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Shortcut Trick

Let the initial Cost Price be 100 units, then the initial Selling Price is 75 units (25% loss).

New Cost Price = 100 + 40% = 140 units.

New Selling Price at 15% loss = 140 × 0.85 = 119 units.

Required percentage increase in Selling Price (y) = [(119 − 75) ÷ 75] × 100.

y = (44 ÷ 75) × 100 = 176 ÷ 3 = 58.67.

∴ The correct answer is 58.67.

Alternate Method

Given:

Initial Loss = 25%

Increase in Cost Price = 40%

New Loss = 15%

Formula Used:

Selling Price (SP) = Cost Price (CP) × (100 − Loss%)/100

Percentage Increase = [(Final Value − Initial Value) ÷ Initial Value] × 100

Calculations:

⇒ Assume initial Cost Price = 100

⇒ Initial Selling Price = 100 − 25% of 100 = 75

⇒ New Cost Price = 100 + 40% of 100 = 140

⇒ New Selling Price = 140 × (100 − 15)/100

⇒ New Selling Price = 140 × 85/100 = 119

⇒ Increase in Selling Price = 119 − 75 = 44

⇒ Percentage increase (y) = (44 ÷ 75) × 100

⇒ y = (44 × 4) ÷ 3 = 176 ÷ 3

⇒ y = 58.666... ≈ 58.67

∴ The correct answer is 58.67.

Additional Information

Profit and Loss Relationship

Selling Price is calculated as CP × (100 + P%)/100 for profit and CP × (100 − L%)/100 for loss.

Percentage Change

The percentage change between two values is always calculated using the original value as the base: [(New − Old) ÷ Old] × 100.

Effective Change

When both CP and SP change, the new profit/loss percentage depends on the final ratio of the updated SP and CP values.

Shortcut Trick

Time for S to complete 3/4 work = 18 days

Time for S to complete full work = 18 × (4/3) = 24 days

Since R is 3 times as efficient as S, he takes 1/3^rd of the time.

Time taken by R = 24 ÷ 3 = 8 days

∴ The correct answer is 8 days.

Alternate Method

Given:

S completes 3/4 part of work in 18 days.

R's Efficiency = 3 × S's Efficiency.

Formula:

Total Time = (Time taken for fraction) ÷ (Fraction of work)

Efficiency × Time = Total Work

⇒ Total days for S = 18 × (4 ÷ 3) = 24 days.

⇒ Let Efficiency of S = 1 unit/day.

⇒ Total Work = Efficiency × Days = 1 × 24 = 24 units.

⇒ Efficiency of R = 3 × Efficiency of S = 3 × 1 = 3 units/day.

⇒ Days taken by R alone = Total Work ÷ Efficiency of R

⇒ Days taken by R alone = 24 ÷ 3 = 8 days.

∴ The correct answer is 8 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is x : y, then the ratio of time taken is y : x.

Total Work Calculation

Total Work = Number of Days × Efficiency. For fractional work, Full Work Time = (Time for Part) × (Reciprocal of the Part).

Combined Efficiency

When multiple people work together, their efficiencies are added: E_total = E₁ + E₂ + ... + E_n.

Shortcut Trick

Total work = LCM(18, 24, 36) = 72 units.

2(M + R + P) combined efficiency = (72 ÷ 18) + (72 ÷ 24) + (72 ÷ 36) = 4 + 3 + 2 = 9.

Combined efficiency of M, R, and P = 9 ÷ 2 = 4.5 units/day.

Work done in 4 days = 4.5 × 4 = 18 units.

Required part of work = 18 ÷ 72 = 1/4.

∴ The correct answer is 1/4.

Alternate Method

Given: M and R = 18 days, R and P = 24 days, P and M = 36 days.

Formula Used: Efficiency = Total Work ÷ Time; Total work done = Combined Efficiency × Time.

⇒ Sum of efficiencies = 2 × (M + R + P) = 4 + 3 + 2 = 9 units/day.

⇒ Efficiency of (M + R + P) working together = 9 ÷ 2 = 4.5 units/day.

⇒ Work completed in 4 days = 4.5 × 4 = 18 units.

⇒ Part of work done = 18 ÷ 72 = 1/4.

∴ The correct answer is 1/4.

Additional Information

Combined Efficiency of Pairs

When people work in pairs (AB, BC, CA), the sum of their individual efficiencies is found by dividing the sum of pair efficiencies by 2.

Work & Wages Principle

Wages are distributed based on the Efficiency × Time product for each person, which represents the actual work done.

Fractional Work

The part of work done in 'n' days is calculated as (Combined Efficiency × n) ÷ Total Work.

Shortcut Trick

Speed of boat in still water is the average of downstream and upstream speeds.

