Quantitative Aptitude PYQs

Shortcut Trick

Apply the BODMAS rule by solving the bracket and division first.

Expression: 81 + 24 − 38 + 17 + 22 − 16

Combine positive terms: 81 + 24 + 17 + 22 = 144

Combine negative terms: 38 + 16 = 54

Final result: 144 − 54 = 90

∴ The correct answer is 90.

Alternate Method

Given: 81 + (24 × 1) − 38 + 17 + 44 ÷ 2 − 16

Formula Used: BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ 81 + 24 − 38 + 17 + 44 ÷ 2 − 16 (Solving bracket 24 × 1 = 24)

⇒ 81 + 24 − 38 + 17 + 22 − 16 (Solving division 44 ÷ 2 = 22)

⇒ 105 − 38 + 17 + 22 − 16 (Addition from left: 81 + 24 = 105)

⇒ 67 + 17 + 22 − 16 (Subtraction: 105 − 38 = 67)

⇒ 84 + 22 − 16 (Addition: 67 + 17 = 84)

⇒ 106 − 16 (Addition: 84 + 22 = 106)

⇒ 90

∴ The correct answer is 90.

Additional Information

BODMAS Rule

It is an acronym to help remember the order of mathematical operations: Brackets (), Orders (exponents/roots), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Left-to-Right Rule

When operators like addition and subtraction (or multiplication and division) appear in the same step, solve the expression from left to right.

Grouping for Speed

In expressions involving only addition and subtraction, grouping all positive terms and negative terms separately before the final subtraction can reduce errors and save time.

Shortcut Trick

The Product of two numbers = HCF × LCM = 24 × 360 = 8640.

Check options for Difference = 48 and Product = 8640.

Option (1): 144 − 96 = 48; 144 × 96 = 13824 (Incorrect).

Option (4): 120 − 72 = 48; 120 × 72 = 8640 (Correct).

∴ The correct answer is 120 and 72.

Alternate Method

Given:

HCF = 24

LCM = 360

Difference (x − y) = 48

Formula Used:

Product of numbers (x × y) = HCF × LCM

(x + y)² = (x − y)² + 4xy

Calculations:

⇒ x × y = 24 × 360 = 8640

⇒ (x + y)² = (48)² + 4 × (8640)

⇒ (x + y)² = 2304 + 34560 = 36864

⇒ x + y = √36864 = 192

⇒ Adding (x + y = 192) and (x − y = 48):

⇒ 2x = 192 + 48 = 240 → x = 120

⇒ Subtracting (x − y = 48) from (x + y = 192):

⇒ 2y = 192 − 48 = 144 → y = 72

∴ The correct answer is 120 and 72.

Additional Information

Fundamental Property of HCF and LCM

For any two numbers a and b, the product of the numbers is always equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

HCF as a Factor of LCM

The HCF of any set of numbers is always a factor of their LCM. If the HCF does not divide the LCM, such numbers cannot exist.

Co-prime Factors

Two numbers can be represented as (HCF × a) and (HCF × b), where 'a' and 'b' are co-prime integers (meaning HCF(a, b) = 1).

Shortcut Trick

Combine the internal terms within the bracket first: 11.85 − (5.45 + 4.60) = 11.85 − 10.05 = 1.80.

Final calculation: 12.1 − 1.80 = 10.3.

∴ The correct answer is 10.3.

Alternate Method

Given Expression: 12.1 − (11.85 − 5.45 − 4.60)

Using the BODMAS rule, we must simplify the terms inside the parentheses first from left to right:

⇒ 12.1 − (6.40 − 4.60)

⇒ 12.1 − 1.80

⇒ 10.3

∴ The correct answer is 10.3.

Additional Information

BODMAS Rule

This rule specifies the priority of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Decimal Subtraction

When subtracting decimals, always align the decimal points and append zeros (e.g., 12.1 as 12.10) to ensure the numbers have the same number of decimal places.

Shortcut Trick

The given expression follows the algebraic identity: a³ − 3a²b + 3ab² − b³ = (a − b)³.

Here, let a = 5.1 and b = 0.1.

Substituting the values: (5.1 − 0.1)³ = (5.0)³.

Calculation: 5 × 5 × 5 = 125.

∴ The correct answer is 125.

