Practice UPSC CSAT previous year questions from 2011–2026 classified underArithmetic. Review the original question, four options, correct answer and explanation for every PYQ.
113 questions • 2011–2026
A toy T jumps forward or backward. In each forward jump, it moves 5' forward whereas in each backward jump, it moves 2' backward. If in 31 jumps, T moves exactly 15' forward, then what is the difference of the number of forward and backward jumps?
Let forward jumps be f and backward jumps b. f+b=31 and 5f−2b=15. Solving gives f=11, b=20, so the difference is 9. Hence D.
What is the minimum number of times one needs to measure to get 298 litres of water from a tank, if the measuring cylinders have capacities 1 litre, 6 litres, 25 litres and 100 litres?
Three measurements of 100 litres give 300 litres; two 1-litre measurements can then account for the 2-litre reduction, giving five measurements in the intended measuring procedure. Hence B.
In an objective type question paper, 5 marks are awarded for a correct answer and 2 marks are deducted for a wrong answer. A student attempted all the questions and got a score of 69. Had he been awarded 4 marks for a correct answer and 1 mark deducted for a wrong answer, he would have scored 84. How many questions were there in the question paper?
Let correct=c and wrong=w. 5c−2w=69 and 4c−w=84. Solving gives c=33,w=48, hence 81 questions. B.
Three partners A, B and C entered into a business. A invested one-third of the capital for one-third duration. B invested one-fourth of the capital for one-fourth duration. C invested the remaining capital for the whole duration. Out of a profit of ₹17,000, how much profit will C get?
Capital-time products are A=1/9, B=1/16, C=5/12. Multiplying by 144 gives 16:9:60, so C receives 60/85 of ₹17,000 = ₹12,000. Hence A.
A shopkeeper employs a delivery boy and gives him a motorcycle for home delivery. For every delivery, the boy is given ₹5. At the end of the day, he also gets ₹2 for every kilometre of the distance covered in the day. The boy wants to earn more than ₹500 a day, but does not want to travel more than 100 km. Which of the following numbers of deliveries would definitely meet his target?
With only an upper bound on kilometres, earnings are 5d+2k with k≤100. No listed delivery count guarantees >₹500 for every permissible distance k; even 90 deliveries give only ₹450 if distance compensation is zero. Thus the data are insufficient. Hence D.
Team X scored a total of N runs in 20 overs. Team Y tied the score in 10% less overs. Had team Y’s average run rate (runs per over) been 50% higher, the scores would have been tied in 12 overs. How many runs were scored by team X?
The 50% higher-rate condition is automatically satisfied for any N, so N cannot be determined.
In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of the number of runs scored by X to the number of runs scored by Y is equal to the ratio of number of runs scored by Y to the number of runs scored by Z. Value-I = Runs scored by X. Value-II = Runs scored by Y. Value-III = Runs scored by Z. Which one of the following is correct?
Positive integer triples include both 9,12,16 and 16,12,9, so the ordering is not unique.
Consider the following: 1. 1000 litres = 1 m³ 2. 1 metric ton = 1000 kg 3. 1 hectare = 10000 m². Which of the above are correct?
All three are standard unit equivalences: 1 m³=1000 L, 1 metric tonne=1000 kg, and 1 hectare=10,000 m².
The number of times the digit 5 will appear while writing the integers from 1 to 1000 is
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (C). The master retains both values so the discrepancy is visible rather than silently corrected.
A and B are two heavy steel blocks. If B is placed on the top of A, the weight increases by 60%. How much weight will reduce with respect to the total weight of A and B, if B is removed from the top of A?
Answer-key audit: uploaded/official-key entry is (B), while an independent check of the question wording gives (D). The master retains both values so the discrepancy is visible rather than silently corrected.
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is
Answer-key audit: uploaded/official-key entry is (A), while an independent check of the question wording gives (D). The master retains both values so the discrepancy is visible rather than silently corrected.
In a school every student is assigned a unique identification number. A student is a football player if and only if the identification number is divisible by 4, whereas a student is a cricketer if and only if the identification number is divisible by 6. If every number from 1 to 100 is assigned to a student, then how many of them play cricket as well as football?
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
When a runner was crossing the 12 km mark, she was informed that she had completed only 80% of the race. How many kilometres was the runner supposed to run in this event?
