divelab/Sys2Bench huggingface dataset:
Question: A train of length 110 meter is running at a speed of 60 kmph. In what time, it will pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
Options:
A)3
B)7
C)6
D)8
E)1
Rationale: Explanation:
Distance = 110 m
Relative speed = 60 + 6 = 66 kmph (Since both the train and the man are in moving in opposite direction)
= m/sec = m/sec
Time taken to pass the man = = 6 s
Answer: C) 6
Answer: C
Question: Two wheels are connected via a conveyor belt. The larger wheel has a 40cm diameter and the smaller wheel has a 35cm diameter. In order for the conveyor belt to work smoothly, each wheel must rotate the exact same number of centimetres per minute. If the larger wheel makes r revolution per minute, how many revolutions does the smaller wheel make per hour in terms of r?
Options:
A)1280π
B)75r
C)48r
D)24r
E)68.6r
Rationale: Interesting to note that the larger wheel has a diameter of 40 (8*5) while the smaller one has a diameter of 35 (7*5)...
If the large wheel has a diameter of 40 and the small wheel, 35, then their circumferences are 40pi and 35pi respectively. In order for them to move the conveyor belt at the same rate, the smaller wheel would need to rotate 1.14 times as fast as the larger wheel. Lets say the large wheel makes 10 revolutions per minute, the smaller wheel would need to make 10*1.14 = 11.4 revolutions per minute. If the large wheel makes 10 revolutions per minute it makes 600 per hour. Therefore, the smaller wheel would need to make 600*1.14 = 686 revolutions per hour.
If r = 10, then the answer choice must be b.
E. 68.6r
Answer: E
Question: The length of a rectangular floor is more than its breadth by 200%. If Rs. 624 is required to paint the floor at the rate of Rs. 4 per sq m, then what would be the length of the floor?
Options:
A)22.8m.
B)21.6m.
C)21.1m.
D)20.6m.
E)22.5m.
Rationale: Let the length and the breadth of the floor be l m and b m respectively.
l = b + 200% of b = l + 2b = 3b
Area of the floor = 624/4= 156 sq m
l b = 156 i.e., l * l/3 = 156
l2 = 468 => l = 21.6m.
ANSWER:B
Answer: B
Question: In a circuit board factory, all circuit boards that pass a verification process are certified. Every board that fails the verification process is indeed faulty, but 1/8 of those that pass are also faulty.
Approximately how many faulty circuit boards exist in a group of 800 circuit boards where 32 fail inspection?
Options:
A)72
B)192
C)200
D)128
E)264
Rationale: Total of 800 boards. All that fail verification are indeed faulty. So the 32 are indeed faulty. 1/8 those that pass are also faulty.
From the 800 we know 32 fail. So 768 must pass. Of these 1/8 are faulty. 768 divided by 8 gives you 96.
What one must do now is to add to the 96 which were not detected the actually detected faulty ones, namely the 32.
Total faulty: 128.
Answer: D
Answer: D
Question: A set S = {x, -8, -7, -6, 6, 8, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?
Options:
A)1
B)0
C)2
D)-1
E)Cannot be determined
Rationale: Median of the set = (-6+6)/2 = 0
As per statement, Mean of the set = 0
Mean of the set
|y|- |x| +23-21 = 0 (where x is negative n y is positive)
|y|- |x| = -2
So the absolute difference between two numbers is 2
Answer C
Answer: C
Question: A part of certain sum of money is invested at 6% per annum and the rest at 15% per annum, if the interest earned in each case for the same period is equal, then ratio of the sums invested is?
Options:
A)4:6
B)4:9
C)4:3
D)5:2
E)4:2
Rationale: 15:6
= 5:2
Answer: D
Answer: D
Question: Look at this series: 1.2, 1.9, 2.6, 3.3, ... What number should come next?
Options:
A)3.8
B)4
C)4.7
D)5
E)5.6
Rationale: B
4
In this simple addition series, each number increases by 0.7.
Answer: B
Question: Six friends go to pizza corner there are 2 types of pizzas. And six different flavors are there they have to select 2 flavors from 6 flavors. In how many ways we can select?
Options:
A)10
B)15
C)20
D)30
E)40
Rationale: 6C2 = 6*5 = 30
Answer D
Answer: D
Question: Gill drives 120 miles from Los Angeles to San Diego to fetch a package. On her way there she drives at 40 miles per hour. On her way back she drives 50% faster. What is Gill's average velocity R for the round trip?
