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Shortcut Trick

Average speed = (Total distance) ÷ (Total time)

Total Distance = 10 + 20 = 30 km

Total Time = (10 ÷ 5) + (20 ÷ 10) = 2 + 2 = 4 hours

Average Speed = 30 ÷ 4 = 7.5 km/h

∴ The correct answer is 7.5 km/h.

Alternate Method

Given:

Distance₁ (d₁) = 10 km, Speed₁ (s₁) = 5 km/h

Distance₂ (d₂) = 20 km, Speed₂ (s₂) = 10 km/h

Formula Used:

Time = Distance ÷ Speed

Average Speed = (d₁ + d₂) ÷ (t₁ + t₂)

⇒ Time taken for the first part (t₁) = 10 ÷ 5 = 2 hours

⇒ Time taken for the second part (t₂) = 20 ÷ 10 = 2 hours

⇒ Total Time = t₁ + t₂ = 2 + 2 = 4 hours

⇒ Total Distance = d₁ + d₂ = 10 + 20 = 30 km

⇒ Average Speed = 30 ÷ 4 = 7.5 km/h

∴ The correct answer is 7.5 km/h.

Additional Information

Equal Distance Case

If a person covers two equal distances at speeds x and y, then Average Speed = 2xy ÷ (x + y).

Equal Time Case

If a person travels for two equal time intervals at speeds x and y, then Average Speed = (x + y) ÷ 2.

Unit Conversion

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

Shortcut Trick

Net walking time = Total time − Stop time = 1.5 hr − 0.5 hr = 1 hr.

Total distance covered = Speed × Time = 4 km/h × 1 hr = 4 km.

Distance to park = Total distance ÷ 2 = 4 ÷ 2 = 2 km.

∴ The correct answer is 2 km.

Alternate Method

Given: Speed = 4 km/h, Stop time = 30 min, Total time = 1.5 hr.

Formula: Distance = Speed × Time

⇒ Time at park = 30 min = 30 ÷ 60 hr = 0.5 hr.

⇒ Actual time spent walking = 1.5 − 0.5 = 1.0 hr.

⇒ Let the distance to the park be D km.

⇒ Time to go = D ÷ 4

⇒ Time to return = D ÷ 4

⇒ Total walking time = (D ÷ 4) + (D ÷ 4) = 1

⇒ 2D ÷ 4 = 1

⇒ D ÷ 2 = 1

⇒ D = 2 km.

∴ The correct answer is 2 km.

Additional Information

Average Speed

When a person travels equal distances at speeds x and y, the average speed for the whole journey is (2 × x × y) ÷ (x + y).

Relative Speed

If two objects move in the same direction, relative speed = S₁ − S₂. If they move in opposite directions, relative speed = S₁ + S₂.

Unit Conversion

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

Shortcut Trick

Selling Price (SP) is (100 − 15) = 85% of the Marked Price (MP).

SP = 85% of ₹1,600

SP = 0.85 × 1,600 = ₹1,360

∴ The correct answer is ₹1360.

Alternate Method

Given:

Original Price (MP) = ₹1,600

Discount Percentage = 15%

Formula Used:

Discount = MP × (Discount% / 100)

Selling Price (SP) = MP − Discount

Calculations:

⇒ Discount = ₹1,600 × (15 / 100)

⇒ Discount = 16 × 15 = ₹240

⇒ Selling Price = ₹1,600 − ₹240

⇒ Selling Price = ₹1,360

∴ The correct answer is ₹1360.

Additional Information

Marked Price & Discount

Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP unless stated otherwise.

Successive Discounts

If two discounts d₁% and d₂% are given, the Equivalent Discount = [d₁ + d₂ − (d₁ × d₂)/100]%.

Profit and Loss Relation

Selling Price = Cost Price + Profit OR Selling Price = Cost Price − Loss. MP × (100 − D%) = CP × (100 + P%).

Shortcut Trick

The overall percentage return can be calculated using the weighted average method based on the given ratios.

Ratio of mutual funds X : Y : Z = 3 : 2 : 5 (Total parts = 10)

Overall Return % = [(3 × 12%) + (2 × 8%) + (5 × 10%)] ÷ 10

Overall Return % = (36 + 16 + 50) ÷ 10 = 102 ÷ 10 = 10.2%

∴ The correct answer is 10.2%.

Alternate Method

Given: Total investment = ₹2,00,000 | Ratio X:Y:Z = 3:2:5 | Returns: X = 12%, Y = 8%, Z = 10%.

Formula Used: Return Amount = Principal × Rate ÷ 100 | Percentage Return = (Total Return ÷ Total Investment) × 100.