Alternate Method

Given: (5.1)² × 5.1 − 3 × 0.1 × (5.1)² + 3 × 5.1 × (0.1)² − (0.1)³

Formula Used: (a − b)³ = a³ − 3a²b + 3ab² − b³

⇒ Let a = 5.1 and b = 0.1.

⇒ a² × a − 3 × b × a² + 3 × a × b² − b³

⇒ a³ − 3a²b + 3ab² − b³

⇒ (a − b)³

⇒ (5.1 − 0.1)³

⇒ (5.0)³

⇒ 5 × 5 × 5 = 125

∴ The correct answer is 125.

Additional Information

Cube of Sum Identity

(a + b)³ = a³ + 3a²b + 3ab² + b³, which can also be written as a³ + b³ + 3ab(a + b).

Sum and Difference of Cubes

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Conditional Identity

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Shortcut Trick

Use the algebraic identity: (a + b)² + (a − b)² = 2(a² + b²).

Here, a = √6 and b = √2.

⇒ Result = 2((√6)² + (√2)²)

⇒ Result = 2(6 + 2) = 2 × 8 = 16.

∴ The correct answer is 16.

Alternate Method

Given: (√6 + √2)² + (√6 − √2)²

Formula Used: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ [(√6)² + (√2)² + 2(√6 × √2)] + [(√6)² + (√2)² − 2(√6 × √2)]

⇒ (6 + 2 + 2√12) + (6 + 2 − 2√12)

⇒ (8 + 2√12) + (8 − 2√12)

⇒ 8 + 8 + 2√12 − 2√12

⇒ 16

∴ The correct answer is 16.

Additional Information

Standard Algebraic Identities

The identity (a + b)² + (a − b)² = 2(a² + b²) is derived by adding the expansions of square of sum and square of difference.

Difference of Identities

Subtracting the two squares gives (a + b)² − (a − b)² = 4ab, which is often used in simplification problems involving roots.

Surds and Squares

For any positive number x, (√x)² = x. This property allows for quick elimination of square root signs in algebraic expressions.

Shortcut Trick

For an odd number of consecutive terms, the Average is always the middle number.

Middle number = Total Sum ÷ Count of numbers = 54 ÷ 3 = 18

Since the numbers are consecutive even, they are 18 − 2, 18, and 18 + 2.

The numbers are 16, 18, and 20.

The largest among them is 20.

∴ The correct answer is 20.

Alternate Method

Given: Sum of 3 consecutive even numbers = 54

Let the three consecutive even numbers be x, x + 2, and x + 4.

⇒ Sum = x + (x + 2) + (x + 4) = 54

⇒ 3x + 6 = 54

⇒ 3x = 54 − 6

⇒ 3x = 48

⇒ x = 48 ÷ 3

⇒ x = 16

⇒ Largest number = x + 4 = 16 + 4

⇒ Largest number = 20

∴ The correct answer is 20.

Additional Information

Consecutive Even/Odd Numbers

Whether consecutive numbers are even or odd, the common difference between them is always 2.

Middle Term Property

In an Arithmetic Progression (AP) with an odd number of terms, the Average is always the middle term (Median).

Sum of AP Series

The sum of an AP is given by S_n = n/2 × (first term + last term) or S_n = n × Average.

Shortcut Trick

The given expression 9³ + 7³ + 3 × 9 × 7 × 16 is in the form of the algebraic identity a³ + b³ + 3ab(a + b) = (a + b)³.

Here, a = 9, b = 7, and (a + b) = 16.

Substituting these values: (9 + 7)³ = 16³.

Evaluating the cube: 16 × 16 × 16 = 4096.

∴ The correct answer is 4096.

Alternate Method

Given:

Expression: 9³ + 7³ + 3 × 9 × 7 × 16

Formula Used:

The algebraic identity for the cube of a sum is: (a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

⇒ Let a = 9 and b = 7.

⇒ Then a + b = 9 + 7 = 16.

⇒ The expression becomes: 9³ + 7³ + 3 × 9 × 7 × (9 + 7).

⇒ According to the identity, this is equal to (9 + 7)³.

⇒ (16)³ = 16 × 16 × 16

⇒ 256 × 16 = 4096.

∴ The correct answer is 4096.

Additional Information

Cube of a Difference Identity

The expansion for the cube of a difference is (a − b)³ = a³ − b³ − 3ab(a − b).

