146

SQUARES OF NUMBER FROM 40 TO 125:Square of any number from 40 to 60All you have to do is to check the difference from 50 and add/subtract it from 25 and by the side of this number you have to write the square of that difference in 2 digits only (in case of 4, write 04).For example: if you have to find square of 44, the first thing you have to do is to find the difference from 50, i.e. 44 is 6 less than 50 so subtract 6 from 25, and you will get 19 and square of (-6) is 36. So44^2 = 1936.If you have to find 58^2, the first thing you have to do is to check difference from 50, i.e. 8, so adding 8 to 25, you get 33, and 8^2=64, so combining both of them, you can say that58^2=3364 Square of any number from 26 to 40 and 61 to 75Like the previous one the only difference here is that the square of the difference from 50 comes out to be a 3-digit number. So add the hundredth digit of the square to the sum obtained by adding/subtracting from 25.For example: if you have to find 62^2, the difference from 50 comes out to be 12 and 12^2=144, so you have to add 1(hundredth place digit) to 25+12 and 25+12+1=38. So62^2=3844If you have to find 36^2, first of all, we have to check the difference from 50, and the difference from 50 is (-14), so subtracting 14 from 25 gives me 11 and (-14)^2=196, so we have to add 1 of 196 to 11(25-14), and we get 11+1=12, now combining every thing we get36^2=1296 Square of any number from 90 to 110All you have to do is to check the difference from 100 and add/subtract double the difference of that number to 100, and by the side of this number you have to write the square of the difference.For example: if you have to calculate 106^2, all you have to do is to calculate the difference 100, (i.e. 106-100=6), now adding 6x2+100, gives me 112, now 6^2=36, so combining 112 and 36, we can say that106^2=11236If you have to calculate 93^2, firstly you have to check the difference from 100, i.e. 93=100-7; so I have to calculate 100-2(7)=86, and (-7)^2=49. So combining 86 and 49, we get93^2=8649. Square of any number from 76 to 89 and 111 to 125Just like the previous part, all you have to do is to check the difference from 100,and add/subtract that difference with 100 and along with that add the hundredth digit of the square of the difference to that sum and along with that write the units and tenth of the square.For example: if you have to calculate 112^2, all you have to do is to calculate the difference from100 and 112-100=12 and 12^2=144, so adding double of the 12 and 1(hundredth place of 144) to 100, we get 100+2(12)+1=125 and putting it with 44, we get112^2=12544.And if we have to calculate 87^2, the difference from 100 is (-13), and (-13)^ 2=169, so adding double of (-13) and 1(hundredth place of 169), we get 100+2(-13)+1=75, and combining 75 and 69, we get 87^2=7569POWERS OF 2 UPTO 15:

2^21/42^11/22^0 12^1 22^2 42^3 82^4 162^5 322^6 642^7 1282^8 2562^9 5122^1010242^1120482^1240962^1381922^14163842^1532768POWERS OF 3 UPTO 6:3^013^1 33^293^3273^4813^52433^6729

Table of Squares and Square Roots, 1-100numbersquaresqrtnumbersquaresqrt

111.000512,6017.141

241.414522,7047.211

391.732532,8097.280

4162.000542,9167.348

5252.236553,0257.416

6362.449563,1367.483

7492.646573,2497.550

8642.828583,3647.616

9813.000593,4817.681

101003.162603,6007.746

111213.317613,7217.810

121443.464623,8447.874

131693.606633,9697.937

141963.742644,0968.000

152253.873654,2258.062

162564.000664,3568.124

172894.123674,4898.185

183244.243684,6248.246

193614.359694,7618.307

204004.472704,9008.367

214414.583715,0418.426

224844.690725,1848.485

235294.796735,3298.544

245764.899745,4768.602

256255.00755,6258.660

266765.099765,7768.718

277295.196775,9298.775

287845.292786,0848.832

298415.385796,2418.888

309005.477806,4008.944

319615.568816,5619.000

321,0245.657826,7249.055

331,0895.745836,8899.110

341,1565.831847,0569.165

351,2255.916857,2259.220

361,2966.000867,3969.274

371,3696.083877,5699.327

381,4446.164887,7449.381

391,5216.245897,9219.434

401,6006.325908,1009.487

411,6816.403918,2819.539

421,7646.481928,4649.592

431,8496.557938,6499.644

441,9366.633948,8369.695

452,0256.708959,0259.747

462,1166.782969,2169.798

472,2096.856979,4099.849

482,3046.928989,6049.899

492,4017.000999,8019.950

502,5007.07110010,00010.000

Division Shortcut MethodsDivision is most frequently used in solving competitive exams problems. By learning shortcut methods of division, you can save crucial time in the exam. Here are some shortcut methods for division.In division, numerator is called Dividend and denominator is called Divisor.Method 1:Division using the factors of the divisorThis method is also called Double Division.Here you can directly divide 75 by 15 and the answer would be 5. But, to understand this method, we are not doing like that.Now, factories divisor.So we can write 15 as 5 3.Now we can divide 75 by 3 which gives us 25.Now we can divide 25 by 5 and that gives 5 as answer.So the answer is 5.Method 2: Division By PartsWrite 75 as 45 + 30So we can write=So the answer is 5.Method 3: Division by 10.Just move the decimal point one place to the left side.So the answer is 1.256Method 4: Division by 100Just move the decimal point two places to the left side.So the answer is 0.1256Method 5: Division by 5Divide the dividend by 100 and multiply by 20.So the answer is 40Method 6: Division by 50Divide the dividend by 100 and multiply it by 2.So the answer is 4.Method 7: Division by 25Divide the dividend by 100 and multiply it by 4.So the answer is 8DIVISIBILITY RULES:Divisible By:IfExamples

2The number is even. Or The last(units) digit of the number is 0, 2, 4, 6 or 884 is divisible by 2.85 is not divisible by 2.

3The sum of the digits of number is divisible by 3.1248 is divisible by 3.(1 + 2 + 4 + 8 = 15)

346 is not divisible by 3.(3 + 4 + 6 = 13)

4The last two digits of the number is divisible by 4. And the numbers having two or more zeros at the end.23456 is divisible by 4.(56 is divisible by 4)

13000 is divisible by 4.(Two or more zeros at the end.)

5The numbers having 0 or 5 at the end.12345 is divisible by 5.(5 is there at the end)

1234 is not divisible by 5.(0 or 5 is not there at the end)

6The number is divisible by both 2 and 3.5358 is divisible by 6.(It is divisible by both 2 and 3)

6782 is not divisible by 6.(It is divisible by 2 but not divisible by 3)

7The difference between twice the units digit and the number formed by other digits is either 0 or divisible by 7.861 is divisible by 7.[86 (1 2)) = 84 which is divisible by 7]

21 is divisible by 7.[2 (1 2)) = 0]

868 is divisible by 7.[86 (8 2)) = 70 which is divisible by 7]

8The number formed by last three digits is divisible by 8. And the numbers having three or more zeros at the end.2056 is divisible by 8.(056 is divisible by 8)

13000 is divisible by 8.(Three zeros at last)

9The sum of all the digits is divisible by 9.5301 is divisible by 9.(5 + 3 + 0 + 1 = 9 which is divisible by 9)

10The number ends with zero.467590 is divisible by 10.(It ends with zero)

11The difference between the sum of digits at even places and sum of digits at odd places is 0 or divisible by 11.10538 is divisible by 11.[(1 + 5 + 8) (0 + 3) = 11 which is divisible by 11]

724867 is divisible by 11.[(7 + 4 + 6) (2 + 8 + 7) = 0]

12The number is divisible by both 3 and 4.5472 is divisible by 12.( The number is divisible by both 3 and 4)

5475 is not divisible by 12.(The number is divisible by 3 but not divisible by 4)

13Method:Multiply last digit of the number by 4 and add it to the remaining number. Continue this process until two digit number is achieved. If this two digit number is divisible by 13 then the number is divisible by 13.182 is divisible by 13.(Multiply last digit by 4 i.e 24 = 8.Add it to the remaining number i.e 18 + 8 = 26.26 is divisible by 13 so 182 is divisible by 13)

2145 is divisible by 13( Multiply last digit by 4 i.e 54 = 20.Add it to the remaining number i.e 214 + 20 = 234 which is not a two digit number so repeat the process.Multiply last digit of 234 by 4 i.e. 44= 16.Add it to the remaining number i.e 23 + 16 = 39 which is divisible by 13 so 2145 is divisible by 13)

14The number is divisible by both 2 and 7.7966 is divisible by 14.(The number is divisible by both 2 and 7)

15The number is divisible by both 3 and 5.3525 is divisible by 15.(The number is divisible by both 3 and 5)

16The number formed by last four digits is divisible by 16.41104 is divisible by 16.(The number formed by last four digits is divisible by 16)

17Method:Multiply last digit with 5 and subtract it from the remaining number. If the result is divisible by 17 then the original number is also divisible by 17. Repeat this process if required.4029 is divisible by 17( Multiply last digit by 5 i.e. 95= 45.Subtract it from the remaining number.402 45 = 357 which is divisible by 17 so 4029 is divisible by 17)

18The number is even and divisible by 9.4428 is divisible by 18.(It is even and divisible by 9)

19Method:Multiply last digit with 2 and add it to the remaining number. If the result is divisible by 19 then original number is also divisible by 19. Repeat this process if required.1235 is divisible by 19.(Multiply last digit with 2 i.e 52 = 10.Add it to the remaining number 123 i.e 123 + 10 = 133 which is divisible by 19 so 1235 is also divisible by 19)

GENERAL SHORTCUT METHOD FOR SQUARE METHOD:You can find square of any number in the world with this method.Lets say the number is two digit number. i.e. AB.So B is units digit and A is tens digit.Step 1: Find Square of BStep 2:Find 2ABStep 3:Find Square of ALets take an example.We want to find square of 37.Step 1:Find square of 7. Square of 7 = 49. So write 9 in the answer and 4 as carry to the second step.Step 2:Find 2(37) 2 (3 7) = 42. 42 + 4(Carry) = 46. Write 6 in the answer and 4 as a carry to the third step.Step 3:Find square of 3 Square of 3 = 9 9 + 4(Carry) = 13. Write 13 in the answer. So the answer is 1369.Now, If the number is of three digit i.e. ABCHere C is units digit, B is tens digit and A is hundredth digit.Step 1:Find Square of CStep 2:Find 2 (B C)Step 3:Find 2 (A C) + B2(NOTE:You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: A and C) and squaring single digit (here B).Step 4:Find 2 (A B)(NOTE:You may observe that whenever there are double digits, we are multiplying it with 2. And whenever there is single digit, we are squaring it.)Step 5:Find square of A (NOTE:Here is single digit, so we are squaring it.)

