mathewhe/aqua_rat huggingface dataset:
Question: Each of the 20 boxes in a certain shipment weighs either 10 pounds or 20 pounds, and the average (arithmetic mean) weight of the boxes in the shipment is 18 pounds. If the average weight of the boxes in the shipment is to be reduced to 12 pounds by removing some of the 20-pound boxes, how many 20-pound boxes must be removed?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 15
Rationale: If the average of 10-pound and 20-pound boxes is 18, the ratio of 10-pound boxes:20-pound boxes is 1:4.
So out of 20 boxes, 4 are 10-pound boxes and 16 are 20-pound boxes.
If the average of 10 and 20-pound boxes is to be 12, the ratio of 10-pound boxes:20-pound boxes should be 4:1.
The number of 10 pound boxes remains the same so we still have 4 of them.
To get a ratio of 4:1, the number of 20-pound boxes must be 1.
We need to remove 15 of the 20-pound boxes.
The answer is E.
Answer: (E) 15
Question: fill in ht blank:(a) the point with coordinates (0,0) is called ____ of a rectangular coordinate system,(b) to find the y-intercept of a line, we let ___ equal ___ and solve for ___; to find x- intercept , we let ____ equal 0 and solve for___
(A) (a) the point with coordinates (0,0) is called axis of a rectangular coordinate system, (b) to find the y-intercept of a line, we let x equal 0 and solve for y ; to find x- intercept , we let y equal 0 and solve for y
(B) (a) the point with coordinates (0,0) is called end of a rectangular coordinate system, (b) to find the y-intercept of a line, we let x equal 0 and solve for x ; to find x- intercept , we let y equal 0 and solve for y
(C) (a) the point with coordinates (0,0) is called origin of a rectangular coordinate system, (b) to find the y-intercept of a line, we let y equal z and solve for x ; to find x- intercept , we let x equal 0 and solve for z
(D) (a) the point with coordinates (0,0) is called origin of a rectangular coordinate system, (b) to find the y-intercept of a line, we let x equal 0 and solve for y ; to find x- intercept , we let y equal 0 and solve for x
(E) none of the above
Rationale: (a) the point with coordinates (0,0) is called origin of a rectangular coordinate system, (b) to find the y-intercept of a line, we let x equal 0 and solve for y ; to find x- intercept , we let y equal 0 and solve for xcorrect answer (D)
Answer: (D) (a) the point with coordinates (0,0) is called origin of a rectangular coordinate system, (b) to find the y-intercept of a line, we let x equal 0 and solve for y ; to find x- intercept , we let y equal 0 and solve for x
Question: Jack and Paul are running 1000m. Jack runs 1.5Km/hr faster than Paul but also finishes the race 1.5 sec earlier. What is the speed of Jack??
(A) 20.61
(B) 21.61
(C) 22.61
(D) 23.61
(E) 24.61
Rationale: distance(given) 1000m
paul speed(let)= x m/s
time(paul take)= 1000/x sec
jack speed= x + 1.5*(5/18) m/s
tym (jack)= 1000/(x+7.5/18) s
tym(paul) - tym (jack) = 1.5
solving for x we get 20.20
speed of jack is 20.20+ 7.5/18=20.61
ANSWER:A
Answer: (A) 20.61
Question: A shop keeper marked 20% above the cost price and offered 15% discount then find it's net profit?
(A) 2%
(B) 5%
(C) 8%
(D) 10%
(E) 15%
Rationale: Net profit = 20-15 + (20*(-15)/100) = 2%
Answer is A
Answer: (A) 2%
Question: If q is not equal to zero, andq+1/q = 3, then what is the value of q^4 + (1/q)^4?
(A) 184
(B) 194
(C) 254
(D) 324
(E) 354
Rationale: q + 1/q=4
we square both sides so we have q^2 + 1/q^2 +2 = 16
or q^2 + 1/q^2= 14
squaring again we have q^4 + 1/q^4 + 2 = 196
or q^4 + 1/q^4 = 194
answer =194 (B)
Answer: (B) 194
Question: Pipes A and B can fill a tank in 9 and 6 hours respectively. Pipe C can empty it in 18 hours. If all the three pipes are opened together, then the tank will be filled in?