Options:
A)24 miles per hour
B)48 miles per hour
C)50 miles per hour
D)53 1/3 miles per hour
E)68 miles per hour
Rationale: Here's an Average Speed question in which the prompt gives you almost all of the immediate numbers needed to work with.
Driving from LA to SD, we have a distance of 120 miles and a speed of 40 miles/hour.
D = (R)(T)
120 = (40)(T)
120/40 = 3 = T
3 hours to drive to SD
On the way back, she drives 50% FASTER. Gill's return speed is (1.5)(40) = 60 miles/hour.
D = (R)(T)
120 = (60)(T)
120/60 = 2 = T
2 hours to drive to SD
Total Distance = 240 miles
Total Time = 3+2 = 5 hours
Average Speed = 240/5 = 48 miles/hour
Final Answer:
B
Answer: B
Question: If goods be purchased for Rs.840 and one-fourth be sold at a loss of 20% at what gain percent should the remainder be sold so as to gain 20% on the whole transaction?
Options:
A)33 1/7%
B)33 8/3%
C)33 1/3%
D)36 1/3%
E)31 1/3%
Rationale: 1/4 CP = 210 SP = 21*(80/100) = 168
SP = 840*(120/100) = 1008
1008 - 168 = 840
3/4 SP = 630
Gain = 210
630 --- 210
100 --- ? => 33 1/3%
Answer:C
Answer: C
Question: 5 OUT OF 2250 PARTS OF EARTH IS SULPHUR. WHAT IS THE PERCENTAGE OF SULPHUR IN EARTH
Options:
A)11/50
B)2/9
C)1/45
D)2/45
E)NONE OF THESE
Rationale: REQUIRED PERCENTAGE=(5/2250*100)%=2/9%
CORRECT OPTIONS: B
Answer: B
Question: What is the area of a triangle with the following vertices L(1, 3), M(4, 1), and N(3, 5) ?
Options:
A)3
B)4
C)5
D)6
E)7
Rationale: There is a direct formula to calculate the are of a triangle based on coordinates of its vertices and one could use it to solve this problem.
Though if you make a diagram minimum simple calculations will be needed:
Answer: C.
Answer: C
Question: Calculate the share of profit made by 2 business partners, who invested $1500 and $3500 respectively in a business if the profit made is $2500.
Options:
A)$750,$1750
B)$710,$1400
C)$750,$1200
D)$753,$1400
E)$950,$1400
Rationale: A:B = 1500:3500
A:B = 3 : 7
A's share = 2500*3/10 = $750
B's share = 2500*7/10 = $1750
Answer is A
Answer: A
Question: A towel, when bleached, lost 30% of its length and 20% of its breadth. What is the percentage decrease in area?
Options:
A)24%
B)30 %
C)44%
D)54%
E)64%
Rationale: percentage change in area
=(−30−20+(30×20)/100)%=−44%
i.e., area is decreased by 44%
ANSWER:C
Answer: C
Question: Find the perimeter and area of a square of side 11 cm.
Options:
A)11 cm²
B)21 cm²
C)22 cm²
D)113 cm²
E)121 cm²
Rationale: We know that the perimeter of square = 4 × side
Side= 11 cm
Therefore, perimeter = 4 × 11 cm = 44 cm
Now, area of the square = (side × side) sq. units
= 11 × 11 cm²
= 121 cm²
ANSWER : OPTION E
Answer: E
Question: Some of 50%-intensity red paint is replaced with 20% solution of red paint such that the new paint intensity is 30%. What fraction of the original paint was replaced?
Options:
A)1/30
B)1/5
C)2/3
D)3/4
E)4/5
Rationale: 30% is 10%-points above 20% and 20%-points below 50%.
Thus the ratio of 25%-solution to 50%-solution is 2:1.
2/3 of the original paint was replaced.
The answer is C.
Answer: C
Question: A and B can do a piece of work in 12 days and 16 days respectively. Both work for 3 days and then A goes away. Find how long will B take to complete the remaining work?
Options:
A)9
B)8
C)7
D)5
E)2
Rationale: 3/12+(3+x)/16 = 1
x = 9 days
ANSWER A
Answer: A
Question: 8 kilograms of oranges contained 98% of water. If the next day the concentration of water decreased by 2%, what was the new weight of the oranges, in kilograms?
Options:
A) 4.9
B) 4.8
C) 4.5
D) 4
E) 0.2
Rationale: Eight kilograms of oranges contained 98% of water. If the next day the concentration of water decreased by 2%, what was the new weight of the oranges, in kilograms?
