Quantitative Aptitude PYQs

Shortcut Trick

Apply the BODMAS rule directly to solve the expression.

Solve Bracket first: (45 + 19) = 64

Solve Multiplication next: 26 x 2 = 52

Perform Addition and Subtraction: 64 − 37 + 52 − 13 = 66

∴ The correct answer is 66.

Alternate Method

Given:

(45 + 19) − 37 + 26 x 2 − 13

Formula Used:

The order of operations is BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

Calculations:

⇒ 64 − 37 + 26 x 2 − 13 (Solving Bracket: 45 + 19)

⇒ 64 − 37 + 52 − 13 (Solving Multiplication: 26 x 2)

⇒ (64 + 52) − (37 + 13) (Grouping positive and negative terms)

⇒ 116 − 50

⇒ 66

∴ The correct answer is 66.

Additional Information

BODMAS Rule

It defines the sequence to solve arithmetic operations: Bracket (), Of, Division (÷), Multiplication (x), Addition (+), and Subtraction (−).

VBODMAS Rule

This includes a Vinculum (bar bracket) which is solved even before standard brackets in an expression.

Sign Convention

Always group positive terms together and negative terms together to simplify the final addition and subtraction steps.

Shortcut Trick

Solve the Of and Division operations first: 18 × 1 = 18 and 33 ÷ 3 = 11.

Substitute these values back: 94 − 38 + 28 + 11 + 21 − 18.

Sum the positive terms: 94 + 28 + 11 + 21 = 154.

Sum the negative terms: 38 + 18 = 56.

Calculate the final difference: 154 − 56 = 98.

∴ The correct answer is 98.

Alternate Method

Given:

94 − 38 + 28 + 33 ÷ 3 + 21 − (18 of 1)

Formula Used:

According to the BODMAS rule, operations are performed in the order: Brackets (B), Of (O), Division (D), Multiplication (M), Addition (A), and Subtraction (S).

Calculations:

⇒ 94 − 38 + 28 + 33 ÷ 3 + 21 − 18

⇒ 94 − 38 + 28 + 11 + 21 − 18

⇒ (94 + 28 + 11 + 21) − (38 + 18)

⇒ 154 − 56

⇒ 98

∴ The correct answer is 98.

Additional Information

BODMAS Rule

It defines the precedence of operators. Brackets are solved first, followed by 'Of' (multiplication), then Division, Multiplication, Addition, and finally Subtraction.

The 'Of' Operator

The term Of is treated as a high-priority multiplication that must be evaluated before the standard Division or Multiplication operators.

Grouping for Addition/Subtraction

To avoid calculation errors, always group all positive numbers together and all negative numbers together before performing the final subtraction.

Shortcut Trick

For proper fractions where the difference between the numerator and denominator is the same, the fraction with the largest numerator is the largest.

Check Gaps: (6 − 5) = 1, (4 − 3) = 1, (3 − 2) = 1, (11 − 10) = 1, (5 − 4) = 1.

Since all gaps are equal to 1, compare numerators: 10 > 5 > 4 > 3 > 2.

Therefore, 10/11 > 5/6.

∴ The correct answer is 10/11.

Alternate Method

Given: Base fraction = 5/6

Formula Used: Decimal Conversion Method

Convert each fraction into its decimal equivalent:

⇒ 5/6 = 5 ÷ 6 ≈ 0.833

⇒ 3/4 = 3 ÷ 4 = 0.75

⇒ 2/3 = 2 ÷ 3 ≈ 0.666

⇒ 10/11 = 10 ÷ 11 ≈ 0.909

⇒ 4/5 = 4 ÷ 5 = 0.8

Comparing the values: 0.909 > 0.833

∴ The correct answer is 10/11.

Additional Information

Cross-Multiplication Method

To compare a/b and c/d, compute (a × d) and (b × c). If a × d > b × c, then a/b > c/d.

Common Denominator Method

Convert all fractions to have the same LCM of denominators. The fraction with the largest numerator is then the greatest.

Fractions with Equal Numerators

If the numerators are the same, the fraction with the smaller denominator is the larger value (e.g., 3/5 > 3/7).

Shortcut Trick

Identify perfect roots: ³√512 = 8 and √225 = 15.

Adjust decimal places: ³√0.512 = 0.8 and √2.25 = 1.5.

Calculate product: 0.8 × 1.5 = 1.2.

∴ The correct answer is 1.2.

