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Shortcut Trick

Equate decimal places: Total numerator places = 13, Total denominator places = 12, resulting in a factor of 10⁻¹.

Identify powers: 6859 = 19³, 54872 = 38³ = (2 × 19)³, and 1156 = 34² = (2 × 17)².

Simplify the ratios: (19/38)³ × (34/17)² × (64000/1600) × 0.1 = (1/8) × 4 × 40 × 0.1 = 2.

Calculate the final root: √2 ≈ 1.414.

∴ The correct answer is 1.414.

Alternate Method

Given Expression: √( (0.6859 × 0.1156 × 0.64000) / (0.289 × 0.1600 × 0.54872) )

⇒ Count total decimal places in Numerator: 4 + 4 + 5 = 13.

⇒ Count total decimal places in Denominator: 3 + 4 + 5 = 12.

⇒ Remove decimals by adjusting powers of 10: √( (6859 × 1156 × 64000) / (289 × 1600 × 54872) × 10⁻¹ )

⇒ Simplify the large numbers using prime factorization: 6859 = 19³, 1156 = 34², 289 = 17², 54872 = 38³.

⇒ √( (19³ × 34² × 40) / (17² × 38³ × 10) )

⇒ √( (19³ × (2 × 17)² × 40) / (17² × (2 × 19)³ × 10) )

⇒ √( (19³ × 4 × 17² × 40) / (17² × 8 × 19³ × 10) )

⇒ Cancel common terms: √( (4 × 40) / (8 × 10) ) = √(160 / 80) = √2

⇒ Square root of 2 up to three decimal places = 1.414

∴ The correct answer is 1.414.

Additional Information

Cubes of Numbers

Memorizing cubes up to 20 is crucial for simplification: 19³ = 6859 and 38³ = 54872.

Square Root Constants

Standard square root values frequently used in exams: √2 ≈ 1.414 and √3 ≈ 1.732.

Power and Indices Rule

For any numbers a and b: (a × b)ⁿ = aⁿ × bⁿ. This helps in simplifying ratios like (34/17)² = 2² = 4.

Shortcut Trick

Relative Speed (opposite direction) = 55 + 65 = 120 km/h

Total distance to be covered = 160 + 140 = 300 m

Time = Total Distance ÷ Relative Speed

Time = 300 ÷ (120 × 5/18) = 300 ÷ (100/3) = 9 sec

∴ The correct answer is 9 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 160 m

Length of Train 2 (L₂) = 140 m

Speed of Train 1 (S₁) = 55 km/h

Speed of Train 2 (S₂) = 65 km/h

Formula Used:

Time taken to cross = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 160 + 140 = 300 m

⇒ Relative Speed = 55 + 65 = 120 km/h

⇒ Convert speed to m/s = 120 × (5/18) m/s

⇒ Relative Speed = 600/18 = 100/3 m/s

⇒ Time = 300 ÷ (100/3)

⇒ Time = (300 × 3) ÷ 100

⇒ Time = 900 ÷ 100 = 9 sec

∴ The correct answer is 9 sec.

Additional Information

Relative Speed in Opposite Directions

When two objects move towards each other or in opposite directions, their relative speed is the sum of their individual speeds: V_rel = S₁ + S₂.

Unit Conversion Factor

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

Crossing a Stationary Object

When a train crosses a pole or a standing man, the distance covered is equal to the length of the train itself.

Shortcut Trick

Given difference (x − y) = 20 and mean proportional √xy = 24.

Square of mean proportional: xy = 24² = 576.

Using the identity: (x + y)² = (x − y)² + 4xy

⇒ (x + y)² = 20² + 4 × 576 = 400 + 2304 = 2704.

⇒ Sum (x + y) = √2704 = 52.

∴ The correct answer is 52.

Alternate Method

Let the two positive numbers be 'a' and 'b' (where a > b).

Given:

Difference between numbers (a − b) = 20

Mean proportional between a and b = 24

Formula Used:

1. Mean Proportional = √(a × b)

2. (a + b)² = (a − b)² + 4ab

⇒ From mean proportional definition: √ab = 24

⇒ Squaring both sides: ab = 24 × 24 = 576

⇒ We know that (a + b)² = (a − b)² + 4ab

⇒ (a + b)² = (20)² + 4 × 576

⇒ (a + b)² = 400 + 2304

⇒ (a + b)² = 2704

⇒ a + b = √2704

⇒ a + b = 52

∴ The correct answer is 52.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is the square root of their product, expressed as x = √ab.

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is called the third proportional, where c = b² ÷ a.

Algebraic Relationship

The relationship between the sum and difference of two numbers is given by the identity (a + b)² − (a − b)² = 4ab.

Shortcut Trick

Combined efficiency of P and Q = 1/16, and efficiency of P alone = 1/24.

