Quantitative Aptitude PYQs

Shortcut Trick

Solve the expression using the BODMAS rule by prioritizing brackets and multiplication.

First, evaluate the bracket: (34 + 18) = 52. Then multiply by 2: 52 × 2 = 104.

Simplify the rest: 87 + 62 − 53 + 104 − 90 = 149 − 53 + 104 − 90 = 96 + 104 − 90 = 200 − 90 = 110.

∴ The correct answer is 110.

Alternate Method

Given:

87 + 62 − 53 + (34 + 18) × 2 − 90

Formula Used:

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

Calculations:

⇒ 87 + 62 − 53 + (52) × 2 − 90

⇒ 87 + 62 − 53 + 104 − 90

⇒ 149 − 53 + 104 − 90

⇒ 96 + 104 − 90

⇒ 200 − 90

⇒ 110

∴ The correct answer is 110.

Additional Information

BODMAS / VBODMAS Rule

Standard order of operations: Vinculum (Bar), Brackets, Of, Division, Multiplication, Addition, and Subtraction. Always solve from left to right for operators of same priority.

Types of Brackets

Solve brackets in the following sequence: Small Brackets (), Curly Brackets {}, and then Square Brackets [].

Precedence of MD and AS

Multiplication and Division have the same priority level, as do Addition and Subtraction; in such cases, calculate from left to right.

Shortcut Trick

Solve (19 of 1) = 19 and 36 ÷ 3 = 12 first.

Rewrite: 96 − 41 + 27 + 12 + 19 − 22

Sum positives: 96 + 27 + 12 + 19 = 154

Sum negatives: 41 + 22 = 63

Final: 154 − 63 = 91

∴ The correct answer is 91.

Alternate Method

Given Expression: 96 − 41 + 27 + 36 ÷ 3 + (19 of 1) − 22

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

⇒ 96 − 41 + 27 + 36 ÷ 3 + 19 − 22 (Solving Of: 19 × 1 = 19)

⇒ 96 − 41 + 27 + 12 + 19 − 22 (Solving Division: 36 ÷ 3 = 12)

⇒ (96 + 27 + 12 + 19) − 41 − 22 (Rearranging Positive terms)

⇒ 154 − 41 − 22 (Performing Addition)

⇒ 154 − 63 (Performing Subtraction of negative totals)

⇒ 91

∴ The correct answer is 91.

Additional Information

BODMAS / VBODMAS Rule

It defines the sequence: V (Vinculum), B (Brackets), O (Of), D (Division), M (Multiplication), A (Addition), S (Subtraction).

The 'Of' Operator

The 'Of' operator is mathematically equivalent to multiplication but must be evaluated before division and standard multiplication.

Addition and Subtraction Priority

Addition and subtraction have the same level of precedence and can be solved from left to right after all higher-priority operations are complete.

Shortcut Trick

Use the ratio relationship: Illumination (I) ∝ Power (P) / Distance² (d²)

I₂ / I₁ = (P₂ / P₁) × (d₁ / d₂)²

I₂ / 15 = (100 / 60) × (2 / 3)²

I₂ / 15 = (5 / 3) × (4 / 9) ⇒ I₂ = 15 × 20 / 27 = 300 / 27

I₂ = 11.11 lux

∴ The correct answer is 11.11 lux.

Alternate Method

Given:

Initial Power (P₁) = 60 W, Distance (d₁) = 2 m, Illumination (I₁) = 15 lux

Final Power (P₂) = 100 W, Distance (d₂) = 3 m

Formula Used:

I = k × (P / d²), where k is the proportionality constant.

Step 1: Find the value of k from the first condition

⇒ 15 = k × (60 / 2²)

⇒ 15 = k × (60 / 4)

⇒ 15 = k × 15

⇒ k = 1

Step 2: Find the illumination for the second condition

⇒ I₂ = k × (P₂ / d₂²)

⇒ I₂ = 1 × (100 / 3²)

⇒ I₂ = 100 / 9

⇒ I₂ = 11.111...

