Quantitative Aptitude PYQs

Shortcut Trick

Identify parts of the expression using BODMAS: Solve brackets and multiplication first.

Value of bracket (24 + 17) = 41

Value of multiplication 35 × 2 = 70

Combine results: 93 − 47 + 69 − 41 + 70 = 144

∴ The correct answer is 144.

Alternate Method

Given: 93 − 47 + 69 − (24 + 17) + 35 × 2

Apply BODMAS (Bracket, Of, Division, Multiplication, Addition, Subtraction):

Step 1: Solve the parenthesis (B)

⇒ 93 − 47 + 69 − 41 + 35 × 2

Step 2: Perform the multiplication (M)

⇒ 93 − 47 + 69 − 41 + 70

Step 3: Perform addition and subtraction from left to right (A/S)

⇒ 46 + 69 − 41 + 70

⇒ 115 − 41 + 70

⇒ 74 + 70

⇒ 144

∴ The correct answer is 144.

Additional Information

BODMAS Hierarchy

Mathematical operations must follow the priority: Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction.

VBODMAS Rule

In complex simplifications, V stands for Vinculum (bar bracket), which is solved even before standard brackets.

Sign Rules

When multiplying signs: (−) × (−) = (+), (+) × (−) = (−), and (−) × (+) = (−).

Shortcut Trick

Solve multiplication and division terms first: (27 × 1) = 27 and 39 ÷ 3 = 13.

Rewrite the expression: 92 + 27 − 41 + 13 + 21 − 17.

Sum all positive numbers: 92 + 27 + 13 + 21 = 153.

Sum all negative numbers: 41 + 17 = 58.

Final value: 153 − 58 = 95.

∴ The correct answer is 95.

Alternate Method

Given: 92 + (27 × 1) − 41 + 39 ÷ 3 + 21 − 17

Method: Apply the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

⇒ 92 + 27 − 41 + 39 ÷ 3 + 21 − 17 (Solving Bracket: 27 × 1)

⇒ 92 + 27 − 41 + 13 + 21 − 17 (Solving Division: 39 ÷ 3)

⇒ 119 − 41 + 13 + 21 − 17 (Solving Addition from left: 92 + 27)

⇒ 78 + 13 + 21 − 17 (Solving Subtraction: 119 − 41)

⇒ 91 + 21 − 17 (Solving Addition: 78 + 13)

⇒ 112 − 17 (Solving Addition: 91 + 21)

⇒ 95 (Final Subtraction)

∴ The correct answer is 95.

Additional Information

BODMAS Rule

It stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction, defining the correct sequence for solving arithmetic operations.

Left-to-Right Associativity

When operations like addition and subtraction (or division and multiplication) appear together, they are generally performed in the order they appear from left to right.

Shortcut Trick

For any set of fractions:

HCF = HCF(Numerators) ÷ LCM(Denominators) = HCF(3, 5, 7) ÷ LCM(8, 12, 18) = 1/72

LCM = LCM(Numerators) ÷ HCF(Denominators) = LCM(3, 5, 7) ÷ HCF(8, 12, 18) = 105/2

∴ The correct answer is HCF = 1/72, LCM = 105/2.

Alternate Method

Given Fractions: 3/8, 5/12, and 7/18

Formula Used:

1. HCF of Fractions = HCF of Numerators ÷ LCM of Denominators
2. LCM of Fractions = LCM of Numerators ÷ HCF of Denominators

Step-by-step Calculations:

⇒ Numerators = 3, 5, 7

⇒ Denominators = 8, 12, 18

⇒ HCF(3, 5, 7) = 1 (Since they are co-prime numbers)

⇒ LCM(3, 5, 7) = 3 × 5 × 7 = 105

⇒ LCM(8, 12, 18):

⇒ 8 = 2³

⇒ 12 = 2² × 3

⇒ 18 = 2 × 3²

⇒ LCM = 2³ × 3² = 8 × 9 = 72

⇒ HCF(8, 12, 18) = 2 (Highest common factor among 8, 12, and 18)

⇒ HCF of Fractions = 1/72

⇒ LCM of Fractions = 105/2

∴ The correct answer is HCF = 1/72, LCM = 105/2.

