Quantitative Aptitude PYQs

Shortcut Trick

Solve using BODMAS: 61 + 45 − 32 + (45) × 2 − 84.

Directly calculate the product: 45 × 2 = 90.

Group positive and negative values: (61 + 45 + 90) − (32 + 84).

Final calculation: 196 − 116 = 80.

∴ The correct answer is 80.

Alternate Method

Given Expression: 61 + 45 − 32 + (28 + 17) × 2 − 84

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

⇒ 61 + 45 − 32 + (45) × 2 − 84

⇒ 61 + 45 − 32 + 90 − 84

⇒ 106 − 32 + 90 − 84

⇒ 74 + 90 − 84

⇒ 164 − 84

⇒ 80

∴ The correct answer is 80.

Additional Information

BODMAS Priority

Operations must be performed in the sequence of Brackets, Orders, Division, Multiplication, Addition, and Subtraction to avoid incorrect results.

Bracket Hierarchy

Always solve the innermost bracket first, usually in the order of small (), curly {}, and then square brackets [].

Addition and Subtraction Associativity

Since addition and subtraction have the same precedence, the expression is typically evaluated from left to right.

Shortcut Trick

Identify the core operations using BODMAS: Bracket/Division and Multiplication.

Solve (27 ÷ 3) = 9 and 28 × 1 = 28.

The simplified expression becomes: 94 − 47 + 31 + 9 + 28 − 19.

Group positive and negative values: (94 + 31 + 9 + 28) − (47 + 19) = 162 − 66 = 96.

∴ The correct answer is 96.

Alternate Method

Given: 94 − 47 + 31 + (27 ÷ 3) + 28 × 1 − 19

Formula Used: The BODMAS rule defines the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

⇒ 94 − 47 + 31 + 9 + 28 × 1 − 19 (Solving Bracket/Division)

⇒ 94 − 47 + 31 + 9 + 28 − 19 (Solving Multiplication)

⇒ 125 + 9 + 28 − 47 − 19 (Applying Addition)

⇒ 162 − 47 − 19 (Continuing Addition)

⇒ 162 − 66 (Combining Subtraction terms)

⇒ 96

∴ The correct answer is 96.

Additional Information

BODMAS Rule

It ensures mathematical expressions are solved consistently by following the sequence: Brackets (), Of/Orders (x², √), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Sign Management

When multiple additions and subtractions are present, it is often faster to group all positive numbers and all negative numbers separately before finding the final difference.

Shortcut Trick

Break the square root into individual components using the property √(a × b) = √a × √b.

Calculate √1.44 = 1.2 and √0.0225 = 0.15.

Multiply the results: 1.2 × 0.15 = 0.18.

∴ The correct answer is 0.18.

Alternate Method

Given:

Expression = √(1.44 × 0.0225)

Formula Used:

√(a × b) = √a × √b

Calculations:

⇒ √(1.44 × 0.0225) = √(144/100 × 225/10000)

⇒ √(144/100) × √(225/10000)

⇒ (12/10) × (15/100)

⇒ 1.2 × 0.15

⇒ 0.18

∴ The correct answer is 0.18.

Additional Information

Square Root of Decimals

When finding the square root of a decimal, the number of decimal places in the result is exactly half the number of decimal places in the original number.

Product Property of Radicals

For any non-negative numbers a and b, the square root of their product is equal to the product of their individual square roots: √(a × b) = √a × √b.

Perfect Squares

Memorizing basic perfect squares (like 12² = 144 and 15² = 225) is essential for solving simplification problems quickly.

Shortcut Trick

Directly evaluate each decimal root based on the number of decimal places:

√0.09 = 0.3 (since 0.3² = 0.09)

³√0.125 = 0.5 (since 0.5³ = 0.125)

⁴√0.0081 = 0.3 (since 0.3⁴ = 0.0081)

√0.64 = 0.8 (since 0.8² = 0.64)

Expression = 0.3 + 0.5 + 0.3 − 0.8 = 1.1 − 0.8 = 0.3

∴ The correct answer is 0.3.

