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Total Marked Price = 25 × 3,000 = ₹75,000

Successive Discount % = 10 + 5 − (10 × 5) ÷ 100 = 14.5%

Net Multiplier = 100% − 14.5% = 85.5% = 0.855

Final Amount = 75,000 × 0.855 = ₹64,125

∴ The correct answer is ₹64,125.

Alternate Method

Given:

Marked Price (MP) per item = ₹3,000

Quantity = 25 items (More than 20 items)

Discounts = 10% (primary) and an additional 5% (successive)

Calculations:

⇒ Total MP = 25 × 3,000 = ₹75,000

⇒ Since quantity is 25 (which is > 20), both 10% and 5% discounts apply.

⇒ Price after 10% Discount = 75,000 − (10% of 75,000)

⇒ Price after 10% Discount = 75,000 − 7,500 = ₹67,500

⇒ Price after Additional 5% Discount = 67,500 − (5% of 67,500)

⇒ Final Amount = 67,500 − 3,375

⇒ Final Amount = ₹64,125

∴ The correct answer is ₹64,125.

Additional Information

Successive Discount Formula

When two discounts d₁ and d₂ are applied successively, the single equivalent discount is calculated as (d₁ + d₂ − (d₁ × d₂) ÷ 100)%.

Net Selling Price Formula

SP = MP × (100 − d₁)/100 × (100 − d₂)/100. This is useful for calculating the final price directly after multiple discounts.

Marked Price vs. Selling Price

Marked Price is the price printed on the item, while Selling Price is the final amount paid after deducting all applicable discounts from the MP.

Shortcut Trick

Total Area = Number of Tiles × (Base × Height)

Total Area = 500 × (8 × 6)

Total Area = 500 × 48 = 24,000 cm²

∴ The correct answer is 24,000 cm².

Alternate Method

Given:

Total identical tiles = 500

Base (b) of each parallelogram-shaped tile = 8 cm

Height (h) of each parallelogram-shaped tile = 6 cm

Formula used:

Area of a parallelogram = Base × Height

Calculations:

⇒ Area of one tile = 8 cm × 6 cm

⇒ Area of one tile = 48 cm²

⇒ Total area of the floor = Number of tiles × Area of one tile

⇒ Total area = 500 × 48 cm²

⇒ Total area = 24,000 cm²

∴ The correct answer is 24,000 cm².

Additional Information

Area of Parallelogram

The area is determined by multiplying the length of the base by the perpendicular height (A = b × h).

Perimeter of Parallelogram

The perimeter is the total length of the boundary, calculated as P = 2 × (a + b), where 'a' and 'b' are adjacent sides.

Properties of Parallelogram

Opposite sides are parallel and equal, opposite angles are equal, and diagonals bisect each other.

Shortcut Trick

Total surface area of a sphere = 4πr².

If 25% is masked, then the Painted Area = 75% of Total Surface Area.

Painted Area = 75% × 4π(10)² = 0.75 × 400π = 300π cm².

∴ The correct answer is 300π cm².

Alternate Method

Given:

Radius of the sphere (r) = 10 cm

Masked percentage = 25%

Formula Used:

Total Surface Area (TSA) of a sphere = 4πr²

Painted Area = TSA − Masked Area = TSA × (100% − Masked %)

Calculations:

⇒ Total Surface Area (TSA) = 4 × π × (10)²

⇒ TSA = 4 × π × 100 = 400π cm²

⇒ Masked Area = 25% of 400π = 0.25 × 400π = 100π cm²

⇒ Painted Area = 400π − 100π

⇒ Painted Area = 300π cm²

∴ The correct answer is 300π cm².

Additional Information

Volume of a Sphere

The amount of space occupied by the sphere is given by Volume = (4/3)πr³.

Hemisphere Surface Area

Curved Surface Area (CSA) = 2πr² and Total Surface Area (TSA) = 3πr².

Surface Area to Volume Ratio

For a sphere, the ratio is (4πr²) ÷ (4/3πr³) = 3 ÷ r.

Shortcut Trick

Total weighted parts = 3 + 2 + 5 = 10 units.

Total required weighted score = 68 × 10 = 680.

Contribution of Physics and Chemistry = (3 × 60) + (2 × 70) = 180 + 140 = 320.

Required contribution from Mathematics = 680 − 320 = 360.

Mathematics marks = 360 ÷ 5 = 72.

∴ The correct answer is 72.