Sum and Difference of Cubes

Factorization formulas are a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

Binomial Square Identities

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

Shortcut Trick

Observe that x and y are reciprocals of each other, hence xy = 1.

For expressions of the form (a+√b)/(a−√b) + (a−√b)/(a+√b), the sum x + y = 2(a² + b) / (a² − b).

Substitute a = 2 and b = 6: x + y = 2(2² + 6) / (2² − 6) = 2(10) / (−2) = −10.

Apply the identity x² + y² = (x + y)² − 2xy.

Calculation: (−10)² − 2(1) = 100 − 2 = 98.

∴ The correct answer is 98.

Alternate Method

Given: x = (2 + √6) / (2 − √6) and y = (2 − √6) / (2 + √6)

Formula Used: (a + b)² = a² + 2ab + b² and Rationalization

Rationalizing x by multiplying with the conjugate (2 + √6):

⇒ x = [(2 + √6) × (2 + √6)] / [(2 − √6) × (2 + √6)]

⇒ x = (4 + 6 + 4√6) / (4 − 6) = (10 + 4√6) / −2

⇒ x = −5 − 2√6

Similarly, rationalizing y by multiplying with the conjugate (2 − √6):

⇒ y = −5 + 2√6

Now, calculate the sum and product:

⇒ x + y = (−5 − 2√6) + (−5 + 2√6) = −10

⇒ xy = (−5 − 2√6) × (−5 + 2√6) = (−5)² − (2√6)² = 25 − 24 = 1

Using the identity for x² + y²:

⇒ x² + y² = (x + y)² − 2xy

⇒ x² + y² = (−10)² − 2(1) = 100 − 2

⇒ x² + y² = 98

∴ The correct answer is 98.

Additional Information

Rationalization of Surds

To rationalize a denominator of the form (a − √b), multiply both the numerator and denominator by the conjugate (a + √b).

Algebraic Identity for Sum of Squares

The expression a² + b² can be derived using the formula (a + b)² − 2ab or (a − b)² + 2ab.

Reciprocal Relations

If x = 1/y, then xy = 1. In such cases, xⁿ + yⁿ is always an integer if (x+y) is an integer.

Shortcut Trick

For a Rectangle, the length, breadth, and diagonal form a right-angled triangle.

Recognize the standard Pythagorean Triplet: (5, 12, 13).

Given: Length = 12 cm and Diagonal (hypotenuse) = 13 cm.

Therefore, the remaining side (Breadth) must be 5 cm.

∴ The correct answer is 5 cm.

Alternate Method

Given:

Length (l) = 12 cm

Diagonal (d) = 13 cm

Formula Used:

In a Rectangle, Diagonal² = Length² + Breadth² (by Pythagoras Theorem)

Calculations:

⇒ 13² = 12² + b²

⇒ 169 = 144 + b²

⇒ b² = 169 − 144

⇒ b² = 25

⇒ b = √25

⇒ b = 5 cm

∴ The correct answer is 5 cm.

Additional Information

Area of Rectangle

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

Perimeter of Rectangle

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

Common Pythagorean Triplets

Frequently used sets of integers satisfying a² + b² = c² include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick

Solve the bracket first using known squares: 25² = 625 and 15² = 225.

Sum of squares = 625 + 225 = 850.

Subtracting 450 from the sum: 850 − 450 = 400.

Multiply the result by 40: 40 × 400 = 16000.

∴ The correct answer is 16000.

Alternate Method

Given expression: 40 × (25² + 15² − 450)

Apply the BODMAS rule by solving terms inside the parentheses first:

⇒ 40 × (625 + 225 − 450)

⇒ 40 × (850 − 450)

⇒ 40 × 400

⇒ 16000

∴ The correct answer is 16000.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Squares ending in 5

To find the square of a number ending in 5 (like n5), multiply n by (n+1) and suffix 25. For example, 25² = (2 × 3) and suffix 25 = 625.

Multiplication with Zeros

When multiplying numbers with trailing zeros, multiply the non-zero digits and then append the total count of zeros to the result (e.g., 40 × 400: 4 × 4 = 16, append three zeros = 16000).

Shortcut Trick

The difference between the sums of two groups with a common element is always equal to the difference in their averages multiplied by the number of terms.

Since m is common, the difference between the averages of (l, m) and (m, n) is simply half the difference between l and n.

