Search Previous Year Questions

Shortcut Trick

Net efficiency = 1/12 + 1/18 − 1/36

Net efficiency = (3 + 2 − 1) / 36 = 4 / 36 = 1 / 9

Since the net efficiency is 1/9, the tank will be filled in 9 hr.

∴ The correct answer is 9 hr.

Alternate Method

Given:

Time taken by Pipe A = 12 hr

Time taken by Pipe B = 18 hr

Time taken by Pipe C = 36 hr (Emptying pipe)

Formula Used:

Total Work = LCM of individual times

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total Work = LCM(12, 18, 36) = 36 units

⇒ Efficiency of Pipe A = 36 ÷ 12 = 3 units/hr

⇒ Efficiency of Pipe B = 36 ÷ 18 = 2 units/hr

⇒ Efficiency of Pipe C = 36 ÷ 36 = 1 unit/hr (Negative as it empties)

⇒ Combined Efficiency (A + B + C) = 3 + 2 − 1 = 4 units/hr

⇒ Time taken to fill the tank = Total Work ÷ Combined Efficiency

⇒ Time = 36 ÷ 4 = 9 hr

∴ The correct answer is 9 hr.

Additional Information

Inlet and Outlet Pipes

An Inlet Pipe fills the tank (positive work) while an Outlet Pipe empties it (negative work). Net work = Σ(Inlet efficiencies) − Σ(Outlet efficiencies).

Time and Efficiency Relationship

Efficiency is inversely proportional to time. If a pipe can fill a tank in x hours, its efficiency is 1/x of the tank per hour.

LCM Method for Work

Taking the Least Common Multiple (LCM) of the given time durations as the total capacity of the tank simplifies calculations by using whole numbers for efficiency.

Shortcut Trick

Ratio of Investment (A : B) = 15,000 : 25,000 = 3 : 5.

Total ratio units = 3 + 5 = 8 units.

Value of 8 units = ₹8,000 ⇒ 1 unit = ₹1,000.

Profit of A = 3 units × ₹1,000 = ₹3,000.

∴ The correct answer is ₹3,000.

Alternate Method

Given:

Investment of A = ₹15,000

Investment of B = ₹25,000

Total Profit = ₹8,000

Formula Used:

Profit Share Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Since time period is not mentioned, it is assumed same for both.

⇒ Profit Ratio (A : B) = 15,000 : 25,000

⇒ Profit Ratio (A : B) = 3 : 5

⇒ Sum of ratio parts = 3 + 5 = 8

⇒ A's share of profit = (Ratio of A ÷ Total Ratio) × Total Profit

⇒ A's share = (3 ÷ 8) × 8,000

⇒ A's share = 3 × 1,000 = ₹3,000

∴ The correct answer is ₹3,000.

Additional Information

Partnership Basic Rule

Profit is always distributed in the ratio of the product of Capital and Time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Types of Partners

An Active Partner manages the business and may receive a salary, while a Sleeping Partner only invests capital and shares profit/loss.

Variable Time Period

If partners join at different times, calculate their effective capital by multiplying Amount × Months they stayed in the business.

Shortcut Trick

Reduced Price per kg = (Total Expenditure × Reduction %) ÷ Extra Quantity

Reduced Price = (640 × 20%) ÷ 6.4

Reduced Price = (640 × 0.2) ÷ 6.4

Reduced Price = 128 ÷ 6.4 = 20

∴ The correct answer is ₹20.

Alternate Method

Given:

Total Expenditure = ₹640

Reduction in Price = 20%

Extra quantity purchased = 6.4 kg

Formula Used:

Price × Quantity = Expenditure (When expenditure is constant, Price ∝ 1/Quantity)

Calculations:

⇒ Price Ratio (Original : New) = 100 : 80 = 5 : 4

⇒ Quantity Ratio (Original : New) = 4 : 5 (Inverse of Price Ratio)

⇒ Difference in quantity parts = 5 − 4 = 1 unit

⇒ 1 unit = 6.4 kg

⇒ New Quantity purchased = 5 units = 5 × 6.4 = 32 kg

⇒ New Price per kg = Total Expenditure ÷ New Quantity

⇒ New Price = 640 ÷ 32 = 20

∴ The correct answer is ₹20.

Additional Information

Original Price Formula

The original price can be calculated as: (Total Amount × Rate %) ÷ [Quantity × (100 ± Rate %)]. Here, Original Price = (640 × 20) ÷ (6.4 × 80) = ₹25.

Price and Consumption Relationship

If the price of a commodity decreases by r%, the increase in consumption to keep the expenditure constant is [r ÷ (100 − r)] × 100%.

Direct Calculation of New Price

The percentage reduction on the total amount directly represents the cost of the extra quantity bought. For example, 20% of ₹640 = ₹128. This ₹128 is the current cost of 6.4 kg, so New Price = 128 ÷ 6.4 = ₹20.

Shortcut Trick

Simplify investments by dividing by 1000: A = (64 × 4 + 100 × 8), B = (96 × 4 + 80 × 8), C = (120 × 4).

Ratio A : B : C = 1056 : 1024 : 480 ⇒ 33 : 32 : 15.