Alternate Method

Given:

Ratio of weights (w₁ : w₂ : w₃) = 3 : 2 : 5

Marks in Physics (x₁) = 60, Marks in Chemistry (x₂) = 70

Weighted Average (A_w) = 68

Formula Used:

Weighted Average (A_w) = (w₁x₁ + w₂x₂ + w₃x₃) ÷ (w₁ + w₂ + w₃)

Calculations:

⇒ 68 = (3 × 60 + 2 × 70 + 5 × x₃) ÷ (3 + 2 + 5)

⇒ 68 = (180 + 140 + 5x₃) ÷ 10

⇒ 68 × 10 = 320 + 5x₃

⇒ 680 = 320 + 5x₃

⇒ 5x₃ = 680 − 320

⇒ 5x₃ = 360

⇒ x₃ = 360 ÷ 5

⇒ x₃ = 72

∴ The correct answer is 72.

Additional Information

Weighted Average Concept

Unlike a simple average where each value contributes equally, a weighted average assigns specific importance (weights) to each value. The sum of the products of weights and values is divided by the sum of the weights.

Net Deviation Method

The sum of weighted deviations from the weighted average is always zero. Σw_i(x_i − A_w) = 0. This can be used to find missing values by balancing positive and negative deviations.

Arithmetic Mean vs Weighted Mean

Arithmetic Mean is a special case of Weighted Mean where all weights are equal (e.g., w₁ = w₂ = w₃ = 1).

Shortcut Trick

Net percentage gap between 10% loss and 5% profit = 10% + 5% = 15%.

According to the question, this 15% difference corresponds to ₹450.

⇒ 15% = ₹450

⇒ 1% = ₹30

⇒ Cost Price (100%) = 30 × 100 = ₹3,000.

∴ The correct answer is ₹3,000.

Alternate Method

Given: Initial Loss = 10%, Final Profit = 5%, Price Increase = ₹450.

⇒ Let the Cost Price (CP) of the watch be ₹x.

⇒ Initial Selling Price (SP₁) = CP − 10% of CP = 0.90x

⇒ New Selling Price (SP₂) = CP + 5% of CP = 1.05x

⇒ According to the problem: SP₂ − SP₁ = 450

⇒ 1.05x − 0.90x = 450

⇒ 0.15x = 450

⇒ x = 450 ÷ 0.15

⇒ x = 45000 ÷ 15

⇒ x = 3000

∴ The correct answer is ₹3,000.

Additional Information

Profit and Loss Percentage

Profit % is always calculated on Cost Price unless stated otherwise. Formula: Profit% = (Profit ÷ CP) × 100.

Marked Price and Discount

Discount is the reduction offered on the Marked Price (MP). SP = MP − Discount.

Successive Percentage Shortcut

For two successive changes of a% and b%, net change = (a + b + ab/100)%.

Shortcut Trick

Average speed = Total distance ÷ Total time

Total distance = 120 + 120 + 100 = 340 km

Total time = (120 ÷ 40) + (120 ÷ 60) + (100 ÷ 80) = 3 + 2 + 1.25 = 6.25 hours

Average speed = 340 ÷ 6.25 = 54.4 km/h

∴ The correct answer is 54.4 km/h.

Alternate Method

Given:

First segment: Distance (d₁) = 120 km, Speed (s₁) = 40 km/h

Second segment: Distance (d₂) = 120 km, Speed (s₂) = 60 km/h

Third segment: Distance (d₃) = 100 km, Speed (s₃) = 80 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Time for first part (t₁) = 120 ÷ 40 = 3 hours

⇒ Time for second part (t₂) = 120 ÷ 60 = 2 hours

⇒ Time for third part (t₃) = 100 ÷ 80 = 1.25 hours

⇒ Total Distance = 120 + 120 + 100 = 340 km

⇒ Total Time = 3 + 2 + 1.25 = 6.25 hours

⇒ Average Speed = 340 ÷ 6.25 = 54.4 km/h

∴ The correct answer is 54.4 km/h.

Additional Information

Average Speed for Equal Distances

If a certain distance is covered at s₁ speed and the same distance is covered at s₂ speed, the average speed is (2 × s₁ × s₂) ÷ (s₁ + s₂).

Speed Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed is s₁ − s₂. In opposite directions, it is s₁ + s₂.

Shortcut Trick

Use the direct distance formula: d = h(cot θ₁ − cot θ₂), where d is the distance moved toward the tower.

Here, d = 100 m, θ₁ = 30°, and θ₂ = 60°.

⇒ 100 = h(cot 30° − cot 60°)

⇒ 100 = h(√3 − 1/√3) ⇒ 100 = h(2/√3)

⇒ h = (100 × √3) / 2 = 50√3 m.

∴ The correct answer is 50√3 m.

Alternate Method

Given: Angle of elevation at point P = 30°, Angle at point Q = 60°, distance PQ = 100 m.

