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Shortcut Trick

Overall average of 12 numbers = 40

Deviation of first 7 numbers from average = 30 − 40 = −10

Total deviation to be balanced = 7 × (−10) = −70

Deviation for remaining 5 numbers = +70

Average of remaining numbers = 40 + (70 ÷ 5) = 40 + 14 = 54

∴ The correct answer is 54.

Alternate Method

Given:

Average of 12 numbers = 40

Average of first 7 numbers = 30

Number of remaining numbers = 12 − 7 = 5

Formula used: Sum = Average × Count

Total: 12 Numbers (Average = 40) First 7 Numbers Average = 30 Remaining 5 Average = ?

Total Sum = 12 × 40 = 480

Calculations:

⇒ Total sum of 12 numbers = 12 × 40 = 480

⇒ Sum of first 7 numbers = 7 × 30 = 210

⇒ Sum of remaining 5 numbers = 480 − 210 = 270

⇒ Average of remaining 5 numbers = 270 ÷ 5 = 54

∴ The correct answer is 54.

Additional Information

Average Formula

Average = (Sum of all observations) ÷ (Total number of observations).

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Effect of Operations on Average

If every number in a data set is increased or decreased by a constant k, the average of the set also increases or decreases by k.

Shortcut Trick

Side of square = √121 = 11 cm.

Perimeter of square = 4 × 11 = 44 cm.

Circumference of circle = Perimeter = 44 cm.

Memory trick: For a circle, if Circumference = 44 cm, then Radius = 7 cm and Area = 154 sq. cm.

∴ The correct answer is 154 sq. cm.

Alternate Method

Given: Area of square = 121 sq. cm.

Formula: Area of square = side²; Perimeter = 4 × side; Circumference = 2πr; Area of circle = πr².

Square (Area = 121)side = 11 cmr = 7Circle (Circum = 44)Area = 154 sq. cm

⇒ Side of square = √121 = 11 cm

⇒ Perimeter of square = 4 × 11 = 44 cm

⇒ Circumference of circle = 2 × (22/7) × r = 44

⇒ r = (44 × 7) ÷ 44 = 7 cm

⇒ Area of circle = (22/7) × 7 × 7 = 154 sq. cm.

∴ The correct answer is 154 sq. cm.

Additional Information

Circle Formulas

Area = πr² and Circumference = 2πr. When r = 7, Area = 154 and Circumference = 44.

Square Formulas

Area = side² and Perimeter = 4 × side. The diagonal of a square is side√2.

Perimeter-Area Comparison

For a given perimeter, a circle has the maximum area compared to any other polygon like a square or rectangle.

Shortcut Trick

For two entities, if (A+B) takes X days and A takes Y days, then B takes (X × Y) ÷ (Y − X) days.

Here, X = 20 days and Y = 24 days.

Days for B = (20 × 24) ÷ (24 − 20)

Days for B = 480 ÷ 4 = 120 days.

∴ The correct answer is 120 days.

Alternate Method

Given: A and B together take 20 days. A alone takes 24 days.

Formula: Efficiency = Total Work ÷ Days

⇒ Total Work = LCM of 20 and 24 = 120 units.

⇒ Efficiency of (A + B) = 120 ÷ 20 = 6 units/day.

⇒ Efficiency of A = 120 ÷ 24 = 5 units/day.

⇒ Efficiency of B = Efficiency of (A + B) − Efficiency of A = 6 − 5 = 1 unit/day.

⇒ Time taken by B = Total Work ÷ Efficiency of B

⇒ Time taken by B = 120 ÷ 1 = 120 days.

∴ The correct answer is 120 days.

Additional Information

Work and Wages

Wages are distributed in the ratio of the efficiency of the workers when they work for the same amount of time.

Pipes and Cisterns

Similar to time and work, but inlet pipes have positive efficiency and outlet (leak) pipes have negative efficiency.

Alternate Day Work

Calculate the work done in one cycle (e.g., 2 days if A and B work alternately) and then find the total cycles needed.

Shortcut Trick

The net equivalent discount for two successive discounts of x% and y% is given by the formula: [x + y − (x × y) ÷ 100]%

Here, the given discounts are x = 30 and y = 40

⇒ Net Discount = [30 + 40 − (30 × 40) ÷ 100]%

⇒ Net Discount = (70 − 12)% = 58%

∴ The correct answer is 58%.

Alternate Method

Given:

Successive discounts = 30% and 40%

Calculations:

⇒ Let the Marked Price (MP) be 100.

