taken by Rohit and Rohan = xy/ x+y   =12*6/(12+6)= 
12*6/(12+6) = 4
4 days   4NSWER:4
Answer: A
If x = y^2, and x and y are positive integers, which of the following numbers could be the value of x?	Choose the most appropriate option:
A) 7396 B) 6847 C) 4262 D) 8798 E) 9563

Solution: The square of an integer can have a units digit of 0, 1, 4, 5, 6, or 9.   The answer is 7396.
Answer: A
Two cars are traveling towards each other. If car A is traveling at a speed of 70 mph and car B is traveling 10% slower, how much time will it take the cars to meet if the initial distance between the two is 705 miles?  Select.
A) Five hours and 30 minutes.  B) Seven hours and 30 minutes.  C) Eight hours and 20 minutes.  D) Nine hours and 15 minutes.  E) Ten hours and 20 minutes.

Solution: Speed of second train= 70*90/100= 
70*90/100 = 63
63   Time= toral distance /total speed   705/ (70+63)= 705/133= 5.___   Five hours and 30 minutes. is the answer
Answer: A
The sum of money will be double itself in 10 years and simple interest find rate of interest? Choose the correct choice from the following answers:	A) A)5%	B) B)10%	C) C)15%	D) D)20%	E) E)25%

Solution: T = 10 years   P= principle amount = x   A = total amount = 2x   SI = simple interest = A-P = 2x-x = x   R = 100SI/PT = 100x/10x =10%   Answer is 10%100 /10= 
100 /10 = 10
10%   Answer is 10%
Answer: B
A person saves 10% of his income. If his income is increased by 20% and he saves 15% of the new income, by what percent his savings will increase?  Choose the correct choice from the following answers:
A) 80%	B) 90%	C) 70%	D) 60%	E) 50%

Solution: Ans. Let previous income = Rs. 100   Previous savings = 10% of Rs. 100 = Rs. 10   Increased income = Rs. 100 + 20% of Rs. 100 = Rs. 120   Increased savings = 15% of Rs. 120 = Rs. 18   Therefore, Increase in savings = Rs. 18 - Rs. 10 = Rs. 8   Therefore, Percent increase in savings = 8/10 x 100 = 80%   80%nswer 80%
Answer: A
If 9 is a factor of 2a, then which of the following may not be an integer? Select
A) 6a/54 + 2a/3	B) (4a-18)/9	C) (2a+27)/9	D) (81-4a^2)/81	E) (2a-3)/3

Solution: I got the right answer A after plugging in 18 for a.   While going through Kaplan's explanation, I couldn't understand the part about choice A.   "6a/54 + 2a/3 = 3/6 * 2a/9 + 2a/3.   Since 9 is a factor of 2a, 3 is also factor of 2a. So 2a/3 is an integer, as is 2a/9. But 3/6 = 1/2, so if 2a/9 is not even, the expression 3/6 * 2a/9 will not be even."   But, isn't 2a/9 always even? 9 is a factor of 2a, which means that we could have 2(9), 2(2)(9), 2(3)(9), and so forth. The 9 in the denominator cancels out the 9 in the numerator. So we are left with 2 * something. So 2a/9 is even.6a/54 + 2a/3
Answer: A
If a and b are both even numbers, which of the following is an even number?  Pick one
A) a + b+3 B) a + b + 1 C) ab D) ab + 1 E) NONE

Solution: Sol.   Product of two odd numbers is always odd and adding 1 gives even.   Answer ab
Answer: C
The present ages of three persons in proportions 4 : 7 : 8. Eight years ago, the sum of their ages was 71. Find their present ages (in years).
Choose the correct choice from the following.
A) 15, 20, 28
B) 24, 28, 36
C) 20, 35, 40
D) 33,42,43
E) of these

Solution: Let their present ages be 4x, 7x and 8x years respectively.   Then, (4x - 8) + (7x - 8) + (8x - 8) = 71   19x = 95   x = 5.   Their present ages are 4x = 20 years, 7x = 35 years and 8x = 40 years respectively.   ANSWER:20, 35, 40
Answer: C
If x+y=290, x-y=200, for integers of x and y, y=?  Pick.
A) 200	B) 245	C) 50	D) 115	E) 150

