quantitative ability - level a

NARENSINSTITUTE FOR COMPETITIVE EXAMINATIONS

QUANTITATIVE ABILITY LEVEL - A

Numerical ReasoningProblems on NumbersProblems on AgesRatio and ProportionAlligation or MixtureChain RulePartnershipVenn Diagram

Numerical ReasoningArea and VolumeProbabilityTime and Work (Pipes)SI and CIAveragePermutation and CombinationPercentage

Numerical ReasoningBoats and StreamsTime and Distance (Trains)Data SufficiencyProfit and LossCalendarClocksData InterpretationCubes

Problems on NumbersDivision Algorithm:Dividend = the number to be divided.Divisor = the number by which it is divided.Dividend / Divisor = Quotient.Quotient * Divisor = Dividend.Quotient * Divisor + Remainder = Dividend.

Problems on NumbersArithmetic Progression: The nth term of A.P. is given by Tn = a + (n 1)d;

Sum of n terms of A.P Sn = n/2 *(a + L) or n/2 *[2a+(n-1)d)]

a = 1st term, n = number of term, d= difference, Tn = nth term

Geometrical Progression: Tn = arn 1.

Sn = a(rn 1)/(r-1); Where a = 1st term , r = 1st term / 2nd term

Basic Formulae1. ( a+b)2 = a2 + b2 + 2ab2. (a-b)2 = a2 +b2 -2ab3. ( a+b)2 - (a b)2 = 4ab4. (a+b)2 + (a b)2 = 2 (a2 +b2)5. (a2 b2) = (a+b) (a-b)6. (a+b+c)2 =a2 +b2 +c2 + 2(ab +bc+ca)7. (a3 +b3) = ( a+b) (a2 +ab +b2)8. (a3 b3) = (a-b) (a2 - ab + b2)

Problems on NumbersThree numbers are in the ratio 3:4:5. the sum of the largest and the smallest equal to the sum of the third and 52. Find the smallest number ?

Problems on NumbersSolution:

Let the numbers be 3x, 4x and 5x

Then 5x+3x = 4x +52 8x 4x = 52 4x = 52 x = 52/4 x = 13

The smallest number = 3x = 3*13 = 39.

Problems on NumbersWhat is one half of two third of three fourths of four fifths of five sixth of six sevenths of seven eights of eight ninth of nine tenths of thirty?

Problems on NumbersSolution:

= * 2/3 *3/4 * 4/5 * 5/6*6/7*7/8*8/9*9/10 *30= 3

Problem on NumbersIf the operation ^ is defined by the equator x ^ y = 2x + y what is the value of a in 2 ^ a = a ^ 3? (Satyam)

Problem on NumbersSolution:2(2) = a ^ 34 + a = 2a + 3 a = 1

Problem on NumbersThere are 150 weight some are 1 kg weight and some are 2 kg weights. The sum of the weights is 260. what is the number of 1 kg weight. (TCS)

Problem on NumbersSolution:X + 2Y = 260 X + Y = 150On Solving Two Equations Y = 110X + Y = 150X = 150 110 = 40 Kg

Problem on NumbersThe cost of 1 pencil, 2 pens and 4 erasers is Rs. 22, while the cost of five pencils, four pens and two eraser is 32. how much will 3 pencils, 3 pens and 3 eraser? (TCS)

Problem on NumbersSolution:Let Pencil be x, Pens be y, Erasers be z x + 2y + 4z = 225x + 4y + 2z = 32Adding we get 6x+6y+6z = 54 3x + 3y + 3z = 273 Pencil, 3 Pens and 3 Eraser is Rs. 27.

Problem on NumbersIf the numerator of a fraction is increased by 25% and denominator decrease by 20%, the new value is 5/4. what is the original value? (TCS)

Problem on NumbersSolution:( x + 25x/100) / (y 20y/100) = 5/4125x / 80y = 5/4x/y = 5/4 * 80/ 125 = 4/5

Problem on NumbersThe difference between two numbers is 1/7 of the sum of these two numbers. What is the ratio of the two numbers? (Wipro)

Problem on Numbers (x- y ) = 1/7 (x+y)7( x- y) =( x + y)7x 7y = x + y6x = 8yx/y = 3 / 4

Problem on NumbersA fraction has a denominator greater than its numerator by 4. but if you add 10 to the denominator, the value of the fraction would then become 1/8, what is the fraction? (Caritor)

Problem on NumbersSolution:x/(x+4+10) = 1/8x/(x+14) = 1/88x =( x + 14)7x = 14; hence x = 2x/(x+4 )= 2/(2+4 )= 2/6

Problems on AgesThe ages of two persons differ by 10 years. If 5 years ago, the elder one be 2 times as old as the younger one, find their present ages.

Problems on AgesSolution:x - y = 10; x = 10 + yx - 5 = 2(y-5)y + 10 -5 = 2y -10y+5 = 2y -102y- y = 15y=15 and x = 25Their present ages are 15 years and 25 years.

Problems on AgesThe present ages of three persons are in the proportion of 4:7:9. Eight years ago, the sum of their ages was 56. Find their present ages ?

Problems on AgesSolution:Three persons ratio = 4:7:9 Total = 4+7+9 = 20Sum of their age = 56, after 8 years their sum = 56 +24 = 80As age = 4/20 *80 = 16Bs age = 7/20 *80 = 28Cs age = 9/20 *80 = 36Their present ages are 16, 28 and 36.

Problems on AgesFathers age is 5 times his son's age.4 years back the father was 9 times older than his son. Find the father's present age? (TCS)

Problems on AgesSolution: F = 5SF 4 = 9(S-4)F 5s = 0F 9S = -36 + 4 = -32 4S = 32S = 8Father age = 40 years

Problems on AgesOne year ago Pandit was three times his sisters age. Next year he will be only twice her age. How old will Pandit be after five years? (TCS)

Problems on AgesSolution:(P-1) = 3(S -1)P + 1 = 2( s+1)P 3S = -3 + 2 = -2P 2S = 2-1 = 1S = 3P 3(3) = -2P 9 = -2P = -2 + 9 = 7After 5 years = 12

Problems on AgesA father is 30 years older than his son, however he will be only thrice as old as his son after 5 years what is fathers present age?

