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Volume of any prism is directly proportional to its height when the base area is constant.

Original volume = 25 × 10 = 250 cm³

Since the height increases by 20%, the volume also increases by exactly 20%.

New volume = 250 × 1.2 = 300 cm³

∴ The correct answer is 300 cm³.

Alternate Method

Given:

Base area of the triangular prism = 25 cm²

Original height = 10 cm

Increase in height = 20%

Formula Used:

Volume of a prism = Base Area × Height

        Original h = 10 cm New h = 12 cm   Base Area = 25 cm² New Volume = 25 × 12 = 300 cm³

Calculations:

⇒ New height = 10 + (20% of 10)

⇒ New height = 10 + 2 = 12 cm

⇒ New volume = Base Area × New height

⇒ New volume = 25 × 12 = 300 cm³

∴ The correct answer is 300 cm³.

Additional Information

Volume of a Prism

The volume of any regular prism (triangular, rectangular, hexagonal, etc.) is always the product of its uniform base area and its vertical height: V = Base Area × h.

Proportionality in 3D Shapes

If only a single dimension (like height) of a 3D figure changes by x%, the total volume also undergoes an exact x% change, provided all other dimensions remain constant.

Total Surface Area of Prism

The Total Surface Area (TSA) is calculated as: TSA = Lateral Surface Area + (2 × Base Area) = (Perimeter of Base × Height) + (2 × Base Area).

Shortcut Trick

For a circular disc, Area ∝ (Radius)².

The net percentage change can be calculated using the successive change formula: x + y + (x × y) ÷ 100.

Since the radius is decreased by 10%, we substitute x = −10 and y = −10.

Net Change = −10 − 10 + [(−10) × (−10)] ÷ 100 = −20 + 1 = −19%.

The negative sign indicates a 19% decrease in the overall area.

∴ The correct answer is 19%.

Alternate Method

Given:

The radius of the circular disc is decreased by 10%.

Formula Used:

Area of a circle = π × r²

Percentage Decrease = [(Initial Area − Final Area) ÷ Initial Area] × 100

r Initial Area = 100%

− 10% Radius

0.9r Final Area = 81%

Net Decrease in Area = 19%

Calculation:

⇒ Let the original radius of the disc be 10 units.

⇒ Initial Area = π × 10² = 100π sq. units.

⇒ New radius = 10 − (10% of 10) = 10 − 1 = 9 units.

⇒ New Area = π × 9² = 81π sq. units.

⇒ Decrease in Area = 100π − 81π = 19π sq. units.

⇒ Percentage Decrease = (19π ÷ 100π) × 100 = 19%.

∴ The correct answer is 19%.

Additional Information

Successive Percentage Change

When a value is subjected to two successive percentage changes of x% and y%, the overall net change is calculated using the formula: Net Change = x + y + (xy ÷ 100). Use negative values for any decrease.

Volumetric Percentage Changes

For 3D shapes like spheres or cubes, if the radius or side changes by x%, the volume change is determined by applying the successive percentage formula twice consecutively to account for all three dimensions.

Constant Expenditure Rule

If the price of an item increases by R%, the consumption must be reduced by [R ÷ (100 + R)] × 100% to keep the total expenditure constant.

Shortcut Trick

For any given perimeter, the ratio of the area of a circle to a square is the constant 4 : π.

Ratio = 4 / π ≈ 4 ÷ 3.1416

Ratio ≈ 1.27 : 1

The total wire length of 100 m is not needed to find this ratio.

∴ The correct answer is 1.27:1.

Alternate Method

Given:

Total length of wire = Perimeter of circle = Perimeter of square = 100 m

Formula Used:

Circumference of circle = 2×π×r, Area of circle = π×r²

Perimeter of square = 4×a, Area of square = a²

r Circle Perimeter = 100 m

a Square Perimeter = 100 m

Calculations:

⇒ Circumference of the circle = 2×π×r = 100

⇒ Radius (r) = 50/π m

⇒ Area of circle = π×(50/π)² = 2500/π m²

⇒ Perimeter of the square = 4×a = 100

⇒ Side (a) = 25 m

⇒ Area of square = 25² = 625 m²

⇒ Ratio of areas = (2500/π) ÷ 625

⇒ Ratio = 4 / π

⇒ Ratio ≈ 4 ÷ 3.1416 = 1.2732...

⇒ Approximate ratio = 1.27 : 1

∴ The correct answer is 1.27:1.

Additional Information

Isoperimetric Theorem

For any given perimeter, the circle is the 2D shape that encloses the maximum possible area among all geometric figures.

Area Ratio for Same Perimeter

If a circle, a square, and an equilateral triangle all have the exact same perimeter, their respective areas are in the approximate ratio of 1 : 0.785 : 0.604.

