Quantitative Aptitude PYQs

Shortcut Trick

Apply BODMAS: Solve the bracket first, then multiplication, then addition/subtraction.

⇒ (25 + 19) = 44

⇒ 44 × 2 = 88

⇒ 71 + 56 − 88 + 92 − 38 = (71 + 56 + 92) − (88 + 38) = 219 − 126 = 93

∴ The correct answer is 93.

Alternate Method

Given Expression: 71 + 56 − (25 + 19) × 2 + 92 − 38

Rule Used: BODMAS (Bracket, Order, Division, Multiplication, Addition, Subtraction)

⇒ 71 + 56 − (44) × 2 + 92 − 38 (Solving Bracket)

⇒ 71 + 56 − 88 + 92 − 38 (Performing Multiplication)

⇒ 127 − 88 + 92 − 38 (Performing Addition from left to right)

⇒ 39 + 92 − 38

⇒ 131 − 38

⇒ 93

∴ The correct answer is 93.

Additional Information

BODMAS Rule Hierarchy

Operations must follow the order: Brackets ((), {}, []), Orders (powers, square roots), Division, Multiplication, Addition, and Subtraction.

Sign Convention for Grouping

To avoid mistakes in large expressions, group all positive terms and negative terms separately before the final subtraction.

Multiplication Priority

Always ensure that a multiplication operation (like 44 × 2) is completed before any addition or subtraction that surrounds it.

Shortcut Trick

Apply the BODMAS rule to solve the expression efficiently.

Solve Of and Brackets first: (27 of 2) = 54 and 18 of 2 = 36.

Next, perform Division: 44 ÷ 4 = 11.

Simplify the expression by canceling out +36 and −36: 89 + 33 − 54 + 11 + 0.

Final calculation: 122 − 54 + 11 = 68 + 11 = 79.

∴ The correct answer is 79.

Alternate Method

Given Expression: 89 + 33 − (27 of 2) + 44 ÷ 4 + 18 of 2 − 36

Formula Used: BODMAS (Bracket, Of, Division, Multiplication, Addition, Subtraction)

⇒ 89 + 33 − 54 + 44 ÷ 4 + 36 − 36 (Solving Bracket and Of operations)

⇒ 89 + 33 − 54 + 11 + 36 − 36 (Performing Division: 44 ÷ 4 = 11)

⇒ 122 − 54 + 11 + 0 (Solving Addition: 89 + 33 = 122 and 36 − 36 = 0)

⇒ 68 + 11 (Solving Subtraction: 122 − 54 = 68)

⇒ 79

∴ The correct answer is 79.

Additional Information

BODMAS Rule

It stands for Bracket (), Of (multiplication), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

The 'Of' Operator

In simplification, Of indicates multiplication and must be calculated before division and regular multiplication.

Left-to-Right Priority

When operations of the same priority (like addition and subtraction) remain, solve them from left to right for accuracy.

Shortcut Trick

Identify the pair in the options that satisfies the sum = 156.

Option 1: 13 + 39 = 52 | Option 2: 65 + 91 = 156

Option 3: 39 + 65 = 104 | Option 4: 78 + 91 = 169

Only 65 and 91 satisfy the sum condition; also their LCM is 455 (13 × 5 × 7).

∴ The correct answer is 65 and 91.

Alternate Method

Given: Sum = 156, HCF = 13, LCM = 455.

⇒ Let the numbers be 13a and 13b, where a and b are co-prime numbers.

⇒ Sum: 13a + 13b = 156 ⇒ 13(a + b) = 156 ⇒ a + b = 12.

⇒ LCM: 13 × a × b = 455 ⇒ ab = 455 ÷ 13 = 35.

⇒ We need two numbers with sum 12 and product 35: (a + b) = 12 and (ab) = 35.

⇒ The factors of 35 are 5 and 7, and their sum 5 + 7 = 12. Thus, a = 5 and b = 7.

⇒ First number = 13 × 5 = 65.

⇒ Second number = 13 × 7 = 91.

∴ The correct answer is 65 and 91.

Additional Information

Fundamental Product Rule

For any two numbers x and y, the product of the numbers is equal to the product of their HCF and LCM: x × y = HCF × LCM.

HCF and Sum Relation

The HCF of two numbers always divides their sum, their difference, and their LCM exactly.

LCM in terms of HCF

If two numbers are represented as Ha and Hb (where H is HCF), then their LCM is H × a × b, where a and b are co-prime.