Problems on AgesSolution:F = S + 30F + 5 = 3(S+5)S+30 + 5 = 3S + 152S = 20S= 10F = 10 + 30 = 40

Problems on AgesA father is three times as old as his son after 15 years the father will be twice as old as his sons age at that time. What is the fathers present age ? (TCS)

Problems on AgesSolution:F = 3SF + 15 = 2(S +15)Fathers age = 45, Sons age = 15

Ratio and Proportion Ratio: The Relationship between two variables is ratio.

Proportion: The relationship between two ratios is proportion.

Ratio and Proportion The two ratios are a : b and the sum nos. is x ax bx -------- and ------- a + b a + b

Similarly for 3 numbers a : b : c

Ratio and ProportionIf Rs. 1260 is divided among A, B, C in the ratio 2 : 3 : 4 what is Cs share?

Ratio and ProportionSolution: Cs Share = 4/9*1260 Cs share = Rs. 560

Ratio and Proportion

To 15 liters of water containing 20% alcohol, we add 5 liters of pure water. What is the % of alcohol?

Ratio and ProportionSolution: 15 lit 20 %20 lit (15+5) x by solving we get = 15% 15% alcohol


What number should be added or subtracted from each term of the ratio 17 : 24 so that it becomes equal to 1 : 2

Ratio and ProportionSolution:

Let the number be x.

17 + x/24 + x = 1/2

Solving the above equation,

The number to be subtracted is 10.


The ratio of white balls and black balls is 1:2. If 9 gray balls are added it becomes 2:4:3. Then what is the number of black balls ?

Ratio and Proportion Solution: Ratio of all the three balls = 2:4:3Ratio of two balls before adding gray = 1:29 gray ratio =3 3 parts = 9 balls 1 part = 9/3 4 parts =? = 9*4/3 =12 Number of black balls is 12

Ratio and ProportionRs. 770 was divided among A, B and C such that A receives 2/ 9th of what B and C together receive. Find As share?

Ratio and ProportionSolution:A = 2/9 (B+C)B+C =9A/2A+B+C = 770A + 9A/2 = 77011A = 770*2A = 140

Alligation or Mixture(Quantity of cheaper / Quantity of costlier) (C.P. of costlier) (Mean price)= -------------------------------------- (Mean price) (C.P. of cheaper)

Alligation or Mixture

Cost of Cheaper Cost of costlier c d

Cost of Mixture m

d-m m-c

(Cheaper quantity) : (Costlier quantity) = (d m) : (m c)


A merchant has 100 kg of salt, part of which he sells at 7% profit and the rest at 17% profit. He gains 10% on the whole. Find the quantity sold at 17% profit?

Alligation or MixtureSolution: 7 17 10 (17-10) (10-7) 7 : 3The ratio is 7:3The quantity of 2nd kind = 3/10 of 100kg = 30kg

Alligation or MixtureA 3-gallon mixture contains one part of S and two parts of R. In order to change it to mixture containing 25% S how much R should be added?

Alligation or MixtureAnswer:

R:S2:175%:25%3:11 gallon of R should be added.


In an examination out of 480 students 85% of the girls and 70% of the boys passed. How many boys appeared in the examination if total pass percentage was 75%

Alligation or MixtureSolution: 85 7510 5

Number of Boys = 480 * 10/15Number of Boys = 320


In two varieties of tea, one costing Rs. 25/kg. and the other costing RS. 30/kg are blended to produce blended variety of tea in ratio 2:3. find the cost price of the mixture ?

Alligation or MixtureSolution: 25 30 x

(30- 28) (28-25) 2 : 3 Let mixed price be xIf you subtract 28 from 30 you will get 2 and if you subtract 25 from 28 you will get 3.


A person has Rs. 5000. He invests a part of it at 3% per annum and the remainder at 8% per annum simple interest. His total income in 3 years is Rs. 750. Find the sum invested at different rates of interest.

Alligation or MixtureSolution:Average rate of interest = 5% per annum3%8% 5%3%2%Investment at 3% per annum = 3x5000/5= 3000Investment at 8% per annum = 2x5000/5=2000

Chain RuleDirect Proportion : A B A BIndirect Proportion: A B A B

Chain RuleA Garrison of 500 men had provision for 27 days. After 3 days, a reinforcement of 300 men arrived. The remaining food will now last for how many days?

Chain RuleSolution:Mendays50024800x 800X = 500x24X =(500x24)/800 =15 days

Chain Rule

If 20 men take 15 days to complete a job. In how many days will 25 men finish the work?

CTS Question

Chain RuleSolution:Men Days 20 15 25 xx/15=20/25x = (20*15)/25 =12They will take 12 days

Chain Rule

If 11.25m of a uniform iron rod weighs 42.75 kg, what will be weight of 6m of the same rod?

Chain RuleSolution: length ( m )weight ( kg ) 11.2542.75

6 xSince it is a direct proportion, x 6 6 x 42.75 --------- = -------- x = --------------- 42.75 11.25 11.25 The weight of rods x = 22.8 kg

Chain RuleA stationary engine has enough fuel to run 12 hours when its tank is 4/5 full. How long will it run when the tank is 1/3 full?

TCS Question

Chain RuleAnswer:Tank hours 4/5 12 1/3 x4/5 x = 12 * 1/3It will run for 5 hours

Chain Rule20 men complete one - third of a piece of work in 20 days. How many more men should be employed to finish the rest of the work in 25 more days?

Chain RuleSolution: Men days work 20 20 1/3 work done = 1/3 x 25 2/3 remaining = 1-1/3=2/3More work, more men (direct proportion)More days, less men (indirect Proportion)1/3 *X = (2/3 )*20*(20 /25) X = 800/25 = 32More men to be employed = (32-20)12 More people needed to finish the job

Chain Rule

15 men take 21 days of 8 hrs each to do a piece of work. How many days of 6 hrs each would 21 women take, if 3 women do as much work as two men?

Chain RuleSolution:3 women = 2 men21 women = 14 menMenDaysHrs 21814 x6x/21=(15/14)/(8/6)x= 30 days

Partnership

Types: A invested Rs.x and B invested Rs.y then A:B = x : yA invested Rs. x and after 3 months B invested Rs. y then the share isA:B = x * 12 : y * 9

Partnership

Sanjiv started a business by investing Rs. 36000. After 3 months Rajiv joined him by investing Rs. 36000. Out of an annual profit of Rs. 37100 find the share of each.Satyam Question

Partnership

Solution :36000 * 12: 36000 * 9 4 : 3Sanjivs share of profit = (4*37100)/7 = 21200 profit = 21200Rajivs share of profit = 15900

Partnership

A sum of money is divided among A, B, C such that for each rupee A gets, B gets 65 paise and c gets 35 paise if cs share is Rs. 560. what is the sum?