∴ The correct answer is 11.11 lux.

Additional Information

Inverse Square Law

The illumination of a surface is inversely proportional to the square of the distance from the source. Formula: I ∝ 1/d².

Luminous Flux vs. Illumination

Luminous flux (Lumen) is the total light emitted by a source, while Illumination (Lux) is the flux received per unit area (1 Lux = 1 Lumen/m²).

Lambert's Cosine Law

The illumination at any point on a surface is proportional to the cosine of the angle between the normal to the surface and the direction of incident light: I = (P/d²) × cosθ.

Shortcut Trick

Sum the whole numbers: 4 + 44 + 444 + 4444 = 4936.

Sum the fractional parts: 2/3 + 2/3 + 2/3 + 2/3 = 8/3.

Convert improper fraction to mixed number: 8/3 = 2 2/3.

Total = 4936 + 2 2/3 = 4938 2/3.

∴ The correct answer is 4938 2/3.

Alternate Method

Given: 4 2/3 + 44 2/3 + 444 2/3 + 4444 2/3

Method: Grouping Integers and Proper Fractions separately.

⇒ (4 + 44 + 444 + 4444) + (2/3 + 2/3 + 2/3 + 2/3)

⇒ 4936 + (4 × 2/3)

⇒ 4936 + 8/3

⇒ 4936 + 2 + 2/3

⇒ 4938 + 2/3

⇒ 4938 2/3

∴ The correct answer is 4938 2/3.

Additional Information

Mixed Fraction Decomposition

A mixed fraction a b/c is identical to the sum a + b/c, where 'a' is the integer and 'b/c' is the proper fraction.

Repeated Digit Addition Shortcut

To sum 4 + 44 + 444 + 4444, use the logic 4 × (1 + 11 + 111 + 1111) = 4 × 1234 = 4936.

Common Denominator Rule

When adding fractions with the same denominator, simply add the numerators and keep the denominator constant.

Shortcut Trick

Compare pairs using cross-multiplication (a c/d if a×d > b×c).

11/16 vs 13/19 ⇒ 11 × 19 = 209 and 16 × 13 = 208 (209 > 208)

11/16 vs 15/22 ⇒ 11 × 22 = 242 and 16 × 15 = 240 (242 > 240)

11/16 vs 7/11 ⇒ 11 × 11 = 121 and 16 × 7 = 112 (121 > 112)

∴ The correct answer is 11/16.

Alternate Method

Given: Fractions 7/11, 11/16, 13/19, and 15/22.

Formula Used: Decimal Conversion (Value = Numerator ÷ Denominator).

⇒ 7 ÷ 11 ≈ 0.6363

⇒ 11 ÷ 16 = 0.6875

⇒ 13 ÷ 19 ≈ 0.6842

⇒ 15 ÷ 22 ≈ 0.6818

⇒ Comparing the values: 0.6875 > 0.6842 > 0.6818 > 0.6363.

∴ The correct answer is 11/16.

Additional Information

Cross-Multiplication Rule

To compare a/b and c/d, check a×d and b×c. If a×d > b×c, then a c/d.

Same Difference Rule

If the difference between numerator and denominator is the same for proper fractions, the one with the larger values is the largest.

Decimal Approximation

Converting fractions to decimals is the most accurate method for comparing values that are very close to each other.

Shortcut Trick

Given Sum of Diagonals (d₁ + d₂) = 62 cm and Area = 336 cm².

Relationship between Side (a) and Diagonals: 4a² = d₁² + d₂².

We know (d₁ + d₂)² = d₁² + d₂² + 2d₁d₂ = 4a² + 4 × Area.

62² = 4a² + 4 × 336 ⇒ 3844 = 4a² + 1344.

