Quantitative Aptitude PYQs

Shortcut Trick

Solve using the BODMAS priority: Brackets first, then Multiplication.

Value of bracket (47 + 11) = 58.

Value of multiplication 49 × 2 = 98.

Final calculation: 53 + 72 − 36 + 98 − 58 = 129.

∴ The correct answer is 129.

Alternate Method

Given: 53 + 72 − 36 + 49 × 2 − (47 + 11)

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

⇒ 53 + 72 − 36 + 49 × 2 − 58 (Solving Bracket)

⇒ 53 + 72 − 36 + 98 − 58 (Performing Multiplication)

⇒ 125 − 36 + 98 − 58 (Addition/Subtraction from left to right)

⇒ 89 + 98 − 58

⇒ 187 − 58

⇒ 129

∴ The correct answer is 129.

Additional Information

BODMAS Rule

Standard order: Brackets (), Of/Orders (x², √), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Priority of Brackets

Always simplify terms inside brackets first. If there are multiple types, follow the order: Parentheses (), then Braces {}, then Brackets [].

VBODMAS

An extended version where 'V' stands for Vinculum (a bar bracket over numbers), which has the highest priority before parentheses.

Shortcut Trick

Perform calculations using the BODMAS order: Brackets, Of, Division, Addition/Subtraction.

Solve 36 ÷ 3 = 12 and 28 of 1 = 28.

Substitute back: 93 − 47 + 19 + 12 + 28 − 21.

Combine positive terms: 93 + 19 + 12 + 28 = 152.

Combine negative terms: −47 − 21 = −68.

Final result: 152 − 68 = 84.

∴ The correct answer is 84.

Alternate Method

Given: 93 − 47 + 19 + 36 ÷ 3 + (28 of 1) − 21

The BODMAS rule states the sequence of operations as Brackets, Of (Orders), Division, Multiplication, Addition, and Subtraction.

⇒ 93 − 47 + 19 + 36 ÷ 3 + 28 − 21 (Solving 'of')

⇒ 93 − 47 + 19 + 12 + 28 − 21 (Solving Division)

⇒ (93 + 19 + 12 + 28) − (47 + 21) (Group Addition and Subtraction separately)

⇒ 152 − 68

⇒ 84

∴ The correct answer is 84.

Additional Information

BODMAS Rule

It is a mnemonic used to remember the order of operations in calculations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

The 'Of' Operation

In mathematical expressions, 'of' acts as multiplication but is calculated after brackets and before division in the priority list.

Order of Addition and Subtraction

Addition and Subtraction are at the same priority level. For consistent results, they should be solved from left to right.

Shortcut Trick

Identify perfect squares: 225 is 15² and 4 is 2².

Equalize decimal places inside the root: √(0.0225 ÷ 0.0400).

Remove decimals: √(225 ÷ 400) = 15 ÷ 20.

Simplify: 15 ÷ 20 = 0.75.

∴ The correct answer is 0.75.

Alternate Method

Given: √(0.0225 ÷ 0.04)

Method: Convert decimals to fractions to simplify the square root expression.

⇒ √((225 ÷ 10000) ÷ (4 ÷ 100))

⇒ √((225 ÷ 10000) × (100 ÷ 4))

⇒ √(225 ÷ 400)

⇒ √225 ÷ √400

⇒ 15 ÷ 20

⇒ 3 ÷ 4

⇒ 0.75

∴ The correct answer is 0.75.

Additional Information

Square Root of Decimals

The square root of a decimal with 'n' decimal places will always result in a number with 'n/2' decimal places, provided the digits form a perfect square.

Division of Decimals

When dividing decimals, it is easier to multiply both the numerator and denominator by a power of 10 (like 100 or 10000) to remove the decimal points before calculation.

Perfect Squares to Remember

Memorizing squares up to 25 (e.g., 15² = 225, 25² = 625) significantly speeds up simplification and root-finding tasks.

Shortcut Trick

Recognize the expression as the algebraic identity: a² + b² + 2ab = (a + b)².

Let a = 4.10 and b = 1.90.

Verify the middle term: 2 × a × b = 2 × 4.10 × 1.90 = 2 × 7.79 = 15.58.

Expression = (4.10 + 1.90)² = (6.00)² = 36.00.

∴ The correct answer is 36.00.

Alternate Method

Given: 4.10 × 4.10 + 1.90 × 1.90 + 15.58

Method: Perform multiplication and addition as per BODMAS rules.