Lets take an example.Find square of 456.Step 1:Find square of 6. Square of 6 = 36.So write 6 in the answer and 3 as a carry to the second step.Step 2:Find 2 (5 6) 2 (5 6) = 60 60 + 3(Carry) = 63 Write 3 in the answer and 6 as a carry to the third step.Step 3:Find 2 (4 6) + 52 2 (4 6) + 52= 73 73 + 6(Carry) = 79 Write 9 in the answer and 7 as a carry to the fourth step.Step 4:Find 2 (4 5) 2 (4 5) = 40 40 + 7 = 47 Write 7 in the answer and 4 as a carry to the fifth step.Step 5:Find square of 4 Square of 4 = 16 16 + 4(Carry) = 20 Write 20 in the answer.So 4562= 207936.Now, If the number is of four digit i.e. ABCDHere D is units digit, C is tens digit, B is hundredth digit and A is thousands digit.Step 1:Find Square of DStep 2:Find 2 (C D)Step 3:Find 2 (B D) + C2(NOTE:You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: B and D) and squaring single remaining digit (here C).Step 4:Find 2 (A D) + 2 (B C)(NOTE:You may observe that where ever there is even digits, we are multiplying end two digits with 2 + remaining two digits with 2.)Step 5:Find 2 (A C) + B2(NOTE:You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: A and C) and squaring single remaining digit (here B).Step 6:Find 2 (A B)(NOTE:You may observe that here even digits so we are multiplying them with 2, and no remaining digits so we are not adding anything.)Step 7:Find square of ALets take an example.Find square of 1234Step 1:Find Square of 4 Square of 4 = 16So write 6 in the answer and 1 as a carry to the second step.Step 2:Find 2 (3 4) 2 (3 4) = 24 24 + 1(Carry) = 25 Write 5 in the answer and 2 as a carry to the third stepStep 3:Find 2 (2 4) + 32 2 (2 4) + 32= 25 25 + 2(Carry) = 27Write 7 in the answer and 2 as a carry to the fourth step.Step 4:Find 2 (1 4) + 2 (2 3) 2 (1 4) + 2 (2 3) = 20 20 + 2(Carry) = 22 Write 2 in the answer and 2 as a carry to the fifth step.Step 5:Find 2 (1 3) + 22 2 (1 3) + 22= 10 10 + 2(Carry) = 12 Write 2 in the answer and 1 as carry to the sixth step.Step 6:Find 2 (1 2) 2 (1 2) = 4 4 + 1(Carry) = 5 Write 5 in the answerStep 7:Find square of 1 Square of 1 = 1 There is no carry so write 1 in the answer.So, 12342= 1522756Square Of The Numbers With Unit Digit As 5This method is to find square of the numbers which has units digit as 5. i.e.: 25, 45, 65, etc.You can find square of these numbers by three easy steps.Step 1:Multiply tens digit with its next number.Step 2:Find square of units digit. i.e.: Square of 5.Step 3:Write answers of step 1 and step 2 to together or side by side.Lets take examples.Find square of 35Step 1:Multiply tens digit with its next number. 3 ( 3 + 1 ) = 3 4 = 12Step 2:Find square of units digit. i.e.: Square of 5. Square of 5 = 25Step 3:Write answers of step 1 and step 2 to together. Answer = 1225Find square of 65Step 1:Multiply tens digit with its next number. 6 ( 6 + 1 ) = 6 7 = 42Step 2:Find square of units digit. i.e.: Square of 5. Square of 5 = 25Step 3:Write answers of step 1 and step 2 to together. Answer = 4225Find square of 95Step 1:Multiply tens digit with its next number. 9 ( 9 + 1 ) = 9 10 = 90Step 2:Find square of units digit. i.e.: Square of 5. Square of 5 = 25Step 3:Write answers of step 1 and step 2 to together. Answer = 9025Find square of 115Step 1: Multiply tens digit with its next number. 11 ( 11 + 1 ) = 11 12 = 132 (Note: We are taking whole 11 as a tens digit.)Step 2:Find square of units digit. i.e.: Square of 5. Square of 5 = 25Step 3:Write answers of step 1 and step 2 to together. Answer = 13225Find square of 215Step 1:Multiply tens digit with its next number. 21 ( 21 + 1 ) = 21 22 = 462 (Note: We are taking whole 21 as a tens digit.)Step 2:Find square of units digit. i.e.: Square of 5. Square of 5 = 25Step 3:Write answers of step 1 and step 2 to together. Answer = 46225Square Of The Numbers In 50sThis method is used to find square of the numbers in 50s i.e. numbers from 51 to 59.You can find square of these numbers in three simple steps.Step 1:Add 25 to the units digit.Step 2:Square the units digit.Step 3:Write the answers of step 1 and step 2 together or side by side.Lets take an example.Square of 56Step 1:Add 25 to the units digit 6 + 25 = 31Step 2:Square the units digit 62= 36Step 3:Write the answers of step 1 and step 2 together. Answer = 3136Square of 59Step 1:Add 25 to the units digit 9 + 25 = 34Step 2:Square the units digit 92= 81Step 3:Write the answers of step 1 and step 2 together. Answer = 3481Square of 53.Step 1:Add 25 to the units digit. 3 + 25 = 28Step 2:Square the units digit. 32= 9Step 3:Write answers of step 1 and step 2 together. Answer: 2809 (NOTE:Whenever square of units digit is on only single digit then we are adding 0 before it.)Square of 52Step 1:Add 25 to the units digit 2 + 25 = 27Step 2:Square the units digit 22= 4Step 3:Write the answers of step 1 and step 2 together. Answer = 2704 (NOTE:Whenever square of units digit is on only single digit then we are adding 0 before it.)Square The Number If You Know Square Of Previous NumberThis method is to find square of the number if you know square of the previous number.You can find answers in three simple steps.Step 1:Find square of the previous number which is known.Step 2:Multiply the number being squared by 2 and subtract 1.Step 3:Add Step 1 and Step 2Lets take some examples.Find square of 31.Step 1:Find square of previous number (30) which is known. 302= 900Step 2:Multiply the number being squared (31) by 2 and subtract 1. (31 2) 1 = 62 1 = 61Step 3:Add Step 1 and Step 2 900 + 61 = 961Find square of 26.Step 1:Find square of previous number (25) which is known. 252= 625Step 2:Multiply the number being squared (31) by 2 and subtract 1. (26 2) 1 = 52 1 = 51Step 3:Add Step 1 and Step 2 625 + 51 = 676Find square of 81.Step 1:Find square of previous number (80) which is known. 802= 6400Step 2:Multiply the number being squared (31) by 2 and subtract 1. (81 2) 1 =162 1 = 161Step 3:Add Step 1 and Step 2 6400 + 161 = 6561Shortcut Method For Finding CubeThis method is to find cube of two digit numbers.Lets understand this method by taking some examples.Find cube of 14.Step 1:Find cube of tens digit and write it down as first digit .Step 2:1 and 4 are in the ratio of 1:4 So write next three numbers in the ratio of 1:4.Step 3:Write double of second and third digit below them.Step 4:Add both these rows as shown below. Starting from right, write 4 of 64 in the answer and 6 as carry. Add 6 + 16 + 32 = 54. Write 4 in the answer and 5 as carry. Add 5 + 4 + 8 = 17 Write 7 in the answer and 1 as carry. Add 1 + 1 = 2 Write 2 in the answer.Answer is 2744.Find cube of 48.Step 1:Find cube of tens digit and write it down as first digit .Step 2:4 and 8 are in the ratio of 1:2 So write next three numbers in the ratio of 1:2.Step 3:Write double of second and third digit below them.Step 4:Add both these rows as shown below. Starting from right, write 2 of 512 in the answer and 51 as carry. Add 51 + 256 + 512 = 819. Write 9 in the answer and 81 as carry. Add 81 + 128 + 256 = 465 Write 5 in the answer and 46 as carry. Add 46 + 64 = 110 Write 110 in the answer.So 483= 110592.Find cube of 63.Step 1:Find cube of tens digit and write it down as first digit .Step 2:6 and 3 are in the ratio of 2:1 So write next three numbers in the ratio of 2:1.Step 3:Write double of second and third digit below them.Step 4:Add both these rows as shown below. Starting from right, write 7 of 27 in the answer and 2 as carry. Add 2 + 54 + 108 = 164. Write 4 in the answer and 16 as carry. Add 16 + 108 + 216 = 340 Write 0 in the answer and 34 as carry. Add 34 + 216 = 250 Write 250 in the answer.So 633= 250047.Finding LCMThis method is to find LCM of given numbers.Lets say we want to find LCM of 20 and 60.First write 4 and 20 as shown below

Divide 20 and 60 by 2

Divide 10 and 30 by 2

We can not divide any of 5 and 15 by 2 perfectly. So divide them by three.Here 5 is not divisible by 3 so 5 is written as it is.

Divide 5 and 5 by 5.

Stop this process when last row has all 1s.So the LCM is 2 2 3 5 = 60HCFHCF is known as Highest Common Factor or Greatest Common Divisor (GCD) or Greatest Common Measure (GCM).

Factor:A number x is said to be factor of y when x exactly divides y. Ex: Lets say x = 3 and y = 15. Here 3(x) can exactly divide 15(y) so 3 is said to be factor of 15.Lets take two numbers 80 and 30 for better understanding. Write factors of both numbers.Factors of 30 :1,2, 3,5, 6,10, 15, 30Factors of 15 :1,2, 4,5, 8,10, 16, 20, 40, 80There are many common factors for 80 and 30. Those are 1, 2, 5 and 10. But 10 is highest of them. So 10 is called Highest Common Factor or HCF of 80 and 30.One thing is to be noted that HCF is less than or equal to the smallest number of the given numbers.Finding HCFThis method is for finding HCF of the given numbers.Step 1: Take two different numbers and divide the bigger number by the smaller number.Step 2: Divide the divisor by the remainder.Step 3: Repeat the process of dividing the divisor by remainder until the remainder is 0.Step 4: The last divisor is the required HCF of the given two numbers.Lets take an example of 80 and 30.