4a² = 2500 ⇒ a² = 625 ⇒ a = 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Sum of diagonals (d₁ + d₂) = 62 cm

Area of rhombus = 336 cm²

Formula Used:

Area of Rhombus = (1/2) × d₁ × d₂

Side of Rhombus (a) = √[(d₁/2)² + (d₂/2)²]

Calculations:

⇒ Area = (1/2) × d₁ × d₂ = 336

⇒ d₁ × d₂ = 672

⇒ We know: (d₁ + d₂)² = d₁² + d₂² + 2d₁d₂

⇒ 62² = d₁² + d₂² + 2(672)

⇒ 3844 = d₁² + d₂² + 1344

⇒ d₁² + d₂² = 3844 − 1344 = 2500

⇒ For a rhombus, 4a² = d₁² + d₂²

⇒ 4a² = 2500

⇒ a² = 2500 ÷ 4 = 625

⇒ a = √625 = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Properties of Rhombus

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

Perimeter of Rhombus

The perimeter is given by 4 × side (a).

Pythagorean Relation

The side and half-diagonals form a right-angled triangle: a² = (d₁/2)² + (d₂/2)².

Shortcut Trick

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

Total ratio parts = 1 + 3 = 4 units.

4 units ⇒ 180°

1 unit ⇒ 180° ÷ 4 = 45°

Larger angle (3 units) ⇒ 3 × 45° = 135°.

∴ The correct answer is 135°.

Alternate Method

Given:

Ratio of adjacent angles = 1 : 3

Formula used:

Sum of adjacent angles in a parallelogram = 180° (Supplementary angles property)

Let the two adjacent angles be 1x and 3x.

⇒ 1x + 3x = 180°

⇒ 4x = 180°

⇒ x = 180° ÷ 4

⇒ x = 45°

⇒ Smaller angle = 1 × 45° = 45°

⇒ Larger angle = 3 × 45° = 135°

∴ The correct answer is 135°.

Additional Information

Properties of a Parallelogram

Opposite angles are equal, and consecutive (adjacent) angles are supplementary (sum to 180°).

Diagonal Properties

The diagonals of a parallelogram bisect each other but are not necessarily equal in length unless it is a rectangle.

Angle Sum Property

The sum of all four interior angles of any quadrilateral, including a parallelogram, is always 360°.

Shortcut Trick

Simplify all fractions to their lowest terms before calculation.

4/7 is already in simplest form.

8/14 → 4/7 (dividing by 2) and 12/21 → 4/7 (dividing by 3).

Since all fractions are now 4/7, their LCM is 4/7.

∴ The correct answer is 4/7.

Alternate Method

Given Fractions: 4/7, 8/14, and 12/21

Formula Used: LCM of Fractions = LCM of Numerators ÷ HCF of Denominators (applied to simplified fractions)

⇒ First, reduce all fractions to their simplest forms:

⇒ Fraction 1 = 4/7

⇒ Fraction 2 = 8/14 = (8 ÷ 2) / (14 ÷ 2) = 4/7

⇒ Fraction 3 = 12/21 = (12 ÷ 3) / (21 ÷ 3) = 4/7

⇒ Now, find the LCM of (4/7, 4/7, 4/7):

⇒ LCM = LCM of (4, 4, 4) ÷ HCF of (7, 7, 7)

⇒ LCM = 4 ÷ 7 = 4/7

∴ The correct answer is 4/7.

Additional Information

LCM of Fractions Formula

LCM of fractions = LCM of Numerators ÷ HCF of Denominators. Ensure fractions are in their simplest form first.

HCF of Fractions Formula

HCF of fractions = HCF of Numerators ÷ LCM of Denominators. This is the inverse of the LCM rule.

Simplification Rule

For any LCM or HCF calculation involving fractions, failing to simplify fractions to their lowest terms (co-prime numerators and denominators) can lead to incorrect results.

Shortcut Trick

Ratio of numbers = 4 : 5

LCM of ratio parts 4 and 5 = 20 units

Given 20 units = 180 ⇒ 1 unit = 9

Sum of ratios = 4 + 5 = 9 units

Total Sum = 9 × 9 = 81

∴ The correct answer is 81.