Alternate Method

Given: ³√0.512 × √2.25

Method: Fractional Simplification

⇒ ³√(512 ÷ 1000) × √(225 ÷ 100)

⇒ (8 ÷ 10) × (15 ÷ 10)

⇒ 0.8 × 1.5

⇒ 1.2

∴ The correct answer is 1.2.

Additional Information

Cube Root Decimal Rule

When calculating the cube root of a decimal, if the original number has 3n decimal places, the result will have n decimal places.

Square Root Decimal Rule

When calculating the square root of a decimal, if the original number has 2n decimal places, the result will have n decimal places.

Multiplication of Decimals

The total number of decimal places in the product is the sum of the decimal places in the numbers being multiplied.

Shortcut Trick

For a Rhombus, the relation between side (a), Area (A), and sum of diagonals (S) is: S² = 4(a² + A).

⇒ S² = 4(41² + 720)

⇒ S² = 4(1681 + 720) = 4 × 2401

⇒ S = √(4 × 2401) = 2 × 49 = 98

∴ The correct answer is 98 cm.

Alternate Method

Given: Area of rhombus = 720 cm², Side (a) = 41 cm.

Formula: Area = (d₁ × d₂)/2 | 4a² = d₁² + d₂²

⇒ d₁ × d₂ = 2 × Area = 2 × 720 = 1440

⇒ d₁² + d₂² = 4 × Side² = 4 × 41² = 4 × 1681 = 6724

⇒ (d₁ + d₂)² = (d₁² + d₂²) + 2(d₁ × d₂)

⇒ (d₁ + d₂)² = 6724 + 2(1440) = 6724 + 2880 = 9604

⇒ d₁ + d₂ = √9604

⇒ d₁ + d₂ = 98 cm

∴ The correct answer is 98 cm.

Additional Information

Properties of Rhombus

A rhombus is a quadrilateral with all four sides equal. Its diagonals bisect each other at 90° and also bisect the internal angles.

Perimeter and Area

Perimeter = 4 × side. Area can also be calculated as base × height if the altitude is known.

Diagonal Relation

The relationship between the side and the diagonals is derived from the Pythagoras theorem: Side = √[(d₁/2)² + (d₂/2)²].

Shortcut Trick

In a parallelogram, any two adjacent angles (consecutive angles) are supplementary.

Sum of adjacent angles = 180°

Given angle = 45°

Adjacent angle = 180° − 45° = 135°

∴ The correct answer is 135°.

Alternate Method

Given:

One angle of the parallelogram = 45°

Formula Used:

Sum of interior angles on the same side of a transversal between parallel lines is 180°.

In parallelogram ABCD, AB is parallel to DC and AD is parallel to BC.

Since AD || BC and AB is the transversal:

⇒ ∠A + ∠B = 180° (Adjacent angles property)

⇒ 45° + ∠B = 180°

⇒ ∠B = 180° − 45°

⇒ ∠B = 135°

∴ The correct answer is 135°.

Additional Information

Opposite Angles of a Parallelogram

The opposite angles of a parallelogram are always equal (∠A = ∠C and ∠B = ∠D).

Sum of All Interior Angles

The total sum of all internal angles in any quadrilateral, including a parallelogram, is always 360°.

Diagonals of a Parallelogram

The diagonals of a parallelogram bisect each other, though they are not necessarily equal in length.

Shortcut Trick

Convert decimal and large fractions into simplest fractions: 0.75 = 3/4 and 25/30 = 5/6.

Apply the fraction formula: LCM = LCM of Numerators ÷ HCF of Denominators.

LCM(3, 5) = 15 and HCF(4, 6) = 2.

Result = 15/2.

∴ The correct answer is 15/2.

Alternate Method

Given: First value = 0.75, Second value = 25/30.

Formula Used: LCM of Fractions = LCM(Numerators) ÷ HCF(Denominators)

⇒ First, simplify 0.75: 0.75 = 75/100 = 3/4.

⇒ Simplify 25/30: Divide both numerator and denominator by 5 → 5/6.

⇒ Now, find the LCM of 3/4 and 5/6.

⇒ LCM of Numerators (3, 5) = 15.

⇒ HCF of Denominators (4, 6) = 2.

⇒ Final LCM = 15/2.

∴ The correct answer is 15/2.

Additional Information

HCF of Fractions

To find the Highest Common Factor of fractions, use the formula: HCF = HCF(Numerators) ÷ LCM(Denominators).