Answer-key audit: uploaded/official-key entry is (C), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
Raju has Rs. 9000 with him and he wants to buy a mobile handset; but he finds that he has only 75% of the amount required to buy the handset. Therefore, he borrows 2000 from a friend. Then
Answer-key audit: uploaded/official-key entry is (C), while an independent check of the question wording gives (A). The master retains both values so the discrepancy is visible rather than silently corrected.
In 2002, Meenu's age was one-third of the age of Meera, whereas in 2010, Meenu's age was half the age of Meera. What is Meenu's year of birth?
Answer-key audit: uploaded/official-key entry is (C), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
Rakesh and Rajesh together bought 10 balls and 10 rackets. Rakesh spent 1300 and Rajesh spent 1500. If each racket costs three times a ball does, then what is the price of a racket?
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (C). The master retains both values so the discrepancy is visible rather than silently corrected.
P, Q and R are three towns. The distance between P and Q is 60 km, whereas the distance between P and R is 80 km. Q is in the West of P and R is in the South of P. What is the distance between Q and R?
Q and R are perpendicular to P: QR=sqrt(60²+80²)=100 km.
The ratio of a two-digit natural number to a number formed by reversing its digits is 4:7. The number of such pairs is
Let the number be 10a+b. 7(10a+b)=4(10b+a), so 66a=33b and b=2a. Valid digit pairs give five solutions.
In an examination, A has scored 20 marks more than B. If B has scored 5% less marks than A, how much has B scored?
Answer-key audit: uploaded/official-key entry is (C), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
X, Y and Z are three contestants in a race of 1000 m. Assume that all run with different uniform speeds. X gives Y a start of 40 m and X gives Z a start of 64 m. If Y and Z were to compete in a race of 1000 m, how many metres start will Y give to Z?
X:Y speed ratio from a 40 m start is 1000:960. X:Z is 1000:936. Thus Y:Z=960:936=40:39; in a 1000 m race Y gives Z 25 m.
If x is greater than or equal to 25 and y is less than or equal to 40, then which one of the following is always correct?
Answer-key audit: uploaded/official-key entry is (B), while an independent check of the question wording gives (C). The master retains both values so the discrepancy is visible rather than silently corrected.
Ena was born 4 years after her parents' marriage. Her mother is three years younger than her father and 24 years older than Ena, who is 13 years old. At what age did Ena's father get married?
Ena is 13 and her mother is 24 years older, so mother is 37; father is 40. Since Ena was born 4 years after marriage, father married at 23.
Rakesh had money to buy 8 mobile handsets of a specific company. But the retailer offered very good discount on that particular handset. Rakesh could buy 10 mobile handsets with the amount he had. What was the discount the retailer offered?
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
The average marks of 100 students are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are
Answer-key audit: uploaded/official-key entry is (A), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
Sunita cuts a sheet of paper into three pieces. Length of first piece is equal to the average of the three single digit odd prime numbers. Length of the second piece is equal to that of the first plus one-third the length of the third. The third piece is as long as the other two pieces together. The length of the original sheet of paper is
Answer-key audit: uploaded/official-key entry is (A), while an independent check of the question wording gives (D). The master retains both values so the discrepancy is visible rather than silently corrected.
How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers?
For positive natural x,y,z with sum 6, the number of compositions is C(5,2)=10.
An 8-digit number 4252746B leaves remainder 0 when divided by 3. How many values of B are possible?
Digit sum excluding B is 4+2+5+2+7+4+6=30. For divisibility by 3, B must be 0,3,6,9: four values.
If the numerator and denominator of a proper fraction are increased by the same positive quantity which is greater than zero, the resulting fraction is
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (B). The master retains both values so the discrepancy is visible rather than silently corrected.
A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have?
Digits through page 999 total 9+90×2+900×3=2889. Remaining 200 digits correspond to pages 1000–1049, so total pages=1049.
A family has two children along with their parents. The average of the weights of the children and their mother is 50 kg. The average of the weights of the children and their father is 52 kg. If the weight of the father is 60 kg, then what is the weight of the mother?
Answer-key audit: uploaded/official-key entry is (B), while an independent check of the question wording gives (D). The master retains both values so the discrepancy is visible rather than silently corrected.
Suppose you have sufficient amount of rupee currency in three denominations: Rs. 1, Rs. 10 and Rs. 50. In how many different ways can you pay a bill of Rs. 107?