(A) 3 9/97 hrs
(B) 4 2/4 hrs
(C) 3 9/17 hrs
(D) 3 9/27 hrs
(E) 3 9/19 hrs
Rationale: Net part filled in 1 hour = 1/9 + 1/6 - 1/18
= 4/18
The tank will be full in 18/4 hrs, i.e., 4 2/4 hrs.
Answer:B
Answer: (B) 4 2/4 hrs
Question: A batsman had a certain average of runs for 16 innings. In the 17th innings, he made a score of 87 runs thereby increasing his average by 3. What is his average after 17 innings?
(A) 39
(B) 88
(C) 266
(D) 278
(E) 221
Rationale: Explanation:
Assume his initial average = xx
His total runs after 16 innings = 16xx
After scoring 87 runs his average got increased by 3 to xx + 3
So his total runs after 17 innings = 17 × (xx+3)
But it was given that the difference in the total scores after 16 innings and 17 innings = 87
Therefore 17×(x+3)−16x=87⇒x=3617×(x+3)−16x=87⇒x=36
His new average = 36 + 3 = 39
Answer:A
Answer: (A) 39
Question: If x ≠ 0 and (y-x)/x=(y+x)/y, which of the following expressions is equal to its opposite?
(A) (x+y) (x-y)
(B) x^2 – y^2
(C) y^2
(D) (y + x)^2
(E) (x – y)^2
Rationale: (y+x)/x=(y+x)/y
=> y^2 + xy = xy + x^2
=> x^2 - y ^2 = 0
=> (x+y) (x-y) = 0
Answer A
Answer: (A) (x+y) (x-y)
Question: Approximately how many cubic feet of water are needed to fill a circular swimming pool that is 70 feet across and 9 feet deep?
(A) 4000
(B) 10000
(C) 20000
(D) 35000
(E) 45000
Rationale: Answer Should be D.
V= \pir^2h= \pi*35^2*9= approximately 35000
Answer: (D) 35000
Question: Three 6 faced dice are thrown together. The probability that all the three show the same number on them is
(A) 1/216
(B) 1/36
(C) 5/9
(D) 5/12
(E) 7/12
Rationale: It all 3 numbers have to be same basically we want triplets. 111, 222, 333, 444, 555 and 666. Those are six in number. Further the three dice can fall in 6 * 6 * 6 = 216 ways.
Hence the probability is 6/216 = 1/36
ANSWER:B
Answer: (B) 1/36
Question: Solve the given two equations and find the inequality of X and Y. (a) x^2-1=0 and (b) y^2+4y+3=0
(A) If x>y
(B) If x>=y
(C) If x < y
(D) If x<=y
(E) x=y or relationship cannot be established
Rationale: Solution:-
X=+1,-1 and Y= -1,-3
Comparing we get X>=Y
B
Answer: (B) If x>=y
Question: After decreasing 50% in the price of an article costs Rs.1200. Find the actual cost of an article?
(A) 1200
(B) 50
(C) 2400
(D) 100
(E) 600
Rationale: CP* (50/100) = 1200
CP= 24 * 100 => CP
= 2400
Answer:C
Answer: (C) 2400
Question: In f is a positive integer smaller than 200 and (14f)/60 is an integer, then f has how many different positive prime factors?
(A) 2
(B) 3
(C) 5
(D) 6
(E) 8
Rationale: 7f/30 tells us that f is a factor of 30, which has 2, 3, and 5 as its prime factors. Each of them is distinct. Moreover, all multiples of 30 less than 200 can be derived by multiplying these prime factors alone. Thus, number of different prime factors is 3.
Answer: B
Answer: (B) 3
Question: The perimeter of a semi circle is 198 cm then the radius is?
