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Total Work = LCM(8, 22, 33) = 264 units.

Efficiencies: Mike = 33, Connor = 12, Floyd = 8.

Adjusted Work = Total Work − (Mike's work) + (Connor's potential work for the 2 days he left early).

Adjusted Work = 264 − (33 × 4) + (12 × 2) = 264 − 132 + 24 = 156 units.

Total Days = Adjusted Work ÷ (Connor's efficiency + Floyd's efficiency) = 156 ÷ (12 + 8) = 7.8 days.

∴ The correct answer is 7.8 days.

Alternate Method

Given: Mike finishes in 8 days, Connor in 22 days, and Floyd in 33 days. Mike leaves after 4 days and Connor leaves 2 days before the end.

Let the total number of days taken to complete the work be T.

⇒ Mike's work + Connor's work + Floyd's work = Total Work

⇒ (33 × 4) + 12 × (T − 2) + (8 × T) = 264

⇒ 132 + 12T − 24 + 8T = 264

⇒ 108 + 20T = 264

⇒ 20T = 156

⇒ T = 156 ÷ 20 = 7.8 days

∴ The correct answer is 7.8 days.

Additional Information

Leaving Work Logic

If a person leaves after x days, subtract their x days of work from the total. If a person leaves before the completion, add their work for those missing days to the total work and treat them as present until the end.

Pipes and Cisterns

Similar to time and work, but outlet pipes have negative efficiency. Total capacity is equivalent to total work.

Work and Wages

Wages are always distributed in the ratio of the work done by each person. If they work for the same amount of time, the ratio of wages is the same as the ratio of their efficiencies.

Shortcut Trick

Since the Original Price is the same for both Article M and Article N (₹10,000), the difference in reduction percentages is proportional to the difference in actual reduction amounts.

Difference in reduction amount = ₹3,000 − ₹2,500 = ₹500.

Difference (R1 − R2) = (Difference in Amount ÷ Common Original Price) × 100.

Difference = (500 ÷ 10,000) × 100 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Original Price (OP) = ₹10,000 (for both articles)

Reduction for Article M = ₹3,000

Reduction for Article N = ₹2,500

Calculations:

⇒ Reduction percentage for Article M (R1) = (Reduction ÷ Original Price) × 100

⇒ R1 = (3,000 ÷ 10,000) × 100 = 30%

⇒ Reduction percentage for Article N (R2) = (Reduction ÷ Original Price) × 100

⇒ R2 = (2,500 ÷ 10,000) × 100 = 25%

⇒ Difference between R1 and R2 = 30% − 25% = 5%

∴ The correct answer is 5.

Additional Information

Percentage Reduction Formula

The rate of reduction or discount is calculated as: (Actual Reduction ÷ Original Marked Price) × 100.

Percentage Difference with Same Base

When comparing two percentages of the same base value, the difference between the percentages is equal to the percentage of the difference between the absolute values.

Successive Discounts

If an article is reduced by a% and then further by b%, the effective reduction is given by the formula: [a + b − (a × b) ÷ 100]%.

Shortcut Trick

Effective rate of compound interest for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%

Effective rate of simple interest for 2 years at 10% = 10 + 10 = 20%

Given that 21% of principal corresponds to ₹220.5.

⇒ 20% of principal = (220.5 ÷ 21) × 20 = 10.5 × 20 = 210

∴ The correct answer is ₹210.

Alternate Method

Given: Rate (R) = 10% per annum, Time (T) = 2 years, Compound Interest (CI) = ₹220.5.

Formula Used: CI = P[(1 + R/100)^T − 1] and SI = (P × R × T) ÷ 100

⇒ 220.5 = P[(1 + 10/100)² − 1]

⇒ 220.5 = P[(1.1)² − 1]

⇒ 220.5 = P[1.21 − 1]

⇒ 220.5 = 0.21P

⇒ P = 220.5 ÷ 0.21 = 1050

⇒ Now, Simple Interest (SI) for 2 years = (1050 × 10 × 2) ÷ 100

⇒ SI = 1050 × 0.2 = 210

∴ The correct answer is ₹210.

Additional Information

Difference between CI and SI for 2 Years

The difference between compound and simple interest for two years is given by D = P × (R/100)².

Effective Interest Rate

For successive percentage increases of x% and y%, the effective change is x + y + (x × y)/100.

Compound Interest Formula

The amount A after n years is A = P × (1 + R/100)ⁿ, where CI = A − P.

Shortcut Trick

Let the Larger Number be L and the Smaller Number be S.

According to the conditions: 2S + 3L = 55 and L + 5 = 2S.

Substitute 2S directly into the first equation: (L + 5) + 3L = 55.

4L + 5 = 55 ⇒ 4L = 50 ⇒ L = 12.5.

∴ The correct answer is 12.5.

Alternate Method

Given:

Let the Smaller Number be x and the Larger Number be y.