Standard Form Requirement

Always convert fractions to their simplest/standard form (where numerator and denominator are co-prime) before applying LCM or HCF formulas.

LCM × HCF Rule

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

Shortcut Trick

The ratio of the numbers is 5 : 3.

Sum of ratio units = 5 + 3 = 8 units.

Since the HCF is the common factor for the ratio units,

Sum = Total units × HCF = 8 × 12 = 96.

∴ The correct answer is 96.

Alternate Method

Given:

HCF = 12

LCM = 180

Ratio of numbers = 5 : 3

Formula Used:

Sum of numbers = (a + b) × HCF (where a and b are coprime ratios)

Calculations:

Let the two numbers be 5x and 3x.

⇒ HCF of 5x and 3x is x (as 5 and 3 are coprime).

⇒ x = 12.

⇒ First number = 5 × 12 = 60.

⇒ Second number = 3 × 12 = 36.

⇒ Sum of numbers = 60 + 36 = 96.

Verification:

⇒ LCM(60, 36) = 12 × 5 × 3 = 180 (Matches the given value).

∴ The correct answer is 96.

Additional Information

Fundamental Property of HCF and LCM

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 × LCM.

HCF as a Factor

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

Coprime Numbers

Two numbers are called coprime if their HCF is 1. If numbers are in the ratio a : b, then a and b must be coprime.

Shortcut Trick

A% of A is equivalent to A² / 100.

Given: A² / 100 = 676

A² = 676 × 100 = 67600

∴ The correct answer is 67600.

Alternate Method

Given:

Percentage value = A%

Base value = A

Result = 676

Formula Used:

Percentage result = (Percentage × Base) ÷ 100

Calculations:

⇒ (A ÷ 100) × A = 676

⇒ (A × A) ÷ 100 = 676

⇒ A² ÷ 100 = 676

⇒ A² = 676 × 100

⇒ A² = 67600

∴ The correct answer is 67600.

Additional Information

Basic Percentage Rule

To find x% of any quantity y, we use the formula (x × y) ÷ 100.

Square of a Number

The square of a number is the result of multiplying the number by itself (A × A = A²).

Perfect Squares

In this problem, 676 is the square of 26, thus 67600 is the square of 260 (A = 260).

Shortcut Trick

Use the unitary method to compare the percentages directly.

Given that 20% corresponds to 80.

1% will correspond to 80 ÷ 20 = 4.

Therefore, 160% will correspond to 160 × 4 = 640.

∴ The correct answer is 640.

Alternate Method

Given:

20% of a number = 80

Formula Used:

Required Value = (Given Value ÷ Given Percentage) × Required Percentage

Calculations:

⇒ Let the number be x.

⇒ 20% of x = 80

⇒ (20 ÷ 100) × x = 80

⇒ x = (80 × 100) ÷ 20

⇒ x = 4 × 100 = 400

⇒ Now, find 160% of 400:

⇒ 160% of 400 = (160 ÷ 100) × 400

⇒ 1.6 × 400 = 640

∴ The correct answer is 640.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide the value by 100 and simplify. Example: 20% = 20/100 = 1/5.

Percentage Change Formula

Percentage change is calculated as (New Value − Old Value) ÷ Old Value × 100.

Successive Percentage Change

If a number changes by a% then b%, the net effective change is (a + b + (ab ÷ 100))%.

Shortcut Trick

If both Cost Price (CP) and Selling Price (SP) are increased or decreased by the same percentage (k%), the ratio of SP to CP remains constant.

Initial Profit = 30% ⇒ Ratio SP : CP = 130 : 100 = 1.3 : 1

Since both are decreased by 40%, the new ratio remains 1.3 :

1. New Profit % = (Ratio − 1) × 100 = (1.3 − 1) × 100 = 30%.

∴ The correct answer is 30%.

Alternate Method

Given: Initial Profit Percentage = 30%, Decrease in CP = 40%, Decrease in SP = 40%.

Formula: New Profit % = [(New SP − New CP) / New CP] × 100

⇒ Let original CP = 100

⇒ Original SP = 100 + 30% of 100 = 130

⇒ New CP = 100 − (40/100) × 100 = 60

⇒ New SP = 130 − (40/100) × 130 = 130 − 52 = 78

⇒ New Profit = 78 − 60 = 18

⇒ New Profit % = (18 / 60) × 100 = 3/10 × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Profit and Loss Percentage

Profit % = (Profit/CP) × 100 and Loss % = (Loss/CP) × 100. Both are always calculated on the Cost Price unless specified otherwise.

