Quantitative Aptitude PYQs

Shortcut Trick

Solve the expression using the BODMAS rule by prioritising the bracket and multiplication.

73 + 88 − 64 + (62) × 2 − 99

161 − 64 + 124 − 99 = 122

∴ The correct answer is 122.

Alternate Method

Given: 73 + 88 − 64 + (47 + 15) × 2 − 99

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

⇒ 73 + 88 − 64 + (62) × 2 − 99

⇒ 73 + 88 − 64 + 124 − 99

⇒ (73 + 88 + 124) − (64 + 99)

⇒ 285 − 163

⇒ 122

∴ The correct answer is 122.

Additional Information

VBODMAS Rule

The acronym VBODMAS helps remember the order: Vinculum (Bar), Bracket, Of, Division, Multiplication, Addition, and Subtraction.

Types of Brackets

Solve brackets in this specific order: Bar/Vinculum → Small Brackets ( ) → Curly Brackets { } → Square Brackets [ ].

Left to Right Rule

For operations of the same priority (like Addition/Subtraction or Multiplication/Division), always calculate from left to right.

Shortcut Trick

Apply BODMAS: Solve Of (33 × 1 = 33) and Division (26 ÷ 2 = 13) first.

Expression becomes: 63 + 41 − 29 + 33 − 18 + 13.

Combine positive terms: 63 + 41 + 33 + 13 = 150.

Combine negative terms: 29 + 18 = 47.

Subtract: 150 − 47 = 103.

∴ The correct answer is 103.

Alternate Method

Given Expression: 63 + 41 − 29 + (33 of 1) − 18 + 26 ÷ 2

Using the BODMAS rule (Bracket, Of, Division, Multiplication, Addition, Subtraction):

⇒ 63 + 41 − 29 + (33 × 1) − 18 + 26 ÷ 2

⇒ 63 + 41 − 29 + 33 − 18 + 26 ÷ 2 (Solving Of)

⇒ 63 + 41 − 29 + 33 − 18 + 13 (Solving Division)

⇒ 104 − 29 + 33 − 18 + 13 (Solving Addition from left)

⇒ 75 + 33 − 18 + 13 (Solving Subtraction)

⇒ 108 − 18 + 13

⇒ 90 + 13

⇒ 103

∴ The correct answer is 103.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets (), Of (multiplication priority), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Priority of Of

The Of operator is essentially a multiplication but must be evaluated before Division and Multiplication according to the hierarchy.

Sign Management

Grouping all positive integers together and all negative integers together can simplify long chains of Addition and Subtraction to prevent errors.

Shortcut Trick

Convert the given fractions into decimal values for faster comparison.

Target Range: 1/4 = 0.25 and 3/4 = 0.75.

Option (1): 7/11 ≈ 0.63 (Lies between 0.25 and 0.75).

Other options: 11/13 ≈ 0.84, 9/11 ≈ 0.81, 15/17 ≈ 0.88 (All > 0.75).

∴ The correct answer is 7/11.

Alternate Method

Given:

Lower limit fraction = 1/4

Upper limit fraction = 3/4

Formula Used:

To check if fraction x lies between a and b, the condition is: a < x < b.

Calculations:

⇒ Boundary 1 = 1 ÷ 4 = 0.25

⇒ Boundary 2 = 3 ÷ 4 = 0.75

⇒ Checking Option 1: 7 ÷ 11 = 0.6363... (0.25 < 0.636 < 0.75) → Valid

⇒ Checking Option 2: 11 ÷ 13 = 0.846... (0.846 > 0.75) → Invalid

⇒ Checking Option 3: 9 ÷ 11 = 0.818... (0.818 > 0.75) → Invalid

⇒ Checking Option 4: 15 ÷ 17 = 0.882... (0.882 > 0.75) → Invalid

∴ The correct answer is 7/11.