Alternate Method

Given:

Ratio of two numbers = 4 : 5

LCM of the numbers = 180

Formula Used:

If two numbers are in the ratio a : b, then the numbers are ax and bx (where x is the HCF).

LCM (ax, bx) = a × b × x (if a and b are coprime).

Calculations:

⇒ Let the two numbers be 4x and 5x.

⇒ LCM of 4x and 5x = 4 × 5 × x = 20x

⇒ According to the question, 20x = 180

⇒ x = 180 ÷ 20

⇒ x = 9 (This is the HCF of the numbers)

⇒ First number = 4 × 9 = 36

⇒ Second number = 5 × 9 = 45

⇒ Sum of the numbers = 36 + 45 = 81

∴ The correct answer is 81.

Additional Information

Relation between 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.

HCF of Ratios

If numbers are in the ratio a : b, then the HCF is the common multiplier x that converts the ratio into actual values ax and bx.

Coprime Numbers

Two numbers are coprime if their HCF is 1. If ratio parts are simplified (like 4 and 5), they are always coprime.

Shortcut Trick

The ratio of the percentages is 30% : 120% = 1 : 4.

Given that 1 part (representing 30%) is equal to 90.

To find 4 parts (representing 120%), we calculate: 4 × 90 = 360.

∴ The correct answer is 360.

Alternate Method

Given: 30% of a number is 90.

Formula: Value = (Total Number × Percentage) ÷ 100

⇒ Let the number be x.

⇒ 30% × x = 90

⇒ (30 ÷ 100) × x = 90

⇒ x = (90 × 100) ÷ 30

⇒ x = 300

⇒ Now, find 120% of the number 300:

⇒ 120% of 300 = (120 ÷ 100) × 300

⇒ 1.2 × 300 = 360

∴ The correct answer is 360.

Additional Information

Unitary Method in Percentage

If p% of a number is 'a', then 1% is a ÷ p, and q% is (a ÷ p) × q.

Percentage to Decimal Conversion

To convert a percentage to a decimal, divide by 100 (e.g., 120% = 120 ÷ 100 = 1.2).

Percentage of a Number

The formula to find p% of x is (p ÷ 100) × x.

Shortcut Trick

Convert percentages into fractions: 40% = 2/5 and 80% = 4/5.

Required value = (2/5) × (4/5) × (5/8) × 4800

Simplify fractions first: (4/5) × (5/8) = 1/2. Then (2/5) × (1/2) = 1/5.

Final calculation: 1/5 × 4800 = 960.

∴ The correct answer is 960.

Alternate Method

Given: Initial value = 4800, fractional part = 5/8, percentages = 80% and 40%.

Formula Used: Final Value = P₁% × P₂% × Fraction × Total Value

⇒ Step 1: Calculate 5/8 of 4800

⇒ (5/8) × 4800 = 5 × 600 = 3000

⇒ Step 2: Calculate 80% of the result

⇒ 80% of 3000 = (80/100) × 3000 = 0.8 × 3000 = 2400

⇒ Step 3: Calculate 40% of the result

⇒ 40% of 2400 = (40/100) × 2400 = 0.4 × 2400 = 960

∴ The correct answer is 960.

Additional Information

The 'Of' Operator

In mathematical expressions, 'of' functions as a multiplication sign and should be prioritized according to BODMAS/VBODMAS rules.

Percentage to Fraction Chart

Memorizing common equivalents helps in quick calculations: 20% = 1/5, 40% = 2/5, 60% = 3/5, 80% = 4/5, and 12.5% = 1/8.

Commutative Property of Multiplication

The order of multiplication does not affect the result. Therefore, (40% of 80% of 5/8) is identical to (5/8 of 40% of 80%).

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) is the inverse of their percentage coefficients.

CP : SP = 50 : 40 = 5 : 4

Since CP (5) > SP (4), there is a Loss of 1 unit.