Constant Ratio Property

When both numerator and denominator of a fraction (SP/CP) are multiplied or divided by the same non-zero constant (or changed by the same %), the value of the ratio remains the same.

Successive Percentage Change

If a value changes by a% and then by b%, the effective change is [a + b + (ab/100)]%.

Shortcut Trick

When the total amount (₹260) is the same, use the quantity ratio.

Ratio of Cost Price (CP) to Selling Price (SP) per article = Inverse of Quantity.

CP : SP = 40 : 30 = 4 : 3

Loss = 4 − 3 = 1 unit

Loss Percentage = (1 / 4) × 100 = 25%

∴ The correct answer is 25% loss.

Alternate Method

Given: Cost Price (CP) of 30 articles = ₹260; Selling Price (SP) of 40 articles = ₹260.

Formula Used: Loss Percentage = [(CP − SP) / CP] × 100

⇒ CP per article = 260 / 30 = ₹26/3

⇒ SP per article = 260 / 40 = ₹26/4

⇒ Loss per article = (26/3) − (26/4)

⇒ Loss per article = (104 − 78) / 12 = 26 / 12 = ₹13/6

⇒ Loss % = [(13/6) / (26/3)] × 100

⇒ Loss % = (13/6 × 3/26) × 100 = (1/2 × 1/2) × 100

⇒ Loss % = 1/4 × 100 = 25%

∴ The correct answer is 25% loss.

Additional Information

Profit and Loss on Quantity

If CP of x articles = SP of y articles, then Profit/Loss % = (x − y) / y × 100. If the result is positive, it is a profit; if negative, it is a loss.

Basic CP and SP Relation

Selling Price (SP) = CP × (100 ± Profit/Loss%) / 100. This formula helps calculate the final value when either CP or percentage is known.

Dishonest Dealer Concept

Gain % = [Error / (True Value − Error)] × 100. This is used when a dealer uses less weight than the actual weight mentioned.

Shortcut Trick

Assume Total Work = LCM of (14, 21, 42) = 42 units.

Efficiency of A = 42 ÷ 14 = 3; B = 42 ÷ 21 = 2; C = 42 ÷ 42 = 1.

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

Time taken together = Total Work ÷ Total Efficiency = 42 ÷ 6 = 7 days.

∴ The correct answer is 7 days.

Alternate Method

Given: Time taken by A = 14 days, B = 21 days, and C = 42 days.

Formula: Total Work = LCM of individual days; Efficiency = Total Work ÷ Days.

⇒ Total Efficiency of A, B, and C working together = 3 + 2 + 1 = 6 units/day.

⇒ Number of days to complete the work together = Total Work ÷ Combined Efficiency

⇒ Time = 42 ÷ 6 = 7 days.

∴ The correct answer is 7 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete a task. If a person completes work in D days, their 1-day work (efficiency) is 1 ÷ D.

Combined Work Formula

For three people taking x, y, and z days, the combined time T is given by: 1÷T = 1÷x + 1÷y + 1÷z.

Work and Wages

In any partnership or group work, wages are always distributed in the direct ratio of the efficiencies of the workers involved.

Shortcut Trick

Individual capacities: 1 man = 1/75, 1 woman = 1/60, 1 boy = 1/48.

Total daily work = (10 men/75) + (12 women/60) + (8 boys/48)

⇒ 2/15 + 1/5 + 1/6 = (4 + 6 + 5) / 30 = 15/30 = 1/2 of work per day.

Total time taken = 2 days.

∴ The correct answer is 2 days.

Alternate Method

Given:

Time for 5 men = 15 days ⇒ 1 man = 75 days.

Time for 6 women = 10 days ⇒ 1 woman = 60 days.

Time for 4 boys = 12 days ⇒ 1 boy = 48 days.

Formula Used:

Total Work = Efficiency × Time

⇒ Efficiency of 10 men = 10 × 16 = 160 units/day.

⇒ Efficiency of 12 women = 12 × 20 = 240 units/day.

⇒ Efficiency of 8 boys = 8 × 25 = 200 units/day.

⇒ Total Efficiency of group = 160 + 240 + 200 = 600 units/day.

⇒ Time required = Total Work ÷ Total Efficiency = 1200 ÷ 600 = 2 days.

∴ The correct answer is 2 days.

Additional Information

Total Work Equation

Total work remains constant and can be expressed as M₁D₁H₁ = M₂D₂H₂ where M is number of workers, D is days, and H is hours.

Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a task (Efficiency = 1 ÷ Time).

Combined Work Formula

When multiple groups work together, their individual efficiencies (work done per day) are added to find the total work rate.

Shortcut Trick

In opposite direction, Relative Speed = (Sum of Lengths) ÷ Time

Total distance = 304 + 304 = 608 m

Since speeds are equal (v), Relative Speed = v + v = 2v

2v = 608 ÷ 16 = 38 m/s

v = 38 ÷ 2 = 19 m/s

∴ The correct answer is 19 m/s.

Alternate Method

Given:

Length of each train (L₁ = L₂) = 304 m

Time to cross each other = 16 sec

Direction = Opposite

Speed of each train = v (same speed)

Formula Used:

Speed = Distance ÷ Time

Relative Speed (Opposite) = (L₁ + L₂) ÷ T

Calculations:

⇒ 2v = (304 + 304) ÷ 16

⇒ 2v = 608 ÷ 16

⇒ 2v = 38

⇒ v = 38 ÷ 2

⇒ v = 19 m/s

∴ The correct answer is 19 m/s.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their relative speed is the sum of their individual speeds: Speed_rel = S₁ + S₂.

Crossing Stationary Objects

When a train crosses a platform or bridge of length P, total distance = L + P. When it crosses a pole or man, distance = L.

Unit Conversion

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

Shortcut Trick

When speed is constant, Time Difference = (Difference in Distance) ÷ Speed.

⇒ Difference in Distance = 143 − 77 = 66 km.

⇒ Time Difference = 66 ÷ 11 = 6 hr.

∴ The correct answer is 6 hr.

Alternate Method

Given:

Distance₁ = 143 km, Distance₂ = 77 km, and Speed = 11 km/hr.

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Time taken for 143 km (T₁) = 143 ÷ 11

⇒ T₁ = 13 hr

⇒ Time taken for 77 km (T₂) = 77 ÷ 11

⇒ T₂ = 7 hr

⇒ Difference in time = T₁ − T₂

⇒ Difference = 13 − 7 = 6 hr

∴ The correct answer is 6 hr.

Additional Information

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds x and y, Average Speed = 2xy ÷ (x + y).

Relative Speed

When moving in the same direction, Relative Speed = S₁ − S₂. When moving in opposite directions, Relative Speed = S₁ + S₂.

Proportionality Rule

If speed is constant, distance is directly proportional to time (D ∝ T). If distance is constant, speed is inversely proportional to time (S ∝ 1/T).

Shortcut Trick

Initial ratio of A : B = 5 : 4

Final ratio of A : B = 6 : 5

Difference in ratio units = (6 − 5) = (5 − 4) = 1 unit

Since 1 unit = 6, then total final units = 6 + 5 = 11 units

Total sum of final values = 11 × 6 = 66

∴ The correct answer is 66.

Alternate Method

Given:

Initial ratio of A and B = 5 : 4

Increment in each number = 6

Final ratio of A and B = 6 : 5

Formula Used:

If numbers are in ratio a : b, they can be represented as ax and bx. After an increment k, the new ratio is (ax + k) / (bx + k).

Calculations:

⇒ Let the numbers be A = 5x and B = 4x

⇒ According to the question: (5x + 6) / (4x + 6) = 6 / 5

⇒ 5 × (5x + 6) = 6 × (4x + 6)

⇒ 25x + 30 = 24x + 36

⇒ 25x − 24x = 36 − 30

⇒ x = 6

⇒ Final value of A = 5 × 6 + 6 = 36

⇒ Final value of B = 4 × 6 + 6 = 30

⇒ Sum of final values = 36 + 30 = 66

∴ The correct answer is 66.

Additional Information

Constant Difference Rule

When the same quantity is added to or subtracted from both terms of a ratio, the absolute difference between the terms remains unchanged only if the ratio parts change by the same units.

Ratio and Proportion

If a/b = c/d, then they are in proportion. The cross-multiplication method (ad = bc) is the most reliable way to solve for an unknown variable x.

Alternative Ratio Balance

If the difference between the ratio parts (5−4 and 6−5) is not the same, we multiply the ratios by the difference of the other ratio to balance them before applying the unitary method.