Additional Information

HCF of Fractions

To find the Highest Common Factor of fractions, calculate HCF(Numerators) ÷ LCM(Denominators). Ensure fractions are in simplest form.

LCM of Fractions

To find the Least Common Multiple of fractions, calculate LCM(Numerators) ÷ HCF(Denominators).

Co-prime Numbers

If a set of numbers are co-prime (like 3, 5, 7), their HCF is always 1 and their LCM is their product.

Shortcut Trick

Calculate the Numerator: 15.625 − 3.375 − 0.25 = 12.

Calculate the Denominator: 6.25 + 2.25 − 7.5 = 1.

Result = 12 ÷ 1 = 12.

∴ The correct answer is 12.

Alternate Method

Given: ((2.5)³ − (1.5)³ − 0.25) ÷ ((2.5)² + (1.5)² − 7.5)

Formula Used: Simplification using powers and basic Arithmetic operations.

⇒ Step 1: Calculate the cubes for the numerator: (2.5)³ = 15.625 and (1.5)³ = 3.375

⇒ Step 2: Simplify Numerator: 15.625 − 3.375 − 0.25 = 12.25 − 0.25 = 12

⇒ Step 3: Calculate the squares for the denominator: (2.5)² = 6.25 and (1.5)² = 2.25

⇒ Step 4: Simplify Denominator: 6.25 + 2.25 − 7.5 = 8.5 − 7.5 = 1

⇒ Final Step: Perform division: 12 ÷ 1 = 12

∴ The correct answer is 12.

Additional Information

BODMAS Rule

Always prioritize operations in the order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Difference of Cubes

The identity a³ − b³ = (a − b)(a² + ab + b²) can be used to simplify decimal expressions involving cubes quickly.

Decimal Exponents

For competitive exams, memorizing squares and cubes of common decimal values like 1.5, 2.5, and 3.5 saves significant calculation time.

Shortcut Trick

Solve for square roots: √289 = 17, √225 = 15, and √144 = 12.

Calculate x: x = (17 × 15) + 12 = 255 + 12 = 267.

Calculate the required value: 2x = 2 × 267 = 534.

∴ The correct answer is 534.

Alternate Method

Given:

Expression: x = √289 × √225 + √144

Method Used:

Apply BODMAS rule and basic Square Roots calculation.

Calculations:

⇒ x = 17 × 15 + 12

⇒ x = 255 + 12

⇒ x = 267

⇒ 2x = 2 × 267

⇒ 2x = 534

∴ The correct answer is 534.

Additional Information

BODMAS Rule

The acronym stands for Brackets, Orders (Powers and Roots), Division, Multiplication, Addition, and Subtraction. It defines the order of operations to solve complex arithmetic expressions.

Perfect Squares

Numbers like 289, 225, and 144 are perfect squares. Memorizing squares up to 30 (e.g., 17² = 289, 15² = 225, 12² = 144) is essential for fast calculations in competitive exams.

Simplification Shortcut

When multiplying numbers ending in 5 (like 17 × 15), use the property (16 + 1)(16 − 1) = 16² − 1 = 256 − 1 = 255 for quicker results.

Shortcut Trick

Identify the highest perfect square factors for all terms: 98 = 49 × 2, 147 = 49 × 3, 18 = 9 × 2, and 27 = 9 × 3.

Rewrite the expression as (7√2 − 7√3) ÷ (3√2 − 3√3).

Factor out the coefficients 7 and 3: [7(√2 − √3)] ÷ [3(√2 − √3)].

Cancel the common factor (√2 − √3) from both the numerator and the denominator.

The remaining fraction is 7/3.

∴ The correct answer is 7/3.