Equation 1: 2x + 3y = 55

Equation 2: y + 5 = 2x

Calculations:

⇒ 2x + 3y = 55 --- (1)

⇒ 2x − y = 5 --- (2)

Substitute the value of 2x from (2) into (1):

⇒ (y + 5) + 3y = 55

⇒ 4y + 5 = 55

⇒ 4y = 55 − 5

⇒ 4y = 50

⇒ y = 50 ÷ 4

⇒ y = 12.5

∴ The correct answer is 12.5.

Additional Information

Linear Equation Substitution

This method involves expressing one variable in terms of another from one equation and substituting it into the second equation to reduce it to a single-variable equation.

Elimination Method

This approach involves multiplying equations by suitable constants so that adding or subtracting them eliminates one variable, allowing for the solving of the remaining one.

Consistency of Equations

A system of Linear Equations has a unique solution if the ratio of the coefficients of the variables is not equal (a₁/a₂ ≠ b₁/b₂).

Shortcut Trick

Calculate semi-perimeter (s) = (51 + 35 + 26) ÷ 2 = 56 cm.

Calculate Area using Heron's formula: √[56 × (56−51) × (56−35) × (56−26)] = √(56 × 5 × 21 × 30) = 420 cm².

Use the area formula: Altitude = (2 × Area) ÷ Base.

Altitude = (2 × 420) ÷ 35 = 840 ÷ 35 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given: Sides of the triangle are a = 51 cm, b = 35 cm, and c = 26 cm.

Formula Used: Heron's Formula for Area = √[s(s − a)(s − b)(s − c)] and Area of Triangle = (1 ÷ 2) × Base × Altitude.

⇒ Semi-perimeter (s) = (a + b + c) ÷ 2

⇒ s = (51 + 35 + 26) ÷ 2 = 112 ÷ 2 = 56 cm

⇒ Area = √[56 × (56 − 51) × (56 − 35) × (56 − 26)]

⇒ Area = √[56 × 5 × 21 × 30] = √[176400] = 420 cm²

⇒ To find the altitude (h) corresponding to the side of 35 cm:

⇒ Area = (1 ÷ 2) × 35 × h

⇒ 420 = (35 × h) ÷ 2

⇒ h = (420 × 2) ÷ 35

⇒ h = 840 ÷ 35 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Heron's Formula

It is used to find the area of a triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.

Altitude of a Triangle

The altitude is the perpendicular distance from a vertex to the opposite side. The area can be expressed as (1/2) × base × corresponding altitude.

Relationship between Sides and Altitudes

In any triangle, the side lengths are inversely proportional to their corresponding altitudes. The longest side has the shortest altitude.

Shortcut Trick

Ratio of the diagonals = 5 : 12

Product of ratio terms = 5 × 12 = 60

We know, (1/2) × Product = Area ⇒ (1/2) × 60x² = 240 ⇒ x² = 8 ⇒ x = 2√2

Required diagonals = 5 × 2√2 = 10√2 cm and 12 × 2√2 = 24√2 cm.

∴ The correct answer is 10√2 cm and 24√2 cm.

Alternate Method

Given:

Ratio of the diagonals of a rhombus = 5 : 12

Area of the rhombus = 240 cm²

Formula Used:

Area of Rhombus = (1/2) × d₁ × d₂

Where d₁ and d₂ are the lengths of the diagonals.

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Shortcut Trick

When two trains of equal lengths cross a pole in t₁ and t₂ seconds, the time taken to cross each other in opposite directions is given by:

Time = (2 × t₁ × t₂) ÷ (t₁ + t₂)

Time = (2 × 12 × 16) ÷ (12 + 16) = 384 ÷ 28

Time = 96 ÷ 7 ≈ 13.71 sec

∴ The correct answer is 13.71 sec.

Alternate Method

Given:

Time taken by Train 1 (t₁) = 12 sec

Time taken by Train 2 (t₂) = 16 sec

Let the length of each train be the LCM of 12 and 16 = 48 units.

Formula Used:

Speed = Distance ÷ Time

Time to cross each other = Total Distance ÷ Relative Speed

Calculations:

⇒ Speed of Train 1 (v₁) = 48 ÷ 12 = 4 units/sec

⇒ Speed of Train 2 (v₂) = 48 ÷ 16 = 3 units/sec

⇒ Total distance to cover = Length of Train 1 + Length of Train 2 = 48 + 48 = 96 units

⇒ Relative Speed (opposite direction) = v₁ + v₂ = 4 + 3 = 7 units/sec

⇒ Required Time = 96 ÷ 7

⇒ Required Time = 13.71428... sec

⇒ Required Time ≈ 13.71 sec

∴ The correct answer is 13.71 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (v₁ + v₂). If they move in the same direction, it is the difference (|v₁ − v₂|).

Distance while Crossing

To cross a stationary point (pole, man, tree), the distance covered is equal to the length of the train. To cross another train or platform, the distance is the sum of their lengths (L₁ + L₂).

Time for Same Direction

If the trains were running in the same direction, the time taken to cross each other would be (2 × t₁ × t₂) ÷ (t₂ − t₁), where t₂ > t₁.

Shortcut Trick

Ratio of Men to Women = 58 : 42 = 29 : 21.

Let the assumed average age be 42 years.

