Quantitative Aptitude PYQs

Shortcut Trick

Apply BODMAS: First solve the bracket (36 + 29) = 65.

Perform multiplication: 65 × 2 = 130.

Solve addition and subtraction from left to right: 88 + 47 − 59 + 130 − 94.

Result: 135 − 59 + 130 − 94 = 76 + 130 − 94 = 206 − 94 = 112.

∴ The correct answer is 112.

Alternate Method

Given: 88 + 47 − 59 + (36 + 29) × 2 − 94

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

⇒ 88 + 47 − 59 + (65) × 2 − 94

⇒ 88 + 47 − 59 + 130 − 94

⇒ 135 − 59 + 130 − 94

⇒ 76 + 130 − 94

⇒ 206 − 94

⇒ 112

∴ The correct answer is 112.

Additional Information

BODMAS Rule

It is an acronym to help remember the order of operations in calculations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Sign Priority

When operations like addition and subtraction are at the same priority level, they are typically solved from left to right to avoid errors.

Brackets Priority

If multiple brackets are present, solve in the order: Round Brackets (), then Curly Brackets {}, and finally Square Brackets [].

Shortcut Trick

Solve using BODMAS: 11 of 2 = 22, 18 ÷ 3 = 6, and 12 × 2 = 24.

Expression simplifies to: 88 − 27 + 24 − 6 + 22.

Combine positive and negative terms: (88 + 24 + 22) − (27 + 6) = 134 − 33.

Final calculation: 134 − 33 = 101.

∴ The correct answer is 101.

Alternate Method

Given: 88 − 27 + 12 × 2 − 18 ÷ 3 + (11 of 2)

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

⇒ 88 − 27 + 12 × 2 − 18 ÷ 3 + 22 (Solving the 'Of' part: 11 × 2)

⇒ 88 − 27 + 12 × 2 − 6 + 22 (Solving Division: 18 ÷ 3 = 6)

⇒ 88 − 27 + 24 − 6 + 22 (Solving Multiplication: 12 × 2 = 24)

⇒ 61 + 24 − 6 + 22 (Solving Subtraction: 88 − 27 = 61)

⇒ 85 − 6 + 22 (Solving Addition: 61 + 24 = 85)

⇒ 79 + 22 = 101 (Final Subtraction and Addition)

∴ The correct answer is 101.

Additional Information

BODMAS Priority

The rule dictates the order: Brackets first, then Of (powers/roots/multiplication), followed by Division and Multiplication (left-to-right), and finally Addition and Subtraction (left-to-right).

The 'Of' Operator

In mathematical expressions, 'Of' acts as a multiplication but must be performed before division and multiplication according to the BODMAS hierarchy.

Simplification Tip

Always group all positive numbers and all negative numbers separately at the end to minimize errors during the final Addition and Subtraction steps.

Shortcut Trick

Apply the BODMAS rule: Multiplication → Addition → Subtraction.

0.5 × 0.3 = 0.15

0.15 + 0.9 = 1.05

1.05 − 0.55 = 0.5

∴ The correct answer is 0.5.

Alternate Method

Given:

0.5 × 0.3 + 0.9 − 0.55

Formula/Rule Used:

BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ Step 1: Perform Multiplication

⇒ 0.5 × 0.3 = 0.15

⇒ Step 2: Perform Addition

⇒ 0.15 + 0.9 = 1.05

⇒ Step 3: Perform Subtraction

⇒ 1.05 − 0.55 = 0.5

∴ The correct answer is 0.5.

Additional Information

BODMAS Rule

Standard order of operations: Brackets, Orders (powers/roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Decimal Multiplication

When multiplying decimals, count the total number of decimal places in the factors and place the decimal point in the product accordingly.

Decimal Addition/Subtraction

Always align the decimal points vertically before performing addition or subtraction to ensure units are matched correctly.

Shortcut Trick

For numbers ending in 0.25, if the integer part is n × (n + 1), the square root is n.5.

√90.25 (9 × 10 = 90) → 9.5

√6.25 (2 × 3 = 6) → 2.5

√2.25 (1 × 2 = 2) → 1.5

√0.25 (0 × 1 = 0) → 0.5

Expression: 9.5 + 2.5 − 1.5 + 0.5 = 11

∴ The correct answer is 11.