Alternate Method

Given: (√98 − √147) ÷ (√18 − √27)

Simplify each radical term separately:

⇒ √98 = √(49 × 2) = 7√2

⇒ √147 = √(49 × 3) = 7√3

⇒ √18 = √(9 × 2) = 3√2

⇒ √27 = √(9 × 3) = 3√3

Substitute these values back into the expression:

⇒ (7√2 − 7√3) ÷ (3√2 − 3√3)

Factor out 7 from the numerator and 3 from the denominator:

⇒ 7(√2 − √3) ÷ 3(√2 − √3)

Since (√2 − √3) ≠ 0, cancel it out:

⇒ 7/3

∴ The correct answer is 7/3.

Additional Information

Properties of Surds

For any positive real numbers a and b, √(a × b) = √a × √b and √(a ÷ b) = √a ÷ √b.

Simplifying Radicals

To simplify a radical, factor the radicand into a perfect square and another factor, then take the square root of the perfect square factor.

Rationalization

If the denominator contains a surd like (√a − √b), it can be rationalized by multiplying both numerator and denominator by its conjugate (√a + √b).

Shortcut Trick

Ratio of Length : Breadth = 5 : 4 (since Breadth = 4/5 of Length).

Product of ratio units = 5 × 4 = 20 units².

20 units² = 320 m² ⇒ 1 unit² = 16 m² ⇒ 1 unit = 4 m.

Difference between Length and Breadth = (5 − 4) = 1 unit.

Difference = 1 × 4 = 4 m.

∴ The correct answer is 4 m.

Alternate Method

Given:

Breadth (B) = 4/5 × Length (L)

Area of the floor = 320 m²

Let the Length be 5x and the Breadth be 4x.

⇒ Area = Length × Breadth

⇒ 320 = 5x × 4x

⇒ 320 = 20x²

⇒ x² = 320 ÷ 20

⇒ x² = 16

⇒ x = √16 = 4 m

⇒ Length = 5 × 4 = 20 m

⇒ Breadth = 4 × 4 = 16 m

⇒ Difference = 20 m − 16 m = 4 m.

∴ The correct answer is 4 m.

Additional Information

Area of a Rectangle

The total space occupied by the rectangle is calculated as Area = Length × Breadth.

Perimeter of a Rectangle

The total boundary length is given by Perimeter = 2 × (Length + Breadth).

Diagonal of a Rectangle

The distance between opposite corners is found using Diagonal = √(Length² + Breadth²).

Shortcut Trick

Recognize the algebraic identity: (a − b)² + 4ab = (a + b)².

Here, a = 325 and b = 275.

Sum of values: 325 + 275 = 600.

Simplified result: 600² = 360000.

∴ The correct answer is 360000.

Alternate Method

Given values: a = 325, b = 275.

Method Used: Identity expansion and simplification.

⇒ (325 − 275)² + 4 × 325 × 275

⇒ Let a = 325 and b = 275.

⇒ The expression is in the form: (a − b)² + 4ab.

⇒ (a² + b² − 2ab) + 4ab

⇒ a² + b² + 2ab = (a + b)².

⇒ (325 + 275)²

⇒ 600² = 360000.

∴ The correct answer is 360000.

Additional Information

Basic Square Identities

Standard expansions for binomial squares: (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab.

Difference and Sum Square Relation

Conversion formula between the sum and difference of squares: (a + b)² − (a − b)² = 4ab.

Sum of Squared Terms

Addition identity for combined squares: (a + b)² + (a − b)² = 2(a² + b²).

Shortcut Trick

Sum of length and breadth = 140 ÷ 2 = 70 m

Difference of dimensions = 10 m

Larger dimension (L) = (70 + 10) ÷ 2 = 40 m

Smaller dimension (B) = (70 − 10) ÷ 2 = 30 m

Area = 40 × 30 = 1200 m²

∴ The correct answer is 1200 m².