(A) 4.9
(B) 4.8
(C) 4.5
(D) 4
(E) 0.2
If D is the answer, then the question means that the concentration of water the newt day became 96%.
Out of 8 kilograms 98%, or 7.84 kilograms was water and 0.16 kilograms was non-water.
The next day, after some water evaporated, oranges became 96% water and 4% of non-water, so the next day 0.16 kilograms of non-water composed 4% of oranges, which means that the new weight of the oranges was x*0.04=0.16 --> x=4 kilograms.
Answer: D.
Answer: D
Question: It takes 7 high school students, working at identical constant individual rates, 10 hours to paint a certain house. At what time will the house be fully painted if 7 students start painting at 9:00 am and one student, working at the same rate, is added per hour starting at 5:00 pm?
Options:
A)5:40 pm
B)6:00 pm
C)6:20 pm
D)6:40 pm
E)7:00 pm
Rationale: Each student paints at a rate of 1/70 of the house per hour.
In 8 hours, the 7 students can paint 56/70 of the house.
From 5 pm to 6 pm, 8 students paint another 8/70 for a total of 64/70.
9 students can paint the remaining 6/70 of the house in 6/9 of an hour = 40 minutes.
The house is completed at 6:40 pm.
The answer is D.
Answer: D
Question: 8 business executives and 7 chairmen meet at a conference. If each business executive shakes the hand of every other business executive and every chairman once, and each chairman shakes the hand of each of the business executives but not the other chairmen, how many handshakes would take place?
Options:
A)144
B)131
C)115
D)84
E)45
Rationale: there are 8 business exec and in each handshake 2 business execs are involved.
hence 8C2=28
Also, each of 8 exec will shake hand with every 7 other chairmen for total of 56 handshake.
total=28+56=84
ANS: D
Answer: D
Question: The sum of five consecutive odd numbers of set p is 435. What is the sum of five consecutive numbers of another set q. Whose largest number is 45 more than the largest number of set p?
Options:
A)530
B)670
C)730
D)770
E)None of these
Rationale: Let the five consecutive odd numbers of set p be 2n - 3, 2n - 1, 2n + 1, 2n + 3, 2n + 5.
Sum of these five numbers
= 2n - 3 + 2n - 1 + 2n + 1 + 2n + 3 + 2n + 5
= 10n + 5 = 435 => n = 43
Largest number of set p = 2(43) + 5 = 91
The largest number of set q = 91 + 45 = 136
=> The five numbers of set q are 132, 133, 134, 135, 136.
Sum of above numbers = 132 + 133 + 134 + 135 + 136 = 670.
ANSWER:B
Answer: B
Question: Two persons A and B can complete a piece of work in 30 days and 45 days respectively. If they work together, what part of the work will be completed in 3 days?
Options:
A)1/9
B)1/7
C)1/6
D)1/2
E)1/1
Rationale: A's one day's work = 1/30
B's one day's work = 1/45
(A + B)'s one day's work = 1/30 + 1/45 = 1/18
The part of the work completed in 3 days = 3 (1/18) = 1/6.
Answer:C
Answer: C
Question: A lent Rs. 5000 to B for 2 years and Rs. 3000 to C for 4 years on simple interest at the same rate of interest and received Rs. 1540 in all from both of them as interest. The rate of interest per annum is?
Options:
A)7%
B)12%
C)74%
D)10%
E)45%
Rationale: Let the rate be R% p.a. Then,
(5000 * R * 2)/100 + (3000 * R * 4)/100 = 1540
100 R + 120 R = 1540
R = 7%
Answer:A
Answer: A
Question: The number of new words that can be formed by rearranging the letters of the word 'COMPUTE' is ?
Options:
A)5040
B)5039
C)5041
D)5554
E)132
Rationale: Number of words which can be formed
= 7! - 1 = 5040 - 1
= 5039
Answer:B
Answer: B
Question: A started a business with an investment of Rs. 70000 and after 6 months B joined him investing Rs. 120000. If the profit at the end of a year is Rs. 78000, then the share of B is?
Options:
A)s.36028
B)s.36000
C)s.36003
D)s.36029
E)s.24029
Rationale: Ratio of investments of A and B is (70000 * 12) : (120000 * 6) = 7 : 6
Total profit = Rs. 78000
Share of B = 6/13 (78000) = Rs.36000
Answer: B
Answer: B
Question: x - y =2
2x = 2y + 4
The system of equations above has how many solutions?