Difference between l and n = 17 × 2 = 34.

∴ The correct answer is 34.

Alternate Method

Given:

Average of l and m = (l + m) / 2

Average of m and n = (m + n) / 2

Difference between averages = 17

Calculations:

⇒ ((l + m) / 2) − ((m + n) / 2) = 17

⇒ (l + m − m − n) / 2 = 17

⇒ (l − n) / 2 = 17

⇒ l − n = 17 × 2

⇒ l − n = 34

∴ The correct answer is 34.

Additional Information

Average Property

The Sum of observations is always equal to the Average × Number of observations.

Overlap Method

When subtracting two averages of the same size, any common term between the sets is cancelled out during the subtraction of their sums.

Linear Transformation

If each term in a set is increased or decreased by a constant k, the average of the set also changes by k.

Shortcut Trick

If Passed students = 30%, then Failed students = 100% − 30% = 70%.

Given that 70% represents 42,000 students.

Applying the Unitary Method: 1% = 42,000 ÷ 70 = 600.

Number of Passed students (30%) = 30 × 600 = 18,000.

∴ The correct answer is 18000.

Alternate Method

Given:

Total appeared students = 100%

Pass Percentage = 30%

Number of Failed Students = 42,000

Calculations:

⇒ Failed Percentage = 100% − 30% = 70%

⇒ Let the Total students be x.

⇒ 70% of x = 42,000

⇒ (70 ÷ 100) × x = 42,000

⇒ x = (42,000 × 100) ÷ 70

⇒ x = 600 × 100 = 60,000

⇒ Passed students = 30% of Total students

⇒ Passed students = 0.30 × 60,000

⇒ Passed students = 18,000

∴ The correct answer is 18000.

Additional Information

Percentage Base Calculation

To find the total value when a percentage part is known: Total = (Given Value ÷ Percentage) × 100.

Complementary Percentages

In any binary outcome (pass/fail, win/loss), the sum of percentages must always equal 100%.

Percentage to Fraction Conversion

Converting percentages to fractions (e.g., 70% = 7/10) can often simplify complex division and multiplication steps.

Shortcut Trick

If the Cost Price (CP) and Selling Price (SP) are both increased or decreased by the same percentage, the ratio between them remains unchanged.

Since the ratio SP : CP remains constant, the Profit Percentage will also remain the same as the initial value.

Initial profit = 30%.

New profit = 30%.

∴ The correct answer is 30%.

Alternate Method

Given: Initial profit = 30%, CP and SP increase = 40%.

Formula Used: Profit Percentage = [(Selling Price − Cost Price) ÷ Cost Price] × 100

⇒ Let the initial Cost Price (CP) = 100

⇒ Since profit is 30%, initial Selling Price (SP) = 100 + 30% of 100 = 130

⇒ New CP = 100 + (40 ÷ 100) × 100 = 140

⇒ New SP = 130 + (40 ÷ 100) × 130 = 130 + 52 = 182

⇒ New Profit = New SP − New CP = 182 − 140 = 42

⇒ New Profit Percentage = (42 ÷ 140) × 100

⇒ New Profit Percentage = (3 ÷ 10) × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Invariant Profit Percentage

If both CP and SP are multiplied by the same factor k, the profit percentage remains constant because (k × SP − k × CP) ÷ (k × CP) = (SP − CP) ÷ CP.

Profit and Loss Formulas

Profit % = (Profit ÷ CP) × 100 | Loss % = (Loss ÷ CP) × 100. Profit or loss is always calculated on the Cost Price unless specified otherwise.

Selling Price Calculation

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

Shortcut Trick

Let total fruits be 100 units.

Sold = 60 units, Remaining = 40 units.

Spoilt = 30% of 40 = 12 units.

Given: 12 units = 48 fruits ⇒ 1 unit = 4 fruits.

Fruits sold = 60 units × 4 = 240.

∴ The correct answer is 240.

Alternate Method

Given:

Percentage sold = 60%

Spoilage rate = 30% of remaining fruits

Total spoilt = 48 fruits

Calculations:

⇒ Let Initial Fruits = x

⇒ Fruits remaining after selling 60% = x − 0.60x = 0.40x

⇒ Spoilt fruits = 30% of 0.40x = 0.30 × 0.40x = 0.12x

⇒ According to the question: 0.12x = 48

⇒ x = 48 ÷ 0.12 = 4800 ÷ 12 = 400

⇒ Number of fruits sold = 60% of 400 = 0.60 × 400 = 240

∴ The correct answer is 240.