(A) 22
(B) 28
(C) 98
(D) 38.5
(E) 13
Rationale: 36/7 r = 198
=> r
= 38.5
Answer: D
Answer: (D) 38.5
Question: . A boy has six trousers and 8 shirts. In how many different ways can he select a trouser and a shirt?
(A) 9
(B) 18
(C) 27
(D) 36
(E) 48
Rationale: Explanation:
The boy can select one trouser in six ways.
The boy can select one shirt in 8 ways.
The number of ways in which he can select one trouser and one shirt is 6 * 8 = 48 ways.
Answer: Option E
Answer: (E) 48
Question: (0.8)(power 3) - (0.5)(power 3)/ (0.8)(power 2) + 0.40 + (0.5)(power 2) is:
(A) 0.6
(B) 0.5
(C) 0.35
(D) 0.3
(E) None of them
Rationale: Given expression
= (0.8) (power 3) - (0.5)(power 3) / (0.8)(power 2) + (0.8 x 0.5) + (0.5)(power 2)
= a (power 3) - b(power 3) / a(power 2) + ab + b(power 2)
= (a - b)
= (0.8- 0.5)
= 0.30
Answer is D.
Answer: (D) 0.3
Question: If the value of a piece of plaza decreases by 10% while the new tax rate on the plaza represent 110% of the original tax rate, what is the effect on the taxes?
(A) a - taxes increase by 10%
(B) b- taxes increase by 1%
(C) c- there is no change in taxes
(D) d- taxes decrease by 1%
(E) c- taxes decrease by 10%
Rationale: Cansider the value of plaza = 100, Original tax rate 10%, hence tax value = 100*10% = 10.
Decreased value of plaza = 90, New tax rate = 110% of 10 = 11%, hence tax value = 90*11% = 9.9
10-9.9 = decrease by 1%.
Answer D.
Answer: (D) d- taxes decrease by 1%
Question: A dealer marks his goods 20% above cost price. He then allows some discount on it and makes a profit of 8%. The rate of discount is:
(A) 16 %
(B) 22 %
(C) 25 %
(D) 10 %
(E) 15 %
Rationale: Explanation:
C.P = Rs. 100
Marked Price = Rs. 120,
S.P = Rs. 108
Discount = (12/120 × 100)% = 10%
ANSWER:D
Answer: (D) 10 %
Question: A solution of cough syrup has 15% alcohol. Another solution has 5% alcohol. How many litres of the second solution must be added to 20 litres of the first solution to make a solution of 10% alcohol?
(A) 10
(B) 5
(C) 15
(D) 20
(E) 25
Rationale: Hence both the types should be added in the ratio of 1 : 1 to obtain the required strength.
Hence 20 litres of first type should be added to the 20 litres of the second type to get the desired solution.
ANSWER:D
Answer: (D) 20
Question: A librarian has a set of nine books, including four different books about Abraham Lincoln. The librarian wants to put the nine books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other five books. How many different arrangements of the nine books are possible?
(A) (9!)/(4!)
(B) (4!)(5!)
(C) (4!)(6!)
(D) (4!)(9!)
(E) (4!)(5!)(9!)
Rationale: You can club the four Lincoln books as one, which leaves us with five other books plus this one to think about. So, these six items can be arranged in 6! ways, and since the four books can be moved around among themselves with each combination giving us a new overall combination, we have an additional 4! ways. Therefore, the answer would be 6!*4!, which is C.
Answer: (C) (4!)(6!)
Question: Which of the following is a prime number ??
(A) 33
(B) 81
(C) 89
(D) 93
(E) 10
Rationale: C
89
Clearly, 89 is a prime number.
Answer: (C) 89
Question: The perimeter of an equilateral triangle is 60. If one of the sides of the equilateral triangle is the side of an isosceles triangle of perimeter 55, then how long is the base of isosceles triangle?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
Rationale: The base of the isosceles triangle is 55-20-20= 15 units
The answer is C.