Alternate Method

Given:

Difference (L − B) = 10 m

Perimeter = 140 m

Formula Used:

Perimeter of rectangle = 2 × (L + B)

Area of rectangle = L × B

Calculations:

⇒ 2 × (L + B) = 140

⇒ L + B = 70 ----(1)

⇒ L − B = 10 ----(2)

Adding equations (1) and (2):

⇒ 2L = 80

⇒ L = 40 m

Substituting L = 40 in (1):

⇒ 40 + B = 70

⇒ B = 30 m

Now, calculate the area:

⇒ Area = 40 × 30

⇒ Area = 1200 m²

∴ The correct answer is 1200 m².

Additional Information

Diagonal of a Rectangle

The distance between opposite vertices is given by the Pythagorean theorem: d = √(L² + B²).

Relationship between Area and Perimeter

For a given perimeter, the area of a rectangle is maximum when it is a square (L = B).

Square Properties

A square is a special rectangle where L = B; its Area = side² and Perimeter = 4 × side.

Shortcut Trick

Weight of B = [Sum of group 1 + Sum of group 2] − [Total sum]

Sum of A and B = 32 × 2 = 64 kg

Sum of B and C = 38 × 2 = 76 kg

Sum of A, B and C = 30 × 3 = 90 kg

Weight of B = (64 + 76) − 90 = 140 − 90 = 50 kg

∴ The correct answer is 50 kg.

Alternate Method

Given:

Average weight of (A + B + C) = 30 kg

Average weight of (A + B) = 32 kg

Average weight of (B + C) = 38 kg

Formula Used:

Total Weight = Average Weight × Number of Persons

Calculations:

⇒ Total weight of A + B + C = 30 × 3 = 90 kg --- (i)

⇒ Total weight of A + B = 32 × 2 = 64 kg --- (ii)

⇒ Total weight of B + C = 38 × 2 = 76 kg --- (iii)

⇒ Adding (ii) and (iii): (A + B) + (B + C) = 64 + 76

⇒ A + 2B + C = 140 kg --- (iv)

⇒ Subtracting (i) from (iv): (A + 2B + C) − (A + B + C) = 140 − 90

⇒ Weight of B = 50 kg

∴ The correct answer is 50 kg.

Additional Information

Definition of Average

The average (Arithmetic Mean) is the sum of all values divided by the total number of values. Formula: Average = Sum of observations ÷ Number of observations.

Net Deviation Method

The sum of deviations of all values from the average is always zero. This is useful for finding missing values in a data set when the average is known.

Combined Average

If two groups have n₁ and n₂ elements with averages A₁ and A₂, the combined average is [n₁A₁ + n₂A₂] ÷ [n₁ + n₂].

Shortcut Trick

Total numbers increased by 50 (from 100 to 150) and the average increased by 20.6 (74.1 − 53.5).

Sum of new numbers = (New Average × Quantity of new numbers) + (Old Quantity × Increase in Average)

Sum = (74.1 × 50) + (100 × 20.6)

Sum = 3705 + 2060 = 5765

∴ The correct answer is 5765.

Alternate Method

Given:

Initial count (n₁) = 100, Initial average (A₁) = 53.5

New added count (n₂) = 50, Total count (n_total) = 150

Final average (A_total) = 74.1

Formula Used:

Sum = Average × Number of terms

Sum of new numbers = (Total Sum of 150 numbers) − (Initial Sum of 100 numbers)

Calculations:

⇒ Initial Sum = 100 × 53.5 = 5350

⇒ Total Sum after adding 50 numbers = 150 × 74.1

⇒ Total Sum = 11115

⇒ Sum of 50 newly added numbers = 11115 − 5350

⇒ Sum = 5765

∴ The correct answer is 5765.

Additional Information

Weighted Average Formula

When two groups are combined, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Effect of Adding/Removing a Term

If a new value X is added to a group of n terms with average A, the new average A′ = (nA + X) ÷ (n + 1).