Alternate Method

Given: √0.09 + ³√0.125 + ⁴√0.0081 − √0.64

Calculation:

⇒ Step 1: Simplify √0.09 = √(9/100) = 3/10 = 0.3

⇒ Step 2: Simplify ³√0.125 = ³√(125/1000) = 5/10 = 0.5

⇒ Step 3: Simplify ⁴√0.0081 = ⁴√(81/10000) = 3/10 = 0.3

⇒ Step 4: Simplify √0.64 = √(64/100) = 8/10 = 0.8

⇒ Step 5: Substitute values: 0.3 + 0.5 + 0.3 − 0.8

⇒ Sum of positive terms = 0.3 + 0.5 + 0.3 = 1.1

⇒ Final calculation = 1.1 − 0.8 = 0.3

∴ The correct answer is 0.3.

Additional Information

Square Roots of Decimals

When taking a square root, the number of decimal places is halved. Example: √0.0001 (4 places) = 0.01 (2 places).

Cube Roots of Decimals

When taking a cube root, the number of decimal places is divided by 3. Example: ³√0.008 (3 places) = 0.2 (1 place).

BODMAS Rule

Always follow the order of Brackets, Orders, Division, Multiplication, Addition, and Subtraction for mixed operator expressions.

Shortcut Trick

For a rhombus, the relation between the side (a), area (A), and the sum of diagonals (d₁+d₂) is given by:

(d₁ + d₂)² = 4a² + 4A

⇒ (d₁ + d₂)² = 4 × 29² + 4 × 840

⇒ (d₁ + d₂)² = 3364 + 3360 = 6724

⇒ d₁ + d₂ = √6724 = 82 cm

∴ The correct answer is 82 cm.

Alternate Method

Given:

⇒ Area of the rhombus = 840 cm²

⇒ Side of the rhombus (a) = 29 cm

Formula used:

⇒ Area = (1/2) × d₁ × d₂

⇒ Side² = (d₁/2)² + (d₂/2)² ⇒ 4a² = d₁² + d₂²

Calculations:

⇒ Area = (1/2) × d₁ × d₂ = 840

⇒ d₁ × d₂ = 1680

⇒ 2 × d₁ × d₂ = 3360

⇒ d₁² + d₂² = 4 × (Side)² = 4 × 29²

⇒ d₁² + d₂² = 4 × 841 = 3364

⇒ (d₁ + d₂)² = d₁² + d₂² + 2d₁d₂

⇒ (d₁ + d₂)² = 3364 + 3360 = 6724

⇒ d₁ + d₂ = √6724

⇒ d₁ + d₂ = 82 cm

∴ The correct answer is 82 cm.

Additional Information

Properties of Rhombus

All sides are equal and opposite sides are parallel. The diagonals bisect each other at 90°.

Perimeter of Rhombus

The perimeter of a rhombus is given by 4 × Side (a).

Relationship between Diagonals and Side

The side and diagonals are related by the formula 4a² = d₁² + d₂², derived from the Pythagoras theorem.

Shortcut Trick

The Area of a parallelogram is directly calculated as the product of its base and its corresponding height.

Area = 12 × 7 = 84 cm²

∴ The correct answer is 84 cm².

Alternate Method

Given:

Base (b) = 12 cm

Height (h) = 7 cm

The formula for the Area of a parallelogram is:

Area = Base × Height

⇒ Area = 12 × 7 ⇒ Area = 84 cm²

∴ The correct answer is 84 cm².

Additional Information

Perimeter of a Parallelogram

The total boundary length is calculated as 2 × (a + b), where a and b are the lengths of adjacent sides.

Relationship with Triangles

A diagonal of a parallelogram divides it into two congruent triangles of equal area. Thus, Area = 2 × (Area of one triangle).

Angle Properties

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (their sum is 180°).

Shortcut Trick

To find the Least Common Multiple (LCM) of fractions, use the formula:

LCM = LCM of Numerators ÷ HCF of Denominators

Numerators: 3, 4, 8 ⇒ LCM = 24

Denominators: 5, 15, 25 ⇒ HCF = 5

Required Result = 24/5

∴ The correct answer is 24/5.

Alternate Method

Given Fractions: 3/5, 4/15, and 8/25

Detailed Calculation:

⇒ Numerators are 3, 4, and 8. The Least Common Multiple of 3, 4, and 8 is 24 (since 24 is the smallest number divisible by 3, 4, and 8).

⇒ Denominators are 5, 15, and 25. The Highest Common Factor (HCF) of 5, 15, and 25 is 5 (since 5 is the largest number that divides 5, 15, and 25 exactly).

