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Shortcut Trick

Combined efficiency of R and S = (1/32 + 1/40) = (5 + 4)/160 = 9/160 per day.

Work completed in 8 days = 8 × (9/160) = 9/20.

∴ The correct answer is 9/20.

Alternate Method

Given: Time taken by R = 32 days, Time taken by S = 40 days, Combined working time = 8 days.

Formula Used: Total Work = LCM of individual times; Efficiency = Total Work ÷ Time.

⇒ Total work (LCM of 32 and 40) = 160 units.

⇒ Combined efficiency of R and S = 5 + 4 = 9 units/day.

⇒ Work completed by both in 8 days = 9 × 8 = 72 units.

⇒ Part of total work completed = 72/160.

⇒ Simplifying the fraction: 72/160 = 9/20.

∴ The correct answer is 9/20.

Additional Information

Remaining Work Calculation

The portion of work left after some part is finished is calculated as (1 − Work Done). In this case, remaining work = 1 − 9/20 = 11/20.

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work (Efficiency ∝ 1/Time). If A is thrice as efficient as B, A will take 1/3^rd of the time taken by B.

Work and Wages

When multiple people work together, the total wages are always distributed in the ratio of the total work actually done by each individual.

Shortcut Trick

Apply the Chain Rule formula: M₁ × D₁ = M₂ × D₂

Initial data: M₁ = 18 persons, D₁ = 45 days

Updated data: M₂ = 18 + 12 = 30 persons

Calculation: 18 × 45 = 30 × D₂ ⇒ D₂ = (18 × 45) ÷ 30 = 27 days

∴ The correct answer is 27 days.

Alternate Method

Given:

Initial group size (M₁) = 18 persons

Time taken (D₁) = 45 days

New members joined = 12 persons

Formula Used:

Total Work = Number of Persons × Number of Days

Calculations:

⇒ Total Work = 18 × 45 = 810 units

⇒ Total number of persons (M₂) = 18 + 12 = 30 persons

⇒ Time required for 30 persons = Total Work ÷ M₂

⇒ Time = 810 ÷ 30

⇒ Time = 27 days

∴ The correct answer is 27 days.

Additional Information

General Chain Rule

For work involving hours and efficiency, use (M₁ × D₁ × H₁ × E₁) / W₁ = (M₂ × D₂ × H₂ × E₂) / W₂.

Inverse Variation

Number of men is inversely proportional to the time taken. If men increase, days decrease in the same ratio.

Man-Day Concept

One man-day represents the amount of work done by one person in one day. Total work is the constant product of Men and Days.

Shortcut Trick

Average Speed = (s₁t₁ + s₂t₂) ÷ (t₁ + t₂)

Average Speed = (40 × 6 + 15 × 4) ÷ (6 + 4)

Average Speed = (240 + 60) ÷ 10 = 300 ÷ 10

Average Speed = 30 km/hr.

∴ The correct answer is 30 km/hr.

Alternate Method

Given:

Speed₁ = 40 km/hr, Time₁ = 6 hr

Speed₂ = 15 km/hr, Time₂ = 4 hr

Formula Used:

Distance = Speed × Time

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Distance covered in first 6 hr = 40 × 6 = 240 km

⇒ Distance covered in next 4 hr = 15 × 4 = 60 km

⇒ Total distance = 240 + 60 = 300 km

⇒ Total time taken = 6 + 4 = 10 hr

⇒ Average speed = 300 ÷ 10 = 30 km/hr

∴ The correct answer is 30 km/hr.

Additional Information

Average Speed (Equal Distances)

When a person travels two equal distances at speeds x and y, the average speed is 2xy ÷ (x + y).

Average Speed (Equal Time)

When a person travels for two equal time intervals at speeds x and y, the average speed is (x + y) ÷ 2.

Unit Conversion

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

Shortcut Trick

The extra distance covered is the length of the platform.

Extra time taken = 40 sec − 20 sec = 20 sec

Speed = (Length of platform) ÷ (Extra time taken)

Speed = 400 ÷ 20 = 20 m/s

∴ The correct answer is 20 m/s.

Alternate Method

Given:

Time to cross pole (t₁) = 20 sec

Time to cross platform (t₂) = 40 sec

Length of platform (L_p) = 400 m

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ Let the length of the train be L.