Sum of First n Natural Numbers

The sum is given by [n(n + 1)] ÷ 2, and their average is (n + 1) ÷ 2.

Shortcut Trick

To find the next simultaneous change, calculate the LCM of the given intervals: 30, 45, and 60 seconds.

LCM(30, 45, 60) = 180 seconds.

Convert 180 seconds to minutes: 180 ÷ 60 = 3 minutes.

Add 3 minutes to the initial time 7:00 a.m. → 7:03 a.m.

∴ The correct answer is 7:03 a.m.

Alternate Method

Given:

Interval of Light 1 = 30 seconds

Interval of Light 2 = 45 seconds

Interval of Light 3 = 60 seconds

Starting Time = 7:00 a.m.

Formula Used:

Next simultaneous event = Initial time + LCM(Intervals)

Calculations:

⇒ Prime factors of 30 = 2 × 3 × 5

⇒ Prime factors of 45 = 3² × 5

⇒ Prime factors of 60 = 2² × 3 × 5

⇒ LCM(30, 45, 60) = 2² × 3² × 5

⇒ LCM = 4 × 9 × 5 = 180 seconds

⇒ Time in minutes = 180 ÷ 60 = 3 minutes

⇒ Next simultaneous change = 7:00 a.m. + 3 minutes = 7:03 a.m.

∴ The correct answer is 7:03 a.m.

Additional Information

Concept of LCM

The Least Common Multiple (LCM) is used to find the next time periodic events (like bells, traffic lights, or racing laps) will occur simultaneously.

Time Conversion

Always ensure units are consistent. Use 1 minute = 60 seconds to convert large second counts into readable time formats.

Relationship between HCF and LCM

For any two numbers a and b: HCF(a, b) × LCM(a, b) = a × b.

Shortcut Trick

Profit 25% = 1/4 (Fractional value)

Ratio of Cost Price (CP) : Selling Price (SP) = 4 : (4 + 1) = 4 : 5

Given 5 units (SP) = ₹5625 ⇒ 1 unit = 5625 ÷ 5 = ₹1125

CP = 4 units = 4 × 1125 = ₹4500

∴ The correct answer is ₹4500.

Alternate Method

Given:

Selling Price (SP) = ₹5625

Profit percentage = 25%

Formula Used:

Cost Price (CP) = [Selling Price × 100] ÷ [100 + Profit %]

Calculations:

⇒ CP = [5625 × 100] ÷ [100 + 25]

⇒ CP = [5625 × 100] ÷ 125

⇒ CP = 5625 ÷ 1.25

⇒ CP = ₹4500

∴ The correct answer is ₹4500.

Additional Information

Cost Price with Loss

When there is a loss, CP = [SP × 100] ÷ [100 − Loss %].

Profit/Loss Basis

Profit or Loss is always calculated on the Cost Price unless otherwise specified in the question.

Marked Price (MP)

MP = CP + Markup. Selling Price is also calculated as MP − Discount.

Shortcut Trick

Convert percentage to ratio: 20% = 1/5

If B is 5 units, then A is (5 − 1) = 4 units.

Given A = 480 ⇒ 4 units = 480

1 unit = 480 ÷ 4 = 120

B = 5 units = 5 × 120 = 600

∴ The correct answer is 600.

Alternate Method

Given:

Value of A = 480

A is 20 percent less than B.

Formula Used:

Value = Base × (100 − % Decrease) ÷ 100

Calculations:

⇒ A = B × (100 − 20) ÷ 100

⇒ 480 = B × (80 ÷ 100)

⇒ 480 = B × 0.8

⇒ B = 480 ÷ 0.8

⇒ B = 4800 ÷ 8

⇒ B = 600

∴ The correct answer is 600.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, memorize common fractions: 20% = 1/5, 25% = 1/4, 33.33% = 1/3, and 50% = 1/2.

Percentage Increase/Decrease

New Value = Original Value × (1 ± r/100), where 'r' is the rate of percentage change.