Additional Information

Comparing Fractions (Cross-Multiplication)

To compare a/b and c/d, check the products (a × d) and (b × c). If ad > bc, then a c/d.

Density Property of Rational Numbers

Between any two rational numbers, there are infinitely many other rational numbers. A simple way to find one is by taking the average: (a + b) ÷ 2.

Decimal Conversion Shortcut

Memorizing common fraction-to-decimal values (like 1/11 ≈ 0.09, 1/13 ≈ 0.077) helps in solving comparison questions within seconds.

Shortcut Trick

The given expression follows the Algebraic Identity: (a + b)² − 4ab = (a − b)².

Here, let a = 9.8 and b = 1.8.

Applying the identity: (9.8 − 1.8)² = 8².

Result = 64.

∴ The correct answer is 64.

Alternate Method

Given: (9.8 + 1.8)² − 4 × 9.8 × 1.8

Method: Direct Calculation using BODMAS and Squares.

⇒ Step 1: Solve the bracket (9.8 + 1.8) = 11.6

⇒ Step 2: Calculate the Square of 11.6 → 11.6 × 11.6 = 134.56

⇒ Step 3: Calculate the product 4 × 9.8 × 1.8

⇒ 4 × 17.64 = 70.56

⇒ Step 4: Subtract the values: 134.56 − 70.56 = 64

∴ The correct answer is 64.

Additional Information

Fundamental Algebraic Identities

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

Relationship between Identities

(a + b)² − (a − b)² = 4ab, which can be rearranged as (a + b)² − 4ab = (a − b)².

Difference of Squares

a² − b² = (a − b)(a + b). This is useful for simplification of large products.

Shortcut Trick

Use the Area formula: Area = (1/2) × d₁ × d₂ ⇒ 336 = (1/2) × 14 × d₂ ⇒ d₂ = 48 cm.

Apply Pythagorean triplet (7, 24, 25) for the right triangle formed by half-diagonals: Side (a) = 25 cm.

Perimeter = 4 × side = 4 × 25 = 100 cm.

∴ The correct answer is 100 cm.

Alternate Method

Given: Area of rhombus = 336 cm² and diagonal d₁ = 14 cm.

Formula Used:

1. Area = (1/2) × d₁ × d₂

2. Side (a) = √[(d₁/2)² + (d₂/2)²]

3. Perimeter = 4 × a

Calculations:

⇒ 336 = (1/2) × 14 × d₂

⇒ 336 = 7 × d₂

⇒ d₂ = 336 ÷ 7 = 48 cm

⇒ Half of diagonals: d₁/2 = 7 cm and d₂/2 = 24 cm

⇒ Side (a) = √[7² + 24²]

⇒ a = √[49 + 576] = √625 = 25 cm

⇒ Perimeter = 4 × 25 = 100 cm

∴ The correct answer is 100 cm.

Additional Information

Properties of a Rhombus

All four sides are equal, and the diagonals bisect each other at right angles (90°).

Relation between Sides and Diagonals

The side 'a' and diagonals 'd₁' and 'd₂' follow the relation: 4a² = d₁² + d₂².

Common Pythagorean Triplets

Triplets like (7, 24, 25), (5, 12, 13), and (8, 15, 17) are frequently used in geometry problems involving right-angled triangles.

Shortcut Trick

In a parallelogram, the sum of adjacent angles is always 180°.

Given ratio = 4 : 5

Total units = 4 + 5 = 9 units

9 units = 180° ⇒ 1 unit = 20°

Larger angle = 5 units = 5 × 20° = 100°

∴ The correct answer is 100°.

Alternate Method

Given:

Ratio of adjacent angles = 4 : 5

Formula Used:

The adjacent angles of a parallelogram are supplementary (sum = 180°).

Calculations:

⇒ Let the measures of the adjacent angles be 4x and 5x.