In right ▵ABQ:

⇒ tan 60° = AB/BQ

⇒ √3 = h/BQ ⇒ BQ = h/√3

In right ▵ABP:

⇒ tan 30° = AB/BP

⇒ 1/√3 = h/(BQ + 100)

⇒ BQ + 100 = h√3

Substituting the value of BQ:

⇒ h/√3 + 100 = h√3

⇒ 100 = h√3 − h/√3

⇒ 100 = (3h − h) / √3

⇒ 100 = 2h / √3

⇒ 2h = 100√3

⇒ h = 50√3 m.

∴ The correct answer is 50√3 m.

Additional Information

Trigonometric Ratios

In a right-angled triangle, tan θ = Perpendicular/Base and cot θ = Base/Perpendicular.

Special Angles

Standard values: tan 30° = 1/√3, tan 60° = √3, and tan 45° = 1.

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking upward at an object.

Shortcut Trick

Effective Compound Interest rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%.

Given 21% of the Principal (y) = ₹630.

Total Amount payable = Principal (100%) + Interest (21%) = 121%.

If 21% → ₹630, then 1% → ₹30.

121% → 121 × 30 = ₹3,630.

∴ The correct answer is ₹3,630.

Alternate Method

Given: Compound Interest (CI) = ₹630, Rate (R) = 10% p.a., Time (T) = 2 years, Principal = ₹y.

Formula: Amount (A) = P(1 + R/100)^T and CI = A − P

⇒ CI = y[(1 + 10/100)² − 1]

⇒ 630 = y[(11/10)² − 1]

⇒ 630 = y[121/100 − 1]

⇒ 630 = y[21/100]

⇒ y = (630 × 100) ÷ 21

⇒ y = 30 × 100 = ₹3,000

⇒ Total Amount (A) = Principal (y) + CI

⇒ Total Amount (A) = 3000 + 630 = ₹3,630

∴ The correct answer is ₹3,630.

Additional Information

Effective Rate of Interest

For a rate R% compounded annually for 2 years, the effective rate is R + R + (R×R)/100 %.

CI and SI Difference (2 Years)

The difference between Compound Interest and Simple Interest for 2 years is given by P(R/100)².

Compound Interest Formula

Amount A = P(1 + r/n)^nt, where n is the number of times interest is compounded per year.

Given:

x = 119^0.49

y = 119^0.62

x^z = y⁶

Formula used:

(a^m)ⁿ = a^{m × n}

Calculation:

Substitute the values of x and y into the given relation:

⇒ (119^0.49)^z = (119^0.62)⁶

⇒ 119^{0.49 × z} = 119^{0.62 × 6}

Since the bases are the same, we can equate the exponents:

⇒ 0.49 × z = 0.62 × 6

⇒ 0.49 × z = 3.72

⇒ z = 3.72 ÷ 0.49

⇒ z ≈ 7.59

∴ The correct answer is option (2).

Shortcut Trick

The difference between a 2-digit number and its reverse is always 9 × (difference of digits).

Given difference = 27 ⇒ 9 × (Digit Difference) = 27 ⇒ Digit Difference = 3.

Sum of digits = 9 (Given). Let digits be x and y.

Larger digit = (9 + 3) ÷ 2 = 6 | Smaller digit = (9 − 3) ÷ 2 = 3.

Since the reversed number exceeds the original, the units digit must be larger ⇒ Number = 36.

∴ The correct answer is 36.

Alternate Method

Given:

Sum of the digits of a 2-digit number = 9

Difference between interchanged number and original number = 27

Formula Used:

A 2-digit number with tens digit x and units digit y is represented as: 10x + y.

Calculations:

⇒ Let the tens digit be x and the units digit be y.

⇒ Original number = 10x + y

⇒ x + y = 9 → (Equation 1)

⇒ Number obtained by interchanging the digits = 10y + x

⇒ (10y + x) − (10x + y) = 27

⇒ 9y − 9x = 27

⇒ y − x = 3 → (Equation 2)

⇒ Adding Equation 1 and Equation 2:

⇒ (x + y) + (y − x) = 9 + 3

⇒ 2y = 12 ⇒ y = 6

⇒ Substituting y = 6 in Equation 1: x + 6 = 9 ⇒ x = 3

⇒ Original Number = 10(3) + 6 = 36

∴ The correct answer is 36.

Additional Information

Number Reversal Property

The difference between a 2-digit number and its reverse is always a multiple of 9, specifically 9 × |tens digit − units digit|.

Sum of Reversed Numbers

The sum of a 2-digit number and its reverse is always a multiple of 11, specifically 11 × (tens digit + units digit).

Place Value Concept

In any n-digit number, the value is the sum of (digit × 10^position−1). For a 2-digit number, it is 10a + b.