Inverse Percentage Relation

If A is x% less than B, then B is [x / (100 − x)] × 100% more than A. In this case, B is 25% more than A.

Shortcut Trick

Calculate the first value: 30% of 500 = 150.

Express 50% more than A as a multiplier: 100% + 50% = 150% or 1.5.

Set up the equation: 1.5 × A = 150 ⇒ A = 150 ÷ 1.5 = 100.

∴ The correct answer is 100.

Alternate Method

Given:

⇒ Target value = 30% of 500

⇒ Condition: Target value is 50% more than A.

Formula Used:

⇒ Value = Base × (1 + Percentage Increase/100)

Calculations:

⇒ First, find 30% of 500:

⇒ (30 ÷ 100) × 500 = 150

⇒ Let the unknown value be A. As per the question:

⇒ 150 = A + 50% of A

⇒ 150 = A + 0.5A

⇒ 150 = 1.5A

⇒ A = 150 ÷ 1.5

⇒ A = 100

∴ The correct answer is 100.

Additional Information

Percentage Multiplier

To increase a value by x%, multiply the base by (1 + x/100). To decrease by x%, multiply by (1 − x/100).

Base Change Rule

If X is r% more than Y, then Y is less than X by [r ÷ (100 + r)] × 100 %.

Fractional Equivalents

50% is equivalent to the fraction 1/2. Thus, 50% more than a value is the same as 3/2 times that value.

Shortcut Trick

Profit is 30% of Selling Price (SP), which equals 3/10.

Let SP = 10 units, then Profit = 3 units.

Cost Price (CP) = SP − Profit = 10 − 3 = 7 units.

Required Ratio of Profit to CP = 3 : 7.

∴ The correct answer is 3 : 7.

Alternate Method

Given:

Profit = 30% of the Selling Price (SP).

Formula Used:

Cost Price (CP) = Selling Price (SP) − Profit

⇒ Let the Selling Price (SP) of the article be ₹100.

⇒ Profit = 30% of ₹100 = (30/100) × 100 = ₹30.

⇒ Cost Price (CP) = SP − Profit

⇒ CP = 100 − 30 = ₹70.

⇒ Ratio of Profit to Cost Price = 30 : 70

⇒ Ratio = 3 : 7.

∴ The correct answer is 3 : 7.

Additional Information

Profit Percentage on Cost Price

Usually, profit is calculated on CP. Profit % = (Profit ÷ CP) × 100.

Selling Price Formula

SP = CP × (100 + Profit%) ÷ 100, where Profit% is based on the Cost Price.

Relationship between SP, CP, and Profit

When SP > CP, there is a profit. Profit = SP − CP.

Shortcut Trick

Time taken to empty one full tank = 48 min

Time taken to empty 1/6 part of the tank = 48 × (1/6) = 8 min

∴ The correct answer is 8 min.

Alternate Method

Given:

Time taken by the pipe to empty the full tank = 48 min

Part of the tank to be emptied = 1/6

Formula Used:

Efficiency = Total Work ÷ Time Taken

Time Taken for partial work = (Part of work) × (Total time for full work)

Calculations:

⇒ Let the total capacity of the tank be 48 units.

⇒ Efficiency of the pipe = 1 unit per minute.

⇒ Work to be done (1/6 of capacity) = 48 × (1/6) = 8 units.

⇒ Time required = 8 units ÷ 1 unit/min = 8 min.

∴ The correct answer is 8 min.

Additional Information

Efficiency of a Pipe

The rate at which a pipe fills or empties a tank is its efficiency, calculated as 1/T where T is the time taken to complete the task alone.

Fractional Work Rule

If a task takes H hours to complete, then 1/n part of the same task will take H ÷ n hours to complete.

Outlet vs Inlet Pipes

Inlet pipes perform positive work (filling), while outlet/leak pipes perform negative work (emptying) in problems involving both.

Shortcut Trick

Total work = LCM of 8 and 16 = 16 units.