Shortcut Trick

Simplify each square root independently: √0.0169 = 0.13, √3.24 = 1.8, and √0.0576 = 0.24.

Substitute the roots back into the given expression: (0.13 ÷ 1.3) × (0.18 ÷ 1.8) × (0.24 ÷ 0.24).

The calculation reduces to 0.1 × 0.1 × 1 = 0.01.

∴ The correct answer is 0.01.

Alternate Method

Given: (√0.0169 ÷ 1.3) × (0.18 ÷ √3.24) × (0.24 ÷ √0.0576)

Step 1: Calculate the square roots of the decimal numbers.

⇒ √0.0169 = √(169 ÷ 10000) = 13 ÷ 100 = 0.13

⇒ √3.24 = √(324 ÷ 100) = 18 ÷ 10 = 1.8

⇒ √0.0576 = √(576 ÷ 10000) = 24 ÷ 100 = 0.24

Step 2: Substitute these values into the original simplification expression.

⇒ (0.13 ÷ 1.3) × (0.18 ÷ 1.8) × (0.24 ÷ 0.24)

Step 3: Solve each term by adjusting the decimal places.

⇒ 0.13 ÷ 1.3 = 13 ÷ 130 = 0.1

⇒ 0.18 ÷ 1.8 = 18 ÷ 180 = 0.1

⇒ 0.24 ÷ 0.24 = 1

Step 4: Multiply the results together.

⇒ 0.1 × 0.1 × 1

⇒ 0.01

∴ The correct answer is 0.01.

Additional Information

Square Root of Decimals

The number of decimal places in the square root is always half the number of decimal places in the square itself (e.g., 4 places in 0.0169 become 2 places in 0.13).

Common Squares

Memorizing basic squares like 13² = 169, 18² = 324, and 24² = 576 helps in solving simplification problems faster.

BODMAS Rules

Always evaluate operations inside brackets and roots before performing multiplication and division in a sequence.

Shortcut Trick

Identify the unit digit of each square root: √169 ends in 3, √81 ends in 9, and √121 ends in 1.

Multiply the unit digits: 3 × 9 × 1 = 27.

The final result must have 7 as its unit digit.

Comparing with the options, only 1287 ends with the digit 7.

∴ The correct answer is 1287.

Alternate Method

Given: x = √169 × √81 × √121

Formula Used: Square root calculation and basic multiplication.

⇒ √169 = 13 (since 13 × 13 = 169)

⇒ √81 = 9 (since 9 × 9 = 81)

⇒ √121 = 11 (since 11 × 11 = 121)

⇒ x = 13 × 9 × 11

⇒ x = 117 × 11

⇒ x = 1287

∴ The correct answer is 1287.

Additional Information

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. For exam speed, memorize squares from 1 to 30.

Multiplication by 11 Shortcut

To multiply a number by 11, such as 117 × 11, write down the last digit (7), add adjacent digits (7+1=8, 1+1=2), and write the first digit (1) to get 1287.

Square Root Properties

The property √(a × b) = √a × √b allows for the simplification of products involving multiple square roots independently.

Shortcut Trick

Breadth = Area ÷ Length = 54 ÷ 9 = 6 cm

Required Ratio (Length : Breadth) = 9 : 6 = 3 : 2

∴ The correct answer is 3 : 2.

Alternate Method

Given:

Area of the Rectangle = 54 cm²

Length of the Rectangle = 9 cm

Formula Used:

Area of Rectangle = Length × Breadth

Ratio = Length : Breadth

Calculations:

⇒ 54 = 9 × Breadth

⇒ Breadth = 54 ÷ 9

⇒ Breadth = 6 cm

⇒ Ratio = Length : Breadth

⇒ Ratio = 9 : 6

⇒ Ratio = 3 : 2

∴ The correct answer is 3 : 2.

Additional Information

Perimeter of a Rectangle

The total distance around the boundary of a rectangle is calculated using the formula: P = 2 × (length + breadth).

Diagonal of a Rectangle

The diagonal (d) of a rectangle is the distance between opposite vertices: d = √(length² + breadth²).

Properties of Rectangle

A rectangle is a parallelogram where all four internal angles are right angles (90°) and opposite sides are equal and parallel.

Shortcut Trick

Smallest 4-digit number = 1000.

Divide 1000 by 17 → 1000 = (17 × 58) + 14 (Remainder).

To find the next multiple: 1000 + (17 − 14) = 1003.

∴ The correct answer is 1003.

Alternate Method

Given: Divisor = 17, Required number = Smallest 4-digit number.

