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Shortcut Trick

Total Work (LCM of 10, 12, 15) = 60 units.

Efficiencies: (A+B) = 6, (B+C) = 5, (A+C) = 4.

Combined efficiency 2(A+B+C) = 6 + 5 + 4 = 15.

Efficiency (A+B+C) = 15 ÷ 2 = 7.5.

Time taken = 60 ÷ 7.5 = 8 days.

∴ The correct answer is 8 days.

Alternate Method

Given: (A+B) takes 10 days, (B+C) takes 12 days, and (A+C) takes 15 days.

Formula Used: Total Work = Time × Efficiency

⇒ Total efficiency of (A+B), (B+C), and (A+C) = 6 + 5 + 4 = 15

⇒ 2 × (Efficiency of A + B + C) = 15

⇒ Efficiency of A, B and C together = 15 ÷ 2 = 7.5

⇒ Time taken by A, B and C = Total Work ÷ Combined Efficiency

⇒ Time taken = 60 ÷ 7.5 = 8 days

∴ The correct answer is 8 days.

Additional Information

Individual Efficiency

To find A's individual efficiency, subtract (B+C) efficiency from (A+B+C) total efficiency: 7.5 − 5 = 2.5.

Work and Wages

Wages are always distributed in the ratio of the work done by each person or the ratio of their efficiencies.

Pipes and Cisterns

This logic is identical to pipes filling a tank, where an outlet pipe performs negative work (efficiency is subtracted).

Shortcut Trick

Total volume of a cuboid formed by joining n identical cubes is simply the sum of the volumes of all individual cubes.

Volume of one cube = side³ = 4³ = 64 cm³

Total Volume = 6 × 64 = 384 cm³

∴ The correct answer is 384 cm³.

Alternate Method

Given: Side of each cube (a) = 4 cm and Number of cubes (n) = 6.

When six cubes are placed adjacent to each other in a row, the dimensions of the resulting cuboid are:

Formula Used: Volume of Cuboid = Length (L) × Breadth (B) × Height (H)

⇒ Length (L) = 6 × 4 = 24 cm

⇒ Breadth (B) = 4 cm

⇒ Height (H) = 4 cm

⇒ Volume = 24 × 4 × 4

⇒ Volume = 24 × 16

⇒ Volume = 384 cm³

∴ The correct answer is 384 cm³.

Additional Information

Total Surface Area of Cuboid

The total area of all six faces of a cuboid is calculated as TSA = 2(LB + BH + HL).

Diagonal of a Cuboid

The length of the longest rod that can be placed inside a cuboid is given by √(L² + B² + H²).

Volume of a Cube

For a cube where all sides are equal (L = B = H = a), the volume is a³ and the lateral surface area is 4a².

Shortcut Trick

Total percentage spent = 25% + 20% + 15% = 60%

Percentage saved = 100% − 60% = 40%

Monthly Savings = 40% of 84,000 = 40 × 840 = 33,600

∴ The correct answer is ₹33,600.

Alternate Method

Given:

Monthly Income of Naveena = ₹84,000

House Rent = 25%

Groceries = 20%

Other Expenses = 15%

Formula used:

Savings = Total Income − Total Expenditure

Savings = Income × [100 − (Spent₁ + Spent₂ + Spent₃)] / 100

Calculations:

⇒ Total Expenditure % = 25% + 20% + 15% = 60%

⇒ Savings % = 100% − 60% = 40%

⇒ Savings Amount = (40 / 100) × 84,000

⇒ Savings Amount = 0.4 × 84,000

⇒ Savings Amount = 33,600

∴ The correct answer is ₹33,600.

Additional Information

Income Expenditure Identity

Total Income is always the sum of Expenditure and Savings. Income = Expenditure + Savings.

Successive Percentage Concept

If the problem states spending on the 'remaining' amount, we use the multiplier method: Final Savings = Income × (100−r₁)/100 × (100−r₂)/100.

Percentage to Fraction Conversion

Quick calculations can be done by converting common percentages to fractions: 25% = 1/4, 20% = 1/5, 40% = 2/5.

Shortcut Trick

Net Simple Interest percentage = Rate × Time

Net Interest % = 5% × 5 = 25%

Simple Interest = 25% of ₹75,000

SI = 75,000 ÷ 4 = ₹18,750

∴ The correct answer is ₹18,750.

Alternate Method

Given:

⇒ Principal (P) = ₹75,000

⇒ Rate (R) = 5% per annum

⇒ Time (T) = 5 years

Formula Used:

⇒ Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ SI = (75,000 × 5 × 5) ÷ 100

⇒ SI = 750 × 25

⇒ SI = ₹18,750

∴ The correct answer is ₹18,750.

Additional Information

Total Amount Formula

The Total Amount (A) is the sum of the principal and the interest earned: A = P + SI.

Rate of Interest Calculation

To find the Rate (R) when other values are known: R = (SI × 100) ÷ (P × T).