Alternate Method

Given:

Expression = √90.25 + √6.25 − √2.25 + √0.25

Calculations:

⇒ √90.25 = √(9025 ÷ 100) = 95 ÷ 10 = 9.5

⇒ √6.25 = √(625 ÷ 100) = 25 ÷ 10 = 2.5

⇒ √2.25 = √(225 ÷ 100) = 15 ÷ 10 = 1.5

⇒ √0.25 = √(25 ÷ 100) = 5 ÷ 10 = 0.5

⇒ Result = 9.5 + 2.5 − 1.5 + 0.5

⇒ Result = 12 − 1.5 + 0.5

⇒ Result = 10.5 + 0.5 = 11

∴ The correct answer is 11.

Additional Information

Square of Numbers Ending in 5

To square a number ending in 5, multiply the digits before 5 by the next consecutive integer and append 25. Example: (45)² = (4 × 5) and 25 = 2025.

Square Root of Decimal Places

When finding the square root of a decimal, the number of decimal places in the result is exactly half of the decimal places in the original number. (e.g., √0.0001 = 0.01).

BODMAS Rule

Always follow the BODMAS order (Brackets, Orders, Division, Multiplication, Addition, Subtraction) for simplification to ensure accuracy.

Shortcut Trick

The perimeter of a square is the total length of its boundary.

Since all four sides of a square are equal, the formula is 4 × side.

Perimeter = 4 × 12 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Side (a) of the square = 12 cm

Formula Used:

Perimeter (P) = 4 × a

Calculations:

⇒ Perimeter = 12 + 12 + 12 + 12

⇒ Perimeter = 4 × 12

⇒ Perimeter = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Area of a Square

The total space occupied inside the boundary of the square, calculated as side × side or a².

Diagonal of a Square

The straight line connecting opposite vertices. Length = Side × √2.

Properties of a Square

All four sides are equal, and each interior angle is exactly 90°.

Shortcut Trick

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

Identify a = 36 and b = 29.

Result = 36 − 29 = 7.

∴ The correct answer is 7.

Alternate Method

Given: √((36 + 29)² − 4 × 36 × 29)

Solve the terms inside the square root step by step:

⇒ √((65)² − (4 × 1044))

⇒ √(4225 − 4176)

⇒ √49

⇒ 7

∴ The correct answer is 7.

Additional Information

Square of a Sum

(a + b)² = a² + b² + 2ab

Square of a Difference

(a − b)² = a² + b² − 2ab

Key Mathematical Relation

(a + b)² − (a − b)² = 4ab, which simplifies the given square root expression directly to (a − b).

Shortcut Trick

Side (a) of rhombus = Perimeter ÷ 4 = 68 ÷ 4 = 17 cm.

One half-diagonal (d₁/2) = 16 ÷ 2 = 8 cm.

Using Pythagorean triplet (8, 15, 17): the other half-diagonal (d₂/2) = 15 cm.

Second diagonal (d₂) = 15 × 2 = 30 cm.

Sum of diagonals = 16 + 30 = 46 cm.

∴ The correct answer is 46 cm.

Alternate Method

Given: Perimeter = 68 cm, Diagonal d₁ = 16 cm.

Formula Used: Side of rhombus a = Perimeter/4; Relationship: a² = (d₁/2)² + (d₂/2)²

⇒ Side a = 68 ÷ 4 = 17 cm

⇒ Let half of the diagonals be h₁ and h₂.

⇒ h₁ = 16 ÷ 2 = 8 cm

⇒ a² = h₁² + h₂²

⇒ 17² = 8² + h₂²

⇒ 289 = 64 + h₂²

⇒ h₂² = 289 − 64 = 225

⇒ h₂ = √225 = 15 cm

⇒ Second diagonal d₂ = 2 × h₂ = 2 × 15 = 30 cm

⇒ Sum of both the diagonals = d₁ + d₂ = 16 + 30 = 46 cm

∴ The correct answer is 46 cm.

Additional Information

Properties of a Rhombus

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

Area of a Rhombus

Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals.