As you can see, last divisor is 10 so HCF of 80 and 30 is 10.Average Concept And Shortcut MethodsAverage is very useful for summarizing any quantity. For example we can say temperature of the city. The city may have different temperature throughout the week. But if we want to give single figure for it or we want to summarize it then we are using average.Basic formula for average is as shown below.For example if we want to find average temperature of the city for the week.Temperatures for the days of the week areSunday: 33 C Monday: 34 CTuesday: 35 C Wednesday: 36 CThursday: 35 C Friday: 35 CSaturday: 36 CHere sum of all the temperature is divided by 7 because there are 7 days.Here are some shortcut methods for averageShortcut Methods For AverageRule 1:If different distance is travelled in different time then,

ExampleIf a car travels 50 Km in 1 hour, another 40 Km in 2 hour and another 70 Km in 3 hour then what is average speed of car.Sol:Total Distance Covered = 50 + 40 + 70 = 160 KmTotal Time Taken = 1 + 2 + 3 = 6 hours.Rule 2:If equal distance is travelled at different speed.If equal distance is travelled at the speed of A and B then,

ExampleA boy goes to his school which is 2 Km away in 10 minutes and returns in 20 mins then what is boys average speed.Sol:

Lets say A = 2/10 = 0.2 km/minAnd B = 2/20 = 0.1 km/min

Rule 3:If equal distance is travelled at the speed of A, B and C then,

ExampleIf a car divides its total journey in three equal parts and travels those distances at speed of 60 kmph, 40 kmph and 80 kmph then what is cars average speed?Sol:

Lets say A = 60, B = 40 and C = 80, thenShortcut Methods For Average - 2When a person leaves the group and another person joins the group in place of that person then,Rule 1:If the average age is increased,Age of new person = Age of separated person + (Increase in average total number of persons)ExampleThe average age of 10 persons is increased by 5 years when one member of age 40 years is replaced by a new person. What is age of new person?Sol:Age of new person = Age of separated person + (Increase in average total number of persons)Age of separated person = 40Increase in average = 5Total number of persons = 10Age of new person = 40 + ( 5 10 ) = 40 + 50 = 90 years.Rule 2:If the average age is decreased,Age of new person = Age of separated person - (Decrease in average total number of persons)ExampleAverage age of 20 persons is decreased by 1 year when a person aged 25 years is replaced by a new person. What is age of new person?Sol:Age of new person = Age of separated person - (Decrease in average total number of persons)Age of separated person = 25 yearsDecrease in average = 1 yearAge of new person = 25 (1 20) = 25 20 = 5 years.Rule 3:New average age of group =

ExampleA group of 10 persons has an average age of 25 years. A person of that group aged 20 years is replaced by a new person aged 30 years. What is new average of the group?Sol:New average age of group =Provious average = 25 yearsNumber of persons = 10Age of leaving person = 20Age of joining person = 30

Shortcut Methods For Average - 3When a person joines the group.Rule 1:In case of increase in averageAge of new member = Previous average + ( Increase in average Number of members including new member)ExampleThe average age of 15 boys is 17 years and is increased by 0.5 years when a new boy is joined in the group. What is the average of new boy?Sol:Age of new member = Previous average + ( Increase in average Number of members including new member)Previous average = 17 yearsIncrease in average = 0.5 yearsNumber of members including new member = 16So,Age of new member = 17 + ( 0.5 16) = 17 + 8 = 25 years.Rule 2:In case of decrease in averageAge of new member = Previous average - ( Decrease in average Number of members including new member)ExampleThe average age of 10 girls is 23 years and is decreased by 0.5 years when a new girl is joined in the group. What is the age of new girl?Sol:Age of new member = Previous average - ( Decrease in average Number of members including new member)Previous average = 23 yearsDecrease in average = 0.5 yearsNumber of members including new member = 11Age of new member = 23 ( 0.5 11) = 23 5.5 = 17.5 yearsRule 3:New average of group =

ExampleA group of 20 members having average age of 25 years has a new member aged 25 years. What is new average of the group?Sol:New average of group =Previous average = 25 yearsNumber of persons excluding new member = 20Age of new member = 25 yearsNumber of persons including new member = 21

Shortcut Methods For Average - 4When a person leaves the groupRule 1:In case of increase in averageAge of leaving member = Previous average - ( Increase in average Number of members excluding leaving member)ExampleThe average weight of 10 mangos in the box is 450 grams. But accidentally one mango fell away out of the box resulting in increase of the average weight by 10 grams. What is the weight of that mango?Sol:Age of leaving member = Previous average - ( Increase in average Number of members excluding leaving member)Previous average = 450 gramsIncrease in average = 10 gramsNumber of members excluding leaving member = 9Age of leaving member = 450 (10 9) = 450 90 = 360 grams.Rule 2:In case of decrease in averageAge of leaving member = Previous average + ( Decrease in average Number of members excluding leaving member)ExampleThe average height of 10 boys is 165 cm. But one boy leaves the class and as a result there is a decrease in average height by 2 cm. What is height of leaving boy?Sol:Age of leaving member = Previous average + ( Decrease in average Number of members excluding leaving member)Previous average = 165 cmDecrease in average = 2 cmNumber of members excluding leaving member = 9Age of leaving member = 165 + ( 2 9) = 165 + 18 = 183 cmRule 3:New average of group =

ExampleA group of 10 persons has average age of 30 years. A person aged 40 years left the group. What is new average of the group?Sol:New average of group =Previous average = 30 yearsNumber of persons including leaving person = 10Age of leaving member = 40 yearsNumber of members excluding leaving member = 9

Average Of Numbers - Shortcut MethodsRule 1: Average of consecutive n natural numbers

ExampleFind average of consecutive 10 natural numbers.Sol:10 Consecutive natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Rule 2: Average of square of consecutive n natural numbers.

ExampleFind average of square of consecutive 5 natural numbers.Sol:Consecutive five natural numbers are 1, 2, 3, 4, 5.

Rule 3: Average of cubes of consecutive n natural numbers.

ExampleFind average of cubes of consecutive 5 natural numbersSol:Consecutive five natural numbers are 1, 2, 3, 4, 5.

Rule 4: Average of n consecutive even numbers.Average = n + 1Example

Find average of 6 consecutive even numbersSol:Six consecutive even numbers are 2, 4, 6, 8, 10, 12Average = 6 + 1 = 7Rule 5: Average of consecutive even numbers till n.

ExampleFind average of consecutive even numbers till 6Sol:Consecutive even numbers till 6 are 2, 4, 6

Rule 6: Average of square of n consecutive even numbers.

Example

Find average of square of 5 consecutive even numbers.Sol:Five consecutive even numbers are 2, 4, 6, 8, 10

Rule 7: Average of square of consecutive even numbers till n.

Example

Find average of square of consecutive even numbers till 4Sol:Consecutive even numbers till 4 are 2, 4

Rule 8: Average of n consecutive odd numbers.Average = nExampleFind average of 10 consecutive odd numbers.Sol:10 consecutive odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.Average = 10

Rule 9: Average of consecutive odd numbers till n.

ExampleFind average of consecutive odd numbers till 9Sol:Consecutive odd numbers till 9 are 1, 3, 5, 7, 9.

Rule 10: Average of square of consecutive odd numbers till n.

ExampleFind average of square of consecutive odd numbers till 5Sol:Consecutive odd numbers till 5 are 1, 3, 5.

Ratio Or FractionComparison of ratio or fraction is most frequently asked question in competitive exams. To find smallest or biggest of the given fraction, two approaches are to be used.Method 1:Make denominator of the given fractions same.Make denominator of the given fractions same.To do it, Take LCM of 4, 6, 8, 10.LCM of 4, 6, 8, 10 is 120.So try to make denominator as 120.(Here We are Multiplying and Dividing by same amount so we are not changing the value of the fraction.)

Now we can easily compare these fractions.

Method 2:Make numerator of the given fractions same.

Make numerator of the given fractions same.To do it, Take LCM of 3, 5, 7, 11.LCM of 3, 5, 7, 11 is 1155.So try to make numerator as 1155.

(Here We are Multiplying and Dividing by same amount so we are not changing the value of the fraction.)

Now we can easily compare these fractions.

The number which has smallest denominator is the biggest number and vise-versa.

PercentagePercent means per every hundred. 5% means 5 per 100.In other words, percentage is a fraction with denominator as 100.Percentage is denoted as P.C. or %. Ex: 70 percent = 70%.

Convert Percentage into Fraction:Step 1: The number is divided by 100.Step 2: % sign is removed.

Convert Fraction Into Percentage:Step 1: Multiply fraction by 100.Step 2: Put a % sign.Convert Percentage into Decimal:

Convert Decimal Into Percentage:0.25 = (0.25 100)% = 25%1.50 = (1.50 100)% = 150%Some Useful Formulas:

Partnership Concept and Shortcut MethodsWhen more than one person agree to invest their money to run a business or firm then this kind of agreement is called partnership. The persons involved in the partnership are called partners.There are two types of partners.1. Sleeping Partner:Sleeping partner is the person who provides only investment but does not take part in running the business.2. Working Partner:Working partner is the person who not only invests the money but also takes part in running the business. For this work he is paid some salary or some percent of profit in addition.There are two types of partnership.1. Simple Partnership: In simple partnership, capitals of partners are invested for the same period of time.2. Compound Partnership: In compound partnership, capitals of partners are invested for the different period of time.Basic FormulasIf two partners A and B are investing their money to run a business then(Simple Partnership)

Capital of A : Capital of B = Profit of A : Profit of BIf two partners A and B are investing their money for different period of time to run a business then(Compound Partnership)

Capital of A Time period of A : Capital of B Time period of B = Profit of A : Profit of BExampleJack and Jill start a business by investing $ 2,000 for 8 months and $ 3,000 for 6 months respectively. If their total profit si $ 510 and then what is profit of Jill?Sol:

Capital of Jack = 2000 and Time period = 8 monthsCapital of Jill = 3000 and Time period = 6 months

So, Profit of Jack : Profit of Jill = 8 : 9So we have 8 + 9 = 17 parts of total profit.Out of this 17 parts, Jack will get 8 parts and Jill will get 9 parts.

Now, total profit is 510So, One Part = 510/17 = 30.