Net deviation from Men = 29 × (57 − 42) = +435.

Total distribution of deviation = 435 ÷ (29 + 21) = 8.7.

Total Average Age = 42 + 8.7 = 50.7 years.

∴ The correct answer is 50.7 years.

Alternate Method

Given:

Percentage of Men = 58% | Percentage of Women = 42%

Average age of Men = 57 years | Average age of Women = 42 years

Formula Used:

Average of groups = (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂)

Calculations:

⇒ Let Total Members = 100

⇒ Total age of Men = 58 × 57 = 3306 years

⇒ Total age of Women = 42 × 42 = 1764 years

⇒ Sum of ages of all members = 3306 + 1764 = 5070 years

⇒ Average age of all members = 5070 ÷ 100

⇒ Average age = 50.7 years

∴ The correct answer is 50.7 years.

Additional Information

Weighted Average Concept

It is used when groups have different sizes. Formula: A_avg = (Sum of all values) ÷ (Total number of items).

Alligation Method

The ratio of quantities (n₁/n₂) is equal to (Average₂ − Average_mix) / (Average_mix − Average₁).

Deviation Method for Averages

Assume a temporary average and calculate the net deviation from each group to find the final result quickly.

Shortcut Trick

Equate the volumes: Radius² × Height of Solid Cylinder = (Outer Radius² − Inner Radius²) × Height of Hollow Cylinder.

6² × 14 = [(2x)² − x²] × 7

36 × 2 = 3x² ⇒ x² = 24 ⇒ x = 2√6 cm

Outer radius = 2x = 2 × 2√6 = 4√6 cm.

∴ The correct answer is 4√6 cm.

Alternate Method

Given:

Solid cylinder: Radius (r₁) = 6 cm, Height (h₁) = 14 cm.

Hollow cylinder: Height (h₂) = 14/2 = 7 cm, Outer radius (R) = 2 × Inner radius (r).

Formula Used:

Volume of Solid Cylinder = πr²h

Volume of Hollow Cylinder = π(R² − r²)h

Calculations:

⇒ Volume of solid cylinder = π × 6² × 14 = 504π cm³

⇒ Let inner radius = x, then outer radius = 2x

⇒ Volume of hollow cylinder = π × [(2x)² − x²] × 7

⇒ 21πx² = 504π

⇒ x² = 504 / 21

⇒ x² = 24

⇒ x = √24 = 2√6 cm

⇒ Outer radius = 2x = 2 × 2√6 = 4√6 cm

∴ The correct answer is 4√6 cm.

Additional Information

Volume Conservation

When a solid is melted and recast into a different shape, the total Volume of the material remains the same.

Hollow Cylinder Properties

The volume of a hollow cylinder with outer radius R, inner radius r, and height h is V = π(R² − r²)h.

Total Surface Area of Hollow Cylinder

The total surface area is given by 2π(R + r)(h + R − r).

Shortcut Trick

Compare effective discount percentages for 10 toys:

Scheme A: 38 + 39 − (38 × 39)/100 = 77 − 14.82 = 62.18%

Scheme B: 42 + 38 − 15.96 = 64.04% → 64.04 + 20 − 12.8 = 71.23%

Scheme C: [ (2 × 11) + (8 × 42) ] ÷ 10 = (22 + 336) ÷ 10 = 35.8%

Scheme D: [ 9 ÷ (10 + 9) ] × 100 ≈ 47.37%

Comparing all: 35.8% (C) is the lowest discount, hence least beneficial.

∴ The correct answer is Scheme C.

Alternate Method

Assume Marked Price (MP) of each toy = ₹100. Total MP for 10 toys = ₹1000.

Detailed Calculations:

⇒ Scheme A: Successive 38% and 39%.

⇒ Net Discount % = 38 + 39 − (38 × 39)/100 = 77 − 14.82 = 62.18%

⇒ Scheme B: Successive 42%, 38% and 20%.

⇒ Net of 42% and 38% = 42 + 38 − (42 × 38)/100 = 80 − 15.96 = 64.04%

⇒ Net with 20% = 64.04 + 20 − (64.04 × 20)/100 = 84.04 − 12.808 = 71.232%

⇒ Scheme C: 11% on first 2, 42% on next 8 toys.

⇒ Total Discount = (2 × 11% of 100) + (8 × 42% of 100) = 22 + 336 = ₹358

⇒ Effective Discount % = (358 ÷ 1000) × 100 = 35.8%

⇒ Scheme D: 9 toys free on buying 10.

⇒ Discount % = [Free Items ÷ Total Items] × 100 = [9 ÷ (10 + 9)] × 100 ≈ 47.37%

⇒ Comparing all discounts: 35.8% is the minimum value.

∴ The correct answer is Scheme C.

Additional Information

Successive Discount Formula

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

Buy X Get Y Free Scheme

The effective discount percentage in a 'Buy X Get Y Free' offer is calculated as [Y ÷ (X + Y)] × 100.

Benefit to Customer

A scheme is most beneficial when the discount percentage is highest and least beneficial when the discount percentage is lowest.