⇒ Applying the Fraction LCM Rule:

⇒ LCM = 24 ÷ 5 = 24/5

∴ The correct answer is 24/5.

Additional Information

HCF of Fractions

The formula to find the HCF of fractions is HCF(Numerators) ÷ LCM(Denominators).

Simplified Form Rule

Always ensure the fractions are in their lowest terms (simplest form) before applying the LCM or HCF formulas.

Product of Numbers Property

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

Shortcut Trick

Ratio of numbers = 3 : 5. Let the numbers be 3x and 5x.

LCM of (3x, 5x) = 15x.

Given 15x = 120 ⇒ x = 8.

Sum of the numbers = (3 + 5) × x = 8 × 8 = 64.

∴ The correct answer is 64.

Alternate Method

Given: Ratio = 3 : 5, LCM = 120.

Formula Used: LCM = Product of ratio parts × HCF (when ratio is in its simplest form).

⇒ Let the two numbers be 3x and 5x.

⇒ LCM of 3x and 5x = 120

⇒ 15x = 120

⇒ x = 120 ÷ 15 = 8

⇒ First number = 3 × 8 = 24

⇒ Second number = 5 × 8 = 40

⇒ Sum of the numbers = 24 + 40 = 64

∴ The correct answer is 64.

Additional Information

HCF and LCM Relation

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

Ratio and Multiples

If the ratio of two numbers is a : b and their HCF is H, then the numbers are aH and bH, and their LCM is abH.

Coprime Numbers

If two numbers are coprime (HCF = 1), then their LCM is simply the product of the numbers.

Shortcut Trick

Combine the percentages first: 30% × 20% = 6% (or 0.06).

Calculate 2/5 of 4500: 4500 ÷ 5 × 2 = 1800.

Find 6% of the result: 6/100 × 1800 = 6 × 18 = 108.

∴ The correct answer is 108.

Alternate Method

Given: Total value = 4500, Fractions/Percentages = 2/5, 20%, 30%

Formula: Value = (P₁/100) × (P₂/100) × (Fraction) × Number

⇒ Result = 30/100 × 20/100 × 2/5 × 4500

⇒ Result = 3/10 × 1/5 × 2/5 × 4500

⇒ Result = 3/10 × 1/5 × 1800

⇒ Result = 3/10 × 360

⇒ Result = 3 × 36

⇒ Result = 108

∴ The correct answer is 108.

Additional Information

Successive Percentages

When calculating multiple percentages of a number, you can multiply the decimal equivalents directly: a% of b% of X = (a × b / 10000) × X.

Fraction-to-Percentage Conversion

Common fractions like 2/5 are equal to 40%. Converting fractions to percentages can often simplify the mental math.

BODMAS in Percentage

The word 'of' in mathematical problems always signifies multiplication and should be solved sequentially or simultaneously in such expressions.

Shortcut Trick

Identify the ratio between B and A directly: B : A = 5X : 2X = 5 : 2.

To find B as a percent of A, use the ratio: (5 ÷ 2) × 100.

Calculation: 2.5 × 100 = 250.

∴ The correct answer is 250 percent.

Alternate Method

Given:

Value of A = 2X

Value of B = 5X

Formula Used:

Percentage = (Value ÷ Base Value) × 100

Calculations:

⇒ Required Percentage = (B ÷ A) × 100

⇒ Required Percentage = (5X ÷ 2X) × 100

⇒ Required Percentage = (5 ÷ 2) × 100

⇒ Required Percentage = 2.5 × 100

⇒ Required Percentage = 250%

∴ The correct answer is 250 percent.

Additional Information

Percentage Base Rule

The value following the word 'of' is always treated as the denominator (base) in percentage calculations.

Percentage More/Less Than

If asked how much percent more B is than A, the formula is: [(B − A) ÷ A] × 100. Here, (5−2)/2 × 100 = 150%.

Fraction to Percentage Conversion

Common fractions: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, and 5/2 = 2.5 = 250%.

Shortcut Trick

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

Initial Profit = 10% ⇒ Ratio CP : SP = 100 : 110 = 10 : 11.

Both values are decreased by 20%, which means they are multiplied by a factor of 0.8.

New Ratio = (10 × 0.8) : (11 × 0.8) = 8 : 8.8 = 10 : 11.