Constant Interest Property

In Simple Interest, the interest amount remains the same for every year if the principal and rate are constant.

Shortcut Trick

Speed of Train 1 = 200 ÷ 10 = 20 m/s

Speed of Train 2 = 60 × 5/18 = 50/3 m/s

Relative Speed (Opposite) = 20 + 50/3 = 110/3 m/s

Required Time = (Total Length) ÷ (Relative Speed) = (200 + 350) ÷ (110/3) = 15 sec

∴ The correct answer is 15 sec.

Alternate Method

Given: Length of first train (L₁) = 200 m, Time to cross man = 10 sec, Length of second train (L₂) = 350 m, Speed of second train (S₂) = 60 km/h.

Formula Used: Speed = Distance ÷ Time | Relative Speed (Opposite) = S₁ + S₂

⇒ Speed of the first train (S₁) = Distance ÷ Time = 200 ÷ 10 = 20 m/s

⇒ Speed of the second train (S₂) in m/s = 60 × (5/18) = 50/3 m/s

⇒ When two trains move in opposite directions, their relative speed is the sum of their individual speeds.

⇒ Relative Speed = S₁ + S₂ = 20 + 50/3 = (60 + 50)/3 = 110/3 m/s

⇒ Total distance to be covered while crossing = L₁ + L₂ = 200 + 350 = 550 m

⇒ Time taken = Total Distance ÷ Relative Speed = 550 ÷ (110/3) = 550 × 3 ÷ 110

⇒ Time taken = 5 × 3 = 15 sec

∴ The correct answer is 15 sec.

Additional Information

Relative Speed Concepts

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

Unit Conversion

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

Train Crossing Distance

To cross a point object (man, pole), distance = length of train. To cross a platform or another train, distance = sum of both lengths.

Shortcut Trick

Profit Ratio = (Capital × Time) Ratio

Kirti (K) : Vaishnavi (V) = (27000 × 12) : (Capital_V × 5)

Given Profit Ratio = 8 : 9

⇒ (27000 × 12) / (Capital_V × 5) = 8 / 9

⇒ Capital_V = (27000 × 12 × 9) / (5 × 8) = 675 × 108 = 72900

∴ The correct answer is ₹72,900.

Alternate Method

Given:

Kirti's Capital (C_K) = ₹27,000

Kirti's investment period (T_K) = 12 months

Vaishnavi joins after 7 months, so investment period (T_V) = 12 − 7 = 5 months

Profit Ratio (P_K : P_V) = 8 : 9

Formula used:

Ratio of Profits = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Calculations:

⇒ (C_K × T_K) / (C_V × T_V) = P_K / P_V

⇒ (27000 × 12) / (C_V × 5) = 8 / 9

⇒ 324000 / (5 × C_V) = 8 / 9

⇒ 5 × C_V × 8 = 324000 × 9

⇒ 40 × C_V = 2916000

⇒ C_V = 2916000 ÷ 40

⇒ C_V = 72900

∴ The correct answer is ₹72,900.

Additional Information

Partnership Profit Distribution

When investments are for different time periods, profit is distributed based on the product of Capital and Time. P₁ : P₂ = C₁T₁ : C₂T₂.

Active vs Sleeping Partner

An Active Partner manages the business and may receive a salary or extra profit share, while a Sleeping Partner only provides capital.

Investment Calculation

To find Capital when Profit and Time are known: Capital = Profit ÷ Time. To find Time: Time = Profit ÷ Capital.

Shortcut Trick

Direct relation: Volume = (Lateral Surface Area)² ÷ (4 × π × Height)

Given LSA = 775.8 cm², Height = 24 cm, π = 3.14

Volume = (775.8 × 775.8) ÷ (4 × 3.14 × 24)

Volume = 601865.64 ÷ 301.44 = 1996.6346...

Rounding to two decimal places, we get 1996.63 cm³.

∴ The correct answer is 1996.63 cm³.

Alternate Method

Given: Lateral Surface Area (LSA) = 775.8 cm², Height (h) = 24 cm, π = 3.14

Formula Used:

1. Lateral Surface Area of a Cylinder = 2 × π × r × h
2. Volume of a Cylinder = π × r² × h

Step 1: Find the radius (r) using the Lateral Surface Area formula.

⇒ 775.8 = 2 × 3.14 × r × 24

⇒ 775.8 = 150.72 × r

⇒ r = 775.8 ÷ 150.72

⇒ r ≈ 5.1473 cm

Step 2: Calculate the Volume of the Cylinder.

⇒ Volume = 3.14 × (5.1473)² × 24

⇒ Volume = 3.14 × 26.4947 × 24

⇒ Volume = 1996.6348...

⇒ Volume ≈ 1996.63 cm³

∴ The correct answer is 1996.63 cm³.