⇒ 4x + 5x = 180°

⇒ 9x = 180°

⇒ x = 180° ÷ 9

⇒ x = 20°

⇒ Larger angle = 5x = 5 × 20° = 100°

⇒ Smaller angle = 4x = 4 × 20° = 80°

∴ The correct answer is 100°.

Additional Information

Properties of Parallelogram

Opposite angles are always equal, and the sum of all interior angles is 360°.

Consecutive Interior Angles

When two parallel lines are cut by a transversal, consecutive interior angles (like adjacent angles in a parallelogram) are supplementary.

Diagonals of a Parallelogram

The diagonals bisect each other but are not necessarily equal in length unless the figure is a rectangle.

Shortcut Trick

Numbers 11 and 30 are co-prime (no common factors except 1).

For co-prime numbers, HCF (Y) = 1 and LCM (X) = Product of the numbers.

X = 11 × 30 = 330 and Y = 1.

Value = X − 2Y = 330 − 2(1) = 328.

∴ The correct answer is 328.

Alternate Method

Given: First number = 11, Second number = 30.

Formula Used: X = LCM (Least Common Multiple) and Y = HCF (Highest Common Factor).

⇒ Prime factorization of 11 = 11¹

⇒ Prime factorization of 30 = 2¹ × 3¹ × 5¹

⇒ Highest Common Factor (Y) = Product of common prime factors with lowest power = 1.

⇒ Least Common Multiple (X) = Product of all prime factors with highest power = 11 × 2 × 3 × 5 = 330.

⇒ Required calculation: X − 2Y

⇒ 330 − 2 × 1 = 330 − 2 = 328.

∴ The correct answer is 328.

Additional Information

Fundamental Relation

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

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. The LCM of co-prime numbers is always equal to their product.

LCM of Fractions

The LCM of fractions is calculated as: LCM of numerators ÷ HCF of denominators.

Shortcut Trick

The Ratio of the two numbers is 9 : 4 and their Highest Common Factor (HCF) is 7.

Total ratio units = 9 + 4 = 13 units.

Since the HCF represents the value of 1 ratio unit, then 1 unit = 7.

Sum of the numbers = 13 units × 7 = 91.

∴ The correct answer is 91.

Alternate Method

Given:

HCF = 7

LCM = 252

Ratio of numbers = 9 : 4

Formula Used:

If ratio is a : b and HCF is x, then the numbers are ax and bx.

Calculations:

⇒ Let the two numbers be 9x and 4x.

⇒ Their HCF is x (as 9 and 4 are coprime).

⇒ Given HCF = 7, so x = 7.

⇒ Verify with LCM: LCM(9x, 4x) = 9 × 4 × x = 36 × 7 = 252 (Matches given LCM).

⇒ First number = 9 × 7 = 63

⇒ Second number = 4 × 7 = 28

⇒ Sum = 63 + 28 = 91.

∴ The correct answer is 91.

Additional Information

Product of Two Numbers

The product of two numbers is always equal to the product of their HCF and LCM. Formula: n₁ × n₂ = HCF × LCM.

LCM and HCF Relationship

LCM is always a multiple of HCF. If the ratio of two numbers is a : b, then LCM = HCF × a × b (where a and b are coprime).

Difference of Numbers

The difference between the two numbers in ratio a : b with HCF x is given by (a − b) × x.

Shortcut Trick

Required Value = (100 − 60)% of (50% of 600)

⇒ 40% × (50% × 600)

⇒ 0.4 × 300 = 120

∴ The correct answer is 120.

Alternate Method

Given:

Base Number = 600

First Step: Find 50% of the number.

Second Step: Find the number which is 60% less than the result of the first step.

Step 1: Calculate 50% of 600

⇒ Value₁ = (50 ÷ 100) × 600

⇒ Value₁ = 0.5 × 600 = 300

Step 2: Calculate 60% less than Value₁

⇒ Final Value = Value₁ − (60% of Value₁)

⇒ Final Value = 300 − (0.6 × 300)

⇒ Final Value = 300 − 180

⇒ Final Value = 120

∴ The correct answer is 120.