Options:
A) None
B) Exactly one
C) Exactly two
D) Infinitely many
E) Exactly three
Rationale: x-y=2 ---------------1
2x=2y+4 ---------------2
Divide equation 2 by 2:
2x/2=(2y+4)/2
x=y+2
x-y=2----------------3
Equation 1 and 3 are equal and thus have infinitely many solutions:
x-y=2
x=4, y=2
x=6, y=4
x=7, y=5
Ans:D
Answer: D
Question: In a ratio which is equal to 7:9, if the antecedent is 42, then the consequent is?
Options:
A)72
B)54
C)64
D)51
E)45
Rationale: We have 7/9 = 42/x
7x = 42*9
x = 54
consequent = 54
Answer is B
Answer: B
Question: Two stations A and B are 110 km apart on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 kmph. Another train starts from B at 8 a.m. and travels towards A at a speed of 25 kmph. At what time will they meet?
Options:
A)11 a.m
B)10 a.m
C)09 a.m
D)08 a.m
E)06 a.m
Rationale: Suppose they meet x hours after 7 a.m.
Distance covered by A in x hours = 20x km.
Distance covered by B in (x - 1) hours = 25(x - 1) km.
Therefore 20x + 25(x - 1) = 110
45x = 135
x = 3.
So, they meet at 10 a.m.
Answer: B
Answer: B
Question: A triangle has a perimeter 15. The two shorter sides have integer lengths equal to x and x + 1. Which of the following could be the length of the other side?
Options:
A)7
B)6
C)8
D)9
E)10
Rationale: The SHORTER sides have integral lengths equal to x and x + 1
Let the longest side be 'a'
So, a + x + (x +1) = 15
a + 2x = 14.......eqn (1)
We know that the sum of the lengths of the shorter sides has to be more than the length of the longer one. i.e 2x+1> a
a =6
B
Answer: B
Question: A can do a work in 15 days and B in 20 days. If they work on it together for 2 days, then the fraction of the work that is left is :
Options:
A)23/20
B)5
C)23/30
D)23/12
E)2
Rationale: Ans is:C
A's 1 day's work =1/15
B's 1 day's work =1/20
(A + B)'s 1 day's work =(1/15+1/20)=7/60
(A + B)'s 2 day's work =(7/60*2)=7/30
Therefore, Remaining work =(1-7/30)=23/30. Ans: C
Answer: C
Question: A committee of three people is to be chosen from five married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee?
Options:
A)16
B)24
C)48
D)64
E)80
Rationale: Each couple can send only one representative to the committee. Let's see in how many ways we can choose 3 couples (as there should be 3 members) out of 5 to send only one representatives to the committee: 5C3=10.
But each of these 3 couples can send two persons (husband or wife): 2*2*2=2^3=8.
Total # of ways: 5C3*2^3=80.
Answer: E.
Answer: E
Question: How many 4 digits number contain the number 2.
Options:
A)3170
B)3172
C)3174
D)3168
E)3167
Rationale: total no. of 4 digit nos are 9000(btwn 1000 and 9999).
consider the case where the nos contain no 2 in it.
so units place can be filled in 9 ways(0,1,3,4,5,6,7,8,9)
smlry,tenths place in 9 ways
hunderedths place in 9 ways
thousands place can be filled in 8 ways only(because 0 cannot come)
so total nos are (9000-(8*9*9*9))=3168
ANSWER:D
Answer: D
Question: 3 candidates in an election and received 1136, 7636 and 11628 votes respectively. What % of the total votes did the winningcandidate got in that election?
Options:
A)30%
B)59%
C)57%
D)64%
E)69%
Rationale: Total number of votes polled = (1136 + 7636 + 11628) = 20400
So, Required percentage = 11628/20400 * 100 = 57%
C
Answer: C
Question: Given positive integer x, which of the following CANNOT be evenly divisible by x?
Options:
A)7x
B)x + 2
C)2x - 1/2
D)3x − 1
E)x + 6
Rationale: This can be easily solved by substituting values:
Assume x to be 1
A. 7x: 7 is evenly divisible by 1
B. x + 2: 3 is evenly divisible by 1
C. 2x - 1/2: 3/2 is not evenly divisible by 1
D. 3x − 1: 2 is evenly divisible by 1
E. x + 6: 7 is not evenly divisible by 1
Option C
Answer: C
Question: If an object travels 80 feet in 2 seconds, what is the object’s approximate speed in miles per hour? (Note: 1 mile = 5280 feet)
Options:
A)21.21
B)24.24
C)27.27
D)30.35
E)33.33
Rationale: 80 feet/2 seconds = 40 feet/second
(40 feet/second)*(3600 seconds/hour)*(1 mile/5280 feet) = 27.27 miles/hour (approximately)
The answer is C.