Difference (A − C) = (33 − 15) = 18 units out of 80 total units.

Profit Difference = (18 ÷ 80) × 3,36,000 = 18 × 4,200 = ₹75,600.

∴ The correct answer is ₹75,600.

Alternate Method

Given: Initial investments of A = ₹64,000 and B = ₹96,000. Changes occur after 4 months. C joins after 8 months with ₹1,20,000. Total time = 12 months.

Formula Used: Profit Share Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

⇒ A's equivalent investment = (64,000 × 4) + (100,000 × 8) = 2,56,000 + 8,00,000 = 10,56,000

⇒ B's equivalent investment = (96,000 × 4) + (80,000 × 8) = 3,84,000 + 6,40,000 = 10,24,000

⇒ C's equivalent investment = 1,20,000 × 4 = 4,80,000

⇒ Ratio A : B : C = 1056 : 1024 : 480 = 33 : 32 : 15

⇒ Sum of ratios = 33 + 32 + 15 = 80 units

⇒ Given Total Profit = ₹3,36,000

⇒ 1 unit value = 3,36,000 ÷ 80 = ₹4,200

⇒ Difference between A and C = (33 − 15) = 18 units

⇒ Required difference = 18 × 4,200 = ₹75,600

∴ The correct answer is ₹75,600.

Additional Information

Partnership Profit Ratio

Profit Share Ratio = (C₁ × T₁) : (C₂ × T₂) : (C₃ × T₃), where C is the capital and T is the time period of investment.

Compound Partnership

When partners invest different amounts for different periods of time, it is called a compound partnership.

Active Partner

A partner who manages the business is an active partner and may receive a salary from the profit before distribution.

Shortcut Trick

When a solid is melted and reshaped, its Volume remains constant.

Volume of Hemisphere = Volume of Cone

(2/3) × π × r_h³ = (1/3) × π × r_c² × h

2 × 7³ = r_c² × 14 ⇒ 2 × 343 = 14 × r_c²

r_c² = 686 ÷ 14 = 49 ⇒ r_c = 7 cm

∴ The correct answer is 7 cm.

Alternate Method

Given:

Radius of Hemisphere (r) = 7 cm

Height of Cone (h) = 14 cm

Formula used:

Volume of Hemisphere = (2/3)πr³

Volume of Cone = (1/3)πr²h

Calculations:

⇒ Volume of Hemisphere = Volume of Cone

⇒ (2/3) × π × (7)³ = (1/3) × π × (Radius of cone)² × 14

⇒ (2/3) × 343 = (1/3) × (r_cone)² × 14

⇒ 2 × 343 = (r_cone)² × 14

⇒ 686 = 14 × (r_cone)²

⇒ (r_cone)² = 686 ÷ 14

⇒ (r_cone)² = 49

⇒ r_cone = √49 = 7 cm

∴ The correct answer is 7 cm.

Additional Information

Volume of Sphere

The volume of a complete Sphere is given by the formula V = (4/3)πr³.

Surface Area of Hemisphere

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

Slant Height of Cone

The slant height (l) of a Cone is calculated using the Pythagorean theorem: l = √(r² + h²).

Shortcut Trick

Identify the Pythagorean triplet for radius 5 and height 12, which gives slant height l = 13.

Total Surface Area (TSA) = πr(l + r) = (22/7) × 5 × (13 + 5).

TSA = (22/7) × 90 = 1980 / 7 ≈ 282.85 cm².

Rounding to the nearest integer, we get 283 cm².

∴ The correct answer is 283 cm².

Alternate Method

Given: Base Radius (r) = 5 cm, Height (h) = 12 cm.

Formula Used:

1. Slant Height (l) = √(r² + h²)
2. Total Surface Area (TSA) = πr(l + r)

⇒ First, calculate the Slant Height (l):

⇒ l = √(5² + 12²)

⇒ l = √(25 + 144) = √169 = 13 cm

⇒ Now, calculate the Total Surface Area:

⇒ TSA = (22/7) × 5 × (13 + 5)

⇒ TSA = (22/7) × 5 × 18

⇒ TSA = (22 × 90) / 7

⇒ TSA = 1980 / 7 ≈ 282.8571 cm²

⇒ Rounding to the nearest integer = 283 cm².

∴ The correct answer is 283 cm².

Additional Information

Curved Surface Area (CSA) of a Cone

The lateral area or curved surface area of a cone is given by the formula: CSA = πrl.

Volume of a Right Circular Cone

The volume represents the space inside the cone and is calculated as: Volume = (1/3)πr²h.

Pythagorean Relation in Cones

In any right circular cone, the height (h), radius (r), and slant height (l) always form a right-angled triangle such that l² = r² + h².

Base Area of a Cone

Since the base of a cone is a circle, its area is calculated using the standard circle area formula: Base Area = πr².

Shortcut Trick

Assume Cost Price (CP) = 100.

Marked Price (MP) = 100 + 20% = 120.

Selling Price (SP) = 100 + 15% = 115.

Discount = MP − SP = 120 − 115 = 5.