⇒ Speed (v) when crossing a pole = L ÷ 20

⇒ Speed (v) when crossing a platform = (L + 400) ÷ 40

⇒ Since speed is constant: L ÷ 20 = (L + 400) ÷ 40

⇒ 40L = 20(L + 400)

⇒ 2L = L + 400

⇒ L = 400 m

⇒ Speed (v) = 400 ÷ 20 = 20 m/s

∴ The correct answer is 20 m/s.

Additional Information

Crossing Different Objects

When a train crosses a stationary point object (pole, man, tree), distance = length of train. When it crosses an object with length (platform, bridge, another train), distance = length of train + length of object.

Unit Conversion

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

Relative Speed

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

Shortcut Trick

Apply the Componendo and Dividendo rule: If (a + b) / (a − b) = c / d, then a / b = (c + d) / (c − d).

Here, a = 4x, b = 3y, c = 5, and d = 2.

4x / 3y = (5 + 2) / (5 − 2) = 7 / 3

x / y = (7 / 3) × (3 / 4) = 7 / 4

∴ The correct answer is 7 : 4.

Alternate Method

Given: The ratio of linear combinations (4x + 3y) : (4x − 3y) = 5 : 2.

Formula Used: Cross-multiplication method for proportions.

⇒ (4x + 3y) / (4x − 3y) = 5 / 2

⇒ 2 × (4x + 3y) = 5 × (4x − 3y)

⇒ 8x + 6y = 20x − 15y

⇒ 6y + 15y = 20x − 8x

⇒ 21y = 12x

⇒ x / y = 21 / 12

⇒ x / y = 7 / 4

∴ The correct answer is 7 : 4.

Additional Information

Componendo and Dividendo Rule

If a/b = c/d, then (a + b) / (a − b) = (c + d) / (c − d). This is used to simplify equations involving ratios of sums and differences.

Invertendo Property

If a : b = c : d, then b : a = d : c. This allows for flipping ratios to solve for the reciprocal of a variable.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b). It is the value x such that a : x = x : b.

Shortcut Trick

Total ratio parts = 1 + 3 + 5 + 4 = 13 units.

Given 13 units = 52 ⇒ 1 unit = 52 ÷ 13 = 4.

Difference between largest (5) and smallest (1) parts = 5 − 1 = 4 units.

Required difference = 4 units × 4 = 16.

∴ The correct answer is 16.

Alternate Method

Given: Ratio = 1 : 3 : 5 : 4 and Sum = 52.

Formula Used: Sum of terms = Total value

⇒ Let the four numbers be 1x, 3x, 5x, and 4x.

⇒ 1x + 3x + 5x + 4x = 52

⇒ 13x = 52

⇒ x = 52 ÷ 13 = 4

⇒ Smallest number (1x) = 1 × 4 = 4

⇒ Largest number (5x) = 5 × 4 = 20

⇒ Difference = 20 − 4 = 16

∴ The correct answer is 16.

Additional Information

Ratio Distribution Rule

To find the value of any part, multiply the Total Value by (Part Ratio ÷ Sum of all Ratio Parts).

Proportionality Constant

The variable x represents the Common Multiplier that scales the ratio parts to their actual values.

Difference between Parts

The difference between any two numbers in a ratio can be calculated directly by finding the difference in their ratio units first.

Shortcut Trick

Let Markup percentage = +20% and Discount percentage = −10% (since SP is 90% of MP).

Net change formula: a + b + (ab/100)

Net Profit % = 20 − 10 − (20 × 10)/100

Net Profit % = 10 − 2 = 8%

∴ The correct answer is 8 percent.

Alternate Method

Given:

Marked Price (MP) = 120% of Cost Price (CP)

Selling Price (SP) = 90% of Marked Price (MP)

Calculations:

⇒ Let Cost Price (CP) = 100

⇒ Marked Price (MP) = 120% of 100 = 120

⇒ Selling Price (SP) = 90% of MP

⇒ SP = (90/100) × 120 = 108

⇒ Profit = SP − CP = 108 − 100 = 8

⇒ Profit Percentage = (Profit/CP) × 100

⇒ Profit % = (8/100) × 100 = 8%

∴ The correct answer is 8 percent.