⇒ Total Ratio Parts = 3 + 2 + 5 = 10 parts

⇒ Value of 1 part = 2,00,000 ÷ 10 = ₹20,000

⇒ Investment in X = 3 × 20,000 = ₹60,000 | Investment in Y = 2 × 20,000 = ₹40,000 | Investment in Z = 5 × 20,000 = ₹1,00,000

⇒ Return from X = 12% of 60,000 = (12 ÷ 100) × 60,000 = ₹7,200

⇒ Return from Y = 8% of 40,000 = (8 ÷ 100) × 40,000 = ₹3,200

⇒ Return from Z = 10% of 1,00,000 = (10 ÷ 100) × 1,00,000 = ₹10,000

⇒ Total Return Amount = 7,200 + 3,200 + 10,000 = ₹20,400

⇒ Overall Return % = (20,400 ÷ 2,00,000) × 100 = 10.2%

∴ The correct answer is 10.2%.

Additional Information

Weighted Average Formula

Average = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n), where w is the weight (ratio) and x is the value (return %).

Percentage Change

Overall Percentage = (Net Change ÷ Initial Value) × 100. This is useful for finding effective returns or successive growth.

Ratio Distribution

To divide a quantity N in ratio a:b:c, the shares are [a/(a+b+c)]×N, [b/(a+b+c)]×N, and [c/(a+b+c)]×N.

Shortcut Trick

Time taken by the mural artist to paint the wall = 16 hr.

The assistant takes twice as long as the mural artist.

Time taken by the assistant = 2 × 16 = 32 hr.

∴ The correct answer is 32 hr.

Alternate Method

Given:

Time taken by Mural Artist = 16 hr.

Time taken by Assistant = 2 × 16 = 32 hr.

Calculations:

⇒ Let the Total Work be the LCM of 16 and 32 = 32 units.

⇒ Efficiency of Assistant = Total Work ÷ Time taken by Assistant.

⇒ Efficiency of Assistant = 32 ÷ 32 = 1 unit/hr.

⇒ Time taken by Assistant to complete the work alone = Total Work ÷ Efficiency.

⇒ Time = 32 ÷ 1 = 32 hr.

∴ The correct answer is 32 hr.

Additional Information

Time and Efficiency Relation

Efficiency is inversely proportional to the Time taken to complete a task. If A is twice as fast as B, B takes twice as much time as A.

Work Equation

The basic formula for time and work is Total Work = Efficiency × Time.

Combined Work

If two people work together, their combined efficiency is the sum of their individual efficiencies: Total Time = Total Work ÷ (Eff₁ + Eff₂).

Shortcut Trick

Total work (LCM of 16, 12, and 4) = 48 units.

Efficiencies: A = 3, B = 4, and A + B + C = 12.

Efficiency of C = 12 − (3 + 4) = 5 units/day.

Time taken by C alone = 48 ÷ 5 = 9.6 days.

∴ The correct answer is 9.6 days.

Alternate Method

Given:

Time for Worker A = 16 days

Time for Worker B = 12 days

Time for A, B, and C together = 4 days

Formula Used:

Efficiency = Total Work ÷ Time taken

Calculations:

⇒ Total efficiency of A and B = 3 + 4 = 7 units/day

⇒ Efficiency of Worker C = Combined Efficiency − (Efficiency of A + B)

⇒ Efficiency of C = 12 − 7 = 5 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time = 48 ÷ 5 = 9.6 days

∴ The correct answer is 9.6 days.

Additional Information

Work and Wages

Wages are distributed in the direct ratio of the efficiencies of the workers if they work for the same duration.

Pipes and Cisterns

The same principle applies where filling pipes have positive efficiency and outlet/leak pipes have negative efficiency.

Combined Time Formula

If two people take x and y days, together they take (x × y) ÷ (x + y) days to complete the work.

Shortcut Trick

Use the direct trigonometric identity: tan θ + cot θ = 1 / (sin θ cos θ).

Given that tan θ + cot θ = 5, we have 1 / (sin θ cos θ) = 5.

The product of sine and cosine is the reciprocal of the given sum.

⇒ sin θ cos θ = 1 / 5.

∴ The correct answer is 1/5.

Alternate Method

Given: The equation tan θ + cot θ = 5 for θ in the first quadrant (0, π/2).

Formula Used: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.

Applying the identities to the given expression:

⇒ (sin θ / cos θ) + (cos θ / sin θ) = 5

⇒ (sin² θ + cos² θ) / (sin θ cos θ) = 5

Using the fundamental identity sin² θ + cos² θ = 1:

⇒ 1 / (sin θ cos θ) = 5

To find the product of sine and cosine, we take the reciprocal:

⇒ sin θ cos θ = 1 / 5

∴ The correct answer is 1/5.