Loss % = (Loss ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given: 40% of CP = 50% of SP

Formula: Loss % = [(CP − SP) ÷ CP] × 100

⇒ 40 ÷ 100 × CP = 50 ÷ 100 × SP

⇒ 4 × CP = 5 × SP

⇒ SP ÷ CP = 4 ÷ 5

⇒ Let CP = 5 and SP = 4

⇒ Loss = 5 − 4 = 1

⇒ Loss % = (1 ÷ 5) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price (SP) > Cost Price (CP). Profit % = [(SP − CP) ÷ CP] × 100.

Inverse Ratio Property

If x% of A = y% of B, then the ratio A : B = y :

x. This is a common shortcut for comparison-based problems.

Marked Price and Discount

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

Shortcut Trick

Profit is 30% on Selling Price (SP) ⇒ Profit : SP = 30 : 100 = 3 : 10

Since Cost Price (CP) = SP − Profit, then CP = 10 − 3 = 7 units

Profit percentage on CP = (Profit ÷ CP) × 100

Profit % = (3 ÷ 7) × 100 = 42.857...% ≈ 42.86%

∴ The correct answer is 42.86 percent.

Alternate Method

Given: Profit on Selling Price = 30%

Formula Used: Profit % on Cost Price = [Profit ÷ (Selling Price − Profit)] × 100

⇒ Let the Selling Price (SP) = 100

⇒ Profit earned = 30% of 100 = 30

⇒ Cost Price (CP) = SP − Profit = 100 − 30 = 70

⇒ Profit Percentage on Cost Price = (Profit ÷ CP) × 100

⇒ Profit % = (30 ÷ 70) × 100

⇒ Profit % = 300 ÷ 7 = 42.857...

⇒ Profit % ≈ 42.86%

∴ The correct answer is 42.86 percent.

Additional Information

Profit Percentage Comparison

Profit % on Cost Price is always numerically higher than Profit % on Selling Price because the denominator (CP) is smaller than SP.

Direct Conversion Rule

If profit on SP is x%, then Profit % on CP = [x ÷ (100 − x)] × 100.

Loss on Selling Price

If loss on SP is y%, then Loss % on CP = [y ÷ (100 + y)] × 100.

Shortcut Trick

Efficiency ratio of R : S = 140 : 100 = 7 : 5.

Time ratio is inversely proportional ⇒ R : S = 5 : 7.

Given S takes 35 days (7 units = 35 days ⇒ 1 unit = 5 days).

Time for R alone = 5 units × 5 = 25 days.

∴ The correct answer is 25 days.

Alternate Method

Given:

Time taken by S = 35 days.

R is 40% more efficient than S.

Formula used:

Total Work = Efficiency × Time

Calculations:

Let the Efficiency of S be 1 unit/day.

⇒ Total Work = 1 × 35 = 35 units.

⇒ Efficiency of R = 1 + 40% of 1 = 1.4 units/day.

⇒ Time taken by R = Total Work ÷ Efficiency of R

⇒ Time for R = 35 ÷ 1.4

⇒ Time for R = 350 ÷ 14 = 25 days.

∴ The correct answer is 25 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a task. If efficiency ratio is a : b, then time ratio is b : a.

Work and Wages

In most competitive exams, wages are distributed in the ratio of the total work done by each individual, which depends on their efficiency and time spent.

Percentage Conversion

To increase a value by x%, multiply the base value by (100 + x) ÷ 100. Here, 40% more means a multiplier of 1.4.

Shortcut Trick

Total hours for R = 10 × 3 = 30 hr | Total hours for S = 6 × 5 = 30 hr.

Since both take 30 hours, their combined time = 30 ÷ 2 = 15 hours.

To finish the work in 4 days, hours per day = 15 ÷ 4 = 3.75 hours.

∴ The correct answer is 3.75 hours.

Alternate Method

Given: R works 10 days (3 hr/day) | S works 6 days (5 hr/day) | Target duration = 4 days.

Formula Used: Total Work = Efficiency × Total Time (in hours).