Work done by P and Q together in 4 days = 4 ÷ 16 = 1/4 of total work.

Remaining work to be finished by P alone = 1 − 1/4 = 3/4.

More days required by P = (3/4) × 24 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given: P and Q can complete the task in 16 days; P alone takes 24 days; they work together for 4 days.

Formula Used: Total Work = LCM of individual days; Efficiency = Total Work ÷ Time.

⇒ Total work (LCM of 16 and 24) = 48 units.

⇒ Efficiency of P + Q = 48 ÷ 16 = 3 units/day.

⇒ Efficiency of P alone = 48 ÷ 24 = 2 units/day.

⇒ Work completed by P and Q in 4 days = 3 units/day × 4 days = 12 units.

⇒ Remaining work = 48 − 12 = 36 units.

⇒ Additional days for P alone = 36 units ÷ 2 units/day = 18 days.

∴ The correct answer is 18 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken (E ∝ 1/T). If a person is twice as efficient, they take half the time.

Combined Efficiency

The total efficiency of multiple workers is the sum of their individual efficiencies (E_Total = E₁ + E₂ + ...).

Pipes and Cisterns Analogy

The same LCM and efficiency method applies to tank-filling problems, where inlet pipes perform positive work and outlet pipes perform negative work.

Shortcut Trick

Net deviation from the overall Average Height (164 cm) must be zero.

Deviation for Class A = (160 − 164) × 24 = −96

Deviation for Class B = (168 − 164) × 20 = +80

Total deviation (A + B) = −16. To balance, Class C must provide a deviation of +16.

Deviation per student in Class C = 165 − 164 = +1.

Number of students in Class C = 16 ÷ 1 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Class A: n_A = 24, Avg_A = 160 cm

Class B: n_B = 20, Avg_B = 168 cm

Class C: n_C = x, Avg_C = 165 cm

Overall Average = 164 cm

Formula Used:

Overall Average = (n_A × Avg_A + n_B × Avg_B + n_C × Avg_C) ÷ (n_A + n_B + n_C)

Calculations:

⇒ 164 = (24 × 160 + 20 × 168 + x × 165) ÷ (24 + 20 + x)

⇒ 164 = (3840 + 3360 + 165x) ÷ (44 + x)

⇒ 164(44 + x) = 7200 + 165x

⇒ 7216 + 164x = 7200 + 165x

⇒ 7216 − 7200 = 165x − 164x

⇒ x = 16

∴ The correct answer is 16.

Additional Information

Weighted Average Formula

The combined average of multiple groups is calculated as A_w = (n₁a₁ + n₂a₂ + ... + n_ka_k) ÷ (n₁ + n₂ + ... + n_k).

Deviation Method

The sum of deviations of all observations from their arithmetic mean is always zero. This property helps in finding missing values or averages quickly.

Average Property

The combined average must always lie between the lowest and the highest individual group averages. In this case, 160 < 164 < 168.

Shortcut Trick

Total Volume = 30 × 20 × 8 = 4800 cm³

Remaining Volume = 90% of 4800 = 4320 cm³

Volume of 1 Cube = 4³ = 64 cm³

Number of Cubes = 4320 ÷ 64 = 67.5 → 67 complete cubes

∴ The correct answer is 67.

Alternate Method

Given:

Dimensions of the solid cuboid = 30 cm × 20 cm × 8 cm

Material lost during melting = 10%

Side of the identical cubes = 4 cm

Calculations:

⇒ Total volume of the cuboid = Length × Breadth × Height

⇒ Volume = 30 × 20 × 8 = 4800 cm³

⇒ Volume of material lost = 10% of 4800 = (10 ÷ 100) × 4800 = 480 cm³

⇒ Volume of remaining metal = 4800 − 480 = 4320 cm³

⇒ Volume of one small cube = (Side)³ = 4³ = 64 cm³

⇒ Number of cubes formed = Remaining Volume ÷ Volume of one cube

⇒ Number of cubes = 4320 ÷ 64

⇒ Number of cubes = 67.5

⇒ Since we need the number of complete cubes, we take the integer part.

∴ The correct answer is 67.

Additional Information

Volume of Solid Shapes

For a cuboid, Volume = L × B × H. For a cube, Volume = a³. During melting and recasting, total volume remains constant unless a percentage loss is specified.

Material Loss Calculation

If x% of material is lost, the usable volume is (100 − x)% of the initial volume. Usable Volume = V_initial × (1 − x/100).

Complete Units Rule

When calculating the number of discrete items formed from a bulk material, always round down to the nearest whole number if the result is a fraction.

Shortcut Trick

Calculate total interest percentage: Interest = 2529.60 − 1860 = 669.60

Total Interest % = (669.60 / 1860) × 100 = 36%

Since the rate is 12% per year, Time (T) = Total % / Rate per annum

T = 36% / 12% = 3 years

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹1,860

Amount (A) = ₹2,529.60

Rate (R) = 12% p.a.