Additional Information

Successive Percentage Application

When a percentage is calculated on a leftover amount, it is a form of successive deduction. Net Percentage = (Remaining%) × (Subsequent%).

Unitary Method in Percentages

Assuming the base value as 100 units simplifies multi-step percentage problems by converting them into basic arithmetic ratios.

Shortcut Trick

Profit on Selling Price (SP) = 25% = 1/4.

This implies if SP = 4 units, then Profit = 1 unit.

Cost Price (CP) = SP − Profit = 4 − 1 = 3 units.

Profit percentage on CP = (Profit ÷ CP) × 100 = (1 ÷ 3) × 100 = 33.33%.

∴ The correct answer is 33.33 percent.

Alternate Method

Given:

Profit earned on Selling Price = 25%

Formula Used:

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

Profit % on CP = (Profit ÷ CP) × 100

Calculations:

⇒ Let the Selling Price (SP) be ₹100.

⇒ Profit = 25% of ₹100 = ₹25.

⇒ Cost Price (CP) = SP − Profit = 100 − 25 = ₹75.

⇒ Profit % on CP = (25 ÷ 75) × 100

⇒ Profit % on CP = (1 ÷ 3) × 100

⇒ Profit % on CP = 33.333...

∴ The correct answer is 33.33 percent.

Additional Information

Relationship between Profit on SP and CP

If a person earns x% profit on SP, the actual profit percentage (on CP) is always higher and is given by [x ÷ (100 − x)] × 100.

Relationship between Loss on SP and CP

If a person incurs x% loss on SP, the actual loss percentage (on CP) is always lower and is given by [x ÷ (100 + x)] × 100.

Markup Percentage

Markup % is the percentage by which the Marked Price is higher than the Cost Price. Formula: Markup % = [(MP − CP) ÷ CP] × 100.

Shortcut Trick

Remaining food for 900 men lasts for (60 − 6) = 54 days.

After 300 men left, remaining men = 900 − 300 = 600.

Ratio of men = 900 : 600 = 3 : 2 ⇒ Ratio of days = 2 : 3 (inverse).

2 units = 54 days ⇒ 1 unit = 27 days.

3 units = 3 × 27 = 81 days.

∴ The correct answer is 81 days.

Alternate Method

Given: Initial men = 900, Total food duration = 60 days, Days passed = 6, Men who left = 300.

Formula Used: M₁ × D₁ = M₂ × D₂ (where work/food is constant).

⇒ Remaining days for 900 men = 60 − 6 = 54 days

⇒ Total remaining food units = 900 × 54

⇒ Remaining men in group = 900 − 300 = 600

⇒ Let the remaining food last for x days for 600 men.

⇒ 600 × x = 900 × 54

⇒ x = (900 × 54) ÷ 600

⇒ x = (3 × 54) ÷ 2

⇒ x = 3 × 27 = 81

∴ The correct answer is 81 days.

Additional Information

Chain Rule Formula

The general formula for work and men is (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where M is men, D is days, H is hours, and W is work.

Inverse Variation

The number of days is inversely proportional to the number of men (D ∝ 1/M). If manpower decreases, the food duration increases proportionally.

Man-Days Concept

Total Food Stock is often measured in Man-Days. Total Work = Strength × Duration.

Shortcut Trick

Efficiency Ratio of A : B = 1 : 2 (since A takes twice as long as B).

Total Work = Efficiency of A × Time of A = 1 × 60 = 60 units.

Efficiency of C = Efficiency of (A + B) = 1 + 2 = 3 units/day.

Total Efficiency (A + B + C) = 1 + 2 + 3 = 6 units/day.

Required Time = Total Work ÷ Total Efficiency = 60 ÷ 6 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given:

Time taken by A = 60 days

Time taken by B = 60 ÷ 2 = 30 days (since A takes twice as long)

Formula Used:

Efficiency = Total Work ÷ Time

Time taken by C = Time taken by (A + B) together

Calculations:

⇒ 1 day work of (A + B) = 1/60 + 1/30 = (1 + 2)/60 = 3/60 = 1/20

⇒ Time taken by (A + B) = 20 days

⇒ Time taken by C = 20 days

⇒ Combined efficiency of A, B, and C = 1 + 2 + 3 = 6 units/day

⇒ Total days taken by A, B, and C together = 60 ÷ 6 = 10 days

∴ The correct answer is 10 days.