⇒ Smallest 4-digit number = 1000

⇒ Divide 1000 by 17:

⇒ 1000 = 17 × 58 + 14

⇒ The smallest 4-digit number divisible by 17 must be the multiple just after 986.

⇒ Next Multiple = 17 × 59

⇒ 17 × (60 − 1) = 1020 − 17

⇒ Required number = 1003

∴ The correct answer is 1003.

Additional Information

Divisibility Rule for 17

To check if a number is divisible by 17, subtract 5 times the last digit from the remaining truncated number. Repeat if necessary. Example: 1003 → 100 − (5 × 3) = 85, which is 17 × 5.

General Formula for Smallest n-digit Multiplier

Smallest n-digit number divisible by x = (Smallest n-digit number) + (x − Remainder), where Remainder is obtained by dividing the smallest n-digit number by x.

Largest n-digit Multiplier

Largest n-digit number divisible by x = (Largest n-digit number) − (Remainder obtained by dividing it by x).

Shortcut Trick

Apply the algebraic identity: a² − b² = (a + b)(a − b).

Let a = 3.452 and b = 2.948.

Calculate the difference: (a − b) = 3.452 − 2.948 = 0.504, which is equal to the denominator.

The expression simplifies to (a + b) × (a − b) ÷ (a − b) = a + b.

Required value = 3.452 + 2.948 = 6.4.

∴ The correct answer is 6.4.

Alternate Method

Given:

Numerator = (3.452)² − (2.948)²

Denominator = 0.504

Formula Used:

The difference of squares formula: a² − b² = (a + b)(a − b)

Calculations:

⇒ Let a = 3.452 and b = 2.948

⇒ a + b = 3.452 + 2.948 = 6.400

⇒ a − b = 3.452 − 2.948 = 0.504

⇒ Numerator = 6.4 × 0.504

⇒ Expression = (6.4 × 0.504) ÷ 0.504

⇒ Expression = 6.4

∴ The correct answer is 6.4.

Additional Information

Difference of Two Squares

The expression a² − b² can always be factored into (a − b)(a + b).

Perfect Square Trinomial (Sum)

The square of a sum is given by (a + b)² = a² + 2ab + b².

Perfect Square Trinomial (Difference)

The square of a difference is given by (a − b)² = a² − 2ab + b².

Shortcut Trick

Sum of Length (L) and Breadth (B) = 60 ÷ 2 = 30 cm.

Ratio L : B = 4 : 1 ⇒ Total 5 units = 30 cm ⇒ 1 unit = 6 cm.

Actual dimensions: L = 24 cm and B = 6 cm.

Diagonal = √(24² + 6²) = 6 × √(4² + 1²) = 6√17 cm.

∴ The correct answer is 6√17 cm.

Alternate Method

Given: Perimeter of rectangle = 60 cm, Length (L) = 4 × Breadth (B).

Formula Used: Perimeter = 2(L + B); Diagonal = √(L² + B²).

⇒ 2(L + B) = 60

⇒ L + B = 30

⇒ Since L = 4B, 4B + B = 30

⇒ 5B = 30 ⇒ Breadth (B) = 6 cm

⇒ Length (L) = 4 × 6 = 24 cm

⇒ Diagonal = √(24² + 6²)

⇒ Diagonal = √(576 + 36) = √612

⇒ Diagonal = √(36 × 17) = 6√17 cm

∴ The correct answer is 6√17 cm.

Additional Information

Diagonal of a Rectangle

The diagonal of a rectangle forms the hypotenuse of a right-angled triangle with the length and breadth: d = √(l² + b²).

Area of a Rectangle

The total space occupied by the rectangle is given by Area = Length × Breadth (l × b).

Pythagorean Triplets

Common sets of integers that satisfy a² + b² = c², such as (3, 4, 5) or (5, 12, 13), help in quick calculation of diagonals.

Shortcut Trick

Total weight (A+B+C) = 30 × 3 = 90 kg

Total of sub-groups (A+B) and (B+C) = (34 × 2) + (28 × 2) = 68 + 56 = 124 kg

Since B is common to both sub-groups, its weight is the difference between the combined sub-groups and the total sum.

Weight of B = 124 − 90 = 34 kg

∴ The correct answer is 34 kg.