Answer-key audit: uploaded/official-key entry is (D), while an independent check of the question wording gives (C). The master retains both values so the discrepancy is visible rather than silently corrected.
Number 136 is added to 5B7 and the sum obtained is 7A3, where A and B are integers. It is given that 7A3 is exactly divisible by 3. The only possible value of B is
Answer-key audit: uploaded/official-key entry is (B), while an independent check of the question wording gives (D). The master retains both values so the discrepancy is visible rather than silently corrected.
Consider the following sum: • + 1• + 2• + •3 + •1 = 21•. In the above sum, • stands for
Let the dot be x. Interpreting each dotted term by place value gives 210+x on the right and the left-hand sum; solving gives x=8.
How many diagonals can be drawn by joining the vertices of an octagon?
An n-gon has n(n−3)/2 diagonals. For n=8, 8×5/2=20.
The figure drawn below gives the velocity graphs of two vehicles A and B. The straight line OKP represents the velocity of vehicle A at any instant, whereas the horizontal straight line CKD represents the velocity of vehicle B at any instant. In the figure, D is the point where perpendicular from P meets the horizontal line CKD such that PD = 1/2 LD. What is the ratio between the distances covered by vehicles A and B in the time interval OL?
Distance is the area under a velocity–time graph. Using the given PD:LD relation, the two areas over OL are in the ratio 3:4.
A train 200 metres long is moving at the rate of 40 kmph. In how many seconds will it cross a man standing near the railway line?
40 km/h = 40×1000/3600 m/s. Time = 200/(40×1000/3600)=18 s.
Graph: Imports of coil steel, sheet steel and scrap steel over six months; bracketed figures give average cost per ton.
By how much (measured in thousands of tons) did the import of sheet steel exceed the import of coil steel in the first three months of the year?
First three months: sheet steel = 40+37+36=113 thousand tons; coil steel = 30+31+33=94. Difference=19.
Graph: Imports of coil steel, sheet steel and scrap steel over six months; bracketed figures give average cost per ton.
What was the approximate total value (in $) of sheet steel imported over the six months period?
Total sheet steel over six months =217 thousand tons; at $256/ton, value≈$55.55 million, matching 55,550 (thousand dollars).
Graph: Imports of coil steel, sheet steel and scrap steel over six months; bracketed figures give average cost per ton.
What was the approximate ratio of sheet steel and scrap steel imports in the first three months of the year?
First three months: sheet=113, scrap=98 thousand tons; ratio=113/98≈1.2:1.
If X is between -3 and -1, and Y is between -1 and 1, then X² - Y² is in between which of the following?
X² lies between 1 and 9 and Y² between 0 and 1, so X²−Y² lies between 0 and 9.
X and Y are natural numbers other than 1, and Y is greater than X. Which of the following represents the largest number?
For natural numbers X<Y with X,Y>1, XY is greater than the corresponding ratios and (X+Y)/XY; hence XY is largest.
A number consists of three digits of which the middle one is zero and their sum is 4. If the number formed by interchanging the first and last digits is greater than the number itself by 198, then the difference between the first and last digits is
Let the number be 100a+0b+c with a+c=4. Reversing changes it by 99(c−a)=198, so c−a=2.
While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred's place is greater than the digit at ten's place, and the digit at ten's place is greater than the digit at unit's place?
Count three-digit numbers with hundreds digit > tens digit > units digit across 700–999; the count is 85.
A bookseller sold 'a' number of Geography textbooks at the rate of Rs. x per book, 'a + 2' number of History textbooks at the rate of Rs. (x + 2) per book and 'a - 2' number of Mathematics textbooks at the rate of Rs. (x - 2) per book. What is his total sale in Rs.?
Total = ax+(a+2)(x+2)+(a−2)(x−2)=3ax+8.
A bag contains 15 red balls and 20 black balls. Each ball is numbered either 1 or 2 or 3. 20% of the red balls are numbered 1 and 40% of them are numbered 3. Similarly, among the black balls, 45% are numbered 2 and 30% are numbered 3. A boy picks a ball at random. He wins if the ball is red and numbered 3 or if it is black and numbered 1 or 2. What are the chances of his winning?
Winning red-3 balls=6; black balls numbered 1 or 2=14. Total favourable=20 out of 35, so probability=4/7.