Speed = (Downstream + Upstream) ÷ 2

Speed = (42 + 36) ÷ 2 = 78 ÷ 2 = 39 km/hr

∴ The correct answer is 39 km/hr.

Alternate Method

Given:

Downstream Speed (D) = 42 km/hr

Upstream Speed (U) = 36 km/hr

Formula Used:

Speed of boat in still water (x) = (D + U) ÷ 2

Speed of stream (y) = (D − U) ÷ 2

Calculations:

⇒ x = (42 + 36) ÷ 2

⇒ x = 78 ÷ 2

⇒ x = 39 km/hr

∴ The correct answer is 39 km/hr.

Additional Information

Speed of Stream

The rate at which the water flows is given by y = (Downstream speed − Upstream speed) ÷ 2. In this case, y = (42 − 36) ÷ 2 = 3 km/hr.

Downstream and Upstream Logic

Downstream Speed = Speed of boat (x) + Speed of stream (y). Upstream Speed = Speed of boat (x) − Speed of stream (y).

Unit Conversion

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

Shortcut Trick

Speed of Stream = Still Water Speed − Upstream Speed

⇒ Speed of Stream = 23 − 17

⇒ Speed of Stream = 6 km/hr

∴ The correct answer is 6 km/hr.

Alternate Method

Given:

Speed of the boat in still water (u) = 23 km/hr

Upstream speed of the boat (u − v) = 17 km/hr

Formula Used:

Upstream Speed = Speed in Still Water − Speed of Stream

Calculations:

Let the speed of the stream be v km/hr.

⇒ 17 = 23 − v

⇒ v = 23 − 17

⇒ v = 6 km/hr

∴ The correct answer is 6 km/hr.

Additional Information

Downstream Speed

It is the net speed of the boat when it travels in the direction of the stream: Downstream Speed = u + v.

Speed in Still Water

If downstream and upstream speeds are known: u = (Downstream + Upstream) ÷ 2.

Speed of Stream

If downstream and upstream speeds are known: v = (Downstream − Upstream) ÷ 2.

Shortcut Trick

Upstream Speed = Downstream Speed − 2 × Stream Speed

Upstream Speed = 35 − 2(5) = 25 km/hr

Total Time = (350 ÷ 35) + (150 ÷ 25) = 10 + 6 = 16 hr

∴ The correct answer is 16 hr.

Alternate Method

Given: Downstream Speed = 35 km/hr, Stream Speed = 5 km/hr, Downstream Distance = 350 km, Upstream Distance = 150 km.

Formula Used: Time = Distance ÷ Speed | Boat Speed (u) = Downstream − Stream | Upstream = u − Stream

⇒ Boat Speed in Still Water (u) = 35 − 5 = 30 km/hr

⇒ Upstream Speed (v) = 30 − 5 = 25 km/hr

⇒ Time taken for Downstream = 350 ÷ 35 = 10 hr

⇒ Time taken for Upstream = 150 ÷ 25 = 6 hr

⇒ Total Time = 10 + 6 = 16 hr

∴ The correct answer is 16 hr.

Additional Information

Boat Speed and Stream Speed Calculation

Boat speed in still water (u) = (Downstream Speed + Upstream Speed) ÷ 2. Stream speed (v) = (Downstream Speed − Upstream Speed) ÷ 2.

Average Speed in Boats and Streams

When the boat covers the same distance downstream and upstream, the Average Speed = (u² − v²) ÷ u, where u is boat speed and v is stream speed.

Upstream and Downstream Logic

Downstream speed is always greater than upstream speed because the stream supports the boat movement in the former and opposes it in the latter.

Shortcut Trick

Use the ratio of Marked Price (MP) to Cost Price (CP):

MP / CP = (100 + Profit%) / (100 − Discount%)

MP / 10000 = (100 + 40) / (100 − 30) = 140 / 70 = 2 / 1

MP = 10000 × 2 = ₹20000

∴ The correct answer is ₹20000.

Alternate Method

Given:

Cost Price (CP) = ₹10000

Profit Percentage = 40%

Discount Percentage = 30%

Formula Used:

Selling Price (SP) = CP × (100 + Profit%)/100

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

Calculations:

⇒ SP = 10000 × (100 + 40) / 100

⇒ SP = 10000 × 140 / 100 = ₹14000

⇒ 14000 = MP × (100 − 30) / 100

⇒ 14000 = MP × 70 / 100

⇒ MP = (14000 × 100) / 70

⇒ MP = 200 × 100 = ₹20000

∴ The correct answer is ₹20000.

Additional Information

Relationship between CP and MP

The direct formula is CP / MP = (100 − Discount%) / (100 + Profit%). If there is a loss, use (100 − Loss%).