Answer: (C) 15
Question: One fourth of one third of two fifth of a number is 15. What will be40% of that number
(A) 140
(B) 150
(C) 180
(D) 200
(E) 300
Rationale: Explanation:
(1/4) * (1/3) * (2/5) * x = 15 then x = 15 * 30 = 450
40% of 450 = 180
Option C
Answer: (C) 180
Question: Pipe A fills a tank of capacity 800 liters at the rate of 40 liters a minute. Another pipe B fills the same tank at the rate of 30 liters a minute. A pipe at the bottom of the tank drains the tank at the rate of 20 liters a minute. If pipe A is kept open for a minute and then closed and pipe B is open for a minute and then closed and then pipe C is open for a minute and then closed and the cycle is repeated, when will the tank be full?
(A) 48 minutes
(B) 14 minutes
(C) 39 minutes
(D) 40 minutes 20 seconds
(E) None of these
Rationale: In one cycle they fill 40+30-20 = 50 liters
800 = 50*n => n = 16
here n = number of cycles.
total time = 16*3 = 48 as in one cycle there are 3 minutes.
thus 48 minutes
ANSWER:A
Answer: (A) 48 minutes
Question: What was the day of the week on, 16th July, 1776?
(A) Saturday
(B) Monday
(C) Tuesday
(D) Friday
(E) Sunday
Rationale: 16th July, 1776 = (1775 years + Period from 1st Jan, 1776 to 16th July, 1776)
Counting of odd days :
1600 years have 0 odd day.
100 years have 5 odd days.
75 years = (18 leap years + 57 ordinary years) = [(18 x 2) + (57 x 1)] = 93 (13 weeks + 2 days) = 2 odd days.
1775 years have (0 + 5 + 2) odd days = 7 odd days = 0 odd day.
Jan Feb Mar Apr May Jun Jul
31 + 29 + 31 + 30 + 31 + 30 + 16 = 198 days= (28 weeks + 2 days)
Total number of odd days = (0 + 2) = 2.
Required day was 'Tuesday'.
Answer is C.
Answer: (C) Tuesday
Question: Each of the positive integers a and c is a four-digit integer. If each of the digits 0 through 9 appears in one of these three integers, what is the maximum possible value of the sum of a and c?
(A) 18695
(B) 18325
(C) 18365
(D) 18395
(E) 18485
Rationale: According to the stem we should use the digits 0 through 9 to construct 2 four-digit integers, so that their sum is as big as possible.
To maximize the sum, maximize the thousands digits of a and c, so make them 9 and 8.
Next, maximize hundreds digits. Make them 7 and 6 .
Next maximize 10's digit place by 5 and 4
Use the remaining digits (3,and 2) for units digits.
So, a would be 9753,and c would be 8642.
9753 + 8642 = 18395
Answer: D.
Answer: (D) 18395
Question: In a series of consecutive numbers, 26 is the eighth smallest number and 25 is the tenth largest number. What is the range of the series?
(A)  32
(B)  30
(C)  31
(D)  16
(E)  14
Rationale: Imp:All are consecutive odd numbers.
26 is the 8th smallest ,there are 7 numbers smaller then 26 ,smallest number :26-14=12
25 is the 10th largest ,so there are 9 numbers larger then 25, largest number:25+18=43
Range:43-12=31
Ans : C
Answer: (C)  31
Question: The radius of a wheel is 22.4 cm. What is the distance covered by the wheel in making 500 resolutions.
(A) 252 m
(B) 704 m
(C) 352 m
(D) 808 m
(E) None of these
Rationale: In one resolution, the distance covered by the wheel is its own circumference. Distance covered in 500 resolutions.
= 500 * 2 * 22/7 * 22.4 = 70400 cm = 704 m
ANSWER:B
Answer: (B) 704 m
Question: Best friends Sprite and Icey have are playing a game of marbles. They invite 16 of their friends to play with them. There are 36 marbles in the bag. If all 18 people are playing, how many marbles does each person get?