Additional Information

Percentage Decrease Formula

To find a value x% less than A, use the formula: Result = A × (100 − x) ÷ 100.

Successive Percentages

When multiple percentage changes are applied, the final multiplier is the product of individual multipliers (e.g., 0.5 × 0.4 = 0.2 of original).

Fraction Equivalents

Memorizing common fractions like 50% = 1/2 and 60% = 3/5 helps in faster mental calculation.

Shortcut Trick

Since 50% of the eggs are rotten, the remaining 50% are NOT rotten.

In mathematics, 50% is equivalent to half of the total value.

Number of eggs NOT rotten = 450 ÷ 2 = 225.

∴ The correct answer is 225 eggs.

Alternate Method

Given:

Total number of eggs = 450

Percentage of rotten eggs = 50%

Formula Used:

Number of items = (Percentage ÷ 100) × Total Quantity

Calculations:

⇒ Total eggs = 450

⇒ Number of rotten eggs = 50% of 450

⇒ Rotten eggs = (50 ÷ 100) × 450 = 225

⇒ Number of eggs NOT rotten = Total eggs − Rotten eggs

⇒ Number of eggs NOT rotten = 450 − 225 = 225

∴ The correct answer is 225 eggs.

Additional Information

Percentage Basics

Percentage means per hundred. To find x% of a value y, the calculation is (x × y) ÷ 100.

Fraction Equivalents

Converting common percentages to fractions can speed up calculations: 50% = 1/2, 25% = 1/4, 75% = 3/4, and 20% = 1/5.

Complementary Percentage

If a certain category is A% of a total, the remaining category is always (100 − A)% of that same total.

Shortcut Trick

If both the Cost Price and the Selling Price are multiplied by the same scaling factor, the Profit Percentage remains constant.

Decreasing both prices by 50 percent is equivalent to multiplying them by a factor of 0.5.

Since both are scaled equally, the ratio of Selling Price to Cost Price does not change.

∴ The correct answer is 40 percent.

Alternate Method

Given:

Initial Profit Percentage = 40 percent

Decrease in Cost Price (CP) = 50 percent

Decrease in Selling Price (SP) = 50 percent

Formula Used:

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

Profit Percentage = (Profit ÷ Cost Price) × 100

Calculations:

⇒ Let the initial Cost Price (CP) = 100 units

⇒ Since Profit is 40%, initial Selling Price (SP) = 100 + (40% of 100) = 140 units

⇒ New CP = 100 − (50% of 100) = 50 units

⇒ New SP = 140 − (50% of 140) = 70 units

⇒ New Profit = New SP − New CP = 70 − 50 = 20 units

⇒ New Profit Percentage = (New Profit ÷ New CP) × 100

⇒ New Profit Percentage = (20 ÷ 50) × 100 = 40 percent

∴ The correct answer is 40 percent.

Additional Information

Profit and Loss Ratios

The profit percentage depends entirely on the ratio of SP to CP. If SP/CP = k, then Profit % = (k − 1) × 100.

Successive Changes in CP and SP

If CP changes by x% and SP changes by y%, the new profit percentage will change unless x = y.

Marked Price Relationship

SP = MP × (1 − Discount/100). Profit is always calculated on the CP unless stated otherwise.

Shortcut Trick

If both the Cost Price (CP) and Selling Price (SP) of an article are increased or decreased by the same percentage, the Profit or Loss percentage remains constant.

This occurs because the ratio between the Selling Price and Cost Price (SP/CP) does not change when both terms are multiplied by the same factor.

Initial loss = 30% ⇒ Final loss = 30%.

∴ The correct answer is 30%.

Alternate Method

Given: Initial loss = 30%, decrease in both CP and SP = 40%.