Markup Percentage

Markup % is the percentage by which the cost price is increased to set the marked price. Markup % = [(MP − CP) / CP] × 100.

Successive Discounts

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

Shortcut Trick

The Marked Price (MP) is calculated as SP ÷ (1 − d), where d is the discount in decimal form.

Compare denominators: 0.9, 0.75, 0.55, and 0.8. A smaller denominator increases the final value.

Option 3: 6000 ÷ 0.55 ≈ 10909.09 (Highest SP with the highest discount %).

Option 4: 6250 ÷ 0.8 = 7812.5 (High SP but much lower discount %).

By inspection, Case 3 has the lowest multiplier (0.55) relative to its high base price.

∴ The correct answer is Rs. 6000 and 45% discount.

Alternate Method

Given: Selling Price (SP) and Discount Percentage for four different cases.

Formula Used: Marked Price (MP) = (SP × 100) ÷ (100 − Discount %)

⇒ Case 1: MP = (4000 × 100) ÷ (100 − 10) = 400000 ÷ 90 ≈ ₹4444.44

⇒ Case 2: MP = (5000 × 100) ÷ (100 − 25) = 500000 ÷ 75 ≈ ₹6666.67

⇒ Case 3: MP = (6000 × 100) ÷ (100 − 45) = 600000 ÷ 55 ≈ ₹10909.09

⇒ Case 4: MP = (6250 × 100) ÷ (100 − 20) = 625000 ÷ 80 = ₹7812.50

Comparing all the results, Case 3 has the highest value.

∴ The correct answer is Rs. 6000 and 45% discount.

Additional Information

Marked Price and Discount

The Marked Price (MP) is also known as the list price or tag price. Discount is always calculated on the MP, and SP = MP − Discount.

Discount Percentage Formula

The percentage of discount is given by: Discount % = (Discount ÷ Marked Price) × 100.

Successive Discounts

When two discounts a% and b% are given, the Effective Discount = (a + b − (a × b) ÷ 100)%.

Shortcut Trick

Net discount % = a + b − (a × b)/100

Net discount = 20 + 30 − (20 × 30)/100 = 50 − 6 = 44%

Selling Price (SP) = 100% − 44% = 56% of Marked Price

56% = Rs. 4480 ⇒ 1% = 80 ⇒ 100% = Rs. 8000

∴ The correct answer is Rs. 8000.

Alternate Method

Given:

First discount (d₁) = 20%

Second discount (d₂) = 30%

Selling Price (SP) = Rs. 4480

Formula Used:

SP = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ SP = MP × (80/100) × (70/100)

⇒ 4480 = MP × (4/5) × (7/10)

⇒ 4480 = MP × (28/50)

⇒ 4480 = MP × (14/25)

⇒ MP = (4480 × 25) ÷ 14

⇒ MP = 320 × 25

⇒ MP = Rs. 8000

∴ The correct answer is Rs. 8000.

Additional Information

Successive Discount Formula

For two successive discounts of x% and y%, the single equivalent discount is (x + y − xy/100)%.

Relationship between MP, CP and Profit/Discount

The ratio of Marked Price to Cost Price is given by: MP/CP = (100 + Profit%)/(100 − Discount%).

Marked Price (MP) Definition

It is the price displayed on the label of an article, also known as the list price. Discount is always calculated on this price.

Shortcut Trick

Ratio of Height to Shadow is constant at the same time.

Boy's Ratio = 1.8 : 1.2 = 3 : 2

For the Building, Shadow = 8 m (which is 2 × 4)

Building Height = 3 × 4 = 12 m

∴ The correct answer is 12 m.

Alternate Method

Given:

Height of the Boy (H₁) = 1.8 m

Shadow of the Boy (S₁) = 1.2 m

Shadow of the Building (S₂) = 8 m

Formula Used:

In Similar Triangles (formed by sun rays), Height₁ / Shadow₁ = Height₂ / Shadow₂

Calculations:

⇒ H₁ / S₁ = H₂ / S₂

⇒ 1.8 / 1.2 = H₂ / 8

⇒ 3 / 2 = H₂ / 8

⇒ H₂ = (3 × 8) / 2

⇒ H₂ = 24 / 2

⇒ H₂ = 12 m

∴ The correct answer is 12 m.

Additional Information

Angle of Elevation

At any given time, the Sun's angle of elevation is the same for all objects in the same vicinity, making the triangles formed by objects and their shadows mathematically similar.

Properties of Similar Triangles

For two triangles to be similar, their corresponding sides must be in the same proportion. Here, Height/Shadow = tan(θ), where θ is the sun's altitude.

Ratio and Proportion

If a : b = c : d, then the product of means equals the product of extremes (ad = bc). This is used to find the unknown height directly.