Jills profit = 30 9 = 270 andJacks profit = 30 8 = 240.If n partners are investing for different period of time thenC1T1: C2T2: C3T3: : CnTn= P1: P2: P3: : PnWhere C is the capital invested, T is time period of capital invested and P is profit earned.ExampleRaju, Kamal and Vinod start a business by investing Rs 5,000 for 12 months, Rs 8,000 for 9 months and Rs 10,000 for 6 months. If at the end of the year their total profit is Rs 2000 then find the profit of each partner.Sol:

Rajus investment is 5000 for 12 months.Kamals investment is 8000 for 9 months.Vinods investment is 10000 for 6 months.

So their ratio of investments is5000 12 : 8000 9 : 10000 660 : 72 : 605 : 6 : 5

So their profit,Raju : Kamal : Vinod = 5 : 6 : 5

So there are 5 + 6 + 5 = 16 parts of profit.Out of these 16 parts, Raju will get 5 parts, Kamal will get 6 parts and Vinod will get 5 parts.

So, Total profit = 2000One part = 2000/16 = 125

Rajus profit = 5 125 = 625Kamals profit = 6 125 = 750Vinods profit = 5 125 = 625Shortcut MethodsRule 1:If two partners are investing their money C1and C2for equal period of time and their total profit is P then their shares of profit are

If these partners are investing their money for different period of time which is T1and T2, then their profits are

ExampleJack and Jill start a business by investing $ 2,000 for 8 months and $ 3,000 for 6 months respectively. If their total profit si $ 510 and then what is profit of Jill?Sol:

Lets Say C1= 2000, T1= 8C2= 3000, T2= 6P = 510

Rule 2:If n partners are investing their money C1, C2, , Cnfor equal period of time and their total profit is P then their shares of profit are

If these partners are investing their money for different period of time which is T1, T2, , Tn then their profits are

ExampleRaju, Kamal and Vinod start a business by investing Rs 5,000 for 12 months, Rs 8,000 for 9 months and Rs 10,000 for 6 months. If at the end of the year their total profit is Rs 2000 then find the profit of each partner.Sol:

Lets Say C1= 5000, T1= 12C2= 8000, T2= 9C3= 10000, T3= 6P = 2000

Simple Interest Shortcut MethodsWhen a person borrows some money from another person then the borrower has to pay some extra money for the use of that money to the lender. This extra money is called Interest.In other words, the amount charged by lender for giving his money for a specific amount of time is called Interest.The amount of money borrowed is known as Principle.Total of Interest and Principle is known as Total Amount.Amount = Principle + Interest.The borrower has to pay interest according to some percent of principle for the fixed period of time. This percentage is known as Interest Rate. This fixed period may be a year, six months, three months or a month and correspondingly the rate of interest is charged annually, half yearly, quarterly or monthly.For example, the rate of interest is 10% per annum means the interest payable on Rs 100 for one year is Rs 10.Some Basic FormulasIf A = AmountP = PrincipleI = InterestT = Time in yearsR = Interest Rate Per Year, thenAmount = Principle + InterestA = P + I

Compound Interest Concept & Shortcut MethodsIn compound interest, the interest for each period is added to the principle before interest is calculated for the next period. With this method the principle grows as the interest is added to it. This method is mostly used in investments such as savings account and bonds.To understand compound interest clearly, lets take an example.1000 is borrowed for three years at 10% compound interest. What is the total amount after three years?You can understand the process of compound interest by image shown below.

YearPrincipleInterest (10%)Amount

1st10001001100

2nd11001101210

3rd12101211331

Difference between Simple Interest and compound interestAfter three years,In simple interest, the total amount would be 1300And in compound interest, the total amount would be 1331.Some Basic FormulasIf A = AmountP = PrincipleC.I. = Compound InterestT = Time in yearsR = Interest Rate Per Year

Shortcut FormulasRule 1:If rate of interest is R1% for first year, R2% for second year and R3% for third year, then

ExampleFind the total amount after three years on Rs 1000 if the compound interest rate for first year is 4%, for second year is 5% and for third year is 10%Sol:P = 1000R1= 4%, R2= 5% and R3= 10%

(From the table given at the bottom of the page)A = 1201.2Rule 2:If principle = P, Rate = R% and Time = T years then1. If the interest is compounded annually:

2. If the interest is compounded half yearly (two times in year):

3. If the interest is compounded quarterly (four times in year):

ExampleFind the total amount on 1000 after 2 years at the rate of 4% if1. The interest is compounded annually2. The interest is compounded half yearly3. The interest is compounded quarterly.Sol:Here P = 1000R = 4%T = 2 yearsIf the interest is compounded annually

(From the table given at the bottom of the page)A = 1081.6If the interest is compounded half yearly

A = 1082.4If the interest is compounded quarterly

A = 1082.9Rule 3: If difference between Simple Interest and Compound Interest is given. If the difference between Simple Interest and Compound Interest on a certain sum of money for 2 years at R% rate is given then

ExampleIf the difference between simple interest and compound interest on a certain sum of money at 10% per annum for 2 years is Rs 2 then find the sum.Sum:

If the difference between Simple Interest and Compound Interest on a certain sum of money for 3 years at R% is given then

ExampleIf the difference between simple interest and compound interest on a certain sum of money at 10% per annum for 3 years is Rs 2 then find the sum.Sol:

Rule 3:If sum A becomes B in T1years at compound interest, then after T2years

ExampleRs 1000 becomes 1100 after 4 years at certain compound interest rate. What will be the sum after 8 years?Sum:Here A = 1000, B = 1100T1= 4, T2= 8

Look up Table

Mixture and Alligation Shortcut MethodsMixture:Mixing of two or more than two type of quantities gives us a mixure.Quantities of these elements can be expressed as percentage or ratio.i.e. Percentage (20% of sugar in water)Fraction ( A solution of sugar and water such that sugar : water = 1:4)Alligation:Alligation is a rule which is used to solve the problems related to mixture and its ingredient.It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.Alligation RuleWhen two elements are mixed to make a mixture and one of the elements is cheaper and other one is costlier then,

Here Mean Price is CP of mixture per unit quantity.Above rule can be written as,

Then,Cheaper Quantity : Costlier Quantity = (D M) : (M C)ExampleExampleIn what proportion must sugar at Rs 40 per kg be mixed with sugar at Rs 60 per kg so that the mixture be Rs 55 per kg?Sol:Here, CP of Cheaper(C) = 40,CP of Costlier(D) = 60 andMean Price(M) = 55So from the rule of alligation we can say that

Proportion of Cheaper Sugar and Costlier Sugar is 1 : 3

Mixture of more than two elements.This method is a bit tricky initially but if you practice it then it becomes quite easy.If the mixture is of more than two ingredients, then write the prices of each ingredient below one another in ascending order. Write the mean price to the left of the list. Now make couples of prices in such a way that one price of the couple is below mean price and another price of the couple is above mean price. Now find the difference between each price and mean price and write it opposite to the price linked to it. This difference is required answer.Dont worry if you dont understand above paragraph. Try to understand above paragraph with the examples given below.ExampleHow must a shop owner mix 4 types of rice worth Rs 95, Rs 60, Rs 90 and Rs 50 per kg so that he can make the mixture of these sugars worth Rs 80 per kg?Sol:Here the prices of sugars are 95, 60, 90 and 50.And the mean price is 80.Now read the above paragraph and the image given below to understand this method.

So the proportion of sugar is50 : 60 : 90 : 95 = 15 : 10 : 20 : 30 or50 : 60 : 90 : 95 = 3 : 2 : 4 : 6

ExampleIn what ratio must a person mix three kind of tea each of which has a price of 70, 80 and 120 rupees per kg, in such a way that the mixture costs him 100 rupees per kg?Sol:Here the prices of tea are 70, 80 and 120And mean price is 100, so

So the proportion of tea is70 : 80 : 120 = 20 : 20 : 50 or70 : 80 : 120 = 2 : 2 : 5

Some Shortcut FormulasRule 1If n different vessels of equal size are filled with the mixture of P and Q in the ratio p1: q1, p2: q2, , pn: qnand content of all these vessels are mixed in one large vessel, then

ExampleThree equal buckets containing the mixture of milk and water are mixed into a bigger bucket. If the proportion of milk and water in the glasses are 3:1, 2:3 and 4:2 then find the proportion of milk and water in the bigger bucket.Sol:Lets say P stands for milk and Q stands for water,So, p1:q1= 3:1p2:q2=2:3p3: q3=4:2

So in bigger bucket,Milk : Water = 109 : 71Rule 2If n different vessels of sizes x1, x2, , xnare filled with the mixture of P and Q in the ratio p1: q1, p2: q2, , pn: qnand content of all these vessels are mixed in one large vessel, then

ExampleThree buckets of size 2 liter, 4 liter and 5 liter containing the mixture of milk and water are mixed into a bigger bucket. If the proportion of milk and water in the glasses are 3:1, 2:3 and 4:2 then find the proportion of milk and water in the bigger bucket.Sol:Lets say P stands for milk and Q stands for water,So, p1:q1= 3:1 , x1= 2p2:q2=2:3 , x2= 4p3: q3=4:2 x3= 5, so

So in bigger bucket,Milk : Water = 193 : 137Rule 3: Removal and ReplacementIf a vessel contains x litres of liquid A and if y litres be withdrawn and replaced by liquid B, then if y litres of the mixture be withdrawn and replaced by liquid B, and the operation is repeated n times in all, then :

ExampleA container is containing 80 liter of wine. 8 liter of wine was taken out from this container and replaced by water. This process was further repeated two times. How much wine is there in the container now?Sol:Here x = 80, y = 8 and n = 3, so

Quantity of wine after 3rd operation = 58.32 liters.Rule 4:p gram of ingredient solution has a% ingredient in it. To increase the ingredient content to b% in the solution

Example125 liter of mixture of milk and water contains 25% of water. How much water must be added to it to make water 30% in the new mixture?Sol:Lets say p = 125, b = 30, a = 25So from the equation

Quantity of water need to be added = 8.92 liter.Profit And LossTo make profit is the basic aim of any business.Cost Price:It is the price at which any article or unit or item is bought. It is abbreviated as CP.Selling Price:It is the price at which any article or unit or item is sold. It is abbreviated as SP.Profit:If Selling Price is greater than Cost Price then seller makes profit. Profit = SP CPLoss:If Cost Price is greater than Selling Price then seller incurs loss. Loss = CP SPBasic Formulas:One point is to be noted that loss or profit is always calculated with reference to CP.Shortcut Method For Profit And Loss