(A) 3
(B) 2
(C) 5
(D) 6
(E) 7
Rationale: Answer= B
36 / 18 = 2
Answer= B
Answer: (B) 2
Question: If a 3-digit integer is selected at random from the integers 100 through 199, inclusive, what is the probability that the second digit is two greater than the first digit and the third digit is less than the second digit?
(A)  1/100
(B)  3/100
(C)  2/103
(D)  2/150
(E)  1/50
Rationale: Hundreds digit has to be 1 . So tens digit needs to be 2 greater than 1
1+2 = 3
The tens digit is 3, and the units digit must be less than the tens digit so the units digit must be less than 3
0, 1, and 2 are all less than 3
140, 141, and 142 are all acceptable numbers giving us 3 possible numbers
Total numbers between 100 and 199 = (199-100)+1 = 100
Probability = 3/100
Answer B
Answer: (B)  3/100
Question: Find the odd man out
31, 37, 41, 45, 55, 58, 61, 67
(A) 31
(B) 41
(C) 67
(D) 58
(E) 61
Rationale: as 58 is only odd number.
ANSWER:D
Answer: (D) 58
Question: The C.P of 15 books is equal to the S.P of 18 books. Find his gain% or loss%?
(A) 16 2/3% loss
(B) 16 2/8% loss
(C) 16 3/3% loss
(D) 36 2/3% loss
(E) 56 2/3% loss
Rationale: 15 CP = 18 SP
18 --- 3 CP loss
100 --- ? => 16 2/3% loss
Answer:A
Answer: (A) 16 2/3% loss
Question: 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?
(A) 227
(B) 350
(C) 881
(D) 277
(E) 271
Rationale: 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: B
Answer: (B) 350
Question: A straight pipe 1 yard in length was marked off in halves and also in thirds. If the pipe was then cut into separate pieces at each of these markings, which of the following gives all the different lengths of the pieces, in fractions of a yard?
(A)  1/6 and 1/4 only
(B)  1/6 and 1/3 only
(C)  1/6, 1/4, and 1/3
(D)  1/12, 1/6 and 1/4
(E)  1/12, 1/6, and 1/3
Rationale: Generally fast way to solve such problem is writing the different marks in ascending/descending order with same denominator:
Here 2th : 0/2, 1/2, 2/2
and 3rd : 0/3, 1/3, 2/3, 3/3
Now with understood common denominator 6 write the numbers : for 2th : 0,3,6 and for 3rd : 0,2,4,6
Now comine : 0,2,3,4,6
Now find the cut with denominator 6 (Substracrt adjacent terms : 1/3, 1/6, 1/6, 1/3 i.e. 1/3 and 1/6 after removing duplicates.
answer : B
Answer: (B)  1/6 and 1/3 only
Question: In a certain game, a large container is filled with red, yellow, green, and blue beads worth, respectively, 7, 5, 3, and 2 points each. A number of beads are then removed from the container. If the product of the point values of the removed beads is 147,000, how many red beads were removed?
(A) 5
(B) 4
(C) 3
(D) 2
(E) 0
Rationale: = 147,000
= 147 x 1000
= (3) x (7 x 7) x (2x2x2) x (5x5x5)
= 2^3 x 3^1 x 5^3 x 7^2
red = 7^2 = 49
ANSWER:D
Answer: (D) 2
Question: Find the sum lend at C.I. at 5 p.c per annum will amount to Rs.441 in 2 years?
(A) 187
(B) 400
(C) 168
(D) 286
(E) 268
Rationale: 441 = P(21/20)2
P = 400
Answer: B
Answer: (B) 400
Question: Two water pumps, working simultaneously at their respective constant rates, took exactly five hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at its constant rate?
(A) 5
(B) 16/3
(C) 11/2
(D) 6
(E) 25/3
Rationale: Let x be the rate of the slower pump. Then 1.5x is the rate of the faster pump.
Both pumps together can fill 1/5 of the pool each hour.
2.5x = 1/5
x = 1/12.5
1.5x = 3/25 so the faster pump could fill the pool in 25/3 hours.
The answer is E.