Since the ratio remains 10 : 11, the profit percentage remains 10%.

∴ The correct answer is 10%.

Alternate Method

Given:

Initial profit percentage = 10%

Reduction in Cost Price (CP) = 20%

Reduction in Selling Price (SP) = 20%

Formula Used:

SP = CP × (100 + Profit%) ÷ 100

Profit% = [(SP − CP) ÷ CP] × 100

⇒ Let the Initial Cost Price (CP₁) = 100 ⇒ Since Profit is 10%, Initial Selling Price (SP₁) = 100 + 10 = 110 ⇒ New Cost Price (CP₂) = 100 − 20% of 100 = 100 − 20 = 80 ⇒ New Selling Price (SP₂) = 110 − 20% of 110 = 110 − 22 = 88 ⇒ New Profit = SP₂ − CP₂ = 88 − 80 = 8 ⇒ New Profit Percentage = (New Profit ÷ New CP) × 100 ⇒ New Profit % = (8 ÷ 80) × 100 = (1 ÷ 10) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Effect of Scaling on Percentages

If all terms in a ratio (like CP and SP) are multiplied or divided by the same non-zero constant, the ratio remains equivalent, and the resulting percentage (Profit or Loss) remains unchanged.

Marked Price and Discount

Selling Price can also be calculated as Marked Price (MP) × (100 − Discount%) ÷ 100. Profit is always calculated on CP, while Discount is calculated on MP.

Net Profit after Successive Changes

If CP changes by x% and SP changes by y%, the new profit percentage must be found by calculating the new absolute values of CP and SP individually.

Shortcut Trick

When the total amount is constant, the ratio of prices is the inverse of the ratio of articles.

Ratio of Cost Price : Selling Price = 25 : 20 = 5 : 4

Loss = 5 − 4 = 1 unit

Loss % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20% loss.

Alternate Method

Given:

Cost Price (CP) of 20 articles = ₹260

Selling Price (SP) of 25 articles = ₹260

Formula Used:

Loss = CP − SP

Loss Percentage = (Loss ÷ CP) × 100

⇒ Cost Price of 1 article = 260 ÷ 20 = ₹13

⇒ Selling Price of 1 article = 260 ÷ 25 = ₹10.4

⇒ Loss = 13 − 10.4 = ₹2.6

⇒ Loss Percentage = (2.6 ÷ 13) × 100

⇒ Loss Percentage = 0.2 × 100 = 20%

∴ The correct answer is 20% loss.

Additional Information

Profit and Loss Basics

Profit occurs when SP > CP. Loss occurs when CP > SP. Both are always calculated on the CP unless stated otherwise.

Relationship Formula

SP = CP × (100 ± Profit or Loss %) ÷ 100. This is the direct way to find SP from CP and percentage.

Inverse Variation in Articles

If CP of 'x' articles = SP of 'y' articles, then Profit or Loss % = (x − y) ÷ y × 100. Positive result is profit, negative is loss.

Shortcut Trick

Total efficiency of A, B and C = 30/10 + 30/15 + 30/30 = 3 + 2 + 1 = 6 units/day.

Total work = LCM(10, 15, 30) = 30 units.

Required work = 30% of 30 = 9 units.

Time taken = 9 ÷ 6 = 3/2 days.

∴ The correct answer is 3/2 days.

Alternate Method

Given: Time taken by A = 10 days, B = 15 days, and C = 30 days.

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

⇒ Total work to be completed = 30 units

⇒ 30% of the work = 30 × (30/100) = 9 units

⇒ Time taken by A, B and C together = Required Work ÷ Total Efficiency

⇒ Time = 9 ÷ 6

⇒ Time = 3/2 days

∴ The correct answer is 3/2 days.

Additional Information

Relationship between Time and Efficiency

Efficiency is inversely proportional to the time taken. If time ratio is x : y, efficiency ratio is y :

x. Total Work Formula

Total Work = Efficiency × Time. This is the fundamental unit for all Time and Work problems.

Combined Efficiency

When people work together, their individual efficiencies are added to find the net rate of work.

Shortcut Trick

Since the work to be done is the same (half part), use the relation: M₁ × D₁ = M₂ × D₂.