Minimum Perimeter for Fixed Area

Conversely, if a fixed area must be enclosed, the circle is the shape that requires the absolute minimum perimeter (or boundary length) to do so.

Shortcut Trick

By the principle of displacement, Volume of oil raised = Volume of immersed hemisphere.

⇒ π × R² × h = ²⁄₃ × π × r³

⇒ R² × 3 = ²⁄₃ × 7³

⇒ R² = 686 ÷ 9

⇒ R = √686 ÷ 3 ≈ 26.19 ÷ 3 ≈ 8.73 cm

∴ The correct answer is 8.73 cm.

Alternate Method

Given:

Radius of the hemispherical stone (r) = 7 cm

Rise in the oil level (h) = 3 cm

Formula Used:

Volume of a cylinder = πR²h

Volume of a hemisphere = ²⁄₃πr³

R = ?

r = 7 cm

h = 3 cm

Rise in Level Original Level

Volume Displaced = Volume of Hemisphere

Calculations:

When the stone is immersed completely, the volume of the oil displaced equals the volume of the stone.

⇒ Volume of displaced oil = Volume of hemisphere

⇒ π × R² × h = ²⁄₃ × π × r³

⇒ π × R² × 3 = ²⁄₃ × π × 7³

Cancelling π from both sides:

⇒ 3 × R² = ²⁄₃ × 343

⇒ R² = 686 ÷ 9

⇒ R = √(686 ÷ 9) = √686 ÷ 3

⇒ R ≈ 26.19 ÷ 3

⇒ R ≈ 8.73 cm

∴ The correct answer is 8.73 cm.

Additional Information

Displacement Principle

When a solid object is fully immersed in a liquid, the volume of the displaced liquid is exactly equal to the volume of the object.

Volume of Cylinder

The total space occupied by a right circular cylinder is calculated as V = πR²h.

Volume of Hemisphere

A hemisphere is half of a sphere. Its volume is given by V = ²⁄₃πr³.

Shortcut Trick

Convert the central angle of the second sector from radians to degrees to easily compare them.

Angle in degrees = (2π ÷ 3) × (180° ÷ π) = 120°.

Both sectors belong to the same circle (radius = 7 cm) and have equal central angles of 120°.

Since both the radius and the central angles are identical, their areas must be exactly equal.

∴ The correct answer is 1:1.

Alternate Method

Given:

Radius of the circle (r) = 7 cm

Central angle of the first sector (θ₁) = 120°

Central angle of the second sector (θ₂) = 2π/3 radians

120° 7 cm Sector 1

2π/3 rad 7 cm Sector 2 :

Formulas Used:

Area of sector (degrees) = (θ ÷ 360°) × πr²

Area of sector (radians) = (1 ÷ 2) × θ × r²

Calculations:

Calculating the area of the first sector (A₁):

⇒ A₁ = (120° ÷ 360°) × π × (7)²

⇒ A₁ = (1 ÷ 3) × 49π

⇒ A₁ = 49π ÷ 3 cm²

Calculating the area of the second sector (A₂):

⇒ A₂ = (1 ÷ 2) × (2π ÷ 3) × (7)²

⇒ A₂ = (π ÷ 3) × 49

⇒ A₂ = 49π ÷ 3 cm²

Finding the ratio of their areas:

⇒ A₁ : A₂ = (49π ÷ 3) : (49π ÷ 3)

⇒ A₁ : A₂ = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Area of a Sector

The area of a sector is directly proportional to its central angle. It represents a specific fraction of the entire area of the circle. Formula: Area = (θ ÷ 360°) × πr²

Radians and Degrees

Degrees and radians are two distinct units for measuring angles. A complete circle equals 360° or 2π radians. Conversion: Angle in Degrees = Angle in Radians × (180° ÷ π).

Arc Length

The length of the boundary arc of a sector relies on the central angle. The formula is: L = (θ ÷ 360°) × 2πr (when in degrees) or simply L = rθ (when in radians).