Time and Work Shortcut MethodsTime and Work problems are most frequently asked problems in quantitative aptitude. To solve these problems very quickly, you should understand the concept of Time and Work and some shortcut methods.If a man can do a piece of work in 5 days, then he will finish 1/5th of the work in one day.If a man can finish 1/5th of the work in one day then he will take 5 days to complete the work.If a man 5/6th of work in one hour then he will take 6/5 hours to complete the full work.If A works three times faster than B then A takes 1/3rd the time taken by B.Here are some shortcut rules which can be very useful while solving Time and Work problems.In total 9 rules are given here.Rule 1: Universal RuleThis rule can be used in almost every problems.If M1persons can do W1work in D1days and M2persons can do W2works in D2days then we can sayM1D1W2= M2D2W1If the persons work T1and T2hours per day respectively then the equation gets modified toM1D1T1W2= M2D2T2W1If the persons has efficiency of E1and E2respectively then,M1D1T1E1W2= M2D2T2E2W1Example5 men can prepare 10 cycles in 6 days working 6 hours a day. Then in how many days can 12 men prepare 16 cycles working 8 hours a day?Sol:Here M1= 5, W1= 10, D1= 6, T1= 6 andM2= 12, W2= 16, T2= 8So from the above ruleM1D1T1W2= M2D2T2W15 6 6 16 = 12 D2 8 10D2= 3 Days.So they will complete the work in 3 days.Rule 2:If A can do a piece of work in n days, thenThe work done by A in one day = 1/nExampleIf A can repair 50 cycles in 5 days then A can repair 50/5 = 10 cycles in one day.Rule 3:If A can do a work in D1days and B can do the same work in D2days then A and B together can do the same work

ExampleContentIf A can do a piece of work in 10 days and B can do the same work in 15 days then how long will they take if they both work together?Sol:A can finish the work in D1= 10 days.B can finish the work in D2= 15 days.

Rule 4:If A is twice as good a workman as B, then A will take half of the time taken by B to complete a piece of work.ExampleA is twice as good a workman as B. Together, they finish the work in 14 days. In how many days can it be done by each separately?Sol:Lets assume that A alone can finish the work in x days.It is given that A is twice as good a workman as B so B alone can finish the work in 2x days

So x = 21 days.So A can finish the work in 21 days and B can finish the work in 42 days.Rule 5:If A is thrice as good a workman as B, then A will take one third of the time taken by B to complete a piece of work.ExampleA is thrice as good a workman as B. Together, they finish the work in 15 days. In how many days can it be done by each separately?Sol:Lets assume that A alone can finish the work in x days.It is given that A is thrice as good a workman as B so B alone can finish the work in 3x days.

So x = 20 days.So A can finish the work in 20 days and B can finish the work in 60 days.Rule 6:If A and B together can do a piece of work in x days, B and C together can do in y days and C and A together can do in z days, then the same work can be done

ExampleA and B can do a piece of work in 30 days while B and C can do the same work in 24 days and C and A in 20 days. Find out the time taken to complete the work by each member working alone. Also find in how many days they will complete the work if they work together.Sol:Here lets assume that x=30, y=24 and z=20.

Rule 7:If A can do a piece of work in D1days, B can do in D2days and C can do in D3days then they together can do the same work in

ExampleContentIf A can do a piece of work in 30 days, B can do in 24 days and C can do in 20 days then they together can do the same work in _____ days.Sol:

Rule 8:If A and B together can do a piece of work in D1days and A alone can do it in D2days, then B alone can do the work in

ExampleContentJack and Jill together can do a piece of work in 10 days. Jack alone can do it in 15 days. In how many days can Jill alone do it?Sol:Lets say D1= 10, D2= 15

Rule 9:If the number of men are changed in the ratio of m:n, then the time taken to complete the work will change in the ratio n:m

Pipes And Cisterns Shortcut MethodsPipe and Cistern problems are similar to time and work problems. A pipe is used to fill or empty the tank or cistern.Inlet Pipe:A pipe used to fill the tank or cistern is known as Inlet Pipe.Outlet Pipe:A pipe used to empty the tank or cistern is known as Outlet Pipe.Some Basic Formulas1. If an inlet pipe can fill the tank in x hours, then the part filled in 1 hour = 1/x2. If an outlet pipe can empty the tank in y hours, then the part of the tank emptied in 1 hour = 1/y3. If both inlet and outlet valves are kept open, then the net part of the tank filled in 1 hour is

Some Shortcut MethodsRule 1:Two pipes can fill (or empty) a cistern in x and y hours while working alone. If both pipes are opened together, then the time taken to fill (or empty) the cistern is given by

ExampleTwo pipes A and B can fill a cistern in 20 and 30 minutes respectively. If both the pipes are opened together, how long will it take to fill the cistern?Sol:Lets say x = 20 and y = 30

So it will take 12 minutes for both the pipes to full the cistern.Rule 2:Three pipes can fill (or empty) a cistern in x, y and z hours while working alone. If all the three pipes are opened together, the time taken to fill (or empty) the cistern is given by

ExampleThree pipes can fill a tank in 20 minutes, 30 minutes and 40 minutes respectively while working alone. If, all the pipes are opened together, how long will it take to fill the tank full?Sol:Lets say x = 20 minutes, y = 30 minutes, z = 40 minutes

So it will take 9.23 minutes to fill the tank full.Rule 3:If a pipe can fill a cistern in x hours and another can fill the same cistern in y hours, but a third one can empty the full tank in z hours, and all of them are opened together, then

ExampleTwo pipes can fill a cistern in 20 minutes and 30 minutes respectively. Third pipe can empty the tank in 40 minutes. If all the three pipes are opened together, how long it will take to fill the tank full?Sol:Lets say x = 20, y = 30 and z = 40

So it will take 17.14 minutes to fill the tank full.Rule 4:A pipe can fill a cistern in x hours. Because of a leak in the bottom, it is filled in y hours. If it is full, the time taken by the leak to empty the cistern is

ExampleA pipe can fill a tank in 3 hours. Because of leak in the bottom, it is filled in 4 hours. If the tank is full, how much time will the leak take to empty it?Sol:

So leak will empty the tank in 12 hours.By formulaLets say x = 3 and y = 4

Time and Distance Shortcut MethodsThe terms time and distance are related to the speed of a moving object.Speed:Speed is defined as the distance covered by an object in unit time.

Some Important FactsDistance travelled is proportional to the speed of the object if the time is kept constant.Distance travelled is proportional to the time taken if speed of object is kept constant.Speed is inversely proportional to the time taken if the distance covered is kept constant.If the ratio of two speeds for same distance is a:b then the ratio of time taken to cover the distance is b:aRelative SpeedIf two objects are moving in same direction with speeds of x and y then their relative speed is (x - y)If two objects are moving is opposite direction with speeds of x and y then their relative speed is (x + y)Unit Conversion

Some Important Shortcut FormulasRule 1:If some distance is travelled at x km/hr and the same distance is travelled at y km/hr then the average speed during the whole journey is given by

ExampleJohn goes from his home to school at the speed of 2 km/hr and returns at the speed of 3 km/hr. What is his average speed during whole journey in m/sec?Sol:Lets say x = 2 km/hrAnd y = 3 km/hr, so

Now, average speed in m/sec

Rule 2:If a person travels a certain distance at x km/hr and returns at y km/hr, if the time taken to the whole journey is T hours then the one way distance is given by

ExampleMr Samson goes to market at the speed of 10 km/hr and returns to his home at the speed of 15 km/hr. If he takes 3 hours in all, what is the distance between his home and market?Sol:Lets say x = 10 km/hry = 10 km/hr, andT = 3 hrs, then

So the distance between home and market is 18 km.Rule 3:If two persons A and B start their journey at the same time from two points P and Q towards each other and after crossing each other they take a and b hours in reaching Q and P respectively, then

ExampleTwo persons Ram and Lakhan start their journey from two different places towards each others place. After crossing each other, they complete their journey in 1 and 4 hours respectively. Find speed of Lakhan if speed of ram is 20 km/hr.Sol:Lets say A = Ram and B = Lakhana = 1 and b = 4, then

Lakhans Speed = 10 km/hrRule 4:If the same distance is covered at two different speeds S1and S2and the time taken to cover the distance are T1and T2, then the distance is given by

ExampleTwo trucks travel the same distance at the speed of 50 kmph and 60 kmph. Find the distance when the distance when the time taken by both trucks has a difference of 1 hour.Sol:Lets say S1= 50 kmph,S2= 60 kmphT1 T2= 1

Trains Shortcut MethodsProblems on trains are most frequently asked questions in any competitive exam.Problems on trains and Time and Distance are almost same. The only difference is we have to consider the length of the train while solving problems on trains.Points To Remember1. Time taken by a train of length of L meters to pass a stationary pole is equal to the time taken by train to cover L meters.2. Time taken by a train of length of L meters to pass a stationary object of length P meters is equal to the time taken by train to cover (L + P) meters.3. If two trains are moving in same direction and their speeds are x km/h and y km/h (x > y) then their relative speed is (x y) km/h.4. If two trains are moving in opposite direction and their speeds are x km/h and y km/h then their relative speed is (x + y) km/h.Unit Conversion

Some Shortcut MethodsRule 1:If two trains of p meters and q meters are moving in same direction at the speed of x m/s and y m/s (x > y) respectively then time taken by the faster train to overtake slower train is given by

ExampleTwo trains of length 130 meter and 70 meter are running in the same direction with the speed of 50 km/h and 70 km/h. How much time will faster train take to overtake the slower train from the moment they meet?Sol:Lets say p = 130 meter = 0. 13 kmq = 70 meter = 0.07 kmx = 70 km/h and y = 50 km/h,So from the equation given above,

0.01 hours = 36 secondSo it will take 36 seconds to overtake.Rule 2:If two trains of p meters and q meters are moving in opposite direction at the speed of x m/s and y m/s respectively then time taken by trains to cross each other is given by

ExampleTwo trains of length 130 meter and 70 meter are running in the opposite direction with the speed of 50 km/h and 70 km/h. How much time will trains take to cross each other from the moment they meet?Sol:Lets say p = 130 meter = 0. 13 kmq = 70 meter = 0.07 kmx = 70 km/h and y = 50 km/h,So from the equation given above,

0.0017 hours = 6 secondsSo it will take 6 seconds to cross each other.Boats and Streams Shortcut MethodsBoats and Streams problems are frequently asked problems in competitive exams.Stream:Moving water of the river is called stream.Still Water:If the water is not moving then it is called still water.Upstream:If a boat or a swimmer moves in the opposite direction of the stream then it is called upstream.Downstream:If a boat or a swimmer moves in the same direction of the stream then it is called downstream.Points to remember When speed of boat or a swimmer is given then it normally means speed in still water.Some Basic FormulasRule 1:If speed of boat or swimmer is x km/h and the speed of stream is y km/h then, Speed of boat or swimmer upstream = (x y) km/h Speed of boat or swimmer downstream = (x + y) km/hRule 2: Speed of boat or swimmer in still water is given by

Speed of stream is given by

Some Shortcut MethodsRule 1:A man can row certain distance downstream in t1hours and returns the same distance upstream in t2hours. If the speed of stream is y km/h, then the speed of man in still water is given by

ExampleA man goes certain distance against the current of the stream in 2 hour and returns with the stream in 20 minutes. If the speed of stream is 4 km/h then how long will it take for the man to go 4 km in still water?Sol:Lets say t1= 20 minutes = 0.33 hours and t2= 1 hours

Y = 4, then mans speed in still water

So mans speed is 7.94 km/h in still water.