Alternate Method

Given:

Average weight of A, B, and C = 30 kg

Average weight of A and B = 34 kg

Average weight of B and C = 28 kg

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total weight of A + B + C = 30 × 3 = 90 kg --- (1)

⇒ Total weight of A + B = 34 × 2 = 68 kg --- (2)

⇒ Total weight of B + C = 28 × 2 = 56 kg --- (3)

⇒ Adding equation (2) and (3):

⇒ (A + B) + (B + C) = 68 + 56

⇒ A + 2B + C = 124 kg

⇒ (A + B + C) + B = 124 kg

⇒ Substituting value from (1):

⇒ 90 + B = 124

⇒ B = 124 − 90 = 34 kg

∴ The correct answer is 34 kg.

Additional Information

Arithmetic Mean (Average)

The average of a set of numbers is the sum of all observations divided by the total number of observations. Formula: Average = Σx ÷ n.

Combined Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the new average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Weighted Average

It is an average where each observation has a different weight. Weighted Average = Σ(weight_i × value_i) ÷ Σweight_i.

Shortcut Trick

Net increase in the average = 42 − 30 = 12

Total increase in sum = Increase in average × Total numbers = 12 × 8 = 96

New number = Replaced number + Total increase = 20 + 96 = 116

∴ The correct answer is 116.

Alternate Method

Given:

Number of observations = 8

Original average = 30

Replaced number = 20

New average = 42

Formula Used:

Total Sum = Average × Number of observations

New Number = New Total Sum − (Original Total Sum − Replaced Number)

⇒ Original total sum = 8 × 30 = 240

⇒ Sum of the other 7 numbers = 240 − 20 = 220

⇒ New total sum = 8 × 42 = 336

⇒ New number = 336 − 220 = 116

∴ The correct answer is 116.

Additional Information

Arithmetic Mean

The average or mean is the sum of all values divided by the total count. Formula: Average = (∑x_i) ÷ n.

Replacement Rule

When one value is replaced, the change in average is equal to (New Value − Old Value) ÷ Total Count.

Scaling Property

If every number in a data set is increased or decreased by a constant value k, the average also increases or decreases by k.

Shortcut Trick

The Highest Common Factor (HCF) is the common multiplier between terms of a ratio.

Numbers = (Ratio Parts) × HCF

First number = 3 × 5 = 15

Second number = 7 × 5 = 35

∴ The correct answer is 15, 35.

Alternate Method

Given:

Ratio of numbers = 3 : 7

HCF = 5

Calculations:

⇒ Let the two numbers be 3x and 7x, where x is a common factor.

⇒ Since 3 and 7 are co-prime (no common factor other than 1), the HCF of 3x and 7x is x.

⇒ From the question, HCF = 5, so x = 5.

⇒ First number = 3x = 3 × 5 = 15.

⇒ Second number = 7x = 7 × 5 = 35.

∴ The correct answer is 15, 35.

Additional Information

Fundamental Relation

For any two numbers, Product of numbers = HCF × LCM.

LCM from Ratio and HCF

If the ratio of two numbers is a : b and their HCF is h, then their Least Common Multiple (LCM) = a × b × h.

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. All prime numbers are co-prime to each other.

Shortcut Trick

In a mixture, 18% weight represents the weight of Element A.

Given that 18% = 90 kg

⇒ 1% = 90 ÷ 18 = 5 kg

⇒ 100% (Total weight) = 5 × 100 = 500 kg

∴ The correct answer is 500 kg.

Alternate Method

Given:

⇒ Weight of Element A in the mixture = 90 kg

⇒ Percentage of Element A by weight = 18%

Formula Used:

⇒ Total Weight = (Weight of component ÷ Percentage of component) × 100

Calculations:

Let the total weight of the mixture be 'W' kg.

⇒ 18% of W = 90

⇒ (18 ÷ 100) × W = 90

⇒ W = (90 × 100) ÷ 18

⇒ W = 5 × 100

⇒ W = 500 kg

∴ The correct answer is 500 kg.

Additional Information

Percentage to Value Conversion

To find the value of a percentage of a total: Value = (Total × Percentage) ÷ 100.

Finding the Base Value

If x% of a number is 'A', then the number itself (100%) is calculated as (A ÷ x) × 100.

Successive Percentage

If a value increases by a% and then by b%, the net percentage change is given by [a + b + (a × b) ÷ 100]%.

Shortcut Trick

Profit percentage 20% corresponds to 1/5 in fraction form.

If 1 unit (Profit) = ₹50

Then 5 units (Cost Price) = 5 × ₹50 = ₹250

∴ The correct answer is ₹250.