Solution: x+y = 290   x-y = 200   2x = 90   x = 45   y = 245   Answer is 245
Answer: B
Worker A takes 8 hours to do a job. Worker B takes 10 hours to do the same Job. How long should it take both A and B, working together but independently, to do the same job?  Choose the correct choice from the following:
A) 70 hours
B) 40/9 hours
C) 9 hours
D) 16 hours
E) 20 hours

Solution: A’s 1 hour's work = 1/8   B's 1 hour's work = 1/10   (A + B)'s 1 hour's work = (1/8) +(1/10)=9/40   Both A and B will finish the work in 40/9 days.   ANSWER 40/9 hours 40/9 hours
Answer: B
What was the day of the week on 4th April, 2006? Pick one.
A) Saturday  B) Thursday  C) Monday  D) Friday  E) Tuesday

Solution: 28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006)   Odd days in 1600 years = 0   Odd days in 400 years = 0   5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2) 6 odd days   Jan. Feb. March April   (31 + 28 + 31 + 4)= 
(31 + 28 + 31 + 4) = 94
94 days   94 days= (13 weeks + 3 day) 1 odd day.   Total number of odd days = (0 + 0 + 6 + 3)= 
(0 + 0 + 6 + 3) = 9
9 0 odd day.   Given day is Tuesday.   Answer: Option Tuesday
Answer: E
Each of the integers from 1 to 17 is written on the a seperate index card and placed in a box. If the cards are drawn from the box at random without replecement, how many cards must be drawn to ENSURE that the product of all the integers drawn is even?	Choose the correct choice from the following choices:
A) 19	B) 12	C) 11	D) 10	E) 3

Solution: Out of the 17 integers: 9 are odd and 8 are even.   If we need to make sure that the product of all the integers withdrawn is even then we need to make sure that we have at least one even number. In the worst case:   1. we will end up picking odd numbers one by one, so we will pick all 9 odd numbers first   2. 10th number will be the first even number   So we need to withdraw at least 10 numbers to make sure that we get one even number and the product of all the integers picked is even.   So, answer will be 10. (10)
Answer: D
In the last set of tests, a student scored marks in each of 5 subjects in the ratio 4:5:6:7:8 and got an overall average score of 54%. If a mark of 50% in a subject is a pass and the maximum marks of each subject is the same, how many subjects did the student pass?	Options:	A) 1	B) 2	C) 3	D) 4	E) 5

Solution: The tests scores are 4k, 5k, 6k, 7k, and 8k.   The sum of the test scores is 30k.   The average test score is 30k/5 = 54. Then k = 9.   The test scores are 36, 45, 54, 63, and 72.   The student passed 3 subjects.   The answer is 3.
Answer: C
The average age of seven persons sitting in a row facing east is 28 years. If the average age of the first three persons is 21 years and the average age of the last three persons is 34 years, then find the age of the person sitting in the middle of the row?	Choose the correct choice from the following answers:
A) 15 years
B) 31 years
C) 27 years
D) 82 years
E) 12 years

Solution: Total age seven persons = (28 * 7)years   Total age of the first three persons and the last three persons are (21 * 3) years and (34 * 3) years respectively.   Age of the person sitting in the middle of the row   = 28 * 7 - 21 * 3 - 34 * 3 = 196 - 63 - 102= 
196 - 63 - 102 = 31
31 years.   Answer: 31 years
Answer: B
Find the simple interest on the Rs. 2000 at 25/4% per annum for the period from 4th Feb 2005 to 18th
April 2005	Pick one.
A) Rs 25	B) Rs 30	C) Rs 35	D) Rs 40	E) None of these