Additional Information

Markup Percentage

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

Discount Percentage

It is the reduction offered on the marked price. Discount % = (MP − SP)/MP × 100.

Relationship between CP and MP

CP/MP = (100 − Discount%)/(100 + Profit%). This direct ratio is useful for finding any one unknown variable.

Shortcut Trick

To find the highest Selling Price (SP), compare the product of Marked Price (MP) and the Selling Factor (1 − Discount%).

1. 9000 × 0.60 = 5400 | 2) 8000 × 0.50 = 4000

2. 5000 × 0.75 = 3750 | 4) 4000 × 0.90 = 3600

The maximum value is 5400.

∴ The correct answer is ₹5400.

Alternate Method

Given: Four cases with different Marked Prices (MP) and Discount percentages.

Formula Used: Selling Price (SP) = MP − (Discount% × MP)

⇒ Case 1: SP = 9000 − (40/100 × 9000) = 9000 − 3600 = ₹5400

⇒ Case 2: SP = 8000 − (50/100 × 8000) = 8000 − 4000 = ₹4000

⇒ Case 3: SP = 5000 − (25/100 × 5000) = 5000 − 1250 = ₹3750

⇒ Case 4: SP = 4000 − (10/100 × 4000) = 4000 − 400 = ₹3600

⇒ Comparing the results: 5400 > 4000 > 3750 > 3600.

∴ The correct answer is ₹5400.

Additional Information

Marked Price (List Price)

It is the price displayed on the product label. Discount is always calculated on the Marked Price.

Selling Price Calculation

SP = MP × (100 − D%)/100. If multiple discounts d₁ and d₂ are given, SP = MP × (100 − d₁)/100 × (100 − d₂)/100.

Successive Discount Shortcut

A single equivalent discount for two successive discounts a% and b% is (a + b − ab/100)%.

Shortcut Trick

Check Option (2): If Discount is 20%, then Selling Price (SP) = 80% of Marked Price (MP).

Applying values: 80% × ₹5000 = (80/100) × 5000 = ₹4000.

Since this matches the given SP of ₹4000, Option (2) is correct.

∴ The correct answer is Selling price = ₹4000 and Discount = 20%.

Alternate Method

Given:

Target Marked Price (MP) = ₹5000

Formula Used:

MP = (SP × 100) / (100 − Discount%)

Step-by-step Calculation for Option (2):

⇒ Selling Price (SP) = ₹4000

⇒ Discount Percentage = 20%

⇒ Marked Price (MP) = (4000 × 100) / (100 − 20)

⇒ MP = 400000 / 80

⇒ MP = ₹5000

This matches the requirement stated in the question.

∴ The correct answer is Selling price = ₹4000 and Discount = 20%.

Additional Information

Relationship between MP and SP

The Selling Price is always calculated by subtracting the Discount from the Marked Price: SP = MP − Discount.

Discount Percentage Formula

Discount% = (Discount / MP) × 100. Always remember that Discount is calculated on the Marked Price, not on the Cost Price.

Net Discount for Successive Rates

If two successive discounts 'a%' and 'b%' are given, the Effective Discount = (a + b − ab/100)%.

Shortcut Trick

Observe the relationship between the second and fourth terms in the proportion.

The fourth term (20) × 2 = The second term (40).

Applying the same factor to the third term: 12 × 2 = 24.

Thus, x + 3 = 24

x = 24 − 3 = 21

∴ The correct answer is 21.

Alternate Method

Given: (x + 3) : 40 :: 12 : 20

Formula Used: In a proportion a : b :: c : d, the product of extremes equals the product of means (a × d = b × c).

⇒ (x + 3) × 20 = 40 × 12

⇒ 20x + 60 = 480

⇒ 20x = 480 − 60

⇒ 20x = 420

⇒ x = 420 ÷ 20

⇒ x = 21

∴ The correct answer is 21.

Additional Information

Extremes and Means

In a proportion a : b = c : d, the terms 'a' and 'd' are called extremes, while 'b' and 'c' are called means. The fundamental property is a × d = b × c.

Mean Proportional

If a, b, c are in continued proportion, then a : b :: b : c, which implies b² = a × c. Here, 'b' is the mean proportional.

Fourth Proportional

If three numbers a, b, and c are given, the fourth proportional 'd' is calculated as d = (b × c) ÷ a.