Answer: (E) 25/3
Question: The speed of a boat in upstream is 120 kmph and the speed of the boat downstream is 154 kmph. Find the speed of the boat in still water and the speed of the stream?
(A) 17
(B) 99
(C) 88
(D) 66
(E) 23
Rationale: Speed of the boat in still water = (120+154)/2 = 137 kmph. Speed of the stream = (154-120)/2 = 17 kmph. Answer:A
Answer: (A) 17
Question: The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 7 minutes, then the area of the park (in sq. m) is:
(A) 153601
(B) 153600
(C) 153602
(D) 117600
(E) 153604
Rationale: Perimeter = Distance covered in 7 min. =(12000/60) x 7m = 1400 m.
Let length = 3x metres and breadth = 2x metres.
Then, 2(3x + 2x) = 1400 or x = 140.
Length = 420 m and Breadth = 280 m.
Area = (420 x 280) m2 = 117600 m2.
Answer: D
Answer: (D) 117600
Question: The sequence a1, a2, … , an, … is such that an=2an−1−x for all positive integers n ≥ 2 and for certain number x. If a5=99 and a3=27, what is the value of x?
(A) 3
(B) 9
(C) 18
(D) 36
(E) 45
Rationale: a5= 2*a4 - x = 99
a4 = 2*a3 - x = 2*27 - x
therefore;
a5 = 2*(54 - x ) -x = 99
108 - 3*x = 99
therefore X = 3
This the answer is "A"
Answer: (A) 3
Question: The combined age of husband and wife on their 20th wedding anniversary was twice than it was at the time of wedding. If the husband is 3 years older than his wife. How old was he at the time of his marriage?
(A) 21.5 years
(B) 21.9 years
(C) 31.5 years
(D) 21.7 years
(E) 61.5 years
Rationale: Explanation:
Let the ages of husband and wife at the time of their 20th wedding anniversary be X years and Y years respectively and X > Y.
Their combined age = (X + Y) years.
The ages of both husband and wife at the time of wedding will be (X – 20) years and (Y – 20) years respectively.
Their combined age = (X – 20) + (Y – 20)
= (X + Y – 40) Years
By first information, we have
(X + Y) = 2( X + Y – 40)
X + Y = 2X + 2Y - 80
Therefore, X + Y = 80-------A
and by second information, we have
X – Y =3 ----------------------B
Solving A and B, we get X = 41.5 years
Therefore, the age of husband at the time of their 20th wedding anniversary X = 41.5 years
Age at the time of marriage = 41.5 years – 20 years = 21.5 years.
ANSWER:A
Answer: (A) 21.5 years
Question: The length of a rectangle is three - fifths of the radius of a circle. The radius of the circle is equal to the side of the square, whose area is 1225 sq.units. What is the area (in sq.units) of the rectangle if the rectangle if the breadth is 10 units?
(A) 240 sq.units
(B) 140 sq.units
(C) 120 sq.units
(D) 130 sq.units
(E) 150 sq.units
Rationale: Given that the area of the square = 1600 sq.units
=> Side of square = √1600 = 40 units
The radius of the circle = side of the square = 35 units Length of the rectangle = 3/5 * 40 = 24units
Given that breadth = 10 units
Area of the rectangle = lb = 24* 10 = 240 sq.units
Answer:A
Answer: (A) 240 sq.units
Question: Find the area of trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.
(A) 227
(B) 299
(C) 285
(D) 161
(E) 212
Rationale: Area of a trapezium = 1/2 (sum of parallel sides) * (perpendicular distance between them)
= 1/2 (20 + 18) * (15)
= 285 cm2
Answer: C
Answer: (C) 285
Question: We have 7 people and 4 chairs. In how many ways chairs can be occupy?
(A) 840
(B) 720
(C) 650
(D) 914
(E) 796
Rationale: no. of ways they can be occupy = 7*6*5*4 = 840
Answer is A
Answer: (A) 840
Question: A mixture of 70 litres of milk and water contains 10% water. How many litres of water should be added to the mixture so that the mixture contains 12 1/2% water?
