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Given : 

 292, 291, 295, 268, 284, 161

Calculation :

 The series follows the following pattern:

 ⇒ 292 - (1)³ = 291

 ⇒ 291+ (2)² = 295

 ⇒ 295- (3)³ = 268

 ⇒ 268+ (4)² = 284

 ⇒ 284 - (5)³ = 159

 The wrong term is 161

 Hence the wrong term in the series is 161

Given :

  27, 48, 80, 134, 221, 355

 Calculation :

 The series follows the following pattern

  ⇒ 27   48   80  134  221  355

  ⇒    21    32   54     87    134     ( difference of numbers 

  ⇒        11    22     33    47          ( Then the double difference is multiple of 11)

  ⇒ 47 is not a multiple of 11 then 44 difference is come here instead of 47

  ⇒ 87 + 44 = 131 then 221 + 131 = 352

 The wrong term is 355

 Hence the wrong term in the series is 355

Given : 

  215, 200, 250,175, 275, 150, 300

 Calculation :

 The series follows the following pattern 

 ⇒ 225 - 25 = 200

 ⇒ 200 + 50 = 250

 ⇒ 250 - 75 = 175

 ⇒ 175+ 100 = 275

 ⇒ 275 - 125 = 150

 ⇒ 150+ 150 = 300

 The wrong term is 215

 Hence the wrong term in the series is 215.

Given :

  25, 50, 85, 130, 185, 245, 325

 Calculation :

 The series follows the following pattern 

 ⇒ 25 + 25 = 50

 ⇒ 50 + 35 = 85

 ⇒ 85 + 45 = 130

 ⇒ 130 + 55 = 185

 ⇒ 185 + 65 = 250

 ⇒ 250 + 75 = 325

 The wrong term is 245

 Hence the wrong term in the series is 245

Given : 

  1, 3, 5, 9, 17, 33, 65

Calculation : 

 The series follows the following pattern 

 ⇒ 2 × 2 - 1 = 3

 ⇒ 3 × 2 - 1 = 5

 ⇒ 5 × 2 - 1 = 9

 ⇒ 9 × 2 - 1 = 17

 ⇒ 17 × 2 - 1 = 33

 ⇒ 33 × 2 - 1 = 65

 The wrong term is 1

 Hence the wrong term in the series is 1

Shortcut Trick

Initial total units = 27 + 29 = 56 units.

56 units = 1008 student → 1 unit = 18.

Initial Boys = 29 × 18 = 522. New Boys = 522 + 9 = 531.

New ratio (Girls : Boys) = 56 : 59. So, 59 units = 531 → 1 unit = 9.

New Girls = 56 × 9 = 504. Initial Girls = 27 × 18 = 486.

New girls admitted = 504 − 486 = 18.

∴ The correct answer is 18 girls.

Alternate Method

Given: Total = 1008, Initial Ratio = 27 : 29, Final Ratio = 56 : 59.

Formula Used: Value = (Ratio Part ÷ Total Parts) × Total Value.

⇒ Initial Girls = (27 ÷ 56) × 1008 = 486

⇒ Initial Boys = (29 ÷ 56) × 1008 = 522

⇒ New Boys = 522 + 9 = 531

⇒ Let new Girls admitted be G. New total girls = 486 + G

⇒ (486 + G) ÷ 531 = 56 ÷ 59

⇒ 486 + G = (56 × 531) ÷ 59

⇒ 486 + G = 56 × 9 = 504

⇒ G = 504 − 486 = 18

∴ The correct answer is 18 girls.

Additional Information

Ratio and Proportion

If a quantity is divided in the ratio a : b, the parts are a/(a+b) × Total and b/(a+b) × Total.

Ratio Change Method

When adding values to terms of a ratio, the cross-multiplication or unitary method helps identify the increase in specific parts effectively.

Shortcut Trick

Ratio of Expenditure to Savings = 75 : 25 = 3 : 1.

Net increase in Income (20%) = [ (Ratio_Exp × %Increase_Exp) + (Ratio_Sav × %Increase_Sav) ] ÷ Total Ratio

20 = [ (3 × 10) + (1 × x) ] ÷ 4

80 = 30 + x ⇒ x = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: Initial Income spending = 75%, Income increase = 20%, Expenditure increase = 10%.

Formula Used: Income = Expenditure + Savings

⇒ Let Initial Income = 100

⇒ Initial Expenditure = 75% of 100 = 75

⇒ Initial Savings = 100 − 75 = 25

⇒ New Income = 100 + 20% of 100 = 120

⇒ New Expenditure = 75 + 10% of 75 = 75 + 7.5 = 82.5

⇒ New Savings = 120 − 82.5 = 37.5

⇒ Increase in Savings = 37.5 − 25 = 12.5

⇒ Percentage Increase = (12.5 ÷ 25) × 100 = 50%

∴ The correct answer is 50%.