Partnership

SolutionA : B : C100 : 65 : 3520 : 13 : 7 Total = 20+13+7 = 40C share = 5607/40 *X =560X= 3200

Partnership

A starts business with Rs.3500 and 5 months after B joins A as his partner. After a year the profits are divided in the ratio of 2:3. How much did B contribute ?

Partnership

Solution: A :B =3500*12 : 7X 42000 : 7X = 2: 3 7X * 2 = 42000 *3 X = 42000 * 3/14 X = 9000 Bs contribution is Rs.9000

Partnership

A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and As share is Rs. 855 what is the total profit?

Partnership

Solution:A : B = 3:2Let Profit be XX 5% of XX- 5X/100 = 95X/100 3/5 * 95X/100 = 28519X= 28500X = 26500/19 = 1500Total profit is Rs.1500

PartnershipA and B enter into partnership for a year. A contributes Rs.1500 and B Rs. 2000. After 4 months, they admit C who contributes Rs. 2250. If B withdraws his contribution after 9 months, find their profit share ratio at the end of the year?

PartnershipSolution:A: B: C = 1500*12: 2000*9: 2250*8 = 18000: 18000: 18000 = 1: 1: 1Profit share at the end of the year,1: 1: 1

Time and WorkIf A can do a piece of work in n days, then As 1 days work = 1/nIf A is thrice as B, then: Ratio of work done by A and B = 3:1

Pipes and Cisterns

P1 fills in x hrs. Then part filled in 1 hr is 1/x

P2 empties in y hrs. Then part emptied in 1 hr is 1/y

Pipes and Cisterns P1 and P2 both working simultaneously which fills in x hrs and empties in y hrs resp ( y>x) then net part filled is 1/x 1/y

P1 can fill a tank in X hours and P2 can empty the full tank in y hours( where x>y), then on opening both pipes, the net part empties in hour 1/y -1/x

Time and WorkOne fast typist types same matter in 2 hours and another slow typist types the same matter in 3 hours. If both do combine in how much time will they finish?TCS Question

Time and WorkSolution:Fast typist = 1/2 ; slow typist = 1/3 ; Together: = 1/2 + 1/3 = 5/6 so 6/5 hrsThe work will be completed in 6/5 Hrs.

Time and Work

A and B can finish a piece of work in 30 days, B and C in 40 days, while C and A in 60 days .In how many days A, B and C together can finish the work ?

Time and WorkSolution:A + B = 30 days = 1/30B + C = 40 days = 1/40C +A = 60 days = 1/60All work together A+B+C+B+C+A = 1/30 +1/40 +1/602(A+B+C) = 1/30+1/40+1/60 = (4+3+2) /120 = 9/120*2 = 9/240 = 3/80 = 26 2/3 A, B and C can finish the work in 26 2/3 days

Time and Work

10 men can complete a piece of work in 15 days and 15 women can complete the same work in 12 days. If all the 10 men and 15 women work together, In how many days will the work get completed ?

Time and WorkSolution:10 men = 15 days means 1day work = 1/1515 men = 12 days means 1 day work = 1/1210 men + 15 women = 1/15 + 1/12 = 4+5/60 = 9/60 = 3/20 20/3 days = 6 2/3 days

The work will be completed in 6 2/3 days.

Time and Work

A work done by two people in 24 minutes. One of them can do this work alone in 40 minutes. How much time is required to do the same work by the second person?TCS Question

Time and WorkSolution :A and B together = 1/24; A = 1/40; B = ? = 1/24 1/40 = 2/120 = 1/60 The second person will complete in 60 minutes.

Time and Work (Pipes)

A cistern has two taps which fill it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the pipes are opened, the empty cistern is filled in 20 minutes. How long will a waste pipe take to empty a full cistern ?

Time and Work (Pipes)Solution:This problem is based on 2nd method.All the tap work together = 1/12 + 1/15 - 1/20 = 5/60 + 4/60 3/60 = 6/60 = 1/10 The waste pipe can empty the cistern in 10 minutes.


A tap can fill a cistern in 8 hours and another can empty it in 16 hours. If both the taps are opened simultaneously, Find the time ( in hours) to fill the cistern

Time and Work (Pipes)Solution:Tap 1 = 1/8 (fill); Tap 2 = 1/16 (empty) = 1/8 1/16 = 1 / 16Total time taken to fill the cistern = 16 hours


A water tank is 2/5th full. Pipe A can fill the tank in 10 minutes and the pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely?

Time and Work (Pipes) Answer :A = 1/10; B = 1/6 = 1/10 -1/6 = - 1/15Empty in 15 minutesTo empty 2/5 of the tank 2/5 * 15 = 6 Time taken (empty)= 6 minutes

Area and VolumeCube:Let each edge of the cube be of length a. then,Volume = a3cubic unitsSurface area= 6a2 sq.units.Diagonal = 3 a units.

Area and VolumeCylinder:Let each of base = r and height ( or length) = h. Volume = r2hSurface area = 2 r h sq. unitsTotal Surface Area = 2 r ( h+ r) units.

Area and VolumeCone:Let radius of base = r and height=h, thenSlant height, l = h2 +r2 unitsVolume = 1/3 r2h cubic unitsCurved surface area = rl sq.unitsTotal surface area = r (l +r)

Area and VolumeSphere:Let the radius of the sphere be r. then,Volume = 4/3 r3Surface area = 4 r2sq.units

Area and VolumeCircle: A= r 2 Circumference = 2 r Square: A= a 2 Perimeter = 4aRectangle: A= l x bPerimeter= 2( l + b)

Area and VolumeTriangle:A = *base*heightEquilateral = 3/4*(side)2 Area of the Scalene Triangle S = (a+b+c)/ 2 A = s*(s-a) * (s-b)* (s-c)

Area and VolumeWhat is the cost of planting the field in the form of the triangle whose base is 2.8 m and height 3.2 m at the rate of Rs.100 / m2

Area and VolumeSolution:Area of triangular field = 1/2 * 3.2 * 2.8 m2 = 4.48 m2 Cost = Rs.100 * 4.48 = Rs.448.