⇒ Let Initial CP = 100

⇒ Initial SP = 100 − 30% of 100 = 70

⇒ New CP = 100 − (40% of 100) = 100 − 40 = 60

⇒ New SP = 70 − (40% of 70) = 70 − 28 = 42

⇒ New Loss = New CP − New SP = 60 − 42 = 18

⇒ New Loss % = (New Loss ÷ New CP) × 100

⇒ New Loss % = (18 ÷ 60) × 100 = 3 ÷ 10 × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Invariant Percentage Rule

When every component of a ratio (like SP and CP) is multiplied or divided by a constant non-zero value, the percentage relationship between them remains unchanged.

Loss Percentage Calculation

Loss % = [(CP − SP) ÷ CP] × 100. Always ensure the Cost Price is the denominator unless the problem specifies calculation on the selling price.

Selling Price for Loss

SP = CP × (100 − Loss%) ÷ 100. This formula helps calculate the price needed to maintain a specific loss threshold.

Shortcut Trick

Use the Successive Remainder method to find the final remaining part.

Remaining fraction after Day 1 = 1 − 2/3 = 1/3

Remaining fraction after Day 2 = 1/3 × (1 − 1/4) = 1/3 × 3/4 = 1/4

Remaining work = Total work × 1/4 = 84 × 1/4 = 21 units.

∴ The correct answer is 21 units.

Alternate Method

Given: Total work = 84 units, Day 1 work = 2/3 of total, Day 2 work = 1/4 of remaining.

⇒ Work done on Day 1 = 2/3 × 84 = 56 units

⇒ Remaining work after Day 1 = 84 − 56 = 28 units

⇒ Work done on Day 2 = 1/4 × 28 = 7 units

⇒ Final remaining work = 28 − 7 = 21 units

∴ The correct answer is 21 units.

Additional Information

Work-Fraction Relation

Total Work is always considered as 1 unit or 100%. If a fraction 'x' is done, then (1 − x) is the remaining work.

Successive Remainder Formula

If fractions a, b, c... are taken from the remaining amount successively, the final remainder is Total × (1−a) × (1−b) × (1−c)...

Work and Efficiency

Total Work = Efficiency × Time. In fractional problems, the denominator usually represents the 'Total units' relative to that specific step.

Shortcut Trick

Ratio of work done by A and B = (Time × Efficiency)_A : (Time × Efficiency)_B

Work Ratio = (5 days × 3 units) : (6 days × 2 units) = 15 : 12 = 5 : 4

Difference in ratio parts = 5 − 4 = 1 part

Given, 1 part = ₹600

Share of A = 5 parts = 5 × 600 = ₹3000

∴ The correct answer is ₹3000.

Alternate Method

Given: A worked for 5 days with 3 units/day efficiency; B worked for 6 days with 2 units/day efficiency. Difference in their shares = ₹600.

Formula: Total Work = Time × Efficiency | Share ∝ Total Work Done

⇒ Work done by A = 5 × 3 = 15 units

⇒ Work done by B = 6 × 2 = 12 units

⇒ Difference in work = 15 − 12 = 3 units

⇒ According to the question, 3 units = ₹600

⇒ 1 unit = 600 ÷ 3 = ₹200

⇒ Share of A = 15 units × 200 = ₹3000

∴ The correct answer is ₹3000.

Additional Information

Work and Wages Rule

Wages are always distributed in the ratio of the actual work performed. If the time worked is different for each person, use the formula: Wages Ratio = (E₁ × T₁) : (E₂ × T₂).

Efficiency and Wage Relationship

If all persons work for the same duration of time, their wages are distributed directly in the ratio of their individual efficiencies.

Total Wage Calculation

Total Wage = Wage per unit work × Total work units. Alternatively, Total Wage = Sum of individual shares.

Shortcut Trick

For a boat moving with the flow, Downstream Speed = Speed in Still Water + Speed of Stream.

Downstream Speed = 17 + 5 = 22 km/hr.