Alternate Method

Given: Profit = ₹50, Profit Percentage = 20%

Formula Used: Profit % = (Profit ÷ Cost Price) × 100

⇒ Profit % = (Profit ÷ Cost Price) × 100

⇒ 20 = (50 ÷ Cost Price) × 100

⇒ Cost Price = (50 × 100) ÷ 20

⇒ Cost Price = 50 × 5

⇒ Cost Price = ₹250

∴ The correct answer is ₹250.

Additional Information

Selling Price (SP)

Selling price is the sum of Cost Price and Profit. SP = CP × (100 + Profit %)/100.

Loss Percentage

When CP is greater than SP, a loss occurs. Loss % = (Loss ÷ CP) × 100, where Loss = CP − SP.

Marked Price (MP)

The price displayed on the article. Discount is always calculated on the MP: SP = MP − Discount.

Shortcut Trick

Assume initial Cost Price (CP) = 100 → Initial SP = 160.

New CP (30% increase) = 130.

New SP with 40% profit = 130 × 1.4 = 182.

Increase in SP (y%) = [(182 − 160) ÷ 160] × 100 = (22 ÷ 160) × 100 = 13.75%.

∴ The correct answer is 13.75.

Alternate Method

Given: Initial Profit = 60%, CP increase = 30%, New Profit = 40%.

⇒ Let Initial CP = 100

⇒ Initial SP = CP + 60% of CP = 100 + 60 = 160

⇒ New CP = 100 + 30% of 100 = 130

⇒ New Profit = 40%

⇒ New SP = 130 + 40% of 130 = 130 + 52 = 182

⇒ Percentage increase in SP (y) = [(New SP − Old SP) ÷ Old SP] × 100

⇒ y = [(182 − 160) ÷ 160] × 100

⇒ y = (22 ÷ 160) × 100 = 13.75

∴ The correct answer is 13.75.

Additional Information

Profit Percentage Formula

Profit % = [(SP − CP) ÷ CP] × 100. It is always calculated on the Cost Price unless specified otherwise.

Successive Percentage Change

If a value increases by a% and then by b%, the net change is [a + b + (a × b) ÷ 100] %.

Relation between CP and SP

SP = CP × [(100 + Profit %) ÷ 100] or SP = CP × [(100 − Loss %) ÷ 100].

Shortcut Trick

Assume initial Income = 100, then Expenditure = 50 and Saving = 50.

Net change: Total increase in Income = 40% of 100 = 40 units.

Increase in Saving = 20% of 50 = 10 units.

Required increase in Expenditure = 40 − 10 = 30 units.

% Increase in Expenditure = (30 ÷ 50) × 100 = 60 percent.

∴ The correct answer is 60 percent.

Alternate Method

Given:

Initial Expenditure = 50% of Income

Increase in Income = 40%

Increase in Saving = 20%

Formula Used:

Income = Expenditure + Saving

Percentage Increase = [(New Value − Old Value) ÷ Old Value] × 100

⇒ Let original Income = 100 units

⇒ Original Expenditure = 50% of 100 = 50 units

⇒ Original Saving = 100 − 50 = 50 units

⇒ New Income = 100 × (1 + 40 ÷ 100) = 140 units

⇒ New Saving = 50 × (1 + 20 ÷ 100) = 60 units

⇒ New Expenditure = 140 − 60 = 80 units

⇒ Change in Expenditure = 80 − 50 = 30 units

⇒ Percentage increase in Expenditure = (30 ÷ 50) × 100 = 60%

∴ The correct answer is 60 percent.

Additional Information

Income Balance Equation

Total Income is always the sum of Expenditure and Savings. Any change in total income must be balanced by the sum of changes in its components: ΔIncome = ΔExpenditure + ΔSaving.

Percentage Change Rule

To find the percentage increase, divide the absolute increase by the original base value and multiply by 100: % Increase = (Increase ÷ Original Value) × 100.

Weighted Average Application

Income % change can also be viewed as a weighted average of Expenditure % change and Saving % change based on their initial shares of income.

Shortcut Trick

If 1/3^rd of the tank is emptied in 45 min,

The complete tank (1 unit) will be emptied in 45 × 3 min.

Total time = 135 min.

∴ The correct answer is 135 min.

Alternate Method

Given:

Part of tank emptied by the pipe = 1/3

Time taken to empty 1/3 part = 45 min

Calculations:

⇒ Let the total capacity of the tank be 1 unit.

⇒ Time to empty 1/3 unit = 45 min

⇒ Time to empty 1 unit = 45 ÷ (1/3)

⇒ Time to empty 1 unit = 45 × 3

⇒ Total time = 135 min

∴ The correct answer is 135 min.