⇒ Price after the first discount of 30% = 100 − (30 ÷ 100 × 100) = 100 − 30 = 70

⇒ Price after the second discount of 40% = 70 − (40 ÷ 100 × 70) = 70 − 28 = 42

⇒ Final Selling Price (SP) = 42

⇒ Total Discount given = MP − SP = 100 − 42 = 58

⇒ Net Discount Percentage = (58 ÷ 100) × 100% = 58%

∴ The correct answer is 58%.

Additional Information

Successive Discount

When two or more discounts are applied one after the other on the Marked Price. For two discounts a% and b%, the combined discount is (a + b − ab ÷ 100)%.

Marked Price (MP) & Selling Price (SP)

Discount is always calculated on the Marked Price (label price). The Selling Price is derived as: SP = MP × (100 − Discount%) ÷ 100.

Multiplier Method for Successive Discounts

To calculate the final SP after multiple successive discounts directly, use: Net SP = MP × (100 − d₁)÷100 × (100 − d₂)÷100 × (100 − d₃)÷100.

Shortcut Trick

Total Ratio parts = 3 + 3 + 7 = 13 units

Total Gain (13 units) = ₹39000

Value of 1 unit = 39000 ÷ 13 = ₹3000

Share of C = 7 units = 7 × 3000 = ₹21000

∴ The correct answer is ₹21000.

Alternate Method

Given: Ratio of profit A : B : C = 3 : 3 : 7. Total gain = ₹39000.

Formula Used: Share of a partner = (Partner ratio ÷ Sum of ratios) × Total profit.

A (3) B (3) C (7) Total Parts = 13

Total Profit = ₹39000

Share C = 7/13 × 39000

⇒ Sum of ratios = 3 + 3 + 7 = 13

⇒ Total gain = ₹39000

⇒ Share of C = (7 ÷ 13) × 39000

⇒ Share of C = 7 × 3000

⇒ Share of C = ₹21000

∴ The correct answer is ₹21000.

Additional Information

Partnership Profit Ratio

Profit is distributed in the ratio of (Investment × Time). If time is same, profit ratio = investment ratio.

Ratio to Value Conversion

To find the value of a part: Value = (Specific Ratio ÷ Sum of all Ratios) × Total Value.

Types of Partners

Active partners manage the business and may receive salary; Sleeping partners only invest capital and share profit.

Shortcut Trick

When two trains of equal length cross the same platform, the total distance (Train + Platform) is constant for both.

If distance is constant, Speed is inversely proportional to the Time taken.

Ratio of Speeds (A  ∶ B) = Inverse Ratio of Times (B  ∶ A) = 20  ∶ 24.

Dividing both by 4, we get the simplified ratio: 5  ∶ 6.

∴ The correct answer is 5 ∶ 6.

Alternate Method

Given:

Time taken by Train A (T_A) = 24 seconds

Time taken by Train B (T_B) = 20 seconds

Length of Train A = Length of Train B = L

Length of the platform = P

Formula Used:

Speed = Distance ÷ Time

Train (L)

Platform (P)

Total Distance (D) = L + P (Same for both Train A and Train B)

Calculations:

⇒ Total distance covered to cross the platform (D) = L + P ⇒ Speed of Train A (S_A) = D ÷ 24 ⇒ Speed of Train B (S_B) = D ÷ 20 ⇒ Ratio of speeds = S_A  ∶ S_B ⇒ S_A  ∶ S_B = (D ÷ 24)  ∶ (D ÷ 20) ⇒ S_A  ∶ S_B = 1/24  ∶ 1/20 ⇒ S_A  ∶ S_B = 20  ∶ 24 ⇒ S_A  ∶ S_B = 5  ∶ 6

∴ The correct answer is 5 ∶ 6.

Additional Information

Train Crossing a Point Object

When a train crosses a pole or a stationary man, the distance covered is exactly equal to the length of the train (D = L).

Relative Speed

If two objects move in the same direction, relative speed = |S₁ − S₂|. If they move in opposite directions, relative speed = S₁ + S₂.

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Shortcut Trick

Initial Ratio M : W = 3 : 2 (Total = 5 units = 60 L ⇒ 1 unit = 12 L)

New Ratio M : W = 1 : 5 (Milk remains constant, so multiply by 3) → 3 : 15

Water increase = 15 − 2 = 13 units

Quantity of water added = 13 units × 12 L/unit = 156 L

∴ The correct answer is 156 L.

Alternate Method

Given: Total Volume = 60 L, Initial Ratio = 3 : 2, Final Ratio = 1 : 5.