Additional Information

Fundamental Identity

The most basic trigonometric identity is sin² θ + cos² θ = 1, which holds for all real values of θ.

Reciprocal Relations

The product tan θ × cot θ is always 1, and similarly sin θ × csc θ = 1 and cos θ × sec θ = 1.

Sum of Tangent and Cotangent

The expression tan θ + cot θ is always equal to sec θ × csc θ or 2 / sin(2θ). This implies its absolute value is always ≥ 2.

Shortcut Trick

Distance traveled = Radius × Angle (in radians)

Radius (r) = Diameter ÷ 2 = 1.4 ÷ 2 = 0.7 unit

Distance = 0.7 × 5π = 3.5 × (22/7) = 0.5 × 22 = 11 units

∴ The correct answer is 11 cm.

Alternate Method

Given:

Diameter of the wheel (d) = 1.4 m (Using magnitude 1.4 for calculation)

Angle of rotation (θ) = 5π radians

Formula Used:

The linear distance (s) traveled by a point on the rim of a rotating wheel is given by the arc length formula: s = r × θ, where r is the radius and θ is the angle in radians.

Calculations:

⇒ Radius (r) = Diameter ÷ 2

⇒ r = 1.4 ÷ 2 = 0.7 unit

⇒ Distance (s) = r × θ

⇒ s = 0.7 × 5π

⇒ s = 3.5 × π

⇒ s = 3.5 × (22/7)

⇒ s = (35/10) × (22/7) = (5/10) × 22

⇒ s = 0.5 × 22 = 11 unit

Matching the units with the given options:

∴ The correct answer is 11 cm.

Additional Information

Radian to Degree Conversion

To convert an angle from radians to degrees, multiply by 180/π. For example, π radians = 180°.

Full Rotations

One complete rotation of a wheel covers a distance equal to its circumference (C = 2πr or πd), which corresponds to an angle of 2π radians.

Area of a Sector

The area of a sector of a circle with radius r and angle θ (in radians) is given by A = (1/2) × r² × θ.

Shortcut Trick

Calculate the semi-perimeter (s): s = (13 + 14 + 15) ÷ 2 = 21 cm.

Apply Heron’s formula: Area = √(21 × 8 × 7 × 6).

Simplify: √(21 × 7 × 8 × 6) = √(147 × 48) = √7056 = 84 cm².

∴ The correct answer is 84 cm².

Alternate Method

Given: Sides of the triangular face are a = 13 cm, b = 14 cm, and c = 15 cm.

Formula Used: Heron’s Formula for Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2.

⇒ Semi-perimeter (s) = (13 + 14 + 15) ÷ 2

⇒ s = 42 ÷ 2 = 21 cm

⇒ Area = √[21 × (21 − 13) × (21 − 14) × (21 − 15)]

⇒ Area = √(21 × 8 × 7 × 6)

⇒ Area = √(168 × 42)

⇒ Area = √7056

⇒ Area = 84 cm²

∴ The correct answer is 84 cm².

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)].

Scalene Triangle

A triangle where all three sides have different lengths. For a triangle with sides 13, 14, 15, the height to the side of 14 cm is exactly 12 cm.

Standard Formula

Area of any triangle can also be calculated as 1/2 × Base × Height if the perpendicular height is known.

Shortcut Trick

For a circle inscribed in a square, the diameter of the circle is always equal to the side of the square.

Side of square (a) = 2 × Radius (r)

a = 2 × 7 = 14 cm

Area of square = a² = 14² = 196 cm²

∴ The correct answer is 196 cm².

Alternate Method

Given:

Radius of the inscribed circle (r) = 7 cm

Formula Used:

1. Diameter (d) = 2 × Radius (r)
2. Side of Square (a) = Diameter of inscribed circle
3. Area of Square = a²

Calculations:

⇒ Diameter of the circle = 2 × 7 cm = 14 cm

⇒ Since the circle is inscribed, Side of the square (a) = 14 cm

⇒ Area of the square = 14 cm × 14 cm

⇒ Area = 196 cm²

∴ The correct answer is 196 cm².

Additional Information

Circle Circumscribing a Square

When a circle is circumscribed about a square, the diagonal of the square is equal to the diameter of the circle (d = a√2).

Area of Inscribed Circle

The area of the circle inscribed in a square of side 'a' is π × (a/2)².

Ratio of Areas

The ratio of the area of a square to the area of its inscribed circle is always 4 : π (approx 14 : 11).