Now, time taken by the man to row 4 km in still water

Rule 2:A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by

ExampleA man can row 4 km/h in still water. When the water is running at 2 km/h, it takes him 2 hours to go to a place and come back. What is the distance between that place and mans initial position?Sol:Lets say x = 4 km/h = mans speed in still water.

y = 2 km/h = waters speed.

t = 2, so

Rule 3:A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then the distance is given by

ExampleA man can row 4 km/h in still water. The water is running at 2 km/h. He travels to a certain distance and comes back. It takes him 2 hours more while travelling against the stream than travelling with the stream. What is the distance?Sol:Lets say x = 4 km/h = mans speed in still water.

y = 2 km/h = waters speed.

t = 2, so

Rule 4:A man can row in still water at x km/h. In a stream flowing at y km/h, if he rows the same distance up and down the stream, then his average speed is given by

ExampleSpeed of boat in still water is 9 km/h and speed of stream is 2 km/h. The boat rows to a place which is 47 km away and comes back in the same path. Find the average speed of boat during whole journey.Sol:Lets say x = 9 km/h = speed in still water

Y = 2 km/h = speed of stream

Work and WagesWork and Wages problems are simpler to solve in quantitative aptitude.Money earned by a person for a certain work is called wage of the person for that work.Points to remember1. Wages are directly proportional to the work done. That means more work more money.2. Wages are inversely proportional to the time taken to complete the work. That means more time less money.Work and Wages problems can be understood by examining some solved examples.ExampleMayank can do a piece of work in 3 days while Sanjay can do the same work in 4 days. The wage for the full work is Rs 350. If they both work together to complete the work then find the earnings of Mayank and Sanjay.Sol:Mayanks 1 days work = 1/3Sanjays 1 days work = 1/4

Wages are directly proportional to the work done.

So, Mayanks Wage : Sanjays Wage = 1/3 : 1/4Mayanks Wage : Sanjays Wage = 4 : 3

Now, total wage is 350 Rs.

ExampleShahrukh alone can finish a work in 4 days while Salman alone can finish it in 6 days. If they both work together to finish it, then out of total wages of 18000, what will be the 20% of Shahrukhs share?Sol:Shahrukhs 1 days work = 1/4Salmans 1 days work = 1/6

Wages are directly proportional to the work done.

So, Shahrukhs share : Salmans share = 1/4 : 1/6Shahrukhs share : Salmans share = 6 : 4

So 20% of Shahrukhs wage = 2160ExampleRahul, Rohit and Rohan can do a piece of work in 3, 4 and 6 days, respectively. Doing that work together, they get an amount of 675 rupees. What is the share of Rohan in that amount.Sol:Rahuls 1 days work = 1/3Rohits 1 days work = 1/4Rohans 1 days work = 1/6

Rahuls share : Rohits share : Rohans share

= 4 : 3 : 2

So, Rohans Wage

Rohans wage is 150 rupees.ExampleSunil can do a piece of work in 2 days while Saurav can do it in 3 days. They work together for a day and rest of the work is done by Sachin in 1 day. If they get 1800 rupees for the whole work then find the wage of Sachin.Sol:Sunils 1 days work = 1/2Sauravs 1 daus work = 1/3

Sunil and Sauravs 1days work = 1/2 + 1/3 = 5/6

Remaining work done by Sachin = 1 5/6 = 1/6(Have doubt? Then refertime and work)

So, Sunils share : Sauravs share : Sachins share

= 3 : 2 : 1So wage of Sachin,

Sachins wage is 300 rupees.TRIGNOMETRY:Types of TrianglesTypes of triangles are defined based on similarity of their sides and angles.Types based on SidesEquilateral Triangle

Equilateral triangle has all the three sides with equal length.Each angle is 60.Isosceles Triangle

Isosceles triangle has two sides with equal length.Angles opposite to the equal sides are also same.Scalene Triangle

Scalene triangle has all sides with unequal length.Angles of scalene triangle are also different.Types based on AngleRight Angle Triangle

Right angle triangle has one angle of 90.Side opposite to right angle is known as hypotenuse.Acute Triangle

Acute triangle has all the angles measuring less than 90.Obtuse Triangle

Obtuse triangle has one angle measuring greater than 90.Triangle PropertiesVertex:The vertex is the corner of the triangle. Triangle has three vertices.Base:Generally, the bottom side of a triangle is called base.Altitude:Altitude is a line through a vertex and perpendicular to the opposite side (or base) of the vertex.Intersection of three altitudes is called the orthocenter of the triangle.Median:A median is a line from the vertex to the midpoint of the opposite side.Intersection of three medians is called the centroid of the triangle.Orthocenter:Intersection of three altitudes is called the orthocenter of the triangle.

As shown in figure, point O is orthocenter of triangle.Centroid:Intersection of three medians is called the centroid of the triangle.

Incircle:The biggest circle which touches all the sides of a triangle is called incircle of a triangle.

The intersection of three angle bisector is center of incircle. It is also called incenter of triangle.Circumcircle:A circle which passes through all the vertices of the triangle is called circumcircle.

The intersection of three perpendicular bisectors is called circumcenter.Interior Angles: A triangle has three interior angles. Sum of three interior angles of a triangle is 180. Shortest side is always opposite to the smallest interior angle. Longest side is always opposite to the biggest interior angle.Exterior Angles:

An exterior (or external) angle is the angle between one side of a triangle and the extension of an adjacent side. Sum of all the exterior angle is 360 An exterior angle of a triangle is equal to the sum of the opposite interior angles. Definition of Sine, Cosine and Tangent There are two right-angle triangles ABC and XYZ shown in the above figure. These two triangles are similar triangles because interior angles of the triangles are same. Now, for right-angle similar triangles, ratio of the corresponding sides is same. These ratios remain same for any similar triangle. So if we keep the angles of the triangle same then these ratio does not change for any size of side. Scientists observed that these ratios always remain constant for the given angle so they decided to give name to these ratios. So they gave name as Sine, Cosine and Tangent to these ratios which are shown below. Now, for the given angle , these sides are opposite, adjacent and hypotenuse, so So from the above equations, we can write that, Trigonometric Ratios Of Standard Angles Trigonometric ratios of standard angles if found out by following simple method. Trigonometric ratio for 30 and 60 As shown in the figure, triangle ABC is equilateral triangle with each side of length 2X. AD is the bisector of angle BAC. As the triangle is equilateral, the length of BD and DC will be X. Triangle ADB is right-angle triangle, so from the Pythagorean theorem we can say that, AD2+ BD2= AB2AD2+ X2= 4X2AD2= 3X2AD = 3X So for the right-angle triangle ADB, we have values of all the sides, AB = 2X, BD = X and AD = 3X So, Simillarly for 60 Trigonometric ratio for 45 As shown in figure, triangle XYZ is right-angle triangle and two sides of the triangle are equal. So the value of the remaining angles are also same which is 45. By Pythagorean theorem, we can say that XY2+ YZ2= XZ2a2+ a2= XZ2XZ2= 2a2XZ = 2a So for the triangle XYZ, we have values of all the sides. XY = a, YZ = a and XZ = 2a So, The table for trigonometric ratio of standard angle is given below.

How To Remember Trigonometric Ratio TableRemembering values of standard angles can be difficult. But with the method shown below you can easily remember the values of standard angles.First of all write down the values of standard angles.

Then write numbers from 0 to 4 below them as shown below.

Then divide all of them by 4.

Take square root of all of them.

Then write resultant value.

Then write those value in reverse order below them.

Then divide values of first line with second line and write resultant values as third line.

Then write sin, cos and tan on the left hand side as shown below.

So, by this way you can easily remember values of standard angles.Values of Cosec, Sec and Cot can be found by taking inverse of Sin, cos and tan respectively for the given angle.Complete table of standard angles is shown below.