Area and Volume

Area of a rhombus is 850 cm2. If one of its diagonal is 34 cm. Find the length of the other diagonal?

Area and VolumeSolution: 850 = * d1 * d2 = * 34 * d2 = 17 d2 d2 = 850 / 17 = 50 cm Second diagonal = 50cm

Area and Volume

A grocer is storing small cereal boxes in large cartons that measure 25 inches by 42 inches by 60 inches. If the measurement of each small cereal box is 7 inches by 6 inches by 5 inches then what is maximum number of small cereal boxes that can be placed in each large carton ?

Area and VolumeSolution: No. of small boxes = (25*42*60 ) / ( 7*6*5 ) = 300300 boxes of cereal box can be placed.

Area and Volume

If the radius of a circle is diminished by 10%, what is the change in the area in percentage?

Area and VolumeSolution:= D1 + D2 D1*D2 /100= 10 + 10 10*10/100= 20 -1= 19% New area changed = 19%.

Area and Volume

A circular wire of radius 42 cm is bent in the form of a rectangle whose sides are in the ratio of 6:5. Find the smaller side of the rectangle?

Area and VolumeSolution:length of wire = 2 r = (22/7*14*14)cm = 264cmPerimeter of Rectangle = 2(6x+5x) cm = 22xcm 22x =264 x = 12 cmSmaller side = (5*12) cm = 60 cm

Area and Volume

A man is running around a rectangle. It takes time, 2 times in traveling length than traveling width and the total perimeter is 300 m. Find the Area?.

Area and VolumeSolution:Breadth = XLength = 2XArea = 2X*X = 2X2Perimeter 6X = 300m X = 50mArea = 2x2 =2*50*50Area = 5000 sq.m.

ProbabilityProbability: P(E) = n(E) / n(S) Addition theorem on probability: n(AUB) = n(A) + n(B) - n(AB)Mutually Exclusive: P(AUB) = P(A) + P(B)Independent Events: P(AB) = P(A) * P(B)

Probability

There are 19 red balls and One black ball. Ten balls are placed in one jar and remaining in one jar. What is probability of getting black ball in right jar ?(Infosys -2008)

ProbabilityAnswer:

Probability is 1/2.

Probability

There are 5 red shoes 4 green shoes. If one draws randomly a shoe what is the probability of getting a red shoe?CTS Question

ProbabilityAnswer:

The probability is 5/9

Probability

A bag contains 2 red, 3 green and 2 blue balls are to be drawn randomly. Two balls are drawn at random. What is the probability that the balls drawn contain only blue balls ?

ProbabilityAnswer :

The probability is 1/21

ProbabilitySam and Jessica are invited to a dance party. If there are 7 men and 7 women in total at the dance and 1 woman and 1 man are chosen to lead the dance, what is the probability that Sam and Jessica will not chosen to lead the dance ?

ProbabilityAnswer:

The Probability of Selecting = 1/7*7 = 1/49The Probability of not Selecting = 1-1/49 = 48/49

Probability

The letters of the word SOCIETY are placed in a row. What is the probability that the three vowels come together?

ProbabilityAnswer:

Required Probability = (5!*3! )/7! = 1/7

Simple / Compound Interest Simple Interest = PNR / 100Amount A = P + PNR / 100When Interest is Compound annually:Amount = P (1 + R / 100)nCompound Interest = A - P

Simple / Compound InterestHalf-yearly C.I.: Amount = P (1+(R/2)/100)2n

Quarterly C.I. :

Amount = P (1+(R/4)/100)4n

Simple/compound interestDifference between C.I and S.I for 2 years = P*(R/100)2.

Difference between C.I and S.I for 3 years = P{(R/100)3+3 (R/100)2 }

Simple / Compound InterestWhat is the S.I. on Rs. 3000 at 18% per annum for the period from 4th Feb 1995 to 18th April 1995Sathyam Question

Simple / Compound InterestAnswer: Time = 24+ 31+17 = 73 days = 73/365 = 1/5P = 3000; R = 18%; = PNR/100 = 3000*1*18/100*5The simple interest is Rs. 108

Simple / Compound InterestA sum of money doubles itself at C.I. in 15 years. In how many years will it become eight times?Satyam Question

Simple / Compound InterestSolution:A = P(1+R/100)152P =P(1+R/100) 15 ; 2 = (1+R/100)15If A = 8P8P = P(1+R/100)n23 = ( 1+R/100)[(1+R/100) 15]3 = (1+R/100)n n = 3*15It will take a period of 45 years.

Simple / Compound InterestRaja borrowed a certain money at a certain rate of S.I. After 5 years, he had to pay back twice the amount that he had borrowed. What was the rate of interest?TCS Question

Simple / Compound InterestSolution:A = 2PA = P + PNR/1002P = P(1+NR/100)2 = (1+5*R/100)1 = R/20 The rate of interest is 20%

Simple/compound interest

In simple interest what sum amounts to Rs. 1120 in 4 years and Rs. 1200 in 5 years?

CTS Question

Simple/Compound interestAnswer : Interest for 1 year = 1200 1120 = 80Interest for 4 year = 80*4 = 320A = 1120P = A P = 1120 320The Principal is Rs. 800

Simple/Compound interest

A simple interest amount for Rs. 5000 for 6 months is Rs. 200. What is the annual rate of interest?

CTS Question

Simple/Compound interestSolution:P = 5000; N = 6/12 = I = 200R = I *100 / P*N =200*100*2/5000*1 = 40/5 = 8% The annual rate of interest is 8%


A man earns Rs. 450 as an interest in 2 years on a certain sum invested with a company at the rate of 12% per annum. Find the sum invested.

Simple/Compound interestSolution:P = I*100/R*N = 450*100/12*2 Principal = Rs. 1875


If Rs. 85 amounts to Rs. 95 in 3 years, whatRs. 102 will amount in 5 years at the same rate percent?

Simple/Compound interestSolution:

Let P = Rs. 85; A = Rs. 95; I = 10/3 in 1 yearRate = I*100/P*N =4% ( app)Amount = P+PNR/100 = 102+20 =122Hence the amount in 5 years = Rs. 122


What will be the difference between S.I and C.I on a sum of Rs. 4500 put for 2 years at 5% per annum?

Simple/Compound interestSolution:

C.I S.I = P (R/100)2 = 4500(5/100)2 = 11.25

Difference = Rs. 11.25


What will be the C.I on Rs. 15625 for 2 years at 4% per annum?