∴ The correct answer is 22 km/hr.

Alternate Method

Given:

Speed of the boat in still water (u) = 17 km/hr

Speed of the stream (v) = 5 km/hr

Formula Used:

Downstream speed (D_s) = u + v

Calculations:

⇒ Downstream speed = 17 + 5

⇒ Downstream speed = 22 km/hr

∴ The correct answer is 22 km/hr.

Additional Information

Upstream Speed

When the boat moves against the flow of the stream: Upstream Speed = Speed in still water − Speed of stream.

Speed of Boat in Still Water

It can be calculated if downstream (D) and upstream (U) speeds are known: Speed in still water = (D + U) ÷ 2.

Speed of Stream

It can be calculated as: Speed of stream = (D − U) ÷ 2.

Shortcut Trick

Initial speed = 36 km ÷ 1 hr = 36 km/hr.

Reduced speed = 36 − 20 = 16 km/hr.

Speed ratio = 36 : 16 = 9 : 4.

Since distance is constant, Time ratio = 4 : 9.

4 units = 60 min ⇒ 9 units = (60 ÷ 4) × 9 = 15 × 9 = 135 min.

∴ The correct answer is 135 min.

Alternate Method

Given:

Distance = 36 km

Initial time = 60 min = 1 hour

Decrease in speed = 20 km/hr

Formula Used:

Speed = Distance ÷ Time

Time = Distance ÷ Speed

Calculations:

⇒ Initial Speed = 36 km ÷ 1 hr = 36 km/hr

⇒ New Speed = 36 − 20 = 16 km/hr

⇒ New Time = Distance ÷ New Speed

⇒ New Time = 36 ÷ 16 hours

⇒ New Time in minutes = (36 ÷ 16) × 60

⇒ New Time = 2.25 × 60 = 135 min

∴ The correct answer is 135 min.

Additional Information

Speed-Time Relationship

When distance is constant, speed is inversely proportional to time (S₁ × T₁ = S₂ × T₂).

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert minutes to hours, divide by 60.

Average Speed

For two equal distances covered at speeds x and y, Average Speed = 2xy ÷ (x + y).

Shortcut Trick

Initial Ratio A : B = 5 : 3 (Difference = 2 units)

Final Ratio A : B = 3 : 2 = 6 : 4 (Multiplied by 2 to make difference = 2 units)

Increase in each part = 6 − 5 = 1 unit → 5

Sum of initial values = 5 + 3 = 8 units

Total Sum = 8 × 5 = 40

∴ The correct answer is 40.

Alternate Method

Given: Initial Ratio of A and B is 5 : 3. Both increased by 5, new Ratio is 3 : 2.

Let the initial values be A = 5x and B = 3x.

According to the question:

⇒ (5x + 5) ÷ (3x + 5) = 3 ÷ 2

⇒ 2 × (5x + 5) = 3 × (3x + 5)

⇒ 10x + 10 = 9x + 15

⇒ 10x − 9x = 15 − 10

⇒ x = 5

⇒ Initial Value of A = 5 × 5 = 25

⇒ Initial Value of B = 3 × 5 = 15

⇒ Sum = 25 + 15 = 40

∴ The correct answer is 40.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is calculated as √(a × b).

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Componendo and Dividendo

If a/b = c/d, then according to the rule, (a + b) / (a − b) = (c + d) / (c − d).

Shortcut Trick

Let Marked Price (MP) = 14 units ⇒ Discount = 5 units.

Selling Price (SP) = 14 − 5 = 9 units.

Given Profit = 40% ⇒ CP = SP ÷ 1.4 = 9 ÷ 1.4 = 45/7 units.

Difference = 14 − 45/7 = 53/7 units.

Percentage less = (53/7 ÷ 14) × 100 = 5300 ÷ 98 ≈ 54.08%.

∴ The correct answer is 54.08%.