Answer: C
Question: A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?
Options:
A)4/7
B)3/7
C)2/7
D)7/7
E)4/3
Rationale: Let number of balls = (6 + 8) = 14.
Number of white balls = 8.
P (drawing a white ball) =8/14=4/7.
Answer: A
Answer: A
Question: Consider a quarter of a circle of radius 49. Let r be the radius of the circle inscribed in this quarter of a circle. Find r.
Options:
A)49*(sqr2 -1)
B)58*(sqr3 -1)
C)40*(sqr7 - 1)
D)12* (sqr7 -1)
E)None of these
Rationale: I got 49/(sqr2 +1) and just forgot to multiply by (sqr2 -1).
Answer is A
Answer: A
Question: An urn contains 10 balls, numbered from 1 to 10. If 2 balls are selected at random with replacement from the urn, what is the probability G that the sum of the 2 numbers on the balls will be even?
Options:
A)25%
B)37.5%
C)50%
D)62.5%
E)75%
Rationale: Correct Answer: C
Since there are 5 even numbered balls and 5 odd numbered balls, the probability G of selecting either an even or an odd is 5/10= 1/2. In order for the sum of the numbers on the two balls to be even, either both numbers must be even or both numbers must be odd; one even and one odd number would add to an odd number. The probability of selecting two even numbers is:
1/2 * 1/2= 1/4
The probability of selecting two odd numbers is also:
1/2 * 1/2= 1/4
Adding both probabilities gives 1/4+1/4= 1/2 , or 50%, as the probability that the sum of the numbers will be even. Thus, the correct answer is (C).
Answer: C
Question: Martin bought 10 concert tickets, some at the full price of $2.00 per ticket, and some at a discounted price of $1.60 per ticket. If he spent a total of $17.20, how many discounted tickets did he buy?
Options:
A)3
B)4
C)5
D)6
E)7
Rationale: Let x be the number of tickets he bought at $2 per ticket.
then 2 x + (10-x)1.6 = 17.2
0.4x = 1.2 => x = 3
discounted tickets = 10 -x = 7
Ans : E
Answer: E
Question: A and B can do a piece of work in 2 days. With the help of C they finish the work in 1 days. C alone can do that piece of work in?
Options:
A)1.5 DAYS
B)4 DAYS
C)3 DAYS
D)1 DAY
E)2 DAYS
Rationale: C = 1/1 – 1/2= 1/2 => 2 days
Answer:E
Answer: E
Question: A can do a piece of work in 4 hours; B and C together can do it in 3 hours, while A and C together can do it in 2 hours. How long will B alone take to do it?
Options:
A)8 hours
B)10 hours
C)12 hours
D)24 hours
E)25 hours
Rationale: Explanation:
A's 1 hour's work =1/4
(B + C)'s 1 hour's work =1/3
(A + C)'s 1 hour's work =1/2
(A + B + C)'s 1 hour's work =1/4 +1/3=7/12
B's 1 hour's work =7/12-1/2=1/12
B alone will take 12 hours to do the work.
ANSWER IS C
Answer: C
Question: If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum Q of the 3 greatest of these temperatures, in degrees Fahrenheit, could be
Options:
A)6x
B)Q=4x
C)Q=5x/3
D)Q=3x/2
E)3x/5
Rationale: OA please. Yeah so 5x is the sum of the 5, then I guess 4x could be the answer (B)
But what about C,D,E? Any ideas why these could be wrong?
Note that we have 5 positive temperatures.
Next, as the average is x then the sum of the temperatures is 5x and as all the temperatures are positive then the sum of the 3 greatest must be more than (or equal to) 3x (as the average of the 3 greatest must be at least x) and less than 5x: 3x<SUM<5x --> only option B fits.
Answer: B.
Answer: B
Question: Find the area of a parallelogram with base 21 cm and height 11 cm?
Options:
A)231 cm2
B)384 cm2
C)672 cm2
D)267 cm2
E)286 cm2
Rationale: Area of a parallelogram
= base * height
= 21 * 11
= 231 cm2
Answer: A
Answer: A
Question: A farmer built a fence around his square plot. He used 27 fence poles on each side of the square. How many poles did he need altogether ?