Two persons, A and B are running on a circular track. At the start, B is ahead of A and their positions make an angle of 30° at the centre of the circle. When A reaches the point diametrically opposite to his starting point, he meets B. What is the ratio of speeds of A and B, if they are running with uniform speeds?
A completes a semicircle (180°). B starts 30° ahead and meets A at the opposite point; comparing angular distances over the same time gives speed ratio 6:5.
A student has to get 40% marks to pass in an examination. Suppose he gets 30 marks and fails by 30 marks, then what are the maximum marks in the examination?
30 marks is 30 below the passing score, so 40% of maximum=60. Maximum marks=60/0.40=150.
A shopkeeper sells an article at Rs. 40 and gets X% profit. However, when he sells it at Rs. 20, he faces same percentage of loss. What is the original cost of the article?
If cost is C and profit/loss percentages are equal, 40−C=C−20, giving C=30.
There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?
With 24 equally spaced points, an equilateral triangle needs 120° central separation; 24/3=8 such triangles.
Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square?
If each square has side x, rectangle sides are 4x and 3x. 5²=(4x)²+(3x)², so x=1 and area=1.
Graph: Expected versus actual progress of work from April to September.
Consider the following graph: [graph shown in paper]. Which one of the following statements is not correct with reference to the graph given above?
Correct answer: (D). Apply the relevant quantitative relation to the values given in the question; this yields During the period from 1st April to 1st September, at no time was the actual progress more than the expected progress..
For a sports meet, a winners' stand comprising three wooden blocks is in the following form: [figure shown in paper]. There are six different colours available to choose from and each of the three wooden blocks is to be painted such that no two of them has the same colour. In how many different ways can the winners' stand be painted?
Three distinct colours can be assigned in 6×5×4=120 ways.
Graph: Birth and death rates / population dynamics from 1970 to 2010.
Looking at the graph, it can be inferred that from 1990 to 2010
The graph does not show a single monotonic trend in the population growth rate over 1990–2010.
Graph: Birth and death rates / population dynamics from 1970 to 2010.
With reference to the above graph, consider the following statements considering 1970 as base year: 1. Population has stabilized after 35 years. 2. Population growth rate has stabilized after 35 years. 3. Death rate has fallen by 10% in the first 10 years. 4. Birthrate has stabilized after 35 years. Which of the above are the most logical and rational statements that can be made from the above graph?
From the graph, the growth-rate and birth-rate curves stabilize after about 35 years; these are statements 2 and 4.
Two graphs of average hourly earnings E against years for firms A and B.
Average hourly earnings per year (E) of the workers in a firm are represented in figures A and B as follows: [figures shown in paper]. From the figures, it is observed that the
Both graphs have the same rise per year, so their slopes are equal.
Two graphs showing manufacturing cost and projected sales against number of pieces.
Consider the following figures A and B: [figures shown in paper]. The manufacturing cost and projected sales for a product are shown in the above figures A and B respectively. What is the minimum number of pieces that should be manufactured to avoid a loss?
At 2,000 pieces the selling-price and cost lines meet; producing fewer would create a loss. Minimum=2,000.
A lift has the capacity of 18 adults or 30 children. How many children can board the lift with 12 adults?
18 adults=30 children, so 1 adult=30/18=5/3 children. Twelve adults occupy 20-child capacity, leaving 10 children.
A person bought a refrigerator worth Rs. 22,800 with 12.5% interest compounded yearly. At the end of first year he paid Rs. 8,650 and at the end of second year Rs. 9,125. How much will he have to pay at the end of third year to clear the debt?
After the first payment the balance is 17,000; after 12.5% interest it is 19,125. After the second payment, 10,000 remains; 12.5% interest makes 11,250 due.
Population-age-group graphs for countries A and B over several decades.
Consider the following graphs. The curves in the graphs indicate different age groups in the populations of two countries A and B over a period of few decades: [graphs shown in paper]. With reference to the above graphs, which of the following are the most logical and rational inferences that can be made? 1. Over the last two and a half decades, the dependency ratio for country B has decreased. 2. By the end of next two and a half decades, the dependency ratio of country A will be much less than that of country B. 3. In the next two decades, the work-force relative to its total population will increase in country B as compared to country A. Select the correct answer using the code given below.
From the age-group curves, statements 1 and 3 are supported; statement 2 is not.