Additional Information

Unitary Method in Time and Work

If a part 'x' of a task is completed in time 'T', then the total time to complete the entire task is T ÷ x.

Efficiency of a Pipe

The efficiency of an emptying pipe is usually taken as negative in combined work problems. Efficiency = Work done ÷ Time taken.

Fractional Work Rule

If a pipe empties the tank in 'H' hours, then the part emptied in 1 minute is 1 ÷ (60 × H).

Shortcut Trick

Work rate of 12 men = 12 ÷ (6 × 18) = 12 ÷ 108 = 1/9

Work rate of 16 women = 16 ÷ (8 × 12) = 16 ÷ 96 = 1/6

Work rate of 10 boys = 10 ÷ (5 × 20) = 10 ÷ 100 = 1/10

Combined efficiency = 1/9 + 1/6 + 1/10 = (20 + 30 + 18) ÷ 180 = 68/180 = 17/45

Required time = Inverse of efficiency = 45/17 days.

∴ The correct answer is 45/17 days.

Alternate Method

Given:

6 men = 18 days, 8 women = 12 days, 5 boys = 20 days.

Formula Used:

Total Work = Time × Efficiency

Calculations:

⇒ Efficiency of 1 man = 10 ÷ 6 = 5/3 units/day

⇒ Efficiency of 1 woman = 15 ÷ 8 units/day

⇒ Efficiency of 1 boy = 9 ÷ 5 units/day

⇒ Efficiency of (12 men + 16 women + 10 boys) = 12 × (5/3) + 16 × (15/8) + 10 × (9/5)

⇒ Total combined efficiency = 20 + 30 + 18 = 68 units/day

⇒ Time required = Total Work ÷ Total Efficiency = 180 ÷ 68 = 45/17 days

∴ The correct answer is 45/17 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work. If the efficiency ratio is a:b, the time ratio is b:a.

Work Equivalence Formula

M₁D₁E₁ = M₂D₂E₂, where M is the number of people, D is days, and E is individual efficiency.

Total Work Calculation

Total work is usually considered as the Least Common Multiple (LCM) of the time taken by different groups to simplify efficiency calculations.

Shortcut Trick

For trains crossing each other in opposite direction, the Relative Speed is the sum of their individual speeds.

Total distance to be covered = Sum of lengths of both trains = 600 m.

Relative Speed = 40 m/s (given as total speed).

Time = Distance ÷ Speed = 600 ÷ 40 = 15 sec.

∴ The correct answer is 15 sec.

Alternate Method

Given:

Total length of Train A and Train B (L_A + L_B) = 600 m

Total speed (Relative Speed in opposite direction) = 40 m/s

Formula Used:

Time = (Sum of Lengths of Trains) ÷ (Relative Speed)

Calculations:

⇒ Total Distance = 600 m

⇒ Relative Speed = 40 m/s

⇒ Time = 600 ÷ 40

⇒ Time = 15 sec

∴ The correct answer is 15 sec.

Additional Information

Relative Speed Concepts

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

Crossing Conditions

To cross a stationary pole or man, distance = Length of Train. To cross a platform or another train, distance = Length of Train + Length of Object.

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 two objects start at the same time and move towards each other, the time (t) taken to meet is constant for both.

Since Distance = Speed × Time, when time is constant, Distance ∝ Speed.

∴ Ratio of distances covered by A and B = Ratio of their speeds = 50 : 30.

Simplified Ratio = 5 : 3.

∴ The correct answer is 5 : 3.

Alternate Method

Given: Speed of A = 50 km/hr, Speed of B = 30 km/hr. Both start simultaneously.

Formula Used: Distance = Speed × Time

⇒ Let the time taken by both A and B to meet at point S be t hours.

⇒ Distance covered by A to reach S = PS = Speed_A × t = 50t

⇒ Distance covered by B to reach S = QS = Speed_B × t = 30t

⇒ Ratio of distances covered = PS : QS

⇒ Ratio = 50t : 30t

⇒ Ratio = 50 : 30 = 5 : 3

∴ The correct answer is 5 : 3.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their Relative Speed is the sum of their individual speeds (S₁ + S₂).

Constant Distance Relation

If the distance is constant, the ratio of speeds is inversely proportional to the ratio of time taken (S₁/S₂ = T₂/T₁).

Meeting Time Formula

Time taken to meet = Total Distance ÷ Relative Speed. This applies when both start at the same time.