Formula Used: Simple Interest (SI) = (P × R × T) / 100

⇒ Simple Interest (SI) = Amount − Principal

⇒ SI = 2529.60 − 1860 = 669.60

⇒ 669.60 = (1860 × 12 × T) / 100

⇒ 669.60 = 18.6 × 12 × T

⇒ 669.60 = 223.2 × T

⇒ T = 669.60 / 223.2

⇒ T = 3 years

∴ The correct answer is 3 years.

Additional Information

Simple Interest Formula

The basic formula is SI = (P × R × T) / 100, where P is Principal, R is Rate, and T is Time.

Amount Calculation

Total Amount (A) = Principal (P) + Simple Interest (SI). It can also be expressed as A = P(1 + RT/100).

Finding Rate of Interest

If time and interest are known, the rate can be calculated as R = (SI × 100) / (P × T).

Shortcut Trick

Apply the Successive Percentage Change formula: Net % change = x + y + (xy/100).

Here, x = +25% (increase) and y = −20% (decrease).

Net change = 25 − 20 + [25 × (−20) / 100] = 5 − 5 = 0%.

Since the net change is 0%, the final salary remains the same as the initial salary.

∴ The correct answer is ₹58,000.

Alternate Method

Given:

Initial Salary = ₹58,000

First Change = 25% Increase

Second Change = 20% Decrease

Step 1: Calculate salary after 25% increase

⇒ Salary after increase = 58,000 + (25/100 × 58,000)

⇒ Salary after increase = 58,000 + 14,500 = ₹72,500

Step 2: Calculate salary after 20% decrease on the new amount

⇒ Final Salary = 72,500 − (20/100 × 72,500)

⇒ Final Salary = 72,500 − 14,500

⇒ Final Salary = ₹58,000

∴ The correct answer is ₹58,000.

Additional Information

Successive Percentage Formula

When a value is changed by x% and then by y%, the effective change is (x + y + xy/100)%. Use positive for increase and negative for decrease.

Multiplier Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). In this case: 58,000 × 1.25 × 0.80 = 58,000 × 1 = 58,000.

Base Change Property

An increase of 25% (1/4) followed by a decrease of 20% (1/5) results in no change because (5/4) × (4/5) = 1.

Shortcut Trick

The Total Surface Area (TSA) of a cube follows the formula 6 × a².

Since the side is 94 cm, the unit digit of 94² is 6 (4 × 4 = 16).

Multiplying by 6, the final unit digit must be 6 (6 × 6 = 36).

Estimating: 94² is slightly less than 100² (10,000), so TSA is slightly less than 60,000.

Calculation: 6 × 8836 = 53016.

∴ The correct answer is 53,016 cm².

Alternate Method

Given:

The side length (a) of the cube is 94 cm.

Formula Used:

Total Surface Area of a cube = 6 × (side)²

Calculations:

⇒ TSA = 6 × (94)²

⇒ TSA = 6 × (94 × 94)

⇒ 94² = (100 − 6)² = 10000 − 1200 + 36 = 8836

⇒ TSA = 6 × 8836

⇒ TSA = 53016 cm²

∴ The correct answer is 53,016 cm².

Additional Information

Volume of a Cube

The volume represents the total capacity of the cube and is calculated as a³.

Lateral Surface Area (LSA)

The LSA refers to the area of only the four side faces (excluding top and bottom), given by 4 × a².

Body Diagonal of a Cube

The length of the longest rod that can be placed inside a cube is a√3.

Face Diagonal of a Cube

The diagonal of any single square face of the cube is calculated as a√2.

Shortcut Trick

Total work = LCM(12, 30) = 60 units.

Efficiency of A = 5 units/day, B = 2 units/day.

Work done by both in 2 days = (5 + 2) × 2 = 14 units.

Remaining work = 60 − 14 = 46 units.

Time taken by B alone = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Alternate Method

Given: A completes work in 12 days and B in 30 days.

⇒ Total work (LCM of 12 and 30) = 60 units.

⇒ Work completed by A and B together in 2 days = (5 + 2) × 2 = 14 units.

⇒ Remaining work to be done by B = 60 − 14 = 46 units.

⇒ Time taken by B to complete the remaining work = Remaining Work ÷ Efficiency of B

⇒ Time = 46 ÷ 2 = 23 days.

∴ The correct answer is 23 days.

Additional Information

Relationship between Efficiency and Time

Efficiency is inversely proportional to Time. If time ratio is x : y, the efficiency ratio is y : x.

Work & Wages

Wages are generally distributed in the ratio of the work done by each person, which is the product of Efficiency × Time.

Total Work Calculation

Total work can be assumed as any multiple of the given days, but LCM is preferred to keep efficiencies in whole numbers.