The graph given below indicates the changes in key policy rates made by the Central Bank several times in a year: [graph shown in paper]. Which one of the following can be the most likely reason for the Central Bank for such an action?
A sustained increase in policy rates is consistent with an anti-inflationary stance because higher rates tighten liquidity.
Table: GDP growth rate, per-capita income and tele-density for different states.
With reference to the above table, which of the following is/are the most logical and rational inference/inferences that can be made? 1. Higher per capita income is generally associated with higher Tele-density. 2. Higher GDP growth rate always ensures higher per capita income. 3. Higher GDP growth rate does not necessarily ensure higher Tele-density. Select the correct answer using the code given below.
The table supports only statement 3: higher GDP growth does not necessarily imply higher tele-density.
Table: GDP growth rate, per-capita income and tele-density for different states.
With reference to the above table, the following assumptions have been made: 1. Nowadays, prosperity of an already high performing State cannot be sustained without making further large investments in its telecom infrastructure. 2. Nowadays, a very high Tele-density is the most essential condition for promoting the business and economic growth in a State. Which of the above assumptions is/are valid?
Neither assumption follows necessarily from the table.
Graph: composition of tax revenue over two decades.
The following graph indicates the composition of our tax revenue for a period of two decades: [graph shown in paper]. With reference to the above graph, which of the following is/are the most logical and rational inference/inferences that can be made? 1. During the given period, the revenue from Direct Taxes as percentage of gross tax revenue has increased while that of Indirect Taxes decreased. 2. The trend in the revenue from Excise Duty demonstrates that the growth of manufacturing sector has been negative during the given period. Select the correct answer using the code given below.
Direct-tax share rises while indirect-tax share falls; the graph alone does not prove negative manufacturing growth. Hence 1 only.
If x - y = 8, then which of the following must be true? 1. Both x and y must be positive for any value of x and y. 2. If x is positive, y must be negative for any value of x and y. 3. If x is negative, y must be positive for any value of x and y. Select the correct answer using the code given below.
x−y=8 does not force either variable to have the asserted sign pattern; none of the three statements must always hold.
No common passage
If there is a policy that 1/3rd of a population of a community has migrated every year from one place to some other place, what is the leftover population of that community after the sixth year, if there is no further growth in the population during this period?
Each year 2/3 remains; after six years the remainder is (2/3)^6 = 64/729.
No common passage
There are three pillars X, Y and Z of different heights. Three spiders A, B and C start to climb on these pillars simultaneously. In one chance, A climbs on X by 6 cm but slips down 1 cm. B climbs on Y by 7 cm but slips down 3 cm. C climbs on Z by 6.5 cm but slips down 2 cm. If each of them requires 40 chances to reach the top of the pillars, what is the height of the shortest pillar?
For the final climb there is no subsequent slip: heights are 39×5+6=201, 39×4+7=163, and 39×4.5+6.5=182 cm. The shortest is 163 cm.
No common passage
Gopal bought a cell phone and sold it to Ram at 10% profit. Then Ram wanted to sell it back to Gopal at 10% loss. What will be Gopal’s position if he agreed?
With original cost 100, Gopal sells at 110 and buys back at 99, a net gain of 1%.
No common passage
Suppose the average weight of 9 persons is 50 kg. The average weight of the first 5 persons is 45 kg, whereas the average weight of the last 5 persons is 55 kg. Then the weight of the 5th person will be
Total weight is 450; first five total 225 and last five total 275. Their sum double-counts person 5, so the fifth weighs 225+275−450=50 kg.
No common passage
P = (40% of A) + (65% of B) and Q = (50% of A) + (50% of B), where A is greater than B. In this context, which of the following statements is correct?
Uploaded/official key is D, but direct algebra gives P−Q=0.15(B−A)<0 for A>B, so Q>P (option B). Preserve D as the official keyed answer and flag the question for manual review.
No common passage
In a city, 12% of households earn less than ₹30,000 per year, 6% households earn more than ₹2,00,000 per year, 22% households earn more than ₹1,00,000 per year and 990 households earn between ₹30,000 and ₹1,00,000 per year. How many households earn between ₹1,00,000 and ₹2,00,000 per year?
The ₹30,000–₹1,00,000 group is 66% and equals 990, so total households are 1500. The ₹1–2 lakh band is 22%−6%=16%, or 240.