Additional Information

Percentage Change Formula

The percentage increase or decrease is calculated as: [(New Value − Old Value) ÷ Old Value] × 100.

Income-Expenditure Relationship

Total Income is always the sum of Expenditure and Savings. If any two components change, the third must adjust to maintain this balance.

Alligation Method in Income

Income % change can be viewed as the weighted average of Expenditure % change and Savings % change based on their respective initial ratios.

Shortcut Trick

Let Cost Price (CP) be 100.

Marked Price (MP) = 100 × 1.5 = 150.

Selling Price (SP) after successive discounts = 150 × 0.80 × 0.85 = 102.

Profit = SP − CP = 102 − 100 = 2 units.

Given 2 units = Rs. 400, then 1 unit = Rs. 200.

CP = 100 units = 100 × 200 = Rs. 20,000.

∴ The correct answer is Rs. 20,000.

Alternate Method

Given: Markup% = 50%, Successive discounts = 20% and 15%, Profit = Rs. 400.

Formulas: SP = MP × (100 − D₁)% × (100 − D₂)%, Profit = SP − CP.

⇒ Let CP = x. Then MP = x + 0.5x = 1.5x.

⇒ SP = 1.5x × (80/100) × (85/100).

⇒ SP = 1.5x × 0.8 × 0.85 = 1.02x.

⇒ Profit = 1.02x − x = 0.02x.

⇒ 0.02x = 400 ⇒ x = 400 ÷ 0.02 = 20,000.

∴ The correct answer is Rs. 20,000.

Additional Information

Effective Discount

For successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Markup and Profit Relationship

Profit occurs if the multiplier of Markup is greater than the multiplier of Discount. Net change = [Markup% − Discount% − (Markup × Discount)/100].

Shortcut Trick

Total work = LCM(18, 27) = 54 units.

Efficiency: Karun = 54 ÷ 27 = 2; Varun = 54 ÷ 18 = 3.

Karun worked for the entire 15 days: Work done = 15 × 2 = 30 units.

Remaining work for Varun = 54 − 30 = 24 units.

Days Varun worked = 24 ÷ 3 = 8 days. Since the total work lasted 15 days, he joined after 15 − 8 = 7 days.

∴ The correct answer is 7 days.

Alternate Method

Given: Varun takes 18 days, Karun takes 27 days. Total time taken = 15 days.

Formula Used: Efficiency = Total Work ÷ Days; Total Work = Efficiency × Time.

⇒ Total work = 54 units (LCM of 18 and 27).

⇒ Karun is present for the full 15 days.

⇒ Work done by Karun = 15 × 2 = 30 units.

⇒ Remaining work to be done by Varun = 54 − 30 = 24 units.

⇒ Number of days Varun actually worked = 24 ÷ 3 = 8 days.

⇒ Time after which Varun joined = Total time − Varun's working time = 15 − 8 = 7 days.

∴ The correct answer is 7 days.

Additional Information

Combined Efficiency Rule

When two people work together, their combined efficiency is the sum of their individual efficiencies. Work Done = (E₁ + E₂) × Time.

Work & Wages

Wages are distributed in the ratio of the actual work done by each individual, which is Efficiency × Time spent on the job.

Leaving/Joining Concept

Always calculate the work done by the person who stays for the entire duration first to find the remaining work for the person who leaves or joins late.

Shortcut Trick

Average Speed for three equal distances = 3 × a × b × c ÷ (ab + bc + ca)

Average Speed = (3 × 10 × 12 × 15) ÷ (120 + 180 + 150) = 5400 ÷ 450 = 12 km/h

Total Distance = Average Speed × Total Time

Total Distance = 12 km/h × (30 ÷ 60) h = 12 × 0.5 = 6 km

∴ The correct answer is 6 km.

Alternate Method

Let each equal distance be 'd' km.

Total time = (Time₁ + Time₂ + Time₃)

⇒ 30 ÷ 60 = (d ÷ 10) + (d ÷ 12) + (d ÷ 15)

⇒ 1 ÷ 2 = (6d + 5d + 4d) ÷ 60

⇒ 1 ÷ 2 = 15d ÷ 60

⇒ 1 ÷ 2 = d ÷ 4

⇒ d = 2 km

⇒ Total Distance = 3 × d = 3 × 2 = 6 km

∴ The correct answer is 6 km.

Additional Information

Average Speed Formula

For two equal distances with speeds x and y, Average Speed = 2xy ÷ (x + y). For three equal distances, it is 3xyz ÷ (xy + yz + zx).

Time, Speed and Distance Relation

Distance = Speed × Time. Ensure units are consistent (e.g., convert minutes to hours by dividing by 60 if speed is in km/h).