⇒ (4.10 × 4.10) + (1.90 × 1.90) + 15.58

⇒ 16.81 + 3.61 + 15.58

⇒ 20.42 + 15.58

⇒ 36.00

∴ The correct answer is 36.00.

Additional Information

Algebraic Identities

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² are fundamental expansion rules.

BODMAS Rule

The order of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Difference of Squares

a² − b² = (a − b)(a + b), which is useful for simplifying products of numbers centered around a mean.

Shortcut Trick

Product of diagonals = 2 × Area

20 × d₂ = 2 × 180

d₂ = 360 ÷ 20 = 18 cm

Ratio = 20 : 18 = 10 : 9

∴ The correct answer is 10 : 9.

Alternate Method

Given:

⇒ Area of rhombus = 180 cm²

⇒ First Diagonal (d₁) = 20 cm

Formula Used:

Area = (1/2) × d₁ × d₂

Calculations:

⇒ 180 = (1/2) × 20 × d₂

⇒ 180 = 10 × d₂

⇒ d₂ = 180 ÷ 10 = 18 cm

⇒ Required Ratio = d₁ : d₂

⇒ Ratio = 20 : 18 = 10 : 9

∴ The correct answer is 10 : 9.

Additional Information

Side of Rhombus

The side 'a' of a rhombus can be calculated using diagonals: a = (1/2) × √(d₁² + d₂²).

Perimeter of Rhombus

Since all sides are equal, the perimeter is given by 4 × side.

Diagonal Properties

The diagonals of a rhombus always bisect each other at right angles (90°).

Shortcut Trick

The side of the rhombus = 100 ÷ 4 = 25 cm.

The half of the given diagonal = 30 ÷ 2 = 15 cm.

Since diagonals bisect at 90°, we use the Pythagorean triple (3, 4, 5) scaled by 5: (15, 20, 25).

The other half-diagonal is 20 cm.

The total length of the other diagonal = 20 × 2 = 40 cm.

∴ The correct answer is 40 cm.

Alternate Method

Given: Perimeter = 100 cm and Diagonal (d₁) = 30 cm.

Formula Used:

1. Side (a) = Perimeter ÷ 4
2. a² = (d₁/2)² + (d₂/2)²

Calculations:

⇒ Side (a) = 100 ÷ 4 = 25 cm

⇒ Half of first diagonal = 30 ÷ 2 = 15 cm

⇒ Let the other half diagonal be x.

⇒ In right-angled triangle AOB: AO² + OB² = AB²

⇒ 15² + x² = 25²

⇒ 225 + x² = 625

⇒ x² = 625 − 225

⇒ x² = 400

⇒ x = √400 = 20 cm

⇒ Length of other diagonal (d₂) = 2 × x = 2 × 20 = 40 cm

∴ The correct answer is 40 cm.

Additional Information

Area of Rhombus

The area can be calculated using diagonals: Area = 1/2 × d₁ × d₂.

Relationship between Side and Diagonals

The formula relating side (a) and diagonals is: 4a² = d₁² + d₂².

Diagonals Property

In a rhombus, diagonals bisect each other at right angles (90°) and also bisect the vertex angles.

Shortcut Trick

To find the Least Common Multiple (LCM) of three numbers, we use the property: LCM(A, B, C) = LCM(LCM(A, B), C).

Given LCM(A, B) = 36 and C = 24.

We simply need to find LCM(36, 24).

List multiples of the larger number (36): 36, 72...

Since 72 is exactly divisible by 24 (24 × 3 = 72), the LCM is 72.

∴ The correct answer is 72.

Alternate Method

Given:

LCM of A and B = 36

Value of C = 24

Formula Used:

LCM(A, B, C) = LCM(LCM(A, B), C)

Calculations:

⇒ We need to calculate the LCM of 36 and 24 using Prime Factorization.

⇒ 36 = 2 × 2 × 3 × 3 = 2² × 3²

⇒ 24 = 2 × 2 × 2 × 3 = 2³ × 3¹

⇒ LCM is the product of the highest powers of all prime factors involved.

⇒ LCM = 2³ × 3²

⇒ LCM = 8 × 9

⇒ LCM = 72

∴ The correct answer is 72.

Additional Information

Relation between LCM and HCF

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(x, y) × LCM(x, y).

LCM of Fractions

LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators).