Side-Diagonal Relationship

4a² = d₁² + d₂², which is derived from the Pythagorean theorem applied to the four internal right-angled triangles.

Shortcut Trick

Side of Rhombus (s) = Perimeter ÷ 4 = 68 ÷ 4 = 17 cm.

Half of the given diagonal (d₁/2) = 30 ÷ 2 = 15 cm.

Using Pythagoras triplet (8, 15, 17), the half of other diagonal (d₂/2) = 8 cm.

Ratio of diagonals is the same as the ratio of their halves.

Required Ratio = 15 : 8.

∴ The correct answer is 15 : 8.

Alternate Method

Given: Perimeter = 68 cm, Diagonal 1 (d₁) = 30 cm.

Formula Used: Side² = (d₁/2)² + (d₂/2)²

⇒ Side of Rhombus = 68 ÷ 4 = 17 cm

⇒ Let half of the diagonals be 15 cm and x cm.

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

⇒ 17² = 15² + x²

⇒ 289 = 225 + x²

⇒ x² = 289 − 225 = 64

⇒ x = √64 = 8 cm

⇒ Second diagonal (d₂) = 2 × 8 = 16 cm

⇒ Ratio of diagonals = d₁ : d₂ = 30 : 16

⇒ Ratio = 15 : 8

∴ The correct answer is 15 : 8.

Additional Information

Area of Rhombus

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

Properties of Diagonals

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

Relation between Side and Diagonals

For a rhombus with side 's', the relation is 4s² = d₁² + d₂².

Shortcut Trick

To find the Highest Common Factor (HCF) of numbers in prime power form, identify the common bases and pick the lowest exponent for each.

Common base 5: The powers are 7 and 5. The minimum power is 5.

Common base 11: The powers are 3 and 6. The minimum power is 3.

HCF = 5⁵ × 11³.

∴ The correct answer is 5⁵ × 11³.

Alternate Method

Given:

Number 1 (N₁) = 5⁷ × 11³

Number 2 (N₂) = 5⁵ × 11⁶

Formula Used:

For any two numbers expressed as product of primes p₁^a × p₂^b and p₁^c × p₂^d:

HCF = p₁^{min(a, c)} × p₂^{min(b, d)}

Calculations:

⇒ Compare exponents of prime factor 5: 7 and 5. The smaller one is 5.

⇒ Compare exponents of prime factor 11: 3 and 6. The smaller one is 3.

⇒ HCF = 5^{min(7, 5)} × 11^{min(3, 6)}

⇒ HCF = 5⁵ × 11³

∴ The correct answer is 5⁵ × 11³.

Additional Information

Least Common Multiple (LCM)

For numbers in prime power form, LCM is found by taking the highest power of every prime factor present across the numbers.

HCF and LCM 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

If the Highest Common Factor of two numbers is 1, they are called co-prime or relatively prime numbers.

Shortcut Trick

The Highest Common Factor (HCF) of any two numbers is always a factor of their Least Common Multiple (LCM).

Therefore, the LCM must be exactly divisible by the HCF without any remainder.

Divide 84 by the given options:

84 ÷ 21 = 4 (Exactly divisible)

Other options (18, 30, 16) do not divide 84 completely.

∴ The correct answer is 21.

Alternate Method

Given: LCM of two numbers = 84

Property used: LCM is always a multiple of HCF. If L is the LCM and H is the HCF, then L/H must be an integer.

Testing each option:

⇒ Case 1: 84 ÷ 18 = 4.66 (Not an integer)

⇒ Case 2: 84 ÷ 21 = 4 (Integer)

⇒ Case 3: 84 ÷ 30 = 2.8 (Not an integer)

⇒ Case 4: 84 ÷ 16 = 5.25 (Not an integer)

⇒ Since only 21 divides 84 completely, it can be the HCF.

∴ The correct answer is 21.

Additional Information

Product Rule of LCM and HCF

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.

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. In this case, their LCM is simply the product of the two numbers.

HCF of Fractions

To find the HCF of fractions, calculate the HCF of numerators and divide it by the LCM of denominators.

Shortcut Trick

Combined efficiency of R and S = (1/32 + 1/40) = (5 + 4)/160 = 9/160 per day.

Work completed in 8 days = 8 × (9/160) = 9/20.