⇒ Total hours for R = 10 × 3 = 30 hr

⇒ Total hours for S = 6 × 5 = 30 hr

⇒ LCM of (30, 30) = 30 units (Total Work)

⇒ Combined Efficiency of (R + S) = 1 + 1 = 2 units/hr

⇒ Total time needed together = 30 ÷ 2 = 15 hours

⇒ Hours per day needed for 4 days = 15 ÷ 4 = 3.75 hours/day

∴ The correct answer is 3.75 hours.

Additional Information

Work and Wages

Wages are always distributed in the ratio of the actual work done by each person, which is (Efficiency × Time).

Combined Efficiency Rule

If two people take 'a' and 'b' hours respectively, they complete the work together in (a × b) ÷ (a + b) hours.

Inverse Proportion

For a constant amount of work, the time taken is inversely proportional to the efficiency of the worker.

Shortcut Trick

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

Speed = 216 × (5/18)

Speed = 12 × 5 = 60 m/s

∴ The correct answer is 60 m/s.

Alternate Method

Given: Speed of the car = 216 km/hr

Formula Used: Speed = Distance ÷ Time

⇒ Distance in 1 hour = 216 km = 216 × 1000 = 216000 meters ⇒ Time taken = 1 hour = 60 × 60 = 3600 seconds ⇒ Speed = 216000 ÷ 3600 ⇒ Speed = 2160 ÷ 36 ⇒ Speed = 60 m/s

∴ The correct answer is 60 m/s.

Additional Information

Speed Conversion Factors

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

Average Speed

Average Speed = Total Distance ÷ Total Time. If a car travels equal distances at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

Relative Speed

When two objects move in opposite directions, their relative speed is the sum (S₁ + S₂); in the same direction, it is the difference (S₁ − S₂).

Shortcut Trick

Speed ratio (Usual : New) = 100% : 40% = 5 : 2.

Since distance is constant, Time ratio is the inverse of speed ratio = 2 : 5.

Difference in time parts = 5 − 2 = 3 units.

Given, 3 units = 15 min ⇒ 1 unit = 5 min.

Usual time = 2 units = 2 × 5 = 10 min.

∴ The correct answer is 10 min.

Alternate Method

Given:

New speed = 40% of Usual speed.

Time delay = 15 minutes.

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Let Usual speed = S and Usual time = T.

⇒ New Speed = 40% of S = 0.4S.

⇒ New Time = T + 15 min.

⇒ Since Distance is same: S × T = 0.4S × (T + 15)

⇒ T = 0.4T + 6

⇒ T − 0.4T = 6

⇒ 0.6T = 6

⇒ T = 6 ÷ 0.6 = 10 min.

∴ The correct answer is 10 min.

Additional Information

Speed-Time Inverse Relationship

When Distance is constant, Speed (S) is inversely proportional to Time (T). Thus, S₁ × T₁ = S₂ × T₂ or S₁/S₂ = T₂/T₁.

Late and Early Concept

If a person is late by time t₁ in one case and late by t₂ in another, the time difference is |t₁ − t₂|. If one is late and one is early, the difference is t₁ + t₂.

Percentage to Ratio Shortcut

Always convert percentage speed to fractions (40% = 2/5). If speed becomes 2/5, then time must become 5/2 of the usual time.

Shortcut Trick

Initial ratio A : B = 7 : 5 (Difference = 2 units)

Final ratio A : B = 4 : 3. To equate differences, multiply by 2 → 8 : 6 (Difference = 2 units)

Increase in each part = 8 − 7 = 1 unit (for A) and 6 − 5 = 1 unit (for B)

Given 1 unit = 4

Initial value of B = 5 units = 5 × 4 = 20

∴ The correct answer is 20.

Alternate Method

Given: Initial ratio of A and B is 7 : 5. Increase in each number is 4. New ratio is 4 : 3.

Formula Used: Ratio → a : b can be written as ax and bx.

⇒ Let the numbers be 7x and 5x.