Co-prime Numbers

If two numbers are co-prime (HCF = 1), their LCM is simply the product of the two numbers.

Shortcut Trick

The Least Common Multiple (LCM) of any set of numbers must always be exactly divisible by their Highest Common Factor (HCF).

Given HCF = 12, we check which option is a multiple of 12.

144 ÷ 12 = 12 (Remains 0).

Since 144 is the only option divisible by 12, it can be the LCM.

∴ The correct answer is 144.

Alternate Method

Given:

HCF of two numbers = 12

Property Used:

If two numbers are 12x and 12y (where x and y are coprime), then their LCM = 12 × x × y.

Calculations:

⇒ Let the numbers be a and b.

⇒ a = 12p and b = 12q (where p, q are integers).

⇒ LCM(a, b) = 12 × LCM(p, q).

⇒ This implies LCM must be divisible by 12.

⇒ Option 1: 130 ÷ 12 = 10.83 (Not an integer)

⇒ Option 2: 110 ÷ 12 = 9.16 (Not an integer)

⇒ Option 3: 144 ÷ 12 = 12 (Integer)

⇒ Option 4: 95 ÷ 12 = 7.91 (Not an integer)

∴ The correct answer is 144.

Additional Information

Product Rule of HCF and LCM

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(a, b) × LCM(a, b).

HCF of Coprime Numbers

If two numbers are coprime, their Highest Common Factor is always 1, and their LCM is simply the product of the two numbers.

HCF and LCM of Fractions

HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators). LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators).

Shortcut Trick

A is 30% less than B, which means A is 70% of B.

70% of B = 630

100% of B = (630 ÷ 70) × 100

Value of B = 9 × 100 = 900

∴ The correct answer is 900.

Alternate Method

Given:

Value of A = 630

Relationship: A = B − 30% of B

Formula used:

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

Calculations:

⇒ 630 = B × (100 − 30) ÷ 100

⇒ 630 = B × (70 ÷ 100)

⇒ 630 = B × 0.7

⇒ B = 630 ÷ 0.7

⇒ B = 900

∴ The correct answer is 900.

Additional Information

Percentage Change Formula

Percentage Change = (Difference ÷ Original Value) × 100.

Base Comparison Rule

If A is x% less than B, then B is [x ÷ (100 − x)] × 100% more than A.

Multiplier Method

A 30% decrease corresponds to a multiplying factor of 0.70, while a 30% increase corresponds to 1.30.

Shortcut Trick

Express the relation as a Ratio: A : B = 1 : 4.

The difference between B and A is 4 − 1 = 3 units.

Percentage less = (Difference / Base value B) × 100.

⇒ (3 / 4) × 100 = 75%.

∴ The correct answer is 75%.

Alternate Method

Given:

Relation: A = (1/4) × B

Formula Used:

Percentage Less = [(Value 1 − Value 2) / Value 1] × 100

Calculations:

⇒ Let the value of B = 100

⇒ Value of A = (1/4) × 100 = 25

⇒ Difference between B and A = 100 − 25 = 75

⇒ Percentage by which A is less than B = (Difference / B) × 100

⇒ Percentage Less = (75 / 100) × 100

⇒ Percentage Less = 75%

∴ The correct answer is 75%.

Additional Information

Percentage Change Formula

Percentage change is calculated as [|New Value − Old Value| / Old Value] × 100. The denominator is always the reference (base) value.

Fraction to Percentage Conversion

Common fractions like 1/4 = 25%, 1/2 = 50%, and 3/4 = 75% are helpful for quick quantitative aptitude calculations.

Base Value Rule

In the phrase 'A is less than B', B is the base value. In 'B is more than A', A becomes the base value.

Shortcut Trick

In a transaction involving a loss, the difference between Cost Price (CP) and Selling Price (SP) is the absolute loss.

Given Loss Percentage = 20% and Difference (Loss) = ₹520.

20% of CP = ₹520

100% of CP = 520 × (100 ÷ 20) = 520 × 5 = ₹2600

∴ The correct answer is ₹2600.

Alternate Method

Given:

Difference (CP − SP) = ₹520

Loss Percentage = 20%

Formula Used:

Loss = CP − SP

Loss % = (Loss ÷ CP) × 100

Calculations:

⇒ Let the Cost Price of the article be ₹x.