⇒ 10 women × 20 days = M₂ × 4 days

⇒ 200 = M₂ × 4

⇒ M₂ = 200 ÷ 4 = 50 women

⇒ Additional women needed = 50 − 10 = 40.

∴ The correct answer is 40 women.

Alternate Method

Given:

Initial workers (M₁) = 10 women

Initial time (D₁) = 20 days

Work completed (W₁) = 1/2 part

Target time (D₂) = 4 days

Remaining work (W₂) = 1 − 1/2 = 1/2 part

Formula Used:

(M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

Calculations:

⇒ Since W₁ = W₂, we have: M₁ × D₁ = M₂ × D₂

⇒ 10 × 20 = M₂ × 4

⇒ 200 = 4 × M₂

⇒ M₂ = 200 ÷ 4

⇒ M₂ = 50 women

⇒ More women required = Total women − Initial women

⇒ More women required = 50 − 10 = 40

∴ The correct answer is 40 women.

Additional Information

The Chain Rule

It is a method to solve problems based on work and time using the formula (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Inverse Proportion

When the work is constant, the number of people required is inversely proportional to the number of days taken (M ∝ 1/D).

Work and Efficiency

Total Work = Number of Persons × Days × Efficiency. If efficiency of each woman is the same, work is directly proportional to the product of workers and days.

Shortcut Trick

Direct conversion: Distance (m) = Speed (km/hr) × (5/18) × Time (sec)

Length = 54 × (5/18) × (6 × 60)

Length = 15 × 360 = 5400 m

∴ The correct answer is 5400 m.

Alternate Method

Given:

Speed of the boy = 54 km/hr

Time taken to cross the bridge = 6 minutes

Formula Used:

Distance = Speed × Time

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

To convert minutes to seconds, multiply by 60.

Calculations:

⇒ Speed in m/s = 54 × (5/18)

⇒ Speed = 3 × 5 = 15 m/s

⇒ Time in seconds = 6 × 60 = 360 seconds

⇒ Length of the bridge = 15 × 360

⇒ Length = 5400 metres

∴ The correct answer is 5400 m.

Additional Information

Unit Conversion

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

Distance-Speed-Time Relation

The fundamental formula is Distance = Speed × Time. Ensure all units are in the same system (e.g., all in meters and seconds) before calculating.

Average Speed

If an object covers equal distances at speeds x and y, the average speed is (2xy) ÷ (x + y).

Shortcut Trick

Speed in m/s = 90 × 5/18 = 25 m/s.

Total distance covered in 54 seconds = 25 × 54 = 1350 m.

Ratio of Train length to Platform length = 1 : 2.

Total ratio parts = 1 + 2 = 3 parts.

Length of the train = (1/3) × 1350 = 450 m.

∴ The correct answer is 450 m.

Alternate Method

Given:

Speed of the train = 90 km/hr

Crossing time = 54 seconds

Platform length = 2 × Train length

Formula Used:

Distance = Speed × Time

Total Distance = Length of Train + Length of Platform

Calculations:

⇒ Speed = 90 × (5/18) m/s = 25 m/s

⇒ Let the length of the train be L metres.

⇒ Length of the platform = 2L metres.

⇒ Total Distance = L + 2L = 3L

⇒ 3L = Speed × Time

⇒ 3L = 25 × 54

⇒ 3L = 1350

⇒ L = 1350 ÷ 3 = 450 m

∴ The correct answer is 450 m.

Additional Information

Speed Conversion

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

Train Crossing Stationary Object

When a train crosses a pole or a standing man, the distance covered equals the length of the train itself.

Relative Speed in Same Direction

If two trains are moving in the same direction, their relative speed is the difference of their individual speeds (V₁ − V₂).

Relative Speed in Opposite Direction

If two trains are moving in opposite directions, their relative speed is the sum of their individual speeds (V₁ + V₂).

Shortcut Trick

Total units in Ratio = 2 + 4 + 5 + 7 = 18 units

Given Sum: 18 units = 54 ⇒ 1 unit = 3

Difference between Largest (7) and Smallest (2) = 7 − 2 = 5 units

Required Difference = 5 × 3 = 15

∴ The correct answer is 15.

Alternate Method

Given:

Ratio of four numbers = 2 : 4 : 5 : 7

Sum of numbers = 54

Let the four numbers be 2x, 4x, 5x, and 7x.

