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When an equal distance is covered at two different speeds x and y, the average speed is given by the Harmonic Mean formula.

Average Speed = (2 × x × y) / (x + y)

Here, x = 5 km/h and y = 4 km/h.

Average Speed = (2 × 5 × 4) / (5 + 4) = 40 / 9 ≈ 4.44 km/h.

∴ The correct answer is 4.4 km/h.

Alternate Method

Given:

Speed from A to B (v₁) = 5 km/h

Speed from B to A (v₂) = 4 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

⇒ Let the distance between A and B be 'd' km.

⇒ Time taken to go from A to B (t₁) = Distance ÷ Speed = d / 5 hours.

⇒ Time taken to return from B to A (t₂) = Distance ÷ Speed = d / 4 hours.

⇒ Total Distance for the whole journey = d + d = 2d km.

⇒ Total Time taken = t₁ + t₂ = d/5 + d/4 = (4d + 5d) / 20 = 9d / 20 hours.

⇒ Average Speed = Total Distance ÷ Total Time = 2d ÷ (9d/20)

⇒ Average Speed = (2d × 20) / 9d = 40 / 9 km/h

⇒ Average Speed ≈ 4.44 km/h

∴ The correct answer is 4.4 km/h.

Additional Information

Average Speed (Equal Time Intervals)

If a person travels at speed v₁ for time t and speed v₂ for the same time t, the average speed is the Arithmetic Mean: (v₁ + v₂) / 2.

General Average Speed Formula

For any journey with different distances d₁, d₂ and speeds v₁, v₂, Average Speed = (d₁ + d₂) / (d₁/v₁ + d₂/v₂).

Relative Speed

When two objects move in opposite directions, relative speed = v₁ + v₂. In the same direction, relative speed = |v₁ − v₂|.

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Winner = 82% of Valid votes → Loser = 18% of Valid votes.

Victory Margin = 82% − 18% = 64% of Valid votes.

64% = 80 → 1% = 1.25 → 100% (Valid) = 125 votes.

Total Votes = Valid + Invalid = 125 + 1780 = 1905 votes.

∴ The correct answer is 1905 votes.

Alternate Method

Given:

Invalid votes = 1,780

Winner's share = 82% of valid votes

Margin of victory = 80 votes

Calculations:

⇒ Let Total Valid Votes = V

⇒ Winner's Votes = 82% of V = 0.82V

⇒ Loser's Votes = (100 − 82)% of V = 18% of V = 0.18V

⇒ Margin = Winner's Votes − Loser's Votes

⇒ 80 = 0.82V − 0.18V

⇒ 80 = 0.64V

⇒ V = 80 ÷ 0.64 = 125

⇒ Total Votes Cast = Valid Votes + Invalid Votes

⇒ Total Votes Cast = 125 + 1780 = 1905

∴ The correct answer is 1905 votes.

Additional Information

Election Margin Calculation

In a two-candidate election, if the winner gets x% of votes, the margin is (x − (100 − x))% = (2x − 100)% of the total valid votes.

Valid vs Invalid Votes

Total votes cast = Valid votes + Invalid votes. Percentages are always calculated on the base of valid votes unless specified otherwise.

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 64% = 64/100 = 16/25. Thus, (16/25) × Valid = 80 ⇒ Valid = 125.

The correct answer is 60 seconds

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Ratio of Distance (Pole : Platform) = 240 : (240 + 360) = 240 : 600 = 2 : 5.

Since Speed is constant, Time is directly proportional to Distance.

2 units of time = 24 seconds ⇒ 1 unit = 12 seconds.

5 units of time = 5 × 12 = 60 seconds.

∴ The correct answer is 60 seconds.

Alternate Method

Given: Length of train = 240 m, Time taken to pass a pole = 24 s, Length of platform = 360 m.

Formula: Speed = Distance ÷ Time; Total Distance = Train Length + Platform Length.

⇒ Speed of the train = Distance ÷ Time = 240 ÷ 24 = 10 m/s.

⇒ Total distance to pass the platform = Length of train + Length of platform.

⇒ Total distance = 240 + 360 = 600 m.

⇒ Time taken to pass the platform = Total distance ÷ Speed.

⇒ Time = 600 ÷ 10 = 60 seconds.

∴ The correct answer is 60 seconds.

Additional Information

Relative Speed

When two trains move in the same direction, relative speed = S₁ − S₂. 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.