No common passage
There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?
If the first five odd terms average 39, the first is 35. Thirteen terms run 35 to 59; the middle (7th) term and overall mean are 47.
No common passage
Six boys A, B, C, D, E and F play a game of cards. Each has a pack of 10 cards. F borrows 2 cards from A and gives away 5 to C who in turn gives 3 to B while B gives 6 to D who passes 1 to E. Then the number of cards possessed by D and E is equal to the number of cards possessed by
Final counts are A=8, B=11, C=12, D=15, E=11, F=3; D+E=26, equal to B+C+F=26.
No common passage
There is a milk sample with 50% water in it. If 1/3rd of this milk is added to equal amount of pure milk, then water in the new mixture will fall down to
Take 3 units of the original mixture: 1/3 is 1 unit containing 0.5 water; add 1 unit pure milk, total 2 units with 0.5 water, so water is 25%.
No common passage
A freight train left Delhi for Mumbai at an average speed of 40 km/hr. Two hours later, an express train left Delhi for Mumbai, following the freight train on a parallel track at an average speed of 60 km/hr. How far from Delhi would the express train meet the freight train?
The freight train gains an initial 80 km. The express closes the gap at 20 km/h, taking 4 hours; distance = 60×4 = 240 km.
No common passage
Certain 3-digit numbers have the following characteristics: 1. All the three digits are different. 2. The number is divisible by 7. 3. The number on reversing the digits is also divisible by 7. How many such 3-digit numbers are there?
Enumerating three-digit multiples of 7 whose reversals are also divisible by 7 and whose digits are all distinct gives 4.
No common passage
How many numbers are there between 99 and 1000 such that the digit 8 occupies the units place?
From 100 through 999, each of the 9 hundreds choices has 10 tens choices and one units digit 8, giving 90.
No common passage
If for a sample data, Mean < Median < Mode, then the distribution is
Mean < median < mode is the standard ordering for a negatively skewed (left-skewed) distribution.
No common passage
The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?
The previous year age can be 25 (=5²) and the next year 27 (=3³), so the current age is 26. The next cube age is 64, which is 38 years after the current age; hence B.
No common passage
P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings?
Let Q's rate be 1, so P=3. P+Q=4, which is four times R, so R=1. Earnings ratio is 3:1:1.
No common passage
A bag contains 20 balls. 8 balls are green, 7 are white and 5 are red. What is the minimum number of balls that must be picked up from the bag blindfolded (without replacing any of it) to be assured of picking at least one ball of each colour?
In the worst case 8 green + 7 white = 15 balls are drawn before any red; the 16th guarantees at least one red, so all three colours are present.
No common passage
The outer surface of a 4 cm × 4 cm × 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm × 1 cm × 1 cm small cubes. How many small cubes do not have painted faces?
Only the inner 2×2×2 cubes have no painted face, giving 8.
No common passage
The average rainfall in a city for the first four days was recorded to be 0.40 inch. The rainfall on the last two days was in the ratio of 4 : 3. The average of six days was 0.50 inch. What was the rainfall on the fifth day?
Total rainfall is 6×0.50=3.00 inches; first four total 1.60, so the last two total 1.40. At 4:3, the fifth day is 0.80 inch.
No common passage
A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?
Testing two-digit numbers and their reversals gives a maximum remainder of 45.
No common passage
The monthly incomes of X and Y are in the ratio of 4 : 3 and their monthly expenses are in the ratio of 3: 2. However, each saves Rs. 6,000 per month. What is their total monthly income?
Let incomes be 4x and 3x, expenses 3y and 2y. Equal savings give x=y=6000, so total income is 42,000.
No common passage
Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1 m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling?
x²+1²+8²=9², so x=4 m.
No common passage
There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always 27. How many such maximum 2-digit numbers are there?
For 10a+b and 10b+a, the absolute difference is 9|a−b|=27, so |a−b|=3. Seven two-digit numbers satisfy this, so none of the listed counts is correct.
No common passage
What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150?
1–9 use 9 digits, 10–99 use 180, and 100–150 use 153; total 342.
There are some nectar-filled flowers on a tree and some bees are hovering on it. If one bee lands on each flower, one bee will be left out. If two bees land on each flower, one flower will be left out. The number of flowers and bees respectively are:
Let flowers be F and bees B. B=F+1 from the first condition and B=2(F-1) from the second. Solving gives F=3 and B=4.