(A) 2
(B) 5
(C) 9
(D) 7
(E) 9
Rationale: Quantity of milk in the mixture = 90/100 (70) = 63 litres.
After adding water, milk would form 87 1/2% of the mixture.
Hence, if quantity of mixture after adding x liters of water, (87 1/2) / 100 x = 63 => x = 72
Hence 72 - 70 = 2 litres of water must be added.
Answer: A
Answer: (A) 2
Question: The function f is defined for all positive integers r by the following rule. f(r) is the number of positive integers each of which is less than r and has no positive factor in common with r other than 1. If x is any prime, number then f(x)=
(A) (x+1)/2
(B) x-2
(C) x-1
(D) (x-1)/2
(E) 2
Rationale: If not the wording the question wouldn't be as tough as it is now. The GMAT often hides some simple concept in complicated way of delivering it.
This question for instance basically asks: how many positive integers are less than given prime number x which have no common factor with p except 1.
Well as x is a prime, all positive numbers less than x have no common factors with x (except common factor 1). So there would be x-1 such numbers (as we are looking number of integers less than x).
For example: if x=6 how many numbers are less than 6 having no common factors with 6: 1, 2, 3, 4, 5, --> 6-1=5.
Answer: C.
Answer: (C) x-1
Question: What is the solution of the following simultaneous equations?
x + y + z = 6, x + 2y + 3z = 14 and x + 3y + z = 10
(A) 3
(B) 7
(C) 8
(D) 1
(E) 2
Rationale: x + y + z = 6 .... (i)
x + 2y + 3z = 14 .... (ii)
x + 3y + z = 10
From (i), we get z = 6 - x - y
Substitute it in (ii) and (iii)
We have from (ii)
x + 2y + 18 - 3x - 3y = 14 or 2x + y = 4
Similarily, from (iii)
x + 3y + 6 - x - y = 10 or 2y = 4 or y = 2
On solving, we get y = 2, x = 1, z = 3.
Answer:A
Answer: (A) 3
Question: If 9 men can reap 80 hectares in 24 days , then how many hectares can 36 men reap in 30 days?
(A) 127
(B) 237
(C) 387
(D) 400
(E) 481
Rationale: Explanation:
Let the required no of hectares be x. Then
Men --- Hectares---Days
9---------80---------24
36---------x---------30
More men, More hectares (Direct proportion)
More days, More hectares (Direct proportion)
x=36/9*30/24*80
x= 400
Answer: D
Answer: (D) 400
Question: When Greenville State University decided to move its fine arts collection to a new library, it had to package the collection in 20-inch by 20-inch by 12-inch boxes. If the university pays $0.50 for every box, and if the university needs 2.4 million cubic inches to package the collection, what is the minimum amount the university must spend on boxes?
(A) $250
(B) $275
(C) $510
(D) $1,250
(E) $2,550
Rationale: The volume of each box is 20*20*12 = 4800 cubic inches.
Number of boxes = 2,400,000/4800 = 500 boxes
Total cost = 500×$0.5=$250
The answer is A.
Answer: (A) $250
Question: Erica is four times as old as Blair. Nicole is three years older than Erica.
The sum of their ages is 21. How old is Erica?
(A) 8
(B) 18
(C) 19
(D) 22
(E) 29
Rationale: The problem concerns three ages. Let x equal Blair’s age. Blair’s age: x
Represent the amounts using the same variable. Erica’s age: 4x
Nicole’s age: 4x 3
Write an equation showing the sum equal to 21. x 4x 4x 3 21
Solve the equation. x 4x 4x 3 21
The variable x is equal to 2, but that 9x 3 21
doesn’t answer the question posed in the 9x 18
problem. The problem asks you to find
Erica’s age, which is equal to 4x, or 4(2). x 2
Answer: Erica is 8 years old.
Check: Blair is 2, Erica is 8, and Nicole is 11. The ages total 21.
correct answer A
Answer: (A) 8