Angle and Distance ExamplesFrom an aeroplane just over a straight road, the angles of depression of two consecutive kilometer stones situated at opposite sides of the aeroplane were found to be 60 and 30 respectively. The height (in km) of the aeroplane from the road at that instant, isSolution:

As shown in image A is the aeroplane. B and C are two consecutive milestones. So distance between them is 1kmSo BC = 1It is given that the aeroplane creates 60 and 30 angle of depression with two milestones. So we can say that angle A = 90. (A + B + C = 180)So for right angle triangle ABC we can say that

Now, we want to find height of the aeroplane from the ground, so we have to find the length of AD.Triangle ABD is right angle triangle.So for triangle ABD, we can say that

So we can say that height of aeroplane from the ground is 3/4 km.A tower standing on a horizontal plane subtends a certain angle at a point 160m apart from the foot of the tower. On advancing 100m towards it, the tower is found to subtend an angle twice as before. The height of the tower isSolution:

As shown in figure, AB is tower. C is the point 160m apart from the tower. is the angle at point C.Now, advancing 100m towards tower we get to the point D. 2 is the angle at point D.We need to find length of AB which is height of tower.For right angle triangle ABC we can say that

And for right angle triangle ABD we can say that

But we know that

So we can write

So we can say that height of tower is 80 meters.The length of a shadow of a vertical tower is 1/3 times its height. The angle of elevation of the Sun isSolution:

As shown in the figure, AB is the tower and lets assume that its length is x.BC is its shadow and its length is (1/3)x.We want to find angle .So for right angle triangle ABC we can say that

Thus, angle of elevation of the Sun is 60.Trigonometric Formulas or IdentitiesBasic identities

Pythagorean Identities

Addition and Subtraction Identities

Single Angle Identity

Double Angle Identities

Triple Angle Identities

Product to sum

Sum to Product

Other Identities

ArticleThe adjectives a, an and the are called articles.There are two types of articles. Indefinite article A/An Definite article TheIndefinite Article:The words a and an are called indefinite articles. You can use them with singular nouns to talk about any single person or thing.The article a is used before words beginning with consonants.Ex: This is a car. This is a uniform. ( Uniforms pronunciation does not start with vowel sound) This is a bat.The article an is usually used before words beginning with vowels (a, e, i, o, u).In simple words, article an is used before words whose pronunciation starts with vowel sound.Ex: Sunil is an intelligent boy. He is an MLA. (MLAs pronunciation start with vowel sound) He is an honest man. (Honests pronunciation start with vowel sound)Rules for using Indefinite article.Rules For Using Indefinite ArticleHere are some rules for using Indefinite articles. These rules can be very useful in solving common errors problems in competitive exams.Rule:Article A is used before a word beginning with a consonant sound.Ex: He is a European. He has a brief case. She has a bicycle.Rule:Article An is used before a word beginning with a vowel sound.Ex: He is an MLA He has an umbrella He has an exe.Rule:Indefinite article A/An is used before a singular countable noun when it is mentioned for the first time and it does not represent specific person or thing.Ex: Kamal lives in a bunglow. Yogeshwar is a lecturer. This is a river.Rule:Indefinite article is used when a singular countable noun represents class of things or persons.Ex: A vehicle should be parked properly. (Vehicle = any vehicle, class of vehicles. A student needs guidance. (Student = any student, class of student) A cow is a friendly animal.Rule:Indefinite article is used when a proper noun is used as common noun.Ex: He is a Hitler. She is a Lata Mangeshker You are a Beauty.Definite article TheThe word the is called the definite article. Use the before a noun when you are talking about a certain or specific person or thing. Virat has won the match The mobile is ringing. The train has arrived.Rules for Using Definite ArticleHere are some rules for using definite articles. These rules can be very useful while solving common error problems in competitive exams.Rule:Definite article the is used when we talk about the person or thing which was mentioned earlier.Ex: Shruti drew a picture. The picture was beautiful. I met a boy. The boy was intelligent. Give me the ball which you bought yesterday.Rule:Definite article the is used when a singular noun is representing a whole class.Ex: The cow is a friendly animal. The rose is a beautiful flower. The lion is a dangerous animal.Rule:Before the names of mountain ranges, group of islands, rivers, oceans, gulfs, desserts, forests etc.Ex: The Himalayas The Ganga The Andamans The Amazon The Pacific Ocean The Sahara The Vrindavan ForestRule:Before the names of newspapers and magazines.Ex: The Hindu The Times of IndiaRule:Before the names of religious and mythological books.Ex: The Ramayana The Mahabharat The BibalRule:Before the name of historical places.Ex: The Tajmahal The LalkillaRule:Before the name of religious community, political party, nationality, trains, ships, government departments.Ex: The Hindus The BJP The Shatabdi Express The Income Tax department The Army The IndianRule:Before the words showing position.Ex: The top The bottom The inside The back The frontRule:The is used in superlative degree.Ex: She is the most beautiful girl in the college. Usain Bolt is the fastest person on the planet. He is the most sincere student in the class.Rule:The is used before the names of natural things.Ex: The Sun The Moon The Planet The River The MountainRule:The is used before the ordinals.Ex: The first The second The lastDifference Between Definite And Indefinite ArticlesMany people have doubt about when to use definite articles and when to use indefinite articles.To clear this doubt we should know the difference between these two types of articles.Indefinite articles are used when you are not talking about something specific or certain.Ex: Give me a ball.Whereas definite article is used when you are talking about something specific or certain.Ex: Give me the red ball.Above examples show that we are talking about specific thing which is red ball. Whereas in the example of indefinite article, we are talking about ball which can be of any colour.NounNoun is a word used to name a person, animal, place or thing.Types of Noun Common Noun Proper Noun Collective Noun Material Noun Abstract NounCommon Noun:Common noun is a name given in general to every person or thing of same kind.Ex: Boy, Country, BirdProper Noun:Proper noun is the name of a particular person or place or thing.Ex: Rahul, India, PeacockCollective Noun:A Collective noun is the name of a group of persons or things taken together and spoken of as a whole or as unit.Ex: Team, Group, CommitteeMaterial Noun:A Material noun is the name of substance or metal, of which things made of.Ex: Gold, Wood, SteelAbstract Noun:Abstract noun in general refers, the name of quality, action or state. An abstract noun is a type of noun that refers to something with which a person cannot physically interact.Ex: Beauty, Joy, Childhood, Laughter, GraspApart from these, there are also other two types of nounsCountable Nouns:Countable Nouns have both singular and plural form and it can be counted.Ex: Table, Banana, RupeeNon Countable Nouns:Non Countable Nouns have only singular form and it cannot be counted.Ex: Air, Furniture, PetrolRules of NounHere are some rules of nouns which should be used while making sentences.These rules will be very helpful in common error problems.Rule:Some nouns are always singular. These are uncountable nouns. Articles A/An are not used with these nouns.These nouns are:Machinery, Work, Wood, Dust, Traffic, Electricity, Scenery, Poetry, Furniture, Advice, Luggage, Information, Luggage, Hair, Money, Language, Business, Mischief, Knowledge, Bread, Stationery, Crockery, Baggage, Postage, Wastage, , Jewellery, Breakage etc.Ex: He gave me information. Sachin transported his furniture by Truck. Rishma has good knowledge of grammar.Rule:Some nouns are always in plural form they dont have singular form. Plural verb is used with them.These nouns are:Cattle, Assets, Alms, Police, Amends, Annals, Archives, Ashes, Arrears, Athletics, Wages, Auspices, Species, Scissors, Gentry, Trousers, Pants, Clippers, Shambles, Bellows, Gallows, Fangs, Measles, Eyeglasses, Tidings, Goggles, Belongings, Breeches, Braces, Binoculars, Dregs, Entrails, Embers, Fireworks, Lees, Odds, Outskirts, Particulars, Proceeds, Proceedings, Riches, Bowels, Remains, Shears, Spectacles, Surroundings, Tactics, Tongs, Vegetables, Valuables, Etc.Ex: Cattle are not allowed to enter that ground. These pants are good. These poultry are mine.Rule:Some nouns have the same form in singular as well as in plural form.These nouns are:Sheep, Fish, Crew, Family, Team, Carp, Pike, Trout, Deer, Aircraft, Counsel, Swine, Vermin, Species Etc.Ex: A deer is grazing in the field.Deer are grazing in the field. Sparrow is now a rare species.There are many species of cow.Rule:Nouns expressing number like hundred, dozen, score etc are used in singular with numerical adjectives.These nouns are:Hundred, Pair, Score, Stone, Dozen, Thousand, Million, Billion, Gross, Etc.Ex: Sumit bought four dozen apples. Samir got five hundred rupees. Deepika has two pair of sandles.Rule:Plural noun is used after One of, Neither of, Either of, and Each of.Ex: One of my friends is an Engineer. Either of them will come. Neither of the students hasRu failed.Rule:Some nouns look plural but have singular meaning. Singular verb is used with them.These nouns are:Summons, News, Politics, Physics, Economics, Mechanics, Mathematics, Measles, Ethics, Rickets, Billiards, Draughts, Innings, etc.Ex: I have a good news. Economics is my favourite subject. It was a good innings by India.Rule:Some nouns look singular but have plural meaning. Plural verb is used with them.These nouns are:Infantry, Children, Cattle, Cavalry, Poultry, Peasantry, Gentry, Police, Clergy, Etc.Ex: Children are playing in the ground. Cattle are not allowed to enter in the ground. Police are coming.Rules for Using ApostropheHere are some rules for the use of Apostrophe 's' or possessive nouns.Rule:Possessive case is used with the nouns of living things.Ex: This is Sachins bat. The cars wheel is punctured (Wrong)The wheel of the car is punctured (Correct)Rule:Non-living things are used in possessive case when they are personified.Ex: Ramu is at deths door. This is earths surface.Rule:Possessive case is used with nouns denoting space, time or weight.Ex: I want a days leave. Shila will be back in a months time.Rule:If two or more noun jointly possess something then possessive sign is put on the latter only.Ex: Sachin and Sauravs partnership was awesome. This is Rahul and Sonams shop.Rule:If else is used after somebody, anybody, nobody etc then apostrophe is used with else.Ex: I obey your orders and nobody elses. This watch is not mine, it is somebody elses.Rule:Apostrophe is not used with possessive pronouns like; his, hers, yours, mine, ours, its, theirs, etcRule:Apostrophe is not used with two consecutive nouns.Ex: Sonals cars colour is very nice. (Wrong)The colour of Sonals car is very nice. (Correct)AdjectiveAdjective are those words which add some more information to the noun. There are different types of adjectives.Some adjectives show size of the things or people.Big, large, small, tiny, huge, thin etc show the size of people or thingsEx: Abigcar.(Here adjective big has added more information to the noun car.) Atinyvillage.Some adjectives show colour of the things.Red, green, blue, orange, yellow etc show colour of the things.Ex: Aredcar. Abluejeans.Some adjectives show quality of the things.Young, kind, beautiful, hot, cool, old, brave, rich, poor etc show quality of the things or persons.Ex: Abeautifulgirl. Ayoungboy. Abeautifulpainting.Some adjectives show quantity of the things.Much, little, some, sufficient, whole etc show quantity of the things.Ex: This issufficientfood. This islittlewhisky.Some adjectives show what things are made of.Steel, wood, iron, plastic, gold, cotton etc show the material the things are made of.Ex: This iswoodentable. Give me aplasticbag. Show me agoldchain.Adjectives have three degrees. ie Positive, Comparative and Superlative. Lets see how to form these.The comparative degree of an adjective is usually formed by adding er and the superlative degree is formed by adding est' to the positive form of adjective.PositiveComparativeSuperlative

HighHigherHighest

LongLongerLongest

SweetSweetersweetest

If the positive degree of an adjective ends in e then only r and st is generally added to make the comparative and superlative degreesPositiveComparativeSuperlative