⇒ Loss = 20% of x = 0.20x

⇒ We know that CP − SP = Loss

⇒ Given, CP − SP = 520

⇒ 0.20x = 520

⇒ x = 520 ÷ 0.20

⇒ x = (520 × 100) ÷ 20

⇒ x = 520 × 5 = 2600

∴ The correct answer is ₹2600.

Additional Information

Profit and Loss Basic Formulas

Profit = Selling Price (SP) − Cost Price (CP) when SP > CP. Loss = Cost Price (CP) − Selling Price (SP) when CP > SP.

Calculating CP from SP and Loss%

CP = [SP ÷ (100 − Loss%)] × 100. This is used when the final selling price and loss percentage are known.

Marked Price and Discount

Selling Price = Marked Price − Discount. Discount % = (Discount ÷ Marked Price) × 100. Discount is always calculated on the Marked Price.

Shortcut Trick

The Loss percentage (16%) represents the difference between Cost Price (CP) and Selling Price (SP).

16% of CP = ₹420

1% of CP = 420 ÷ 16 = 26.25

100% (Cost Price) = 26.25 × 100 = ₹2625

∴ The correct answer is ₹2625.

Alternate Method

Given:

Difference (CP − SP) = ₹420

Loss Percentage = 16%

Formula Used:

Loss = Cost Price − Selling Price

Loss% = (Loss ÷ CP) × 100

Calculations:

⇒ Loss % = (Loss ÷ CP) × 100

⇒ 16 = (420 ÷ CP) × 100

⇒ CP = (420 × 100) ÷ 16

⇒ CP = 42000 ÷ 16

⇒ CP = ₹2625

∴ The correct answer is ₹2625.

Additional Information

Profit Percentage

Profit occurs when SP > CP. It is calculated as Profit% = [(SP − CP) ÷ CP] × 100.

Selling Price in Case of Loss

When an article is sold at a loss, the Selling Price is given by: SP = CP × (100 − Loss%) ÷ 100.

Relation between CP and Loss

The absolute Loss value is always the product of the Cost Price and the Loss fraction (Loss% ÷ 100).

Shortcut Trick

Daily efficiency of A = 40% ÷ 8 = 5% per day.

Work completed in 15 days = 15 × 5% = 75%.

∴ The correct answer is 75%.

Alternate Method

Given: A completes 40% work in 8 days.

Total time for 100% work = 8 × (100 ÷ 40) = 20 days.

⇒ Work done by A in 15 days = Efficiency × Time

⇒ Work = 5 × 15 = 75 units

⇒ Percentage = (75 ÷ 100) × 100% = 75%

∴ The correct answer is 75%.

Additional Information

Work Rate Formula

Total Work = Efficiency × Time. Efficiency is the amount of work done in 1 unit of time.

Unitary Method in Work

If a person does 1/n part of a work in one day, they will take 'n' days to complete the full work.

Percentage Work Relation

Work done is directly proportional to the time taken when efficiency remains constant (W₁/T₁ = W₂/T₂).

Shortcut Trick

Ratio of work done by A and B = (6 × 3) : (4 × 5) = 18 : 20 = 9 : 10

Total wage units = 9 + 10 = 19 units

Given, 19 units = ₹4408

Difference in shares = (10 − 9) = 1 unit = 4408 ÷ 19 = ₹232

∴ The correct answer is ₹232.

Alternate Method

Given:

Days worked by A = 6 days, Days worked by B = 4 days

Efficiency of A = 3 units/day, Efficiency of B = 5 units/day

Total wages = ₹4408

Formula Used:

Total Work = Time × Efficiency

Share of Wage ∝ Total Work Done

Calculations:

⇒ Total work done by A = 6 × 3 = 18 units

⇒ Total work done by B = 4 × 5 = 20 units

⇒ Total work = 18 + 20 = 38 units

⇒ Wage for 38 units = ₹4408

⇒ Wage for 1 unit = 4408 ÷ 38 = ₹116

⇒ Difference in shares = 20 − 18 = 2 units

⇒ Required difference = 2 × 116 = ₹232

∴ The correct answer is ₹232.

Additional Information

Wage Distribution Principle

Wages are always distributed in the ratio of the total work done by each person, which is the product of their efficiency and the time they worked.

Efficiency and Time Relationship

If the work is constant, efficiency is inversely proportional to the time taken. Efficiency = Work ÷ Time.

Work & Wages in Partnership

Similar to partnership investments, where Profit = Capital × Time, in work problems, Wage = Efficiency × Time.

