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Shortcut Trick

Assume Total Work = LCM(12, 15, 20) = 60 units.

Efficiency of 2(A + B + C) = Efficiency(A+B) + Efficiency(B+C) + Efficiency(C+A) = 5 + 4 + 3 = 12 units/day.

Efficiency of (A + B + C) = 12 ÷ 2 = 6 units/day.

Time taken by A, B, and C together = 60 ÷ 6 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given:

Time taken by (A + B) = 12 days

Time taken by (B + C) = 15 days

Time taken by (C + A) = 20 days

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total Efficiency of 2(A + B + C) = 5 + 4 + 3 = 12

⇒ Efficiency of (A + B + C) = 12 ÷ 2 = 6

⇒ Days required for (A + B + C) = 60 ÷ 6 = 10 days

∴ The correct answer is 10 days.

Additional Information

Individual Efficiency Calculation

To find A's efficiency, subtract (B + C) from (A + B + C): 6 − 4 = 2. Similarly, B = 6 − 3 = 3 and C = 6 − 5 = 1.

Work and Time Relationship

Time is inversely proportional to efficiency. If efficiency doubles, the time required to complete the same work is halved.

Combined Rate Formula

For such cyclical pairs, 1/Time_total = 1/2 × (1/T₁ + 1/T₂ + 1/T₃).

Given : 

A car travels 60 KM in the first hour

And 40 KM in the second hour

Concept used:

Average speed = total distance / total time

Calculation:

Total distance travelled by the car = (60 + 40) = 100 km

Total time taken = (1 + 1) = 2 hr

So, average speed = 100/2 = 50 km/hr

∴ The correct answer is 50 km/hr

Given : 

Ravi sold an article for Rs. Sells for 1530

Loss of 15%

Calculation:

Let's assume that the CP of the article be k, then ATQ

⇒ k × 85/100 = 1530

⇒ k = (1530 × 100)/85

⇒ k = 1800

∴ The correct answer is 1800

Given : 

Radius of the top of the circular shape (r) = 20 CM.

Height of the cylinder (h) = 160 CM.

Formula used:

The outer surface area of the hemisphere = 2πr²

The curved surface area of the cylinder = 2πrh

where, r = Radius, and h = Height

Calculation:

The total surface area of ​​the bird bath pot equal to 

Curved surface area of the cylinder + Inner surface area of the hemisphere

The curved surface area of ​​the bird bath pot

⇒ 2πr² + 2πrh

⇒ 2 × π ×  202 + 2π × 20 × 160 

⇒ 6400π + 800π = 7200π cm2

∴ The correct answer is 7200 π cm2.

Shortcut Trick Distance gap at 4:00 am = 50 km/h × 1 hour = 50 km. Relative speed (same direction) = 75 − 50 = 25 km/h. Time to catch = Distance gap ÷ Relative speed = 50 ÷ 25 = 2 hours. Catch time = 4:00 am + 2 hours = 6:00 a.m. ∴ The correct answer is 6:00 a.m.

Alternate Method Given: Speed of the thief = 50 km/h Speed of the owner = 75 km/h Time difference (Head start) = 4:00 am − 3:00 am = 1 hour Formula Used: Distance = Speed × Time Time to catch = Initial distance gap ÷ Relative Speed

⇒ Distance covered by thief in 1 hour = 50 × 1 = 50 km. ⇒ Relative speed (Owner and Thief) = 75 − 50 = 25 km/h. ⇒ Time taken by owner to cover 50 km gap = 50 ÷ 25 = 2 hours. ⇒ Owner starts at 4:00 am, so he will catch the thief at = 4:00 am + 2 hours = 6:00 a.m. ∴ The correct answer is 6:00 a.m.

Additional Information Relative Speed in Same Direction When two objects move in the same direction with speeds u and v (u > v), the relative speed is (u − v). Relative Speed in Opposite Direction When two objects move toward each other or away from each other, the relative speed is (u + v). Catching Time Concept The time required to catch up is always calculated from the moment the second object starts moving. Unit Conversion To convert km/h to m/s, multiply the speed by 5/18. To convert m/s to km/h, multiply by 18/5.

Given : 

Surface area of sphere = 616 cm²

Concept used:

Surface area of sphere = 4 × π × r² 

Calculation:

According to the question,

⇒ 4 × π × r2 = 616

⇒ r2 =  616 × 7/22 × 1/4

⇒ r² = 49

⇒ r = √49

⇒ r = 7

∴ The correct answer is 7 cm

Shortcut Trick

For a closed cylinder, Total Surface Area = 2πr(r + h)

Radius (r) = 140 ÷ 2 = 70 cm = 0.7 m, Height (h) = 1 m

Area = 2 × (22/7) × 0.7 × (0.7 + 1)

Area = 44 × 0.1 × 1.7 = 7.48

∴ The correct answer is 7.48 m².