Shortcut Trick

Selling Price (SP) at 30% discount = 70% of Marked Price (MP) = Rs. 420. ⇒ 100% MP = (420 ÷ 70) × 100 = Rs. 600. If no discount is given, new SP = MP = Rs. 600. Profit of 50% means 150% of Cost Price (CP) = Rs. 600. ⇒ CP = 600 ÷ 1.5 = Rs. 400. ∴ The correct answer is Rs. 400.

Alternate Method

Given:

Selling Price (SP₁) = Rs. 420

Discount = 30% on Marked Price (MP)

Profit = 50% (if SP₂ = MP)

Formula Used:

MP = [SP × 100] ÷ [100 − Discount%]

CP = [SP × 100] ÷ [100 + Profit%]

Calculations:

⇒ First, find the Marked Price (MP) using the given discount and SP.

⇒ MP = [420 × 100] ÷ [100 − 30]

⇒ MP = [420 × 100] ÷ 70 = 6 × 100 = Rs. 600

⇒ When no discount is given, the Marked Price becomes the new Selling Price.

⇒ New SP = Rs. 600

⇒ Now, find the Cost Price (CP) using the 50% profit condition.

⇒ CP = [600 × 100] ÷ [100 + 50]

⇒ CP = [600 × 100] ÷ 150

⇒ CP = 4 × 100 = Rs. 400

∴ The correct answer is Rs. 400.

Additional Information

Relationship between CP and MP

The direct ratio between Cost Price and Marked Price is given by CP ÷ MP = (100 − D%) ÷ (100 + P%).

Marked Price (List Price)

The price at which an article is listed or labeled by the retailer is called the Marked Price or List Price.

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%.

Shortcut Trick

Let Cost Price (CP) = 10 units ⇒ Selling Price (SP) = 11 units (as Profit = 10%).

Discount = 3/11 × SP = 3/11 × 11 = 3 units.

Marked Price (MP) = SP + Discount = 11 + 3 = 14 units.

Required % = [(14 − 10) / 14] × 100 = 4/14 × 100 = 2/7 × 100 = 28.57%.

∴ The correct answer is 28.57%.

Alternate Method

Given: Profit = 10%, Discount = (3/11) of Selling Price.

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

⇒ Selling Price (SP) = CP × (100 + Profit%)/100 = 100 × (110/100) = ₹110.

⇒ Discount = (3/11) × SP = (3/11) × 110 = ₹30.

⇒ Marked Price (MP) = SP + Discount = 110 + 30 = ₹140.

⇒ Difference between MP and CP = 140 − 100 = ₹40.

⇒ Required percentage = (Difference / MP) × 100 = (40 / 140) × 100.

⇒ Required percentage = 2/7 × 100 ≈ 28.57%.

∴ The correct answer is 28.57%.

Additional Information

Relationship between CP and MP

The direct ratio between Marked Price and Cost Price is given by MP/CP = (100 + Profit%) / (100 − Discount%), where discount % is calculated on MP.

Discount Calculation

Discount = MP − SP. Discount percentage is always calculated on the Marked Price unless otherwise specified.

Profit Percentage

Profit% = [(SP − CP) / CP] × 100. It represents the gain over the original Cost Price.

Shortcut Trick

To find the number to be added to a, b, c, d for proportion, use the formula: x = (bc − ad) ÷ [(a + d) − (b + c)]

Given a = 12, b = 20, c = 28, d = 44.

x = (20 × 28 − 12 × 44) ÷ [(12 + 44) − (20 + 28)]

x = (560 − 528) ÷ (56 − 48)

x = 32 ÷ 8 = 4

∴ The correct answer is 4.

Alternate Method

Given: Numbers 12, 20, 28, and 44.

Formula Used: If (a + x), (b + x), (c + x), and (d + x) are in proportion, then (a + x) ÷ (b + x) = (c + x) ÷ (d + x).

Let the number to be added be x.

⇒ (12 + x) ÷ (20 + x) = (28 + x) ÷ (44 + x)

Cross-multiplying the terms:

⇒ (12 + x)(44 + x) = (20 + x)(28 + x)

⇒ 528 + 12x + 44x + x² = 560 + 20x + 28x + x²

⇒ 528 + 56x = 560 + 48x

⇒ 56x − 48x = 560 − 528

⇒ 8x = 32

⇒ x = 32 ÷ 8 = 4

∴ The correct answer is 4.

Additional Information

Continued Proportion

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

Fourth Proportional

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

Mean Proportional

The mean proportional between two numbers a and b is given by the square root of their product, √(a × b).

Invertendo Property

If a:b = c:d, then by the invertendo property, b:a = d:c.