Shortcut Trick

Total Distance covered when a train crosses a bridge = Length of Train + Length of Bridge.

Distance = Speed × Time

Total Distance = 15 m/s × 65 s = 975 m.

Length of Train = Total Distance − Bridge Length = 975 − 500 = 475 m.

∴ The correct answer is 475 m.

Alternate Method

Given:

Speed of the train = 15 m/s

Length of the Bridge = 500 m

Time taken to cross the bridge = 65 s

Formula Used:

Total Distance (D) = Speed (s) × Time (t)

When a train crosses a platform or a bridge, the total distance is the sum of the length of the train and the bridge.

Calculations:

⇒ Total Distance = Speed × Time

⇒ (Length of Train + Length of Bridge) = 15 × 65

⇒ Length of Train + 500 = 975

⇒ Length of Train = 975 − 500

⇒ Length of Train = 475 m

∴ The correct answer is 475 m.

Additional Information

Crossing a Stationary Point Object

When a train crosses a pole, tree, or a stationary man, the distance covered is equal to the length of the train itself.

Speed Conversion

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

Relative Speed

When two objects move in the opposite direction, relative speed is the sum of their speeds (s₁ + s₂). In the same direction, it is the difference (s₁ − s₂).

Shortcut Trick

Upstream Speed = 80 ÷ 8 = 10 km/hr

Downstream Speed = 80 ÷ 4 = 20 km/hr

Speed of Boat in Still Water = (Downstream Speed + Upstream Speed) ÷ 2

Still Water Speed = (20 + 10) ÷ 2 = 30 ÷ 2 = 15 km/hr

∴ The correct answer is 15 km/hr.

Alternate Method

Given:

Distance (D) = 80 km

Upstream Time (t_u) = 8 hr

Downstream Time (t_d) = 4 hr

Formula Used:

Speed = Distance ÷ Time

If speed of boat is x and speed of stream is y:

Downstream Speed = x + y

Upstream Speed = x − y

⇒ Upstream speed (x − y) = Distance ÷ Upstream Time

⇒ x − y = 80 ÷ 8 = 10 km/hr ---- (i)

⇒ Downstream speed (x + y) = Distance ÷ Downstream Time

⇒ x + y = 80 ÷ 4 = 20 km/hr ---- (ii)

⇒ Adding (i) and (ii): (x − y) + (x + y) = 10 + 20

⇒ 2x = 30

⇒ x = 30 ÷ 2 = 15 km/hr

∴ The correct answer is 15 km/hr.

Additional Information

Speed of Current (Stream)

The speed of the current can be calculated as: y = (Downstream Speed − Upstream Speed) ÷ 2.

Relative Speed in Water

While moving Downstream, the speeds of the boat and the water add up, whereas in Upstream, the water speed is subtracted from the boat speed.

Average Speed of Boat

Average speed for the total journey = (2 × Downstream × Upstream) ÷ (Downstream + Upstream).

Shortcut Trick

Initial Ratio A : B = 7 : 5 (Difference = 2 parts)

New Ratio A : B = 4 : 3 (Difference = 1 part)

Multiply new ratio by 2 to equate difference: 8 : 6

Increase in each part = 8 − 7 = 1 unit → 3

Sum of initial parts = 7 + 5 = 12 units

Total Initial Sum = 12 × 3 = 36

∴ The correct answer is 36.

Alternate Method

Given initial ratio of Numbers A and B is 7 : 5.

Let the common multiplier be x. So, A = 7x and B = 5x.

According to the question, when both numbers are increased by 3:

⇒ (7x + 3) / (5x + 3) = 4 / 3

⇒ 3 × (7x + 3) = 4 × (5x + 3)

⇒ 21x + 9 = 20x + 12

⇒ 21x − 20x = 12 − 9

⇒ x = 3

Initial Values:

⇒ A = 7 × 3 = 21

⇒ B = 5 × 3 = 15

⇒ Sum of A and B = 21 + 15 = 36

∴ The correct answer is 36.

Additional Information

Ratio Balancing Method

When the same value is added or subtracted from both terms, the difference between the terms remains proportional. Equating the differences simplifies calculation.

Compound Ratio

The compound ratio of two ratios (a : b) and (c : d) is (ac : bd), obtained by multiplying corresponding terms.