Discount % = (5 ÷ 120) × 100 = 25/6 % = 4 1/6 %.

∴ The correct answer is 4 1/6 %.

Alternate Method

Given:

Markup Percentage = 20%

Profit Percentage = 15%

Formula Used:

SP = CP × (1 + Profit%/100)

SP = MP × (1 − Discount%/100)

Calculations:

⇒ Let the Cost Price (CP) be 100.

⇒ Marked Price (MP) = 100 + (20% of 100) = 120.

⇒ Desired Profit = 15%.

⇒ Selling Price (SP) = 100 + (15% of 100) = 115.

⇒ Discount = MP − SP = 120 − 115 = 5.

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (5 ÷ 120) × 100

⇒ Discount % = 500 ÷ 120 = 25/6 = 4 1/6 %.

∴ The correct answer is 4 1/6 %.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price can be expressed as: CP / MP = (100 − Discount%) / (100 + Profit%).

Markup Percentage

Markup% = [(MP − CP) ÷ CP] × 100. It is always calculated on the Cost Price.

Successive Discounts

If two discounts d₁ and d₂ are given, the effective discount is d₁ + d₂ − (d₁ × d₂ ÷ 100).

Shortcut Trick

Total Time = 1 hr 40 min = 100 minutes.

Speed in km/min = 7200 ÷ 100 = 72 km/min.

Speed in km/h = 72 × 60 = 4320 km/h.

∴ The correct answer is 4320 km/h.

Alternate Method

Given:

Distance (D) = 7200 km

Time (T) = 1 hr 40 min

Formula Used:

Speed (S) = Distance ÷ Time

Calculations:

⇒ Convert Time to hours: 1 hr + (40/60) hr

⇒ Time = 1 + 2/3 = 5/3 hours

⇒ Speed = 7200 ÷ (5/3)

⇒ Speed = 7200 × (3/5)

⇒ Speed = 1440 × 3

⇒ Speed = 4320 km/h

∴ The correct answer is 4320 km/h.

Additional Information

Speed 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.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distance segments at speeds x and y, the Average Speed is 2xy ÷ (x + y).

Relative Speed

When two objects move in the Same Direction, Relative Speed = S₁ − S₂. In Opposite Directions, Relative Speed = S₁ + S₂.

Shortcut Trick

Since the investment time for A and B is the same (1 year), the Profit Ratio is equal to the Investment Ratio.

Ratio of A : B = 5 : 6

Total ratio parts = 5 + 6 = 11 units

11 units = ₹95,172 ⇒ 1 unit = ₹8,652

B's share = 6 units = 6 × 8,652 = ₹51,912

∴ The correct answer is ₹51,912.

Alternate Method

Given:

Investment ratio of A and B = 5 : 6

Total profit = ₹95,172

Formula Used:

Profit = Investment × Time

Calculations:

⇒ Let the profit share of A be 5x and B be 6x.

⇒ Total Profit = 5x + 6x = 11x

⇒ 11x = 95,172

⇒ x = 95,172 ÷ 11

⇒ x = 8,652

⇒ Profit share of B = 6x = 6 × 8,652

⇒ Profit share of B = ₹51,912

∴ The correct answer is ₹51,912.

Additional Information

Ratio of Profit

In a partnership, the profit is distributed in the ratio of (Capital₁ × Time₁) : (Capital₂ × Time₂).

Simple vs Compound Partnership

A Simple Partnership involves capitals invested for the same time, while Compound Partnership involves different capitals for different time periods.

Calculating Investment

If the profit ratio and time ratio are known, the Investment Ratio = (Profit Ratio ÷ Time Ratio).

Shortcut Trick

The time taken to complete a task is inversely proportional to the rate (efficiency).

Rate ratio of Pipe A to Pipe B = 1 : 2

Time ratio of Pipe A to Pipe B = 2 : 1

Given that 2 units of time for Pipe A = 8 hr

Therefore, 1 unit of time for Pipe B = 8 ÷ 2 = 4 hr

∴ The correct answer is 4 hr.

Alternate Method

Given:

Capacity of the tank = 200 liters

Time taken by Pipe A to empty = 8 hr

Rate of Pipe B = 2 × Rate of Pipe A

Calculations:

⇒ Emptying rate of Pipe A = Capacity ÷ Time = 200 ÷ 8 = 25 L/hr

⇒ Emptying rate of Pipe B = 2 × Rate of Pipe A = 2 × 25 = 50 L/hr

⇒ Time taken by Pipe B = Capacity ÷ Rate of Pipe B

⇒ Time taken by Pipe B = 200 ÷ 50 = 4 hr

∴ The correct answer is 4 hr.

Additional Information

Efficiency and Time Relationship

Efficiency is defined as the amount of work done in a unit of time. It is inversely proportional to the time taken to complete a specific task. If efficiency doubles, time is halved.

Combined Rate for Pipes

If two pipes with times t₁ and t₂ work together to empty a tank, the combined time (T) is given by 1/T = 1/t₁ + 1/t₂.

Negative Work in Pipes and Cisterns

Inlet pipes perform positive work (filling), while outlet or leak pipes perform negative work (emptying).