∴ The correct answer is 9/20.

Alternate Method

Given: Time taken by R = 32 days, Time taken by S = 40 days, Combined working time = 8 days.

Formula Used: Total Work = LCM of individual times; Efficiency = Total Work ÷ Time.

⇒ Total work (LCM of 32 and 40) = 160 units.

⇒ Combined efficiency of R and S = 5 + 4 = 9 units/day.

⇒ Work completed by both in 8 days = 9 × 8 = 72 units.

⇒ Part of total work completed = 72/160.

⇒ Simplifying the fraction: 72/160 = 9/20.

∴ The correct answer is 9/20.

Additional Information

Remaining Work Calculation

The portion of work left after some part is finished is calculated as (1 − Work Done). In this case, remaining work = 1 − 9/20 = 11/20.

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a work (Efficiency ∝ 1/Time). If A is thrice as efficient as B, A will take 1/3^rd of the time taken by B.

Work and Wages

When multiple people work together, the total wages are always distributed in the ratio of the total work actually done by each individual.

Shortcut Trick

Apply the Chain Rule formula: M₁ × D₁ = M₂ × D₂

Initial data: M₁ = 18 persons, D₁ = 45 days

Updated data: M₂ = 18 + 12 = 30 persons

Calculation: 18 × 45 = 30 × D₂ ⇒ D₂ = (18 × 45) ÷ 30 = 27 days

∴ The correct answer is 27 days.

Alternate Method

Given:

Initial group size (M₁) = 18 persons

Time taken (D₁) = 45 days

New members joined = 12 persons

Formula Used:

Total Work = Number of Persons × Number of Days

Calculations:

⇒ Total Work = 18 × 45 = 810 units

⇒ Total number of persons (M₂) = 18 + 12 = 30 persons

⇒ Time required for 30 persons = Total Work ÷ M₂

⇒ Time = 810 ÷ 30

⇒ Time = 27 days

∴ The correct answer is 27 days.

Additional Information

General Chain Rule

For work involving hours and efficiency, use (M₁ × D₁ × H₁ × E₁) / W₁ = (M₂ × D₂ × H₂ × E₂) / W₂.

Inverse Variation

Number of men is inversely proportional to the time taken. If men increase, days decrease in the same ratio.

Man-Day Concept

One man-day represents the amount of work done by one person in one day. Total work is the constant product of Men and Days.

Shortcut Trick

Average Speed = (s₁t₁ + s₂t₂) ÷ (t₁ + t₂)

Average Speed = (40 × 6 + 15 × 4) ÷ (6 + 4)

Average Speed = (240 + 60) ÷ 10 = 300 ÷ 10

Average Speed = 30 km/hr.

∴ The correct answer is 30 km/hr.

Alternate Method

Given:

Speed₁ = 40 km/hr, Time₁ = 6 hr

Speed₂ = 15 km/hr, Time₂ = 4 hr

Formula Used:

Distance = Speed × Time

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Distance covered in first 6 hr = 40 × 6 = 240 km

⇒ Distance covered in next 4 hr = 15 × 4 = 60 km

⇒ Total distance = 240 + 60 = 300 km

⇒ Total time taken = 6 + 4 = 10 hr

⇒ Average speed = 300 ÷ 10 = 30 km/hr

∴ The correct answer is 30 km/hr.

Additional Information

Average Speed (Equal Distances)

When a person travels two equal distances at speeds x and y, the average speed is 2xy ÷ (x + y).

Average Speed (Equal Time)

When a person travels for two equal time intervals at speeds x and y, the average speed is (x + y) ÷ 2.

Unit Conversion

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

Shortcut Trick

The extra distance covered is the length of the platform.

Extra time taken = 40 sec − 20 sec = 20 sec

Speed = (Length of platform) ÷ (Extra time taken)

Speed = 400 ÷ 20 = 20 m/s

∴ The correct answer is 20 m/s.

Alternate Method

Given:

Time to cross pole (t₁) = 20 sec

Time to cross platform (t₂) = 40 sec

Length of platform (L_p) = 400 m

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ Let the length of the train be L.