⇒ According to the question: (7x + 4) / (5x + 4) = 4 / 3

⇒ 3 × (7x + 4) = 4 × (5x + 4)

⇒ 21x + 12 = 20x + 16

⇒ 21x − 20x = 16 − 12

⇒ x = 4

⇒ Initial value of B = 5x = 5 × 4 = 20

∴ The correct answer is 20.

Additional Information

Concept of Ratio

A ratio a : b represents the relative size of two quantities. Multiplying both terms by the same constant k gives an equivalent ratio ak : bk.

Equating Differences

If the same value is added to or subtracted from both terms of a ratio, the absolute difference between the parts must be adjusted to be consistent across both ratios to solve using the unit method.

Proportionality Rule

If a/b = c/d, then by cross-multiplication, ad = bc. This is the fundamental property used to solve linear equations involving ratios.

Shortcut Trick

The difference in Selling Price is equal to the difference in Discount Percentages applied on the Marked Price.

Difference in discount % = 25% − 18% = 7%

Given, 7% of Marked Price = Rs. 70

1% of Marked Price = Rs. 10

⇒ 100% (Marked Price) = 10 × 100 = Rs. 1000

∴ The correct answer is Rs. 1000.

Alternate Method

Given:

Discount₁ = 25% | Discount₂ = 18% | Difference in SP = Rs. 70

Formula Used:

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100

Calculations:

Let the Marked Price (MP) of the article be Rs.

x. ⇒ SP₁ = x × (100 − 25) ÷ 100 = 0.75x

⇒ SP₂ = x × (100 − 18) ÷ 100 = 0.82x

⇒ Difference = SP₂ − SP₁ = 0.82x − 0.75x

⇒ 70 = 0.07x

⇒ x = 70 ÷ 0.07

⇒ x = 1000

∴ The correct answer is Rs. 1000.

Additional Information

Relationship between MP and CP

Marked Price (MP) ÷ Cost Price (CP) = (100 + Profit%) ÷ (100 − Discount%). This is used when both profit and discount percentages are known.

Successive Discounts

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

Net Profit after Discount

Profit = [MP × (100 − D%)/100] − CP. If SP is less than CP, it results in a loss.

Shortcut Trick

Quickly calculate the Discount ÷ Marked Price ratio for all options:

1. 200/1200 = 1/6 ≈ 16.67%

2. 300/1500 = 1/5 = 20%

3. 800/2300 = 8/23 ≈ 34.78%

4. 600/2200 = 3/11 ≈ 27.27%

Comparing the values, 34.78% is the highest.

∴ The correct answer is Selling price = Rs. 1500 and Marked price = Rs. 2300.

Alternate Method

Given: Selling Price (SP) and Marked Price (MP) for four different scenarios.

Formula Used: Discount % = [(MP − SP) ÷ MP] × 100

Calculating discount percentage for each case:

Case (1): MP = 1200, SP = 1000

⇒ Discount % = [(1200 − 1000) ÷ 1200] × 100 = (200 ÷ 1200) × 100 = 16.66%

Case (2): MP = 1500, SP = 1200

⇒ Discount % = [(1500 − 1200) ÷ 1500] × 100 = (300 ÷ 1500) × 100 = 20%

Case (3): MP = 2300, SP = 1500

⇒ Discount % = [(2300 − 1500) ÷ 2300] × 100 = (800 ÷ 2300) × 100 ≈ 34.78%

Case (4): MP = 2200, SP = 1600

⇒ Discount % = [(2200 − 1600) ÷ 2200] × 100 = (600 ÷ 2200) × 100 ≈ 27.27%

By comparison, Case (3) provides the highest discount percentage.

∴ The correct answer is Selling price = Rs. 1500 and Marked price = Rs. 2300.

Additional Information

Successive Discount Formula

For two successive discounts of x% and y%, the single equivalent discount is (x + y − xy/100)%.

Marked Price (List Price)

The price printed on the label of an article is called the Marked Price. Discounts are always computed on this value unless specified otherwise.

Relationship between MP and CP

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