Distance Rule

When a train crosses a stationary object with length (like a bridge or tunnel), the total distance covered is the sum of their lengths.

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Volume of cylinder = π × r² × h

Given: radius (r) = 5 cm, height (h) = 10 cm, π = 3.14

Volume = 3.14 × 5² × 10

Volume = 3.14 × 25 × 10 = 785 cm³

∴ The correct answer is 785 cm³.

Alternate Method

Given:

Radius (r) = 5 cm

Height (h) = 10 cm

π = 3.14

Formula Used:

Volume of Cylinder = πr²h

Calculations:

⇒ Volume = 3.14 × (5)² × 10

⇒ Volume = 3.14 × 25 × 10

⇒ Volume = 78.5 × 10

⇒ Volume = 785 cm³

∴ The correct answer is 785 cm³.

Additional Information

Curved Surface Area (CSA) of Cylinder

The area of the side surface of the cylinder is given by 2πrh.

Total Surface Area (TSA) of Cylinder

The sum of the CSA and the areas of the two circular bases: 2πr(h + r).

Volume of a Cone

A cone with the same radius and height as a cylinder has exactly one-third of its volume: (1/3)πr²h.

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For a solid hemisphere, equating Volume and Total Surface Area: (2/3) π r³ = 3 π r².

Simplifying the terms, we get r = 3 × (3/2) = 4.5 cm.

The Diameter is 2 × radius = 2 × 4.5 = 9 cm.

∴ The correct answer is 9 cm.

Alternate Method

Given:

The Volume of the solid hemisphere is numerically equal to its Surface Area.

Formula Used:

1. Volume of Hemisphere = (2/3) π r³
2. Total Surface Area of Solid Hemisphere = 3 π r²
3. Diameter (d) = 2 × Radius (r)

Calculations:

⇒ Volume = Surface Area

⇒ (2/3) π r³ = 3 π r²

⇒ (2/3) r = 3 (By dividing both sides by π r²)

⇒ r = 3 × 3 / 2

⇒ r = 9 / 2 = 4.5 cm

⇒ Diameter (d) = 2 × r

⇒ d = 2 × 4.5

⇒ d = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Curved Surface Area (CSA) of Hemisphere

The area of the curved part only is 2 π r². For a hollow hemisphere, this is the only surface area.

Total Surface Area (TSA) of Solid Hemisphere

It includes the curved part plus the flat circular base: 2 π r² + π r² = 3 π r².

Properties of a Sphere

A full sphere has a Volume of (4/3) π r³ and a Surface Area of 4 π r².

Numerical Equality Concept

When values are numerically equal, we ignore the units (cm² vs cm³) and equate the magnitude.

Shortcut Trick

Average Selling Price (SP) = (80 × 10 + 20 × 6) / 100 = 9.2

Cost Price (CP) = 8

Profit % = [(9.2 − 8) / 8] × 100

Profit % = (1.2 / 8) × 100 = 15%

∴ The correct answer is 15% profit.

Alternate Method

Given:

Total oranges = 100

Cost Price (CP) per orange = ₹8

Selling Price (SP) of 80 oranges = ₹10 each

Selling Price (SP) of 20 oranges = ₹6 each

Formula Used:

Total CP = Total Quantity × CP per unit

Total SP = (Quantity₁ × SP₁) + (Quantity₂ × SP₂)

Profit % = [(Total SP − Total CP) / Total CP] × 100

⇒ Total CP = 100 × 8 = ₹800

⇒ SP of 80 oranges = 80 × 10 = ₹800

⇒ SP of 20 oranges = 20 × 6 = ₹120

⇒ Total SP = 800 + 120 = ₹920

⇒ Profit = Total SP − Total CP = 920 − 800 = ₹120

⇒ Profit % = (120 / 800) × 100

⇒ Profit % = 15%

∴ The correct answer is 15% profit.

Additional Information

Profit and Loss Basics

Profit occurs when SP > CP, and Loss occurs when CP > SP. Both percentages are always calculated on the CP.

Weighted Average in SP

When items are sold at different prices, the overall SP per unit is the weighted average: (n₁s₁ + n₂s₂) / (n₁ + n₂).

Marked Price and Discount

Selling Price = Marked Price − Discount. Discount percentage is always calculated on the Marked Price (MP).