WiseWiserWisest

LargeLargerLargest

SafeSaferSafest

If the positive degree of an adjective ends with consonant + y then y is replaced by ier in comparative degree and iest in superlative degree.PositiveComparativeSuperlative

LovelyLovelierLoveliest

PrettyPrettierPrettiest

EasyEasierEasiest

If the positive form of an adjective ends in a single consonant and is of one syllable which is preceded by vowel then the consonant is doubled before adding er' or est' to form comparative or superlative degree.PositiveComparativeSuperlative

WetWetterWettest

HotHotterHottest

BigBiggerBiggest

If the adjectives have two or more than two syllables. Comparative and superlative degrees of these adjectives are formed by adding more and most before the positive form.PositiveComparativeSuperlative

BeautifulMore beautifulMost beautiful

ColourfulMore colourfulMost colourful

CarelessMore carelessMost careless

Some adjectives have irregular forms of comparative and superlative degrees.PositiveComparativeSuperlative

GoodBetterBest

BadWorseWorst

LateLaterLast

AdverbAn adverb is a word which adds meaning to the verb. Adverbs can modify adjectives, nouns and other adverbs also. Adverbs tell us how, when, where etc something was done.According to their meaning, there are following types of adverbs.Adverbs of MannerAdverb of manner tell us how an action takes place.Carefully, badly, quickly, bravely, friendly way, etc are adverbs of manner.Ex: You must drivecarefully.(How? = carefully) The teacher teaches us in afriendly way.(How? = in a friendly way)Adverbs of PlaceAdverb of place tell us where an action takes place.Abroad, down there, everywhere, here, there, downstairs etc are adverbs of place.Ex: It is rainingeverywhere.(Where? = everywhere) He is goingabroad.(Where? = abroad)Adverbs of TimeAdverb of time tell us when an action takes place.Now, then, Monday, yesterday, daily, tonight etc are adverbs of time.Ex: Simran went to Delhiyesterday.(When? = yesterday) We are going to watch a movietonight.(When? = tonight)Adverbs of FrequencyAdverb of frequency tell us how often an action takes place.Always, often, sometimes, twice a month, monthly etc are adverbs of frequency.Ex: Ialwaysdo yoga.(How often? = always) I eat chocolatesometimes.(How often? = sometimes)Adverbs of Quantity or DegreeAdverb of Quantity or Degree tell us how much or in what degree something has happned.Very much, only, almost, quite, hardly etc are adverbs of quantity or degree.Ex: Hehardlycomes to the ground.(How much? = hardly) I like mangosvery much.(How much? = very much)Adverbs of DurationAdverb of duration tell us how long an action takes place.For three days, for a moment, over an hour, for ages, all night etc are adverbs of duration.Ex: He has been the king of this areafor ages.(How long? = for ages) The rain lastedfor two days.(How long? = for two days)Adverb of ReasonAdverb of Reason tell us why an action takes place.Hence, therefore, so etc are adverb of reason.Ex: Thereforehe went to clinic.(Why? = therefore) He washencegoing to that way.(Why? = hence)Rules for using adverbs.VerbVerb is a word which describes action. It shows what people or things are doing.Here are some common verbs which are used normally.Drink, Speak, Eat, write, jump, sing, look, walk, learn, swim etc are commonly used verbs.The tense of a verb indicates time. The use of singular and plural forms indicates the quantity of things acting in a sentence.Have a look at the following sentence.I amplayingcricket.Here play is the verb and shows what action the subject is doing.The form of verb must agree the subject. If the subject and verb agrees than only the sentence is grammatically correct.Click on below link to see the rules of subject and verb agreement.Subject - Verb AgreementWhen the subject and the verb of any sentence match each other or have same form then it is said that subject and verb agree with each other. If the subject is singular, the verb must be singular. If the subject is plural, the verb must also be plural.Subject and verb must agree in following ways. In person:First person, second person and third person. In number:Singular or plural.Now, see following examples. In following examples, subjects are in bold and verbs are in colour. Sankethplaystennis.(Here subject Sanketh is third person singular so verb plays is also in singular form. )(verb + s/es = singular,here play + s = plays) Theyplayfootball.(Here subject They is third person plural so verb play is in plural form.) Iam eating.(Here subject I is first person so we are using am with the verb) Youare dancing.(Here subject you is second person so we are using are with the verb) My friend Supreethgoesto the same gym as I do.(Here subject My friend Supreeth is third person singular so verb goes is also in singular form. )(verb + s/es = singular,here go + es = goes) The planefliesto London every day.(Here subject The plane is third person singular so verb flies is also in singular form. )(verb + s/es = singular,here fly + es = flies)Rules of s, es and iesMany of the students have confusion in when to use s or es or ies while working with verbs.Here are some simple rules by which you can remove your confusion.When to use es?Es is used in two cases.When the word ends with vowel o.Ex: Do: does Go: goesWhen the word ends with ch, sh, ss, zz, xEx: Assess: Assesses Teach: Teaches Buzz: Buzzes Wash: Washes Fix: FixesWhen to use ies?When the word ends with consonant + y then replace y with ies.Ex: Fly: Flies (here l is consonant and the word ends with l + y) Apply: Applies (consonant l + y) Copy: Copies (consonant p + y) Carry: Carries (consonant r + y)But there is an exception in this case.If the word ends with vowel + y then simply s is added to the word.Ex: Boy: Boys (here o is vowel and the word ends with o + y) Play: Plays (vowel a + y) Enjoy: Enjoys (vowel o + y)When to use s?For all the cases other than above shown, we use s after the word.Ex: Rain: Rains Eat: Eats Start: Starts. Etc etc etcRules of Subject - Verb AgreementHere are some rules of subject verb agreement which can be very useful in finding errors sentence in competitive exams.Subject is shown in bold and verb is shown in colour in examples of rules.Rule:If the subject is singular, the verb must be singular. If the subject is plural, the verb must also be plural.Ex: Simranplayschess.(Here subject Simran is singular so verb plays is also in singular form.)(verb + s/es = singular,here play + s = plays) Theyplaycricket.(Here subject They is plural so verb play is also in plural form.) Mom and Dadloveus.(Here subject Mom and Dad is plural (more than one) so verb love is also in plural form.)Rule:If two singular noun refer to the same person or thing then the verb must be in singular form.Ex: The director and producerlovesthis food.(Here the director and producer refers to the same person so verb is in singular form.)(Here same person is the producer as well as the director of the film.We can say that The director and producer Karan Johar.Here Karan Johar is diractor of the film and he is also producer of the film. So director and producer refers to the same person.) The captain and wicket keeperplayshere.(Here the captain and wicket keeper refers to the same person so the verb is in singular form.)(MS Dhoni is captain and wicket keeper of Indian cricket team)If you observe above examples, article is used only once in the sentence. If the article is used more than once than it does not refer to same person and verb must be in plural form.Ex: The director and the producerlovethis food.(Here article the is used more than once so it does not refer to same person so there are two different persons ie the director and the producer so verb love is in plural form. The captain and the wicket keeperplayhere.(Here the captain and the wicket keeper refers to two different persons so verb is in plural form)Rule:If two or more subjects are joined by with, like, besides, as well as, together with, along with, in addition to etc, the verb is used according to first subject.Ex: The captain as well as team memberslovesthis food.(Here two subjects the captain and team members are joined with as well as but first subject, the captain, is singular so the verb loves is also in singular form.) The team members as well as captain of the teamlovethis food.(Here two subjects the team members and the captain of the team are joined with as well as but first subject, the team members is in plural so the verb love is also in plural form.)Rule:When two subjects are connected by either...or, neither...nor, not only...but also, or, nor, the subject which is nearest to the verb decides whether the verb will be singular or plural.Ex: Neither Sumitra nor her family memberswere presentin the party.(Here two subjects sumitra and her family members are joined with neither... nor. And subject her family members is nearest to the verb present. Subject her family members is plural so the verb were is also in plural) Either you or Iam drivingthe car.(Here two subjects you and I are joined with either or. And subject I is nearest to the verb driving. So verb has taken the form am driving according to the subject I.)Rule:Plural verb is used to show wish, regret, unlikely condition etc.Ex: I wish Iwerea soldier.(Here I were is used instead of I am because it is a wish)Rule:Singular verb must be followed by each, every, anyone, someone, either, neither, etc.Ex: Neither of this cityknowsyou. Each of this classisa scholar.Rule:Collective nouns like Team, Family, Jury, Crowd, Class, Committee, Army, Assembly, Fleet, Majority, Mob, Government, Parliament Council, Staff, etc., the verb used can be singular or plural according to the meaning of the sentence. If the collective noun is used as a unit then the verb will be singular, but if the collective noun is not working as unit then the verb used will be plural.Ex: The committeehasarrived.(Here subject the committee is used as a unit so verb has is in plural form) The committeehavedifferent opinions.(Here subject the committee is divided and does not work as a unit so verb have is in plural form)Rule:Some nouns like glasses, shoes, pants, trousers, spectacles etc take plural verb.Ex: My pantsarenot ready to wear. My spectaclesaremissing.Rule:Some nouns like News, Gallows, Billiards, Innings, Wages, Alms, Physics etc sounds like plural noun but they are singular in meaning and take singular verb.Ex: Physicswasmy favourite subject. The newsisvery good.PronounA pronoun is a word which is used in place of a noun. There are different kinds of pronouns. Personal Pronoun Reflexive Pronoun Possessive Pronoun Demonstrative Pronoun Interrogative Pronoun Indefinite PronounPersonal PronounI, you, he, she, me, her, him them etc are personal pronoun. Personal pronoun are used in subjective as well as objective way.Subjective CaseI, you, he, she, it, we and they can all be used as the subject of a verb. Study the following two sentence:Alia likes sandals.Shehas ten sandals.In the first sentence, the proper noun Alia is the subject of the verb likes. In the second sentence, the pronoun she is the subject of the verb has.Here are some examples which shows personal pronouns used as subjects of verbs. Sachin is a cricketer.Helikes to bat. My name is Rohit.Iam 21. Sunil and I are going to watch movie.Welike Amitabh Bachchan. Smriti and Shruti are going to watch a cricket match.Theylike MS Dhoni. Deepica,youare a sweet girl.Objective CaseThe personal pronouns me, you, him, her, it, us and them can all be used as the object of a verb. Look at the following two sentence:Alia likes sandals. She likes to wearthem.In the first s