Duplicate and Triplicate Ratios

The duplicate ratio of a : b is a² : b², and the triplicate ratio is a³ : b³.

Shortcut Trick

Let Selling Price (SP) = 11 units.

Discount = 9/11 × 11 = 9 units ⇒ Marked Price (MP) = 11 + 9 = 20 units.

Profit = 20% ⇒ Cost Price (CP) = 11 ÷ 1.2 = 55/6 units.

Ratio MP : CP = 20 : (55/6) = 120 : 55 = 24 : 11.

∴ The correct answer is 24 : 11.

Alternate Method

Given:

Profit % = 20%

Discount = 9/11 × Selling Price (SP)

Formula Used:

SP = CP × (100 + Profit%)/100

MP = SP + Discount

Calculations:

⇒ Let CP = 100x

⇒ SP = 100x + 20% of 100x = 120x

⇒ Discount = (9/11) × 120x = 1080x/11

⇒ MP = SP + Discount

⇒ MP = 120x + 1080x/11

⇒ MP = (1320x + 1080x) ÷ 11 = 2400x/11

⇒ Ratio MP : CP = (2400x/11) ÷ 100x

⇒ Ratio MP : CP = 2400 ÷ 1100 = 24 : 11

∴ The correct answer is 24 : 11.

Additional Information

Relationship between CP and MP

The ratio of Marked Price to Cost Price can also be calculated using the formula: MP/CP = (100 + Profit%)/(100 − Discount%).

Discount Percentage

Discount% is always calculated on the Marked Price: Discount% = (Discount ÷ MP) × 100.

Profit and Selling Price

Selling Price is the final amount after adding Profit to CP or subtracting Discount from MP.

Shortcut Trick

Price after the first 10% discount = 3000 − (10% of 3000) = ₹2700.

The article is finally sold for ₹1890.

Value of second discount (A) = [(2700 − 1890) ÷ 2700] × 100.

A = (810 ÷ 2700) × 100 = 30.

∴ The correct answer is 30.

Alternate Method

Given:

Marked Price (MP) = ₹3000

Final Selling Price (SP) = ₹1890

Successive Discounts = 10% and A%

Formula:

SP = MP × (1 − d₁/100) × (1 − d₂/100)

Calculations:

⇒ 1890 = 3000 × (1 − 10/100) × (1 − A/100)

⇒ 1890 = 3000 × (90/100) × (100 − A)/100

⇒ 1890 = 3000 × 0.9 × (100 − A)/100

⇒ 1890 = 2700 × (100 − A)/100

⇒ 1890 = 27 × (100 − A)

⇒ 100 − A = 1890 ÷ 27

⇒ 100 − A = 70

⇒ A = 100 − 70 = 30

∴ The correct answer is 30.

Additional Information

Single Equivalent Discount

For two successive discounts d₁% and d₂%, the single equivalent discount is calculated as (d₁ + d₂ − d₁d₂/100)%.

Relationship between MP and SP

Selling Price is always calculated on the Marked Price after deducting the discount: SP = MP × (100 − Discount%) / 100.

Successive Multiplier Method

Final SP = MP × f₁ × f₂, where f is the multiplying factor (e.g., for 10% discount, f = 0.9).

Shortcut Trick

For four numbers in proportion (a : b :: c : d), the fourth proportional is found using: d = (b × c) ÷ a.

X = (5 × 20) ÷ 25

X = 100 ÷ 25 = 4

∴ The correct answer is 4.

Alternate Method

Given: The numbers 25, 5, 20, and X are in proportion.

Formula Used: If four numbers a, b, c, and d are in proportion, then the ratio of the first two is equal to the ratio of the last two: a/b = c/d (also expressed as product of extremes = product of means).

Calculations:

⇒ 25 ÷ 5 = 20 ÷ X

⇒ 5 = 20 ÷ X

⇒ 5 × X = 20

⇒ X = 20 ÷ 5

⇒ X = 4

∴ The correct answer is 4.

Additional Information

Fourth Proportion

In the proportion a : b :: c : d, the term d is called the fourth proportional, calculated as d = (b × c) ÷ a.

Mean Proportion

The mean proportional between two numbers a and b is the square root of their product: Mean = √(a × b).

Continued Proportion

Three numbers a, b, and c are in continued proportion if a : b = b : c, which implies b² = ac.

Third Proportion

In a continued proportion a : b :: b : c, c is the third proportional and is given by c = b² ÷ a.