⇒ Speed (v) when crossing a pole = L ÷ 20

⇒ Speed (v) when crossing a platform = (L + 400) ÷ 40

⇒ Since speed is constant: L ÷ 20 = (L + 400) ÷ 40

⇒ 40L = 20(L + 400)

⇒ 2L = L + 400

⇒ L = 400 m

⇒ Speed (v) = 400 ÷ 20 = 20 m/s

∴ The correct answer is 20 m/s.

Additional Information

Crossing Different Objects

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

Unit Conversion

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

Relative Speed

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

Shortcut Trick

Apply the Componendo and Dividendo rule: If (a + b) / (a − b) = c / d, then a / b = (c + d) / (c − d).

Here, a = 4x, b = 3y, c = 5, and d = 2.

4x / 3y = (5 + 2) / (5 − 2) = 7 / 3

x / y = (7 / 3) × (3 / 4) = 7 / 4

∴ The correct answer is 7 : 4.

Alternate Method

Given: The ratio of linear combinations (4x + 3y) : (4x − 3y) = 5 : 2.

Formula Used: Cross-multiplication method for proportions.

⇒ (4x + 3y) / (4x − 3y) = 5 / 2

⇒ 2 × (4x + 3y) = 5 × (4x − 3y)

⇒ 8x + 6y = 20x − 15y

⇒ 6y + 15y = 20x − 8x

⇒ 21y = 12x

⇒ x / y = 21 / 12

⇒ x / y = 7 / 4

∴ The correct answer is 7 : 4.

Additional Information

Componendo and Dividendo Rule

If a/b = c/d, then (a + b) / (a − b) = (c + d) / (c − d). This is used to simplify equations involving ratios of sums and differences.

Invertendo Property

If a : b = c : d, then b : a = d : c. This allows for flipping ratios to solve for the reciprocal of a variable.

Mean Proportional

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

Shortcut Trick

Total ratio parts = 1 + 3 + 5 + 4 = 13 units.

Given 13 units = 52 ⇒ 1 unit = 52 ÷ 13 = 4.

Difference between largest (5) and smallest (1) parts = 5 − 1 = 4 units.

Required difference = 4 units × 4 = 16.

∴ The correct answer is 16.

Alternate Method

Given: Ratio = 1 : 3 : 5 : 4 and Sum = 52.

Formula Used: Sum of terms = Total value

⇒ Let the four numbers be 1x, 3x, 5x, and 4x.

⇒ 1x + 3x + 5x + 4x = 52

⇒ 13x = 52

⇒ x = 52 ÷ 13 = 4

⇒ Smallest number (1x) = 1 × 4 = 4

⇒ Largest number (5x) = 5 × 4 = 20

⇒ Difference = 20 − 4 = 16

∴ The correct answer is 16.

Additional Information

Ratio Distribution Rule

To find the value of any part, multiply the Total Value by (Part Ratio ÷ Sum of all Ratio Parts).

Proportionality Constant

The variable x represents the Common Multiplier that scales the ratio parts to their actual values.

Difference between Parts

The difference between any two numbers in a ratio can be calculated directly by finding the difference in their ratio units first.

Shortcut Trick

Let Markup percentage = +20% and Discount percentage = −10% (since SP is 90% of MP).

Net change formula: a + b + (ab/100)

Net Profit % = 20 − 10 − (20 × 10)/100

Net Profit % = 10 − 2 = 8%

∴ The correct answer is 8 percent.

Alternate Method

Given:

Marked Price (MP) = 120% of Cost Price (CP)

Selling Price (SP) = 90% of Marked Price (MP)

Calculations:

⇒ Let Cost Price (CP) = 100

⇒ Marked Price (MP) = 120% of 100 = 120

⇒ Selling Price (SP) = 90% of MP

⇒ SP = (90/100) × 120 = 108

⇒ Profit = SP − CP = 108 − 100 = 8

⇒ Profit Percentage = (Profit/CP) × 100

⇒ Profit % = (8/100) × 100 = 8%

∴ The correct answer is 8 percent.

Additional Information

Markup Percentage

It is the percentage by which the cost price is increased to set the marked price. Markup % = (MP − CP)/CP × 100.

Discount Percentage

It is the reduction offered on the marked price. Discount % = (MP − SP)/MP × 100.