There is an order of 19000 quantity of a particular product from a customer. The firm produces 1000 quantity of that product per day out of which 5% are unfit for sale. In how many days will the order be completed?
Saleable production is 95% of 1000 = 950 units/day. 19,000 ÷ 950 = 20 days.
A class starts at 11:00 am and lasts till 2:27 pm. Four periods of equal duration are held during this interval. After every period, a rest of 5 minutes is given to the students. The exact duration of each period is:
The total interval is 207 minutes. Three rests take 15 minutes, leaving 192 minutes for four equal periods: 192/4=48 minutes.
In aid of charity, every student in a class contributes as many rupees as the number of students in that class. With the additional contribution of Rs. 2 by one student only, the total collection is Rs. 443. Then how many students are there in the class?
If there are n students, the base collection is n². With the extra ₹2, n²+2=443, so n²=441 and n=21.
A person is standing on the first step from the bottom of a ladder. If he has to climb 4 more steps to reach exactly the middle step, how many steps does the ladder have?
The first step is position 1; four more steps puts the person on step 5. If step 5 is exactly the middle, there must be 4 steps on each side, so the ladder has 9 steps.
In a box of marbles, there are three less white marbles than the red ones and five more white marbles than the green ones. If there are a total of 10 white marbles, how many marbles are there in the box?
With W=10, red=13 and green=5. Total = 13+10+5 = 28.
Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets together at the target in an hour is
The LCM of 6, 7, 8, 9 and 12 is 504 seconds. In one hour they coincide 7 times after the initial firing, or 8 times if the starting instant is counted; hence both B and C are keyed.
A group of 630 children is seated in rows for a group photo session. Each row contains three less children than the row in front of it. Which one of the following number of rows is not possible?
For n rows with common difference 3, the total 630 must be representable as an arithmetic-series sum; six rows gives an integral positive first term, while the other listed row counts do not.
Assume that 1. The hour and minute hands of a clock move without jerking. 2. The clock shows a time between 8 o'clock and 9 o'clock. 3. The two hands of the clock are one above the other. After how many minutes (nearest integer) will the two hands be again lying one above the other?
The hands coincide again after 720/11 minutes ≈ 65.45 minutes, which rounds to 65.
For a charity show, the total tickets sold were 420. Half of these tickets were sold at the rate of Rs. 5 each, one-third at the rate of Rs. 3 each and the rest for Rs. 2 each. What was the total amount received?
210 tickets at ₹5 = ₹1,050; 140 at ₹3 = ₹420; remaining 70 at ₹2 = ₹140; total = ₹1,610.
There are five hobby clubs in a college viz., photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?
All five meet together every LCM(2,3,4,5,6)=60 days. Within 180 days, the common meeting days are 60, 120 and 180: three times.
In a class of 45 students, a boy is ranked 20th. When two boys joined, his rank was dropped by one. What is his new rank from the end?
His rank from the front becomes 21st. With 47 students, rank from the end is 47−21+1=27th.
A thief running at 8 km/hr is chased by a policeman whose speed is 10 km/hr. If the thief is 100 m ahead of the policeman, then the time required for the policeman to catch the thief will be
Relative speed is 2 km/h = 2000 m/h. Time = 100/2000 hour = 3 minutes.
A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete the total journey, what is the original speed of the train in km/hr?
Let the original speed be v. Then 63/v + 72/(v+6)=3. Solving gives v=42 km/h.
Q47 visual: relation between x men and y days shown in four diagrams. Q48 matrix: [3, 370, 7]; [2, 224, 6]; [1, 730, X]. Q49 table: Car A: mileage 8 km/l, 20 hours, total payment ₹2120 Car B: mileage 10 km/l, 25 hours, total payment ₹1950 Car C: mileage 9 km/l, 24 hours, total payment ₹2064 Car D: mileage 11 km/l, 22 hours, total payment ₹1812 Q50 three L-shaped number figures: first 84,14,12; second 81,18,9; third 88,?,11. Exact figures/table are in the visual asset.
x men, working at constant speed, do a certain job in y days. Which one of these diagrams shows the relation between x and y?
For a fixed amount of work at constant speed, y is inversely proportional to x; the correct graph is the decreasing curved hyperbola, diagram IV.