Simple/Compound interestSolution:A = P(1+R/100)nA = 15625 [(1+4/100)2 ( 1+(4*1/2) / 100)] = 17238C.I = A P C.I = 17238 - 15625Compound interest = Rs. 1613

AverageAverage is a simple way of representing an entire group in a single value.Average of a group is defined as: X = (Sum of items) / (No of items)

AverageTotal temperature for the month of September is 840C. If the average temperature of that month is 28C. Find out the days in the month of September?

AverageSolution

Number of days= 840/28=30 days

AverageThe painter is paid x rupees for painting every 10m of a wall and y rupees for painting every extra meter. During one week, he painted 10m on Monday, 13m on Tuesday, 12m on Wednesday, 11m on Thursday and 12m on Friday. What is his average daily earning in rupees for the five day week?

AverageSolution:Day 1 = x, Day 2 = x+3y, Day 3 = x+2y, Day 4 = x+y, Day 5 = x+2y

Average = (x+ x+3y+ x+2y+ x+y+ x+2y) / 5 = 5x+8y / 5 = 5x/5 + 8y / 5Average for 5 days is x+ (8y/5)

Average

The average of 11 observations is 60. If the average of 1st five observations is 58 and that of last five is 56, find sixth observation?

AverageSolution:5 observations average = 58Sum = 58*5 = 290Last 5 observation average = 56Sum = 56*5 = 280Total sum of 10 numbers = 570 (290 + 280)Total sum of 11 numbers = 660 (11*60)6th number = 90 (660 -570)

Average

The average of age of 30 students is 9 years. If the age of their teacher included, it becomes 10 years. Find the age of the teacher?

AverageSolution:Total age of 30 students = 30*9 = 270

Including the teachers age = 31*10 = 310

Difference is = 310-270 = 40 years

Permutations and CombinationsFactorial Notation: n! = n(n-1)(n-2).3.2.1 Number of Permutations: n!/(n-r)!

Combinations: n!/r!(n r)!

Permutations and CombinationsA foot race will be held on Saturday. How many different arrangements of medal winners are possible if medals will be for first, second and third place, if there are 10 runners in the race

Permutations and CombinationsSolution:n = 10r = 3n P r = n!/(n-r)! = 10! / (10-3)! = 10! / 7! = 8*9*10 = 720 Number of ways is 720.

To fill a number of vacancies, an Employer must hire 3 programmers from 6 applicants, and two managers from 4 applicants. What is total number of ways in which she can make her selection ?Permutations and Combinations

Permutations and CombinationsSolution:It is selection so use combination formulaProgrammers and managers = 6C3 * 4C2 = 20 * 6 = 120Total number of ways = 120 ways.

Permutations and CombinationsIn how many ways can the letters of the word BALLOON be arranged so that two Ls do not come together?

Permutations and CombinationsSolution:

Total arrangement = 7! / 2!*2! (L and O occurred twice) =1260Ls come together (BAOON) (LL) = 6! / 2! = 3* 4* 5*6* = 360Ls not come together1260 360 = 900Number of ways = 900.

Permutations and CombinationsA man has 7 friends. In how many ways can he invite one or more of them to a party?

Permutations and CombinationsSolution: In this problem, the person is going to select his friends for party, he can select one or more person, so = 7C1 + 7C2+7C3 +7C4 +7C5 +7C6 +7C7 = 127 Number of ways is 127

Percentage By a certain Percent, we mean that many hundredths. Thus, x Percent means x hundredths, written as x%

PercentageTwo numbers are respectively 30% and 40% less than a third number. What is the second number as a percentage of first?

PercentageSolution:Let 3rd number be x.1st number = x 30% of x = x 30x/100 = 70x/ 100 = 7x/102nd number = x 40% of x = x 40x/100 = 60x/ 100 = 6x/10Suppose 2nd number = y% of 1st number6x / 10 = (y/100 )* 7x /10 y = 600 / 7 y = 85 5/7 Hence it is 85 5/7%

Percentage After having spent 35% of the money on machinery, 40% on raw material, 10% on staff, a person is left with Rs.60,000. What is the total amount of money spent on machinery and the raw materials?

PercentageSolution:Let total salary =100%Spending:Machinery + Raw material + staff = 35%+40%+10% = 85%Remaining percentage = 100 % 85% = 15%15 % of X = 60000X = 4, 00,000In this 4, 00,000 75% for machinery and raw material = 4, 00,000* 75/100 = 3, 00,000

Percentage

If the number is 20% more than the other, how much percent is the second number less than the first?

Percentage

Solution:Let X =20 = X / (100+X) *100% = 20 /120 *100% =16 2/3% The percentage is 16 2/3%

PercentageAn empty fuel tank of a car was filled with A type petrol. When the tank was half empty, it was filled with B type petrol. Again when the tank was half empty, it was filled with A type petrol. When the tank was half empty again, it was filled with B type petrol. What is the percentage of A type petrol at present in the tank?

PercentageSolution:Let capacity of the tank be 100 liters. Then,Initially: A type petrol = 100 litersAfter 1st operation:A = 100/2 = 50 liters, B = 50 litersAfter 2nd operation:A = 50 / 2+50 = 75 liters, B = 50/2 = 25 litersAfter 3rd operation:A = 75/2= 37.5 liters, B = 25/2 +50 = 62.5 litersRequired Percentage of Type A is = 37.5%

Percentage

Find the percentage increase in the area of a Rectangle whose length is increased by 20% and breadth is increased by 10%

PercentageAnswer:Percentage of Area Change=( X +Y+ XY/100)% =20+10+20*10/100 =32%

Percentage

If As income is 40% less than Bs income, then how much percent is Bs income more than As income?

PercentageAnswer:Percentage = R*100%/(100-R) = (40*100)/ (100-40) =66 2/3%

Percentage

One side of a square is increased by 30%. To maintain the same area by how much percentage the other side will have to be decreased?

PercentageAnswer:Percentage = r*100%/(100+r)= (30*100) / 130 = 23 1/3%

Boats and streamsUp stream against the streamDown stream along the streamu = speed of the boat in still water v = speed of stream Down stream speed (a)= u+v km / hrUp stream speed (b)=u-v km / hru = (a+b) km/hrV = (a-b) km / hr

Boats and streams

A man can row Upstream at 12 kmph and Downstream at 16 kmph. Find the mans rate in still water and rate of the current?