Solution: Explanation:   One thing which is tricky in this question is to calculate the number of days.   Always remember that the day on which money is deposited is not counted while the day on which   money is withdrawn is counted.   So lets calculate the number of days now,   Time = (24+31+18)= 
(24+31+18) = 73
73/365 years= 1/5 years   P = 2000   R = 25/4%   S.I. = =2000×254×5×100=25   Rs 25nswer: Rs 25
Answer: A
If 'A' means '-', 'B' means '/', 'C' means '+', and 'D' means 'X' then
Find the value of the following:
21 B 7 C 9 D 10 A 13 Select the correct option
A) 80 B) 40 C) 60 D) 70 E) 50

Solution: 80   80   Explanation:   21 B 7 C 9 D 10 80 13   = 21 / 7 + 9 X 10 - 13   = 3 + 90 - 13= 
3 + 90 - 13 = 80
80
Answer: A
At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years	Choose the correct option.
A) 3%  B) 4%  C) 5%  D) 6%  E) 7%

Solution: Explanation:   Let Rate will be R%   1200(1+R/100)2=134832 /100   (1+R/100)2=134832/120000   (1+R/100)2=11236/10000   (1+R/100)=106/100   =>R=6%   Option 6%
Answer: D
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5/4 w widgets in 3 days, how many days would it take machine X alone to produce 5w widgets? Choose the correct choice from the following: A) 4 B) 30 C) 8 D) 10 E) 12

Solution: Let Y produce w widgets in y Days   Hence, in 1 day Y will produce w/y widgets.   Also, X will produce w widgets in y+2 days (given, X takes two more days)   Hence, in 1 day X will produce w/y+2 widgets.   Hence together X and Y in 1 day will produce {w/y + w/y+2} widgets.   Together X and Y in 3 days will produce = 3 * [{w/y + w/y+2}] widgets.   It is given that in 3 days together they produce (5/4)w widgets.   Equating, 3 * [{w/y + w/y+2}] = (5/4)w   Take out w common and move 3 to denominator of RHS   w{1/y + 1/(y+2)} = (5/12)w   Canceling w from both sides   {1/y + 1/(y+2)} = 5/12   2y+2/y(y+2) = 5/12   24y+24=5y^2 + 10y   5y^2-14y-24=0   5y^2-20y+6y-24=0   5y(y-4)+6(y-4)=0   (5y+6)+(y-4)=0   y=-6/5 or y=4   Discarding y=-6/5 as no of days cannot be negative   y=4   hence it takes Y, 4 days to produce w widgets.   Therefore, it will take X  (4+2)= 
 (4+2) = 6
6 days to produce w widgets.   Hence it will take X 5*6= 
5*6 = 30
30 days to produce 2w widgets.   Answer : 30
Answer: B
In the quadratic equation ax2+ 1x - 6 = 0, if the sum of two roots is -1, what is the product of the two roots?	Choices:
A) 4
B) 5
C) 6
D) -6
E) 8

Solution: Explanation:   The sum of the roots of the quadratic equation ax2 + bx + c = 0 are (-b/a) and the product of the roots are (c/a).   Thus, in the equation ax2+ x - 6 = 0, where a = a, b = 1 and c = -6.   we get, sum of the roots = -1 / a = -1   a = 1   Product of the roots = -6 / 1 = -6   ANSWER: -6
Answer: D
If r>0.9, which of the following options can be the value of r?
Pick one:	A) 0.9^2	B) √(0.9)	C) 0.09	D) 0.9	E) 0.9^3

Solution: Lets take a few examples   When .2 is squared it gives .04   when .5 is squared it gives .25   When .9 is squared it gives .81   So raising a fraction to its power of 2 or 3 only reduces the value of the fraction   This is because   .2 = 2/10   .2^2= 4/100   .2^3=8/1000   Since the denominator gets an extra 0 each time u increase the power , it is going to drag the value down   √(0.9)ut if you just think about the opposite case :   Then we can see , if squaring a fraction diminish its value, then taking the square root of it will again increase its value   that is √(0.04) will give you back 0.2   so √(0.04) > 0.04   so obviously diminishing the power will appreciate the value   So r √(0.9) > 0.9   Hence Correct Answer is (√(0.9))
Answer: B