Additional Information

Relationship between Time and Efficiency

Efficiency is inversely proportional to the time taken. If the ratio of time taken by A and B is x : y, then the ratio of their efficiencies is y :

x. Combined Work Formula

If A can do a work in 'x' days and B can do it in 'y' days, then together they can finish it in (x × y) ÷ (x + y) days.

Work and Wages

Wages are always distributed in the ratio of the work done by each person, which is equivalent to the ratio of their efficiencies if they work for the same duration.

Shortcut Trick

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

Shortcut: 270 × (5/18) = 15 × 5 = 75.

∴ The correct answer is 75 m/s.

Alternate Method

Given:

Speed of the car = 270 km/hr

Formula Used:

Speed (m/s) = Speed (km/hr) × (5 ÷ 18)

Calculations:

⇒ Speed in m/s = 270 × (5 ÷ 18)

⇒ Speed in m/s = (270 ÷ 18) × 5

⇒ Speed in m/s = 15 × 5

⇒ Speed in m/s = 75 m/s

∴ The correct answer is 75 m/s.

Additional Information

Unit Conversion Factor

1 km/hr = 1000 m ÷ 3600 s = 5/18 m/s. To convert m/s to km/hr, multiply by 18/5.

Speed, Distance and Time

Speed = Distance ÷ Time. If distance is in meters and time is in seconds, speed will be in m/s.

Average Speed

Average Speed = Total Distance ÷ Total Time. This is useful when the object moves with different speeds over different intervals.

Shortcut Trick

Since the speed is constant, the distance covered is directly proportional to the time taken.

Distance 1 (Length of Train) corresponds to the time to cross a man = 5 sec.

Distance 2 (Length of Train + Platform) corresponds to the time to cross the platform = 12 sec.

The required ratio of lengths is equal to the ratio of the respective times taken.

Required Ratio = 5 : 12.

∴ The correct answer is 5 : 12.

Alternate Method

Given:

Time to cross a standing man = 5 sec

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

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Let the length of the train be L and its speed be S.

⇒ When crossing a man, Distance = L. ∴ L = S × 5

⇒ When crossing a platform of length y, Distance = L + y. ∴ L + y = S × 12

⇒ We need to find the ratio of length of train (L) to the total length (L + y).

⇒ Ratio = L : (L + y)

⇒ Ratio = (S × 5) : (S × 12)

⇒ Ratio = 5 : 12

∴ The correct answer is 5 : 12.

Additional Information

Point Object vs Extended Object

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

Speed Conversion

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

Relative Speed

For two objects moving in the same direction, relative speed = S₁ − S₂. For opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Assume Marked Price (X) = 100 units.

Purchased Price (Y) = 80% of 100 = 80 units (after 20% discount).

Selling Price (Z) = 120% of 80 = 96 units (20% more than Y).

Ratio of Z : X = 96 : 100 = 24 : 25.

∴ The correct answer is 24 : 25.

Alternate Method

Given:

Marked Price = Rs. X

Discount = 20%

Purchased Price = Rs. Y

Selling Price = Rs. Z (where Z is 20% more than Y)

Calculations:

⇒ Y = X − 20% of X

⇒ Y = 0.8X

⇒ Z = Y + 20% of Y

⇒ Z = 1.2Y

⇒ Substitute Y = 0.8X in the equation for Z:

⇒ Z = 1.2 × (0.8X)

⇒ Z = 0.96X

⇒ Z/X = 0.96

⇒ Z/X = 96/100

⇒ Z/X = 24/25

∴ The correct answer is 24 : 25.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the net effective change is given by the formula: [a + b + (a × b)/100]%.

Relationship between MP, CP, and SP

Marked Price (MP) × (100 − Discount%)/100 = Selling Price (SP). Also, Cost Price (CP) × (100 + Profit%)/100 = Selling Price (SP).

Fraction to Ratio Conversion

To find a ratio from a decimal like 0.96, write it as a fraction (96/100) and simplify it to its lowest terms by dividing by the Greatest Common Divisor (GCD).