Shortcut Trick Substitute approximate decimal values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 ⇒ x ≈ 1.414 + 1.732 = 3.146 ⇒ y ≈ 2.236 + 1 = 3.236 ⇒ z ≈ 2 × 1.414 + 1 = 3.828 ⇒ Comparing values: 3.146 < 3.236 < 3.828 ⇒ x < y < z ∴ The correct answer is x < y < z. Alternate Method Given: x = √2 + √3 y = √5 + √1 z = 2√2 + 1 Calculations: ⇒ Squaring all the given terms to compare their magnitudes: ⇒ x2 = (√2 + √3)2 = 2 + 3 + 2√6 = 5 + √24 ⇒ y2 = (√5 + 1)2 = 5 + 1 + 2√5 = 6 + √20 ⇒ z2 = (2√2 + 1)2 = 8 + 1 + 4√2 = 9 + √32 ⇒ Using approximations for easier comparison: √20 ≈ 4.47, √24 ≈ 4.89, √32 ≈ 5.65 ⇒ x2 ≈ 5 + 4.89 = 9.89 ⇒ y2 ≈ 6 + 4.47 = 10.47 ⇒ z2 ≈ 9 + 5.65 = 14.65 ⇒ Comparing the calculated squares gives: 9.89 < 10.47 < 14.65 ⇒ x2 < y2 < z2 ⇒ Since x, y, and z are all positive numerical values, their order remains the same: x < y < z. ∴ The correct answer is x < y < z. Additional Information Comparing Surds To compare irrational numbers (surds), squaring both sides is a standard method. It removes the square roots and helps compare the resulting rational and irrational parts more easily. Important Square Roots Memorising common square roots helps quickly estimate and simplify values in exams: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449. Inequality Property of Squares For any positive real numbers a and b, if a2 > b2, it directly implies a > b. This fundamental property allows us to determine the order of numbers by ordering their squares.

Shortcut Trick

Net percentage change can be calculated directly using the weighted average of the given ratios.

Net % Change = [(Ratio₁ × %₁) + (Ratio₂ × %₂) + (Ratio₃ × %₃)] ÷ Total Ratio parts

Net % Change = [5 × 5 + 3 × 10 + 2 × (−20)] ÷ (5 + 3 + 2)

Net % Change = (25 + 30 − 40) ÷ 10 = 15 ÷ 10

Net % Change = +1.5% (A positive value denotes an increase)

∴ The correct answer is 1.5% increase.

Alternate Method

Given:

Ratio of cost of land acquisition : building materials : labor = 5 : 3 : 2

Change in land acquisition cost = +5%

Change in building materials cost = +10%

Change in labor cost = −20%

Formula Used:

Percentage change = (Net Change ÷ Initial Total) × 100

Land (500)+5% (→ +25)Materials (300)+10% (→ +30)Labor (200)−20% (→ −40)Total Base = 1000Net Change = +25 + 30 − 40 = +15

Calculation:

⇒ Let the initial cost of land acquisition, building materials, and labor be 500, 300, and 200 respectively.

⇒ Total initial project cost = 500 + 300 + 200 = 1000

⇒ Change in land acquisition cost = 5% of 500 = +25

⇒ Change in building materials cost = 10% of 300 = +30

⇒ Change in labor cost = −20% of 200 = −40

⇒ Net change in total cost = 25 + 30 − 40 = +15

⇒ Percentage change = (15 ÷ 1000) × 100

⇒ Percentage change = 1.5%

⇒ Since the net change is positive, there is an overall increase in the total cost.

∴ The correct answer is 1.5% increase.

Additional Information

Percentage Change

Percentage change is calculated as [(New Value − Old Value) ÷ Old Value] × 100. A positive result denotes an increase, and a negative result denotes a decrease.

Weighted Average Method

When components of a total are subjected to different percentage changes, the net percentage change can be found using the weighted average formula: Net % = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n).

Successive Percentage Change

If a single quantity changes by a% and then further changes by b%, the effective percentage change is given by: Net Change = a + b + (a × b)÷100 %.

Shortcut Trick

According to BODMAS, solve inner brackets first: (1 − 1/3) = 2/3

Multiply inner terms: 1/2 × 2/3 = 1/3

Solve curly brackets: 1/6 + 1/3 = 1/2

Solve square brackets: 2/3 ÷ 1/2 = 2/3 × 2 = 4/3

Final subtraction: 6 − 4/3 = 14/3

∴ The correct answer is 14/3.

Alternate Method

Given:

Expression: 6 − [2/3 ÷ {1/6 + (1/2 × (1 − 1/3))}]

Formula Used:

BODMAS Rule: Brackets → Of → Division → Multiplication → Addition → Subtraction

Calculations:

⇒ Step 1: Solve the innermost round bracket ( )

⇒ 1 − 1/3 = (3 − 1)/3 = 2/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + (1/2 × 2/3)}]

⇒ Step 2: Solve the multiplication inside the remaining round bracket

⇒ 1/2 × 2/3 = 2/6 = 1/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + 1/3}]

⇒ Step 3: Solve the addition inside the curly bracket { }

⇒ 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2

⇒ The expression becomes: 6 − [2/3 ÷ 1/2]

⇒ Step 4: Solve the division inside the square bracket [ ]

⇒ 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3

⇒ The expression becomes: 6 − 4/3

⇒ Step 5: Perform the final subtraction

⇒ 6 − 4/3 = (18 − 4)/3 = 14/3

∴ The correct answer is 14/3.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction. It sets the standard order of operations to solve mathematical expressions correctly.