Boats and streamsSolution:

Rate in still water = 1/2 (16 + 12) = 14 Kmph

Rate of Current = 1/2 (16 12 ) = 2 Kmph

Boats and streamsA Boat is rowed down a river 40 km in 5 hr and up a river 21 km in 7 hr. Find the speed of the boat and the river?

Boats and streamsSolution:Speed of the Boat Downstream = 40/7 = 8 (a)Speed of the Boat Upstream = 21/7 = 3 (b)Speed of the Boat = 1/2 ( a + b ) = 1/2 ( 8+3 ) = 5.5 KmphSpeed of the River = 1/2 ( a b ) = 1/2 (8 3)= 2.5 kmph

Boats and streamsA boats crew rowed down a stream from A to B and up again in 7 hours. If the stream flows at 3km/hr and speed of boat in still water is 5 km/hr. find the distance from A to B ?

Boats and streamsSolution: Down Stream = Sp. of the boat + Sp. of the stream = 5 +3 =8Up Stream = Sp. of the boat Sp. of the stream = 5-3 = 2Let distance be XDistance/Speed = Time X/8 + X/2 = 7 X/8 +4X/8 = 15/2 5X / 8 = 15/2 5X = 15/2 * 8 5X = 60 X =12

Boats and Streams

A boat goes 40 km upstream in 8 hours and 36 km downstream in 6 hours. Find the speed of the boat in still water in km/hr?

Boats and StreamsSolution:Speed of the boat in upstream = 40/8 = 5 km/hrSpeed of the boat in downstream= 36/6 =6 km/hr Speed of the boat in still water = (5+6 ) / 2 = 5.5 km/hr

Boats and Streams

A man rows to place 48 km distant and back in 14 hours. He finds that he can row 4 kmph with the stream in the same time as 3 Kmph against the stream. Find the rate of the stream?

Boats and StreamsSolution:Down stream 4 km in x hours. Then,Speed Downstream = 4/x km/hr, Speed Upstream = 3/x km/hr48/ (4/x) + 48/(3/x) = 14x = 1/2Speed of Downstream = 8,Speed of upstream = 6Rate of the stream =1/2 (8-6) km/hr = 1 km/hr

Time and DistanceSpeed:- Distance covered per unit time is called speed.Speed = distance/time (or)Distance = speed*time (or)Time = distance/speed

Time and Distance Distance covered Time (direct variation).

Distance covered speed (direct variation).

Time 1/speed (inverse variation).

Time and DistanceSpeed from km/hr to m/sec - (Multi by 5/18).

Speed from m/sec to km/h, - (Multi by 18/5).

Average Speed:-

Average speed = Total distance traveled Total time taken

Time and Distance

From height of 8 m a ball fell down and each time it bounces half the distance back. What will be the distance traveled?Sathyam Question

Time and DistanceSolution: = 8 + 4 + 4+2+2+1+1+0.5+0.5 +.etc.The total distance traveled is 24 m

Time and Distance

Two cars are 15 km apart. One is running at a speed of 50 kmph and the other at 40 kmph. How much time will it take for the two cars to meet?Sathyam Question

Time and DistanceSolution:Time taken =Distance / (S1 S2) = 15 / (50 40) = 15 / 10 = 1.5It will take 1 hours.

Time and Distance

The center of a storm shifts 22.5 miles in 1 hour. At the same rate what time will it take to move 60 miles?TCS Question

Time and DistanceAnswer:

For 22.5 miles it takes 1 hourIt means for 60 miles T = D / STime taken = 60 / 22.5It will take 2 2/3 hours.

Time and DistanceBy walking at of his usual speed, a man reaches office 20 minutes later than usual. Find his usual time?

Time and DistanceSolution: Usual time = Numerator * late time = 3*20 = 60

Time and Distance

A man on motorcycle rides 110 miles in 330 minutes. What is his average speed in miles per hour?

TCS Question

Time and DistanceAnswer: Speed = D / T =110*60 /330 The average speed = 20 miles/hour

Time and Distance (Trains)A train starts from Delhi to Madurai and at the same time another train starts from Madurai to Delhi after passing each other they complete their journeys in 9 and 16 hours, respectively. At what speed does second train travels if first train travels at 160 km/hr ?

Time and Distance (Trains)Solution:Let x be the speed of the second trainS1 / S2 = T2/T1160/x = 16/9160/x = 4/3 x = 120The speed of second train is 120km/hr.

Time and Distance (Trains)Two hours after a freight train leaves Delhi a passenger train leaves the same station traveling in the same direction at an average speed of 16 km/hr. After traveling 4 hours the passenger train overtakes the freight train. What was the average speed of the freight train?Wipro Question

Time and Distance (Trains)Solution :Speed of Passenger train = 16 kmphDistance = 16*4 = 64Speed of freight train = Distance / ( S1 + S2 )= 64 / (4+2)= 64/6 = 10.6 km/hrThe average speed = 10.6 km/hr

Time and Distance (Trains)

There are 20 poles with a constant distance between each pole. A train takes 24 sec to reach the 12 pole. How much time will it take to reach the last pole ?

Time and Distance (Trains)Solution:To cross 11 poles it is taking 24 secTo cross 19 poles it will take x time Poles time 11 24 19 x11x = 19 * 24 x = 19* 24 /11 x = 41.45 sec It reaches the last pole in 41.45sec

Time and Distance (Trains)120 m long train crosses the pole after 2 sec. Find how much time it takes to cross 140 m long platform?Caritor Question

Time and Distance (Trains)Solution:To cross 120 m it is taking 2 sec. (5/2sec)To cross (120 +140)=260 m it will take x sec120x = 260*5/2 (apply chain rule) = 5 5/12 It takes 5 5/12 seconds.

Time and Distance (Trains)

A train X speeding with 120 kmph crossed another train Y, running in the same direction, in 2 minutes. If the lengths of the trains X and Y be 100 m and 200m respectively, what is the speed of train Y?

Time and Distance (Trains)Solution:Let the speed of train Y be x km/hrRelative Speed of X to Y = (120 x) km/hr = [(120 x)*5/18] m/sec =( 600 5x) / 18 m/secT = D / Rel. Speed300 / (600 5x /18) = 120 ( 2 Minutes ) 5400 = 120 (600 -5x) x = 111 m/sec.