Additional Information

Total Surface Area of Cylinder

The Total Surface Area (TSA) includes both the lateral area and the area of the two circular bases: TSA = 2πr(r + h).

Volume of a Hollow Cylinder

If a cylinder is hollow with external radius R and internal radius r, the volume of material is: V = πh(R² − r²).

Base Area and Perimeter

The area of the circular base is πr² and its perimeter (circumference) is 2πr. LSA can also be written as Perimeter of Base × Height.

Shortcut Trick

Apply the Empirical formula: 2 × Mean = 3 × Median − Mode

2 × Mean = 3 × 65 − 60

2 × Mean = 195 − 60

2 × Mean = 135

Mean = 135 ÷ 2 = 67.5

∴ The correct answer is 67.5.

Alternate Method

Given:

Median = 65

Mode = 60

Formula Used:

The relationship between Mean, Median, and Mode is given by: Mode = 3(Median) − 2(Mean)

Calculations:

⇒ Mode = 3 × Median − 2 × Mean

⇒ 60 = 3 × 65 − 2 × Mean

⇒ 60 = 195 − 2 × Mean

⇒ 2 × Mean = 195 − 60

⇒ 2 × Mean = 135

⇒ Mean = 135 ÷ 2

⇒ Mean = 67.5

∴ The correct answer is 67.5.

Additional Information

Empirical Formula

For a moderately skewed distribution, the relationship is Mean − Mode = 3 × (Mean − Median).

Symmetrical Distribution

In a perfectly symmetrical distribution, the values of Mean, Median, and Mode are all equal (Mean = Median = Mode).

Skewness Types

If Mean > Median > Mode, the distribution is positively skewed. If Mode > Median > Mean, it is negatively skewed.

Shortcut Trick

Solve the expression by breaking it into two major square brackets using BODMAS.

First bracket: [ (120 − 4) × 2 − (10 + 6) ] → [ 232 − 16 ] = 216.

Second bracket: [ (18 + 24 − 8) − (3 + 5) ] → [ 34 − 8 ] = 26.

Total: 216 + 26 = 242.

∴ The correct answer is 242.

Alternate Method

Given: [(120 − 24 ÷ (4 + 2)) × 2 − (50 ÷ 5 + 6)] + [(18 + 12 × 2 − 8) − (9 ÷ 3 + 5)]

Formula Used: VBODMAS rule (Viniculum, Brackets, Of, Division, Multiplication, Addition, Subtraction)

⇒ [(120 − 24 ÷ 6) × 2 − (10 + 6)] + [(18 + 24 − 8) − (3 + 5)]

⇒ [(120 − 4) × 2 − 16] + [(42 − 8) − 8]

⇒ [116 × 2 − 16] + [34 − 8]

⇒ [232 − 16] + 26

⇒ 216 + 26

⇒ 242

∴ The correct answer is 242.

Additional Information

BODMAS Rule

It specifies the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction. Always follow this hierarchy to avoid errors in simplification.

Types of Brackets

Standard order of solving is: Circular Brackets ( ), then Curly Brackets { }, and finally Square Brackets [ ].

Signs Multiplication Rule

(+) × (+) = (+), (−) × (−) = (+), (+) × (−) = (−), and (−) × (+) = (−). This applies to division as well.

Shortcut Trick

Total Initial Income = 12,000 + 8,000 + 15,000 = ₹35,000

Net increase in Income = 15% of 35,000 = ₹5,250

Net increase in Expenditure = (10% of 12,000) + (20% of 8,000) = 1,200 + 1,600 = ₹2,800

Increase in Savings = Income increase − Expenditure increase = 5,250 − 2,800 = ₹2,450

New Savings = 15,000 + 2,450 = ₹17,450

∴ The correct answer is ₹17,450.

Alternate Method

Given:

Initial Food Exp = ₹12,000 | Initial Rent Exp = ₹8,000 | Initial Savings = ₹15,000

Growth rates: Income +15%, Food +10%, Rent +20%

Formula Used:

Income = Food Expenditure + Rent Expenditure + Savings

⇒ Total Initial Income = 12,000 + 8,000 + 15,000 = 35,000

⇒ New Income = 35,000 × (115/100) = 35,000 × 1.15 = 40,250

⇒ New Food Exp = 12,000 × (110/100) = 13,200

⇒ New Rent Exp = 8,000 × (120/100) = 9,600

⇒ Total New Exp = 13,200 + 9,600 = 22,800

⇒ New Savings = New Income − Total New Exp

⇒ New Savings = 40,250 − 22,800 = 17,450

∴ The correct answer is ₹17,450.

Additional Information

Income-Expenditure Relationship

Income is always the sum of total Expenditure and Savings. Formula: I = E + S.

Percentage Change Formula

New Value = Old Value × (100 ± Change%)/100. Use (+) for increase and (−) for decrease.

Net Change in Savings

Change in Savings (ΔS) = Change in Income (ΔI) − Change in Expenditure (ΔE).