Options:
A)100
B)104
C)108
D)112
E)None of these
Rationale: one side=2 corners( 1 nd 1 each) plus 25 ...
1 25 1
25 25
1 25 1
der4 ans=25*4+4=104
ANSWER:B
Answer: B
Question: Ram professes to sell his goods at the cost price but he made use of 500 grms instead of a kg, what is the gain percent?
A. 11%
Options:
A)11 1/9%
B)11 1/8%
C)11 9/9%
D)100%
E)11 1/5%
Rationale: 500 --- 500
100 --- ? => 100%
Answer:D
Answer: D
Question: Two pipes A and B can separately fill a tank in 12 and 15 minutes respectively. A third pipe C can drain off 45 liters of water per minute. If all the pipes are opened, the tank can be filled in 15 minutes. What is the capacity of the tank?
Options:
A)542
B)540
C)542
D)829
E)279
Rationale: 1/12 + 1/15 - 1/x = 1/15
x = 12
12 * 45 = 540
Answer: B
Answer: B
Question: If x : y = 3 : 5, find the value of (2x + 4y) : (4x – y)
Options:
A)29:7
B)22:7
C)26:7
D)20:7
E)26:5
Rationale: Explanation:
Given: a/b = 3/5
(2*3 + 4*5) : (4*3 – 5)
= 26:7
ANSWER: C
Answer: C
Question: The average marks of 14 students was calculated as 71. But, it was discovered later that the marks of one student was taken as 42 instead of 56 and the marks of another student was wrongly taken as 74 while it was actually 32. If these corrections are now made then what is the average of the 14 students?
Options:
A)67
B)68
C)69
D)80
E)71
Rationale: total marks =14*71 = 994
marks with out that two studnts = 994-42-74 = 878
total marks ith corrections = 878+56+32 = 966
the average marks = 966/14 =69
ANSWER:C
Answer: C
Question: If s is the product of the integers from 40 to 50 inclusive, and if t is the product of the integers from 40 to 51 inclusive, what is 1/s + 1/t in terms of t?
Options:
A)32/t
B)42/t
C)52/t
D)62/t
E)72/t
Rationale: Similarly in the actual problem
s =t/51. 1/s = 1/(t/51) or 1/s = 51/t
1/s + 1/t = 51/t + 1/t = 52/t
C
Answer: C
Question: A train starts full of passengers at the first station it drops 1/3 of the passengers and takes 280 more at the second station it drops one half the new total and takes twelve more .on arriving at the third station it is found to have 248 passengers. Find the no of passengers in the beginning?
Options:
A)258
B)288
C)248
D)278
E)238
Rationale: Let no of passengers in the beginning be x
After first station no passengers=(x-x/3)+280=2x/3 +280
After second station no passengers =1/2(2x/3+280)+12
½(2x/3+280)+12=248
2x/3+280=2*236
2x/3=192
x=288
Answer is B.
Answer: B
Question: 19/6 = 4/27. Which method did you use to determine whether this proportion is true or false?
Options:
A)24
B)77
C)88
D)26
E)91
Rationale: 19/6 = 4/27.
Cross multiplication:
19 × 27 = 513.
6 × 4 = 24.
Answer:A
We see that 513 are not equal to 24.
Therefore, we determine 19/6 = 4/27 is not proportion so, the answer is false.
Answer: A
Question: Balls of equal size are arranged in rows to form an equilateral triangle. the top most row consists of one ball, the 2nd row of two balls and so on. If 253 balls are added, then all the balls can be arranged in the shape of square and each of the sides of the square contain 8 balls less than the each side of the triangle did. How many balls made up the triangle?
Options:
A)903
B)2209
C)2878
D)1210
E)1560
Rationale: As expected, this question boils down to 2 equation,
Consider total number of Balls in Triangle = T and number of balls in last row = x.
1+2+3+...+x = T
x(x+1)/2 = T ----(a)
As mentioned in the question, side of a square will be (x-8) and total number of Balls in square will be (T+253)
(x-8)^2 = T+253 -----(b)
Now the hardest part of the question will be to solve these 2 equations and this looks like time consuming BUT the easy way will be plug and play. Also, we've to find a value of T (from 5 optiosn given below) which can make a square of a a number. One we know this, it will be a cake walk.
We can see that option A fits this criteria in eq (b). Add - 903+253 = 1156 = 34^2 = (x-8)^2 Hence, x = 42.
Cross check by putting in eq (a) = x(x+1)/2 = T = > 42*43/2=903
Hence, answer is A.
Answer: A