Profit and Loss

Cost Price. - CP

Selling Price. - SP

Profit or Gain. - P = SP CP

Loss. - L = CP - SP

Profit and Loss.Gain% = [(Gain*100)/C.P.]

Loss% = [(Loss*100)/C.P.]

S.P. = ((100 + Gain%)/100)C.P.

S.P. = ((100 Loss%)/100)C.P.

Profit and Loss

Anu bought a necklace for Rs. 750 and sold it for Rs. 675. Find her Loss percentage?.

Profit and LossSolution:CP = 750, SP = 675 L = 750 675 = 75Loss% = loss /CP *100 = 75*100/750 = 10%Loss 10%

Profit and Loss

A shopkeeper bought a watch for Rs. 400 and sold it for Rs. 500. What is his profit percentage?TCS Question

Profit and LossSolution:CP = 400; SP = 500 P = 500 400 = 100Profit % = Profit /CP *100 = 100*100/400 = 25%Profit 25%

Profit and Loss

By Selling 15 Mangoes , a Fruit vendor recovers the cost price of 20 Mangoes. Find the profit percentage?

Profit and LossSolution:The expenditure and the revenue are equated,Percentage of profit =Goods left*100Goods sold = ( 5*100 ) /15 = 33.3%

Profit and Loss

A shopkeeper loses 7% by selling a cricket ball for Rs. 31. for how much should he sell the ball so as to gain 5%

Profit and LossSolution:First case S.P = Rs. 31 and Loss% = 7%C.P = [100/(100 loss%)]*S.P = (100*31) / (100-7) = 100 / 3Second case, C.P = Rs 100 / 3 and gain% = 5%S.P = [(100+gain 5%) / 100]*C.P = [ (100+5 ) / 100] * 100/3 = Rs. 35

Profit and LossWhat is the selling price of a Toy car if the cost of the car is Rs. 60 and a profit of 10% over selling price is earned?

CTS Question

Profit and LossSolution:Profit = 60*10/100 = 6Selling Price = C.P + Profit = 60 + 6Selling Price = Rs. 66

Profit and Loss

Find the single discount to a series discount 20%, 10% and 5%.

Profit and LossAnswer:SP = [( 80*90*95 )/ 100*100*100 ]* CP = 0.684 CPDiscount = (1 0.684) * 100% = 0.316 *100 %Discount = 31.6%

CalendarOdd days:0 = Sunday1 = Monday2 = Tuesday3 = Wednesday4 = Thursday5 = Friday6 = Saturday

CalendarMonth code: Ordinary yearJ = 0 F = 3M = 3 A = 6M = 1 J = 4J = 6 A = 2S = 5 O = 0N = 3 D = 5 Month code for leap year after Feb. add 1.

CalendarOrdinary year = (A + B + C + D )-2 -----------------------take remainder 7

Leap year = (A + B + C + D) 3 ------------------------- take remainder 7

Calendar

What is the day of the week on 30/09/2007?

CalendarSolution:A = 2007 / 7 = 5B = 2007 / 4 = 501 / 7 = 4C = 30 / 7 = 2D = 5 ( A + B + C + D )-2 = ----------------------- 7 = ( 5 + 4 + 2 + 5) -2 ----------------------- = 14/7 = 0 = Sunday 7

CalendarWhat was the day of the week on 13th May, 1984?

CalendarSolution:A = 1984 / 7 = 3B = 1984 / 4 = 496 / 7 = 6C = 13 / 7 = 6D = 2 ( A + B + C + D) -3 = ----------------------- 7 = 14/7= 0, Sunday.

Calendar

On what dates of April 2005 did Sunday fall?

CalendarSolution: You should find for 1st April 2005 and then you find the Sundays date.A = 2005 / 7 = 3B = 2005 / 4 = 501 / 7 = 4C = 1 / 7 = 1D = 6 (A + B + C + D) -2 = ----------------------- 7 3 + 4 + 1 + 6 -2 ----------------------- = 12 / 7 = 5 = Friday. 7 1st is Friday means Sunday falls on 3, 10, 17, 24

Calendar

What was the day on 5th January 1986?

CalendarSolution:

A = 1986 / 7 = 5B = 1986 / 4 = 496/7 = 6C = 5 / 7 = 5D = 0 (A + B + C + D) -2 = ----------------------- 7 5 + 6 + 5 + 0-2 = ----------------------- = 14 / 7 = Sunday 7

Clocks Clock:In every Hour, the minute hand gains 55 minutes on the hour handIn every hour both the hands coincide once. = (11m/2) 30h (hour hand to min hand) = 30h (11m/2) (min hand to hour hand)If you get answer in minus, you have to subtract your answer with 360 o

Clocks

Find the angle between the minute hand and hour hand of a clock when the time is 7:20.

Clocks Solution: = 30h (11m/2) = 30 (7) 11 20/2 = 210 110 = 100Angle between 7: 20 is 100o

Clocks

How many times in a day, the hands of a clock are straight?

ClocksSolution:

In 12 hours, the hands coincide or are in opposite direction 22 times a day.


Clocks

How many times do the hands of a clock coincide in a day?

ClocksSolution:



Clocks

At what time between 7 and 8 oclock will the hands of a clock be in the same straight line but, not together?

ClocksSolution:h = 7 = 30h 11m/2180 = 30 * 7 11 m/2On simplifying we get ,5 5/11 min past 7

Clocks

At what time between 5 and 6 oclock will the hands of a clock be at right angles?

ClocksSolution:h = 5 90 = 30 * 5 11m/2Solving10 10/11 minutes past 5

Clocks

Find the angle between the two hands of a clock at 15 minutes past 4 oclock

ClocksSolution:

Angle = 30h 11m/2 = 30*4 11*15 / 2The angle is 37.5o

Clocks

At what time between 5 and 6 oclock are the hands of a clock together?

ClocksSolution: h = 5

O = 30 * 5 11m/2 m = 27 3/11Solving

27 3/11 minutes past 5

Data InterpretationIn interpretation of data, a chart or a graph is given. Some questions are given below this chart or graph with some probable answers. The candidate has to choose the correct answer from the given probable answers.