Alternate Method

Given:

Height of the closed cylindrical tank (h) = 1 m

Base diameter (d) = 140 cm = 1.4 m

⇒ Radius (r) = 1.4 ÷ 2 = 0.7 m

Formula Used:

Total Surface Area of a closed cylinder = 2πr(r + h)

Calculations:

⇒ Total Surface Area = 2 × (22/7) × 0.7 × (0.7 + 1)

⇒ Total Surface Area = 44 × 0.1 × 1.7

⇒ Total Surface Area = 4.4 × 1.7 = 7.48 m²

∴ The correct answer is 7.48 m².

Additional Information

Curved Surface Area (CSA) of a Cylinder

The area of the curved sides excluding the top and bottom circular bases. Formula: CSA = 2πrh.

Open vs Closed Cylinder

A closed cylinder includes both bases where TSA = 2πr(r + h), whereas an open cylinder has no top and its area is calculated as 2πrh + πr².

Volume of a Cylinder

The total space occupied by the cylinder. Formula: Volume = πr²h.

Given : 

Quantity of cement in 216 kg of the mixture is 84 kg.

Calculation:

The quantity of cement is 84 kg in the 216 kg mixture, then the quantity of sand = (216 - 84) = 132 kg

So, the quantity of sand is 132 kg in 216 kg of mixture 

Then, 231 kg of sand, the quantity of mixture will be = 216/132 × 231 = 378 kg

∴ The correct answer is 378 kg

Shortcut Trick

For a markup percentage a and discount percentage b, the overall profit or loss percentage can be calculated using the successive percentage change formula:

Net Change % = a + b + (a × b) ÷ 100

Here, a = +35 (Markup) and b = −20 (Discount).

⇒ Profit % = 35 − 20 + [35 × (−20)] ÷ 100

⇒ Profit % = 15 − 7 = 8%

∴ The correct answer is 8%.

Alternate Method

Given:

Markup % = 35%

Discount % = 20%

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100

Profit % = [(SP − CP) ÷ CP] × 100

Calculations:

⇒ Let the Cost Price (CP) of the bed sheet be 100.

⇒ Marked Price (MP) = 100 + 35% of 100 = 135

⇒ Selling Price (SP) = MP − 20% of MP

⇒ SP = 135 − (20 ÷ 100) × 135

⇒ SP = 135 − 27 = 108

⇒ Profit = SP − CP = 108 − 100 = 8

⇒ Profit % = (8 ÷ 100) × 100 = 8%

∴ The correct answer is 8%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP), never on the Cost Price (CP). Selling Price = MP − Discount.

Relation between CP and MP

A direct relation connecting CP and MP when profit % and discount % are given is: CP ÷ MP = (100 − Discount %) ÷ (100 + Profit %).

Successive Discount

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂) ÷ 100]%.

Shortcut Trick

When squares of side x are cut from corners of a sheet L × W, the volume of the resulting tray is V = (L − 2x) × (W − 2x) × x.

Given: L = 15 cm, W = 10 cm, and x = 2 cm.

New dimensions: (15 − 4), (10 − 4), and 2.

⇒ Volume = 11 × 6 × 2 = 132 cm³.

∴ The correct answer is 132 cm³.

Alternate Method

Given:

Length of rectangular sheet = 15 cm

Width of rectangular sheet = 10 cm

Side of the square cut from each corner = 2 cm

When the corners are cut and the sides are folded up:

⇒ Length of the tray (l) = 15 − 2 − 2 = 11 cm

⇒ Width of the tray (w) = 10 − 2 − 2 = 6 cm

⇒ Depth (height) of the tray (h) = 2 cm

Formula Used:

Volume of a rectangular tray (Cuboid) = Length × Width × Height

⇒ Volume = l × w × h

⇒ Volume = 11 × 6 × 2

⇒ Volume = 66 × 2

⇒ Volume = 132 cm³

∴ The correct answer is 132 cm³.

Additional Information

Volume of a Cuboid

The space occupied by a three-dimensional rectangular object is given by Volume = Length × Breadth × Height.

Surface Area of an Open Tray

The total area of the sheet used to make the open tray is (l × w) + 2(l × h) + 2(w × h), which is also equal to the original sheet area minus the area of the 4 cut-out squares.

Diagonal of a Cuboid

The longest rod that can be placed in a cuboidal box is the diagonal, calculated as √(l² + w² + h²).