Formula: Quantity = [Ratio Part ÷ Total Parts] × Total Volume

Initial (60 L) Milk: 36 L Water: 24 L

• Water (x)

Final Ratio 1 : 5 Milk: 36 L (1 unit) Water: 180 L (5 units) x = 180 − 24 = 156 L

⇒ Initial Milk = (3/5) × 60 = 36 L

⇒ Initial Water = (2/5) × 60 = 24 L

⇒ Let x be the amount of water added.

⇒ New Ratio = Milk / (Water + x) = 1 / 5

⇒ 36 / (24 + x) = 1 / 5

⇒ 36 × 5 = 24 + x

⇒ 180 = 24 + x

⇒ x = 180 − 24 = 156 L

∴ The correct answer is 156 L.

Additional Information

Constant Part Concept

When one component is added or removed from a mixture, the quantity of the other component remains constant. Equating the constant part in both ratios simplifies the calculation.

Alligation Method

For adding pure water to a mixture, the concentration of water in the added part is 100%. Alligation cross-difference helps find the ratio of initial mixture to added water.

Percentage Base Change

Initial % of milk × Initial Volume = Final % of milk × Final Volume. This formula works efficiently when only one component changes.

Shortcut Trick

If Income and Expenditure both increase by the same percentage (x%), then Savings will also increase by the same percentage (x%).

Given increase in Income = 10%

Given increase in Expenditure = 10%

Since both parts of the income equation increase by 10%, the balancing part (Savings) must also change by 10%.

∴ The correct answer is 10% increase.

Alternate Method

Given: Initial Income = 100%, Expenditure = 80%, Income & Spending increase = 10%.

Formula Used: Income = Expenditure + Savings; % Change = (Change ÷ Original) × 100

Initial State Exp: 80 Sav: 20 New State (+10%) Exp: 88 Sav: 22 Savings: 20 → 22 (10% Increase)

⇒ Let Initial Income = 100

⇒ Initial Expenditure = 80% of 100 = 80

⇒ Initial Savings = 100 − 80 = 20

⇒ New Income = 100 + (10% of 100) = 110

⇒ New Expenditure = 80 + (10% of 80) = 88

⇒ New Savings = 110 − 88 = 22

⇒ Increase in Savings = 22 − 20 = 2

⇒ % Change in Savings = (2 ÷ 20) × 100 = 10%

∴ The correct answer is 10% increase.

Additional Information

Income-Expenditure Relationship

The fundamental relation is Income = Expenditure + Savings. Any change in one part must be balanced by others to maintain the equality.

Percentage Change Formula

Percentage Change = [(Final Value − Initial Value) ÷ Initial Value] × 100. This determines the relative growth or decline.

Weighted Average Concept

If income changes by I%, expenditure by E%, and savings by S%, then Income × I% = Expenditure × E% + Savings × S%.

Concept used:

Calculation:

Our equation 29 – [13 + {12 ÷ (8 – 2)}] × 5 ÷ 3

⇒ 29 – [13 + {12 ÷ 6}] × 5 ÷ 3

⇒ 29 – [13 + 2] × 5 ÷ 3

⇒ 29 – 15 × 5 ÷ 3

⇒ 29 – 15 × 5 × 1/3

⇒ 29 – 25 = 4

∴ The correct answer is 4

Shortcut Trick

Net deviation = Correct value − Incorrect value = 46 − 53 = −7

Correction in average = Net deviation ÷ Number of items = −7 ÷ 10 = −0.7

Correct Average = Old Average + Correction = 46 − 0.7 = 45.3

∴ The correct answer is 45.3.

Alternate Method

Given:

Number of observations (n) = 10

Incorrect Average = 46

Incorrect value recorded = 53

Correct value to be recorded = 46

Formula Used:

Average = Sum of observations ÷ Total number of observations

Incorrect Sum (10 × 46) = 460

Wrong: 53 Correct: 46

Difference = −7 Correct Sum = 460 − 7 = 453

⇒ Incorrect Sum = 10 × 46 = 460

⇒ Correct Sum = Incorrect Sum − Incorrect Value + Correct Value

⇒ Correct Sum = 460 − 53 + 46 = 453

⇒ Correct Average = Correct Sum ÷ n

⇒ Correct Average = 453 ÷ 10 = 45.3

∴ The correct answer is 45.3.

Additional Information

Correction of Average

Correct Average = Incorrect Average + [(Sum of Correct values − Sum of Incorrect values) ÷ n].

Change in Observations

If every observation is increased or decreased by a constant 'k', the new average also increases or decreases by 'k'.

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the new average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).