Efficiency of (P + Q) = 16 ÷ 8 = 2 units/day.

Efficiency of R = 16 ÷ 16 = 1 unit/day.

Combined efficiency of (P + Q + R) = 2 + 1 = 3 units/day.

Time for 1/2 part of work = (16 ÷ 2) ÷ 3 = 8/3 days.

∴ The correct answer is 8/3 days.

Alternate Method

Given:

Time taken by (P + Q) = 8 days

Time taken by R = 16 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Total work assumed (LCM of 8, 16) = 16 units

⇒ Efficiency of (P + Q + R) = Efficiency of (P + Q) + Efficiency of R

⇒ Combined Efficiency = 2 + 1 = 3 units/day

⇒ 1/2 part of total work = 16 ÷ 2 = 8 units

⇒ Time taken by (P + Q + R) to complete 8 units = 8 ÷ 3 days

∴ The correct answer is 8/3 days.

Additional Information

Total Work Concept

Total work is always constant for the same task and is usually taken as the LCM of the days to simplify calculations.

Efficiency Relationship

Efficiency is inversely proportional to Time taken when work is constant (E₁/E₂ = T₂/T₁).

Work and Wages

When people work together for the same duration, their wages are distributed in the direct ratio of their efficiencies.

Combined Work Formula

If A takes 'a' days and B takes 'b' days, together they take (a × b) ÷ (a + b) days to finish the total work.

Shortcut Trick

When two objects move in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 40 m/s

Total distance to cover = Sum of lengths of Train A and Train B = 800 m.

Time = Total Distance ÷ Relative Speed = 800 ÷ 40 = 20 seconds.

∴ The correct answer is 20 sec.

Alternate Method

Given:

Sum of lengths (L₁ + L₂) = 800 m

Sum of speeds (S_A + S_B) = 40 m/s

Formula Used:

Time (t) = Total Distance ÷ Relative Speed

Calculations:

⇒ Total Distance = 800 m

⇒ Relative Speed = 40 m/s

⇒ Crossing Time (t) = 800 ÷ 40

⇒ t = 20 sec

∴ The correct answer is 20 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, the relative speed is the sum (S₁ + S₂). If they move in the same direction, the relative speed is the difference (|S₁ − S₂|).

Crossing Criteria

When a train crosses a pole or a man, distance equals the length of the train. When it crosses a bridge or platform, distance equals (Length of train + Length of platform).

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

When speed is constant, the Distance covered is directly proportional to the Time taken.

Time to cover Train Length (L) = 4 sec

Time to cover Total Length (Train + Platform) = 12 sec

Required Ratio = 4 : 12

⇒ Ratio = 1 : 3

∴ The correct answer is 1 : 3.

Alternate Method

Given:

Time taken to cross a standing man = 4 sec

Time taken to cross a platform of length 'y' = 12 sec

Let the Length of the Train be L and its Speed be S.

Case 1: Crossing a standing man

⇒ Speed (S) = Distance ÷ Time = L ÷ 4

Case 2: Crossing a platform of length y

⇒ Speed (S) = (L + y) ÷ 12

Equating the speeds:

⇒ L ÷ 4 = (L + y) ÷ 12

⇒ 3 × L = L + y

⇒ y = 2 × L

Total length of Train and Platform = L + y = L + 2L = 3L

Required Ratio = Length of Train : Total Length

⇒ Ratio = L : 3L

⇒ Ratio = 1 : 3

∴ The correct answer is 1 : 3.

Additional Information

Crossing Objects

When a train crosses a pole or a standing man, the distance covered equals the Length of the Train. When it crosses a platform or bridge, the distance equals Length of Train + Length of Platform.

Relative Speed in Same Direction

If two trains of lengths L₁ and L₂ move in the same direction at speeds S₁ and S₂, the time to cross is (L₁ + L₂) ÷ (S₁ − S₂).

Speed Conversion Factor

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