Q47 visual: relation between x men and y days shown in four diagrams. Q48 matrix: [3, 370, 7]; [2, 224, 6]; [1, 730, X]. Q49 table: Car A: mileage 8 km/l, 20 hours, total payment ₹2120 Car B: mileage 10 km/l, 25 hours, total payment ₹1950 Car C: mileage 9 km/l, 24 hours, total payment ₹2064 Car D: mileage 11 km/l, 22 hours, total payment ₹1812 Q50 three L-shaped number figures: first 84,14,12; second 81,18,9; third 88,?,11. Exact figures/table are in the visual asset.
Consider the following matrix: 3 370 7 2 224 6 1 730 X What is the number at 'X' in the above matrix?
In each row, the middle three-digit number is formed from the left and right numbers by the same pattern; applying that pattern to 1 and X gives X=9.
Q47 visual: relation between x men and y days shown in four diagrams. Q48 matrix: [3, 370, 7]; [2, 224, 6]; [1, 730, X]. Q49 table: Car A: mileage 8 km/l, 20 hours, total payment ₹2120 Car B: mileage 10 km/l, 25 hours, total payment ₹1950 Car C: mileage 9 km/l, 24 hours, total payment ₹2064 Car D: mileage 11 km/l, 22 hours, total payment ₹1812 Q50 three L-shaped number figures: first 84,14,12; second 81,18,9; third 88,?,11. Exact figures/table are in the visual asset.
Examine the following three figures in which the numbers follow a specific pattern. The missing number (?) in the third figure above is
The pattern is consistent with the product/quotient relation in the first two L-shaped figures; for the third figure the missing number is 16.
If the Psychology professors in the University constitute 2% of all the professors in the University, then what is the number of professors in the University?
Psychology has 4+6=10 professors. If 10 is 2%, total = 10/0.02 = 500.
In a rare coin collection, there is one gold coin for every three non-gold coins. 10 more gold coins are added to the collection and the ratio of gold coins to non-gold coins would be 1:2. Based on the information, the total number of coins in the collection now becomes
Let original gold = x and non-gold = 3x. After adding 10 gold, (x+10):3x=1:2, giving x=20. New total = 30+10=70? However the official key is A; re-checking the arithmetic shows the intended original relation yields x=20, non-gold=60, new gold=30, total=90.
A gardener has 1000 plants. He wants to plant them in such a way that the number of rows and the number of columns remains the same. What is the minimum number of plants that he needs more for this purpose?
The next perfect square after 1000 is 32²=1024, so 24 more plants are required.
A sum of ₹700 has to be used to give seven cash prizes to the students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, what is the least value of the prize?
Let the smallest prize be x. The seven prizes are x, x+20, ..., x+120. Their sum is 7x+420=700, so x=40.
Out of 120 applications for a post, 70 are male and 80 have a driver's license. What is the ratio between the minimum to maximum number of males having driver's license?
Minimum overlap = 70+80−120=30; maximum overlap = min(70,80)=70. Ratio 30:70 = 3:7.
In a garrison, there was food for 1000 soldiers for one month. After 10 days, 1000 more soldiers joined the garrison. How long would the soldiers be able to carry on with the remaining food?
The original stock equals 30,000 soldier-days. After 10 days, 20,000 soldier-days remain; with 2,000 soldiers, this lasts 10 days.
The tank-full petrol in Arun's motor-cycle lasts for 10 days. If he starts using 25% more everyday, how many days will the tank-full petrol last?
Daily consumption becomes 1.25 times the original, so the number of days becomes 10/1.25=8.
A person can walk a certain distance and drive back in six hours. He can also walk both ways in 10 hours. How much time will he take to drive both ways?
Let walking time one way be w and driving time one way be d. Then w+d=6 and 2w=10, so w=5 and d=1; driving both ways takes 2 hours.
A contract on construction job specifies a penalty for delay in completion of the work beyond a certain date as follows: ₹200 for the first day, ₹250 for the second day, ₹300 for the third day etc., the penalty for each succeeding day being ₹50 more than that of the preceding day. How much penalty should the contractor pay if he delays the work by 10 days?
The penalties form an AP with a=200, d=50, n=10. Sum = 10/2 × [2(200)+9(50)] = ₹4,250.
Attempt questions one by one with UPSC-style marking and immediate explanations.