Types of Brackets

Brackets must be solved in a specific order: Line bracket or Vinculum first, followed by Round brackets ( ), Curly brackets { }, and finally Square brackets [ ].

Fraction Division Rule

When dividing two fractions, you must multiply the first fraction by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

Since the time period is identical for all, the profit ratio equals the investment ratio.

Investment Ratio of X : Y : Z = 25000 : 35000 : 40000 = 5 : 7 : 8

Total profit parts = 5 + 7 + 8 = 20 parts

Y's share = (7 ÷ 20) × 30000 = 7 × 1500 = 10500

∴ The correct answer is ₹10,500.

Alternate Method

Given:

Investment of X = ₹25,000

Investment of Y = ₹35,000

Investment of Z = ₹40,000

Total profit at the end of the year = ₹30,000

Formula Used:

Profit Ratio = Capital Investment × Time Period

Partnership Investment & Profit Ratio

₹25,000 5 parts Partner X

₹35,000 7 parts Partner Y

₹40,000 8 parts Partner Z

Total Profit = ₹30,000 (20 parts)

Calculations:

⇒ Profit Ratio of X : Y : Z = (25000 × 1) : (35000 × 1) : (40000 × 1)

⇒ X : Y : Z = 25 : 35 : 40

⇒ X : Y : Z = 5 : 7 : 8

⇒ Sum of profit ratios = 5 + 7 + 8 = 20 parts

⇒ 20 parts = ₹30,000

⇒ 1 part = 30000 ÷ 20 = ₹1,500

⇒ Y's share = 7 parts = 7 × 1500 = ₹10,500

∴ The correct answer is ₹10,500.

Additional Information

Simple Partnership

When all partners invest their money for the exact same duration, the profit or loss is distributed purely in the ratio of their original investment amounts.

Compound Partnership

When partners invest their capital for different time periods, the profit ratio is calculated as the product of their investment and their respective time duration (Profit = Capital × Time).

Active vs Sleeping Partner

An active partner manages the business operations and generally receives a fixed percentage or salary from the profit before the remaining profit is distributed among all partners (including sleeping partners) based on their capital ratio.

Shortcut Trick

Ratio of Profit = (Capital_A × Time_A) : (Capital_B × Time_B)

Profit Ratio = [(48 × 6) + (24 × 6)] : [(72 × 6) + (96 × 6)]

Profit Ratio = [6 × (48 + 24)] : [6 × (72 + 96)] = 72 : 168 = 3 : 7

B's share = (7 ÷ 10) × 55000 = ₹38,500

∴ The correct answer is ₹38,500.

Alternate Method

Given:

Initial capital of A = ₹48,000

Initial capital of B = ₹72,000

Total time of investment = 1 year = 12 months

Total profit earned at the end of the year = ₹55,000

A

₹48k × 6m

₹24k × 6m = 432k

B

₹72k × 6m

₹96k × 6m = 1008k

Profit Ratio (A : B) = 432 : 1008 = 3 : 7

Formula Used:

Profit Ratio = Product of Capital and Time Period = C₁T₁ : C₂T₂

Calculations:

⇒ Equivalent capital of A = (₹48,000 × 6 months) + (₹24,000 × 6 months)

⇒ Equivalent capital of A = ₹2,88,000 + ₹1,44,000 = ₹4,32,000

⇒ Equivalent capital of B = (₹72,000 × 6 months) + (₹96,000 × 6 months)

⇒ Equivalent capital of B = ₹4,32,000 + ₹5,76,000 = ₹10,08,000

⇒ Ratio of profit (A : B) = 4,32,000 : 10,08,000

⇒ Ratio of profit (A : B) = 432 : 1008 = 3 : 7

⇒ Sum of ratio parts = 3 + 7 = 10 parts

⇒ B's share in profit = (7 ÷ 10) × Total Profit

⇒ B's share = (7 ÷ 10) × 55,000

⇒ B's share = 7 × 5,500 = ₹38,500

∴ The correct answer is ₹38,500.

Additional Information

Partnership Profit Ratio

When investments are made for unequal time periods, the profit is distributed in the ratio of the product of capital and corresponding time period: C₁T₁ : C₂T₂ : C₃T₃.

Sleeping vs Working Partner

A working partner manages the business and usually receives a fixed salary or a certain percentage of the total profit before the remaining profit is distributed according to capital ratios.

Joining Mid-Way

If a partner joins x months after the start of a year, their investment time period is taken as (12 − x) months when calculating the effective capital for profit distribution.