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Direct relation: r² = (3 × Volume) / (π × h)

r² = (3 × 48π) / (10π) = 144 / 10 = 14.4

r = √14.4

Since 3.8² = 14.44, then r ≈ 3.8 cm.

∴ The correct answer is 3.8 cm.

Alternate Method

Given:

Volume of cone (V) = 48π cm³

Height of cone (h) = 10 cm

Formula Used:

Volume of a cone = (1/3) × π × r² × h

Calculations:

⇒ 48π = (1/3) × π × r² × 10

⇒ 48 = (10/3) × r²

⇒ r² = (48 × 3) / 10

⇒ r² = 144 / 10

⇒ r² = 14.4

⇒ r = √14.4

⇒ r ≈ 3.7947 cm

⇒ r ≈ 3.8 cm (rounding to nearest one decimal place)

∴ The correct answer is 3.8 cm.

Additional Information

Slant Height of a Cone

The slant height (l) is the distance from the apex to any point on the base circumference: l = √(r² + h²).

Surface Area of a Cone

Curved Surface Area (CSA) = πrl and Total Surface Area (TSA) = πr(l + r).

Relationship with Cylinder

For the same radius and height, the volume of a cylinder is exactly three times the volume of a cone (V_cyl = πr²h).

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Total age of 5 members = 5 × 20 = 100 years

Total age of 6 members = 6 × 18 = 108 years

New member's age = 108 − 100

⇒ 8 years

∴ The correct answer is 8 years.

Alternate Method

Given:

Average age of 5 members = 20 years

Average age of 6 members = 18 years

Formula Used:

Total Age = Average × Number of Members

5 Members = 100 6 Members = 108 + New Member 108 − 100 = 8

Calculations:

⇒ Total age of 5 members = 5 × 20 = 100 years

⇒ Total age of 6 members = 6 × 18 = 108 years

⇒ Age of new member = 108 − 100

⇒ = 8 years

∴ The correct answer is 8 years.

Additional Information

Average Formula

Average = Total Sum ÷ Number of Observations.

Adding a New Member

New Value = New Total Sum − Old Total Sum.

Average Decrease Concept

If the average decreases after adding a member, the new member's value is below the original average.

Shortcut Trick

Using the Ratio Method:

P : Q = 120 : 100 = 6 : 5

R : Q = 80 : 100 = 4 : 5

Combining them, P : R = 6 : 4 = 3 : 2

P is more than R by (3 − 2) = 1 unit over base 2.

Required % = (1 / 2) × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given:

P = Q + 20% of Q

R = Q − 20% of Q

Formula Used:

Percentage More = [(Value₁ − Value₂) / Value₂] × 100

Calculations:

⇒ Let the value of Q be 100.

⇒ P = 100 + 20% of 100 = 100 + 20 = 120.

⇒ R = 100 − 20% of 100 = 100 − 20 = 80.

⇒ Difference between P and R = 120 − 80 = 40.

⇒ Required Percentage = (Difference / R) × 100

⇒ Required Percentage = (40 / 80) × 100

⇒ Required Percentage = 1/2 × 100 = 50%.

∴ The correct answer is 50%.

Additional Information

Percentage Change Formula

The percentage change is calculated as [(Final Value − Initial Value) / Initial Value] × 100. Always identify the correct base value (the value after 'than').

Base Comparison Rule

If A is x% more than B, then B is [x / (100 + x)] × 100% less than A. This is useful for quick inverse percentage calculations.

Fraction to Percentage Conversion

Common fractions help in speed: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, and 1/8 = 12.5%.

Shortcut Trick

Discount of 20% means the Sale Price is 80% of the original price.

Sale Price = 80% of 400

Sale Price = 0.8 × 400

Sale Price = 320

∴ The correct answer is 320.

Alternate Method

Given:

Original Price (Marked Price) = 400

Discount Percentage = 20%

Formula Used:

Sale Price = Original Price − (Discount% × Original Price)

Calculations:

⇒ Discount Amount = 20% of 400

⇒ Discount Amount = (20/100) × 400 = 80

⇒ Sale Price = 400 − 80

⇒ Sale Price = 320

∴ The correct answer is 320.

Additional Information

Marked Price (MP)

The price at which an article is listed or printed on the label is called the Marked Price or List Price.

Discount Formula

Discount is always calculated on the Marked Price. Formula: Discount = MP − SP.

Successive Discounts

If two discounts a% and b% are given successively, the single equivalent discount is (a + b − ab/100)%.