1. The following table gives the distribution of students according to professional courses:__________________________________________________________________ Courses Faculty ___________________________________ Commerce Science Total Boys girls Boys girls___________________________________________________________Part time management 30 10 50 10 100C. A. only 150 8 16 6 180Costing only 90 10 37 3 140C. A. and Costing 70 2 7 1 80__________________________________________________________________On the basis of the above table, answer the following questions:

Data InterpretationThe percentage of all science students over Commerce students in all courses is approximately: (a) 20.5 (b) 49.4 (c) 61.3 (d) 35.1

Data InterpretationAnswer:

Percentage of science students over commerce students in all courses = 35.1%

Data InterpretationWhat is the average number of girls in all courses ? (a) 15 (b) 12.5 (c) 16 (d) 11


Average number of girls in all courses = 50 / 4 = 12.5

Data InterpretationWhat is the percentage of boys in all courses over the total students? (a) 90 (b) 80 (c) 70 (d) 76


Percentage of boys over all students = (450 x 100) / 500 = 90%

Data SufficiencyFind given data is sufficient to solve the problem or not.If statement I alone is sufficient but statement II alone is not sufficientIf statement II alone is sufficient but statement I alone is not sufficientIf both statements together are sufficient but neither of statement alone is sufficient.If both together are not sufficient

Data SufficiencyWhat is Johns age?In 15 years will be twice as old as Dias would beDias was born 5 years ago. (Wipro)

Data SufficiencyAnswer: c) If both statements together are sufficient but neither of statement alone is sufficient.

Data SufficiencyWhat is the distance from city A to city C in kms?City A is 90 kms from city B.City B is 30 kms from city C

Data SufficiencyAnswer:

d) If both together are not sufficient

Data SufficiencyIf A, B, C are real numbers, Is A = C? A B = B CA 2C = C 2B

Data SufficiencyAnswer:D . If both together are not sufficient

Data SufficiencyWhat is the 30th term of a given sequence?The first two term of the sequence are 1, The common difference is -1/2

Data SufficiencyAnswer:If statement I alone is sufficient but statement II alone is not sufficient

Data SufficiencyWas Avinash early, on time or late for work?He thought his watch was 10 minute fast.Actually his watch was 5 minutes slow.

Data SufficiencyAnswer:D. If both together are not sufficient

Data SufficiencyWhat is the value of A if A is an integer?A4 = 1A3 + 1 = 0

Data SufficiencyAnswer:B. If statement II alone is sufficient but statement I alone is not sufficient

CubesA cube object 3*3*3 is painted with green in all the outer surfaces. If the cube is cut into cubes of 1*1*1, how many 1 cubes will have at least one surface painted?

CubesAnswer:3*3*3 = 27All the outer surface are painted with colour. 26 One inch cubes are painted at least one surface.

CubesA cube of 12 mm is painted on all its sides. If it is made up of small cubes of size 3mm, and if the big cube is split into those small cubes, the number of cubes that remain unpainted is

CubesAnswer:

= 8

CubesA cube of side 5 cm is divided into 125 cubes of equal size. It is painted on all 6 sides.How many cubes are coloured on only one side?How many cubes are coloured on only two side? How many cubes are coloured on only three side?How many cubes are not coloured?

CubesAnswer:5436827

CubesA cube of 4 cm is divided into 64 cubes of equal size. One side and its opposite side is coloured painted with orange. A side adjacent to this and opposite side is coloured red. A side adjacent to this and opposite side is coloured green?Cont..

CubesHow many cubes are coloured with Red alone?How many cubes are coloured orange and Red alone?How many cubes are coloured with three different colours?How many cubes are not coloured?How many cubes are coloured green and Red alone?

CubesAnswer:88888

CubesA 10*10*10 cube is split into small cubes of equal size 2*2*2 each. A side and adjacent to it is coloured Pink. A side adjacent to Pink and opposite side is coloured Blue. The remaining sides are coloured yellow.Find the no. of cubes not coloured?Find the no. of cubes coloured blue alone?Find the no. of cubes coloured blue & pink & yellow?Find the no. of cubes coloured blue & pink ? Find the no. of cubes coloured yellow & pink ?

CubesAnswer:271841212

Venn DiagramIf X and Y are two sets such that X u Y has 18 elements, X has 8 elements, and Y has 15 elements, how many element does X n Y have?

Venn DiagramSolution:We are given n (X uY) = 18, n (X) = 8, n (Y) =15. using the formula. n( X n Y) = n (X) + n (Y) - n ( X u Y) n( X n Y) = 8 + 15 18n( X n Y) = 5

Venn DiagramIf S and T are two sets such that S has 21elemnets, T has 32 elements, and S n T has 11 elements, how many element elements does S u T have?

Venn DiagramAnswer:n (s) = 21, n (T) = 32, n ( S n T) = 11, n (S u T) = ?n (S u T) = n (S) + n( T) n (S n T) = 21 + 32 11 = 42

Venn DiagramIf A and B are two sets such that A has 40 elements, A u B has 60 elements and A n B has 10 elements, how many element elements does B have?

Venn DiagramAnswer: n ( A) = 40, n ( n B) = 60 and n ( A n B) = 10, n ( A u B) = n ( A) + n (B) n ( A n B) 60 = 40 + n (B) 10 n (B) = 30

Venn DiagramIn a group of 1000 people, there are 750 people who can speak Hindi and 400 who can speak English. How many can Speak Hindi only?

Answer: n( H u E) = 1000, n (H) = 750, n (E) = 400, n( H u E) = n (H) + n (E) n( H n E)1000 = 750 +400 n ( H n E) n ( H n E) = 1150 100 = 150No. of people can speak Hindi only _ = n ( H n E) = n ( H) n( H n E) = 750 150 = 600

Venn DiagramIn a class of 100 students, the number of students passed in English only is 46, in maths only is 46, in commerce only is 58. the number who passes in English and Maths is 16, Maths and commerce is 24 and English and commerce is 26, and the number who passed in all the subject is 7. find the number of the students who failed in all the subjects.

Venn DiagramSolution:No. of students who passed in one or more subjects = 11+ 9 + 13 + 17 + 15 + 19 + 7 = 91No of students who failed in all the subjects = 100 -91 = 9

Venn DiagramIn a group of 15, 7 have studied Latin, 8 have studied Greek, and 3 have not studied either. How many of these studied both Latin and Greek?

Venn DiagramAnswer: 3 of them studied both Latin and Greek.

THANK YOUSreenivasa Chowdary Nellore M.B.A, [Director]NARENS Institute for Competitive Examinations,HYDERABAD.E-mail: [email protected]: + 91 99123 31989, + 91 90593 14623

quantitative ability - level a

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