Relationship between CP and MP

CP/MP = (100 − Discount%)/(100 + Profit%). This direct ratio is useful for finding any one unknown variable.

Shortcut Trick

To find the highest Selling Price (SP), compare the product of Marked Price (MP) and the Selling Factor (1 − Discount%).

1. 9000 × 0.60 = 5400 | 2) 8000 × 0.50 = 4000

2. 5000 × 0.75 = 3750 | 4) 4000 × 0.90 = 3600

The maximum value is 5400.

∴ The correct answer is ₹5400.

Alternate Method

Given: Four cases with different Marked Prices (MP) and Discount percentages.

Formula Used: Selling Price (SP) = MP − (Discount% × MP)

⇒ Case 1: SP = 9000 − (40/100 × 9000) = 9000 − 3600 = ₹5400

⇒ Case 2: SP = 8000 − (50/100 × 8000) = 8000 − 4000 = ₹4000

⇒ Case 3: SP = 5000 − (25/100 × 5000) = 5000 − 1250 = ₹3750

⇒ Case 4: SP = 4000 − (10/100 × 4000) = 4000 − 400 = ₹3600

⇒ Comparing the results: 5400 > 4000 > 3750 > 3600.

∴ The correct answer is ₹5400.

Additional Information

Marked Price (List Price)

It is the price displayed on the product label. Discount is always calculated on the Marked Price.

Selling Price Calculation

SP = MP × (100 − D%)/100. If multiple discounts d₁ and d₂ are given, SP = MP × (100 − d₁)/100 × (100 − d₂)/100.

Successive Discount Shortcut

A single equivalent discount for two successive discounts a% and b% is (a + b − ab/100)%.

Shortcut Trick

Check Option (2): If Discount is 20%, then Selling Price (SP) = 80% of Marked Price (MP).

Applying values: 80% × ₹5000 = (80/100) × 5000 = ₹4000.

Since this matches the given SP of ₹4000, Option (2) is correct.

∴ The correct answer is Selling price = ₹4000 and Discount = 20%.

Alternate Method

Given:

Target Marked Price (MP) = ₹5000

Formula Used:

MP = (SP × 100) / (100 − Discount%)

Step-by-step Calculation for Option (2):

⇒ Selling Price (SP) = ₹4000

⇒ Discount Percentage = 20%

⇒ Marked Price (MP) = (4000 × 100) / (100 − 20)

⇒ MP = 400000 / 80

⇒ MP = ₹5000

This matches the requirement stated in the question.

∴ The correct answer is Selling price = ₹4000 and Discount = 20%.

Additional Information

Relationship between MP and SP

The Selling Price is always calculated by subtracting the Discount from the Marked Price: SP = MP − Discount.

Discount Percentage Formula

Discount% = (Discount / MP) × 100. Always remember that Discount is calculated on the Marked Price, not on the Cost Price.

Net Discount for Successive Rates

If two successive discounts 'a%' and 'b%' are given, the Effective Discount = (a + b − ab/100)%.

Shortcut Trick

Observe the relationship between the second and fourth terms in the proportion.

The fourth term (20) × 2 = The second term (40).

Applying the same factor to the third term: 12 × 2 = 24.

Thus, x + 3 = 24

x = 24 − 3 = 21

∴ The correct answer is 21.

Alternate Method

Given: (x + 3) : 40 :: 12 : 20

Formula Used: In a proportion a : b :: c : d, the product of extremes equals the product of means (a × d = b × c).

⇒ (x + 3) × 20 = 40 × 12

⇒ 20x + 60 = 480

⇒ 20x = 480 − 60

⇒ 20x = 420

⇒ x = 420 ÷ 20

⇒ x = 21

∴ The correct answer is 21.

Additional Information

Extremes and Means

In a proportion a : b = c : d, the terms 'a' and 'd' are called extremes, while 'b' and 'c' are called means. The fundamental property is a × d = b × c.

Mean Proportional

If a, b, c are in continued proportion, then a : b :: b : c, which implies b² = a × c. Here, 'b' is the mean proportional.

Fourth Proportional

If three numbers a, b, and c are given, the fourth proportional 'd' is calculated as d = (b × c) ÷ a.