Sum of numbers = 2x + 4x + 5x + 7x = 54

⇒ 18x = 54

⇒ x = 54 ÷ 18

⇒ x = 3

Smallest number = 2x = 2 × 3 = 6

Largest number = 7x = 7 × 3 = 21

Difference = 21 − 6 = 15

∴ The correct answer is 15.

Additional Information

Ratio and Proportional Parts

To divide a quantity into parts based on a ratio a : b : c, the individual parts are (a/Sum) × Total, (b/Sum) × Total, etc., where Sum = a + b + c.

Mean Proportional

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

Compound Ratio

The compound ratio of two ratios a : b and c : d is found by multiplying corresponding terms: (a × c) : (b × d).

Shortcut Trick

Selling Price (SP) = 125% of Cost Price (CP) = 1.25 × 3000 = ₹3750

Discount = Marked Price (MP) − SP = 5000 − 3750 = ₹1250

Discount Percentage = (Discount / MP) × 100 = (1250 / 5000) × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Marked Price (MP) = ₹5000

Cost Price (CP) = ₹3000

Profit Percentage = 25%

Formula Used:

1. Selling Price (SP) = CP × (100 + Profit %)/100
2. Discount Percentage = [(MP − SP) / MP] × 100

Calculations:

⇒ SP = 3000 × (125 / 100)

⇒ SP = ₹3750

⇒ Discount = MP − SP = 5000 − 3750 = ₹1250

⇒ Discount Percentage = (1250 / 5000) × 100

⇒ Discount Percentage = 25%

∴ The correct answer is 25%.

Additional Information

Relationship between CP and MP

The ratio of Marked Price to Cost Price is given by: MP / CP = (100 + Profit %) / (100 − Discount %).

Markup Percentage

Markup % is the extra price added to CP to get MP: Markup % = [(MP − CP) / CP] × 100.

Successive Discounts

If two successive discounts of a% and b% are given, the net discount is (a + b − ab/100)%.

Shortcut Trick

Let the Cost Price (CP) be 100.

Since the Marked Price (MP) is 25% more, MP = 125.

To ensure no loss, the Selling Price (SP) must be at least equal to the CP (100).

Maximum Discount allowed = 125 − 100 = 25.

Maximum Discount % = (25 ÷ 125) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Markup = 25% above Cost Price (CP).

Formula Used:

Discount % = (Discount ÷ Marked Price) × 100

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

Calculations:

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

⇒ Marked Price (MP) = x + 25% of x = 1.25x.

⇒ For no loss, the minimum Selling Price (SP) must be equal to CP.

⇒ SP = x.

⇒ Maximum Discount = MP − SP = 1.25x − x = 0.25x.

⇒ Maximum Discount % = (0.25x ÷ 1.25x) × 100.

⇒ Discount % = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Relationship between CP, MP, Profit, and Discount

The ratio of Marked Price to Cost Price is given by: MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %).

Markup Percentage

It is the percentage by which the Cost Price is increased to set the Marked Price. Markup % = [(MP − CP) ÷ CP] × 100.

Condition for No Profit No Loss

A 'no profit no loss' scenario occurs when the Selling Price (SP) is exactly equal to the Cost Price (CP).

Shortcut Trick

The fourth proportion (x) of three numbers a, b, and c is calculated using the formula: x = (b × c) ÷ a.

Given: a = 8, b = 10, and c = 16.

x = (10 × 16) ÷ 8

x = 160 ÷ 8 = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Three numbers: 8, 10, and 16.

Formula Used:

If four numbers a, b, c, and d are in proportion, then a : b :: c : d, which means (a ÷ b) = (c ÷ d) or Product of Extremes = Product of Means (a × d = b × c).

Let the fourth proportion be x.

⇒ 8 : 10 :: 16 : x

⇒ 8 ÷ 10 = 16 ÷ x

⇒ 8 × x = 10 × 16

⇒ 8x = 160

⇒ x = 160 ÷ 8

⇒ x = 20

∴ The correct answer is 20.

Additional Information

Mean Proportion

The mean proportion between two numbers a and b is given by √(a × b).

Third Proportion

The third proportion of two numbers a and b is the value x such that a : b :: b : x, which is x = b² ÷ a.

Continued Proportion

Three quantities a, b, and c are in continued proportion if a : b = b : c.