Alternate Method

Given:

Profit percentage = 40%

Discount = 5/14 of Marked Price (MP)

⇒ Let Marked Price (MP) = 14x

⇒ Discount = (5/14) × 14x = 5x

⇒ Selling Price (SP) = MP − Discount = 14x − 5x = 9x

⇒ Profit = 40%, so SP = 1.4 × Cost Price (CP)

⇒ 9x = 1.4 × CP

⇒ CP = 9x ÷ 1.4 = 90x ÷ 14 = 45x/7

⇒ Difference between MP and CP = 14x − 45x/7 = (98x − 45x)/7 = 53x/7

⇒ Required Percentage = (Difference ÷ MP) × 100

⇒ Required Percentage = (53x/7 ÷ 14x) × 100 = 53/98 × 100

⇒ Required Percentage = 5400 ÷ 98 ≈ 54.08%

∴ The correct answer is 54.08%.

Additional Information

Relationship between CP and MP

The ratio of Marked Price to Cost Price can be expressed as MP/CP = (100 + Profit%)/(100 − Discount%).

Markup Percentage

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

Successive Discounts

If two discounts d₁ and d₂ are given, the total effective discount is d₁ + d₂ − (d₁ × d₂)/100.

Shortcut Trick

Ratio CP : MP = 100 : 150 = 2 : 3.

Ratio CP : SP = 100 : 125 = 4 : 5 (since Profit is 25%).

Equating CP in both ratios: CP : SP : MP = 4 : 5 : 6.

Discount = MP − SP = 6 − 5 = 1 unit.

Discount % = (1 ÷ 6) × 100 = 16.67%.

∴ The correct answer is 16.67 percent.

Alternate Method

Given:

Markup Percentage = 50%

Profit Percentage = 25%

Formula Used:

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

Discount = Marked Price (MP) − Selling Price (SP)

Discount % = (Discount ÷ Marked Price) × 100

Calculations:

⇒ Let the Cost Price (CP) of the article be 100.

⇒ Marked Price (MP) = 100 + 50% of 100 = 150.

⇒ Selling Price (SP) = 100 + 25% of 100 = 125.

⇒ Discount Amount = 150 − 125 = 25.

⇒ Discount Percentage = (25 ÷ 150) × 100.

⇒ Discount Percentage = (1 ÷ 6) × 100 = 16.666...%

⇒ Discount Percentage ≈ 16.67%.

∴ The correct answer is 16.67 percent.

Additional Information

Markup and Profit Relation

The relationship between Marked Price (MP) and Cost Price (CP) can be expressed as: MP / CP = (100 + Profit %) / (100 − Discount %).

Effective Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is given by: (a + b − ab/100)%.

Profit and Discount Calculation

Profit is always calculated on the Cost Price (CP), whereas the Discount is always calculated on the Marked Price (MP).

Shortcut Trick

The Fourth Proportional (d) of three numbers a, b, and c is given by the formula:

d = (b × c) ÷ a

Here, a = 18, b = 4, and c = 45.

⇒ Fourth Proportional = (4 × 45) ÷ 18 = 180 ÷ 18 = 10

∴ The correct answer is 10.

Alternate Method

Given: The three numbers are 18, 4, and 45.

Formula Used: If four numbers a, b, c, and d are in proportion, then a : b :: c : d, which implies a ÷ b = c ÷ d or a × d = b × c.

Let the fourth proportion be x.

⇒ 18 : 4 :: 45 : x

⇒ 18 ÷ 4 = 45 ÷ x

⇒ 18 × x = 45 × 4

⇒ 18x = 180

⇒ x = 180 ÷ 18

⇒ x = 10

∴ The correct answer is 10.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is calculated as √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a : b = b : c), then c is the third proportional, given by c = b² ÷ a.

Invertendo and Alternendo

If a : b = c : d, then b : a = d : c (Invertendo) and a : c = b : d (Alternendo).