Quantitative Aptitude PYQs

Shortcut Trick

Solve the expression using the BODMAS rule: Brackets first, then Multiplication, then Addition/Subtraction.

Step 1 (Brackets): (38 + 19) = 57.

Step 2 (Multiplication): 57 × 2 = 114.

Step 3 (Simplify remaining): 82 + 41 − 27 + 114 − 77 = 133.

∴ The correct answer is 133.

Alternate Method

Given:

82 + 41 − 27 + (38 + 19) × 2 − 77

Formula Used:

Operations are performed in the order defined by BODMAS:

B: Brackets | O: Of (Orders/Powers) | D: Division | M: Multiplication | A: Addition | S: Subtraction

Calculations:

⇒ 82 + 41 − 27 + (57) × 2 − 77 (Solving Bracket)

⇒ 82 + 41 − 27 + 114 − 77 (Solving Multiplication)

⇒ 123 − 27 + 114 − 77 (Solving Addition from left)

⇒ 96 + 114 − 77 (Solving Subtraction)

⇒ 210 − 77 (Solving Addition)

⇒ 133 (Final Subtraction)

∴ The correct answer is 133.

Additional Information

BODMAS Priority

Division and Multiplication hold the same priority level, as do Addition and Subtraction. When they appear together, solve from left to right.

Types of Brackets

Always solve inside-out: Small Brackets ( ) → Curly Brackets { } → Square Brackets [ ].

VBODMAS

V stands for Vinculum (Bar bracket). If a bar is present over numbers, solve that part before any other bracket or operation.

Shortcut Trick

Expression: 92 + 63 − 36 ÷ 6 + (33 of 2) − 58 + 29

Solve 'Of' and Division first: 33 × 2 = 66 and 36 ÷ 6 = 6.

Simplified: 92 + 63 − 6 + 66 − 58 + 29

Group positive and negative terms: (92 + 63 + 66 + 29) − (6 + 58) = 250 − 64 = 186.

∴ The correct answer is 186.

Alternate Method

Given: 92 + 63 − 36 ÷ 6 + (33 of 2) − 58 + 29

Formula Used: Apply BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) rule.

⇒ 92 + 63 − 36 ÷ 6 + (33 × 2) − 58 + 29

⇒ 92 + 63 − 36 ÷ 6 + 66 − 58 + 29 (Solving 'Of')

⇒ 92 + 63 − 6 + 66 − 58 + 29 (Solving Division)

⇒ 155 − 6 + 66 − 58 + 29 (Solving Addition step 1)

⇒ 221 − 6 − 58 + 29 (Solving Addition step 2)

⇒ 250 − 6 − 58 (Solving Addition step 3)

⇒ 244 − 58 (Solving Subtraction step 1)

⇒ 186

∴ The correct answer is 186.

Additional Information

BODMAS Rule

It defines the sequence of operations: Brackets (), then Of/Orders (square roots/indices), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Operator Priority of 'Of'

The operator 'of' acts as multiplication but has a higher priority than division and multiplication in mathematical expressions.

Grouping Terms

In long addition and subtraction chains, grouping all positive numbers and all negative numbers separately before the final subtraction reduces the chance of calculation errors.

Shortcut Trick

Convert decimal values into simple fractions for faster calculation.

0.625 = 5/8 | 0.4 = 2/5 | 0.25 = 1/4 | 0.8 = 4/5 | 0.18 = 18/100

⇒ [(5/8 × 2/5) + (1/4 × 4/5)] ÷ (18/100) = [1/4 + 1/5] ÷ (9/50)

⇒ (9/20) ÷ (9/50) = (9/20) × (50/9) = 5/2 = 2.5

∴ The correct answer is 2.5.

Alternate Method

Given: ((0.625 × 0.4) + (0.25 × 0.8)) / (0.18)

Following the BODMAS rule, we solve the brackets and multiplication first:

⇒ 0.625 × 0.4 = 0.25

⇒ 0.25 × 0.8 = 0.2

Now, add the terms in the numerator:

⇒ 0.25 + 0.2 = 0.45

Now, perform the division by the denominator:

⇒ 0.45 ÷ 0.18

⇒ 45 ÷ 18 (By multiplying both by 100 to remove decimals)

⇒ 5 ÷ 2 = 2.5

∴ The correct answer is 2.5.

Additional Information

BODMAS Rule

Simplification follows the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Decimal to Fraction Conversion

Identifying standard fractions like 0.625 = 5/8 or 0.125 = 1/8 helps in reducing calculation time significantly.

Division of Decimals

To divide two decimals, shift the decimal point in both divisor and dividend to the right until the divisor becomes a whole number.

Shortcut Trick

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

Here, a = 4.25 and b = 3.75.

The middle term 2ab = 2 × 4.25 × 3.75 = 8.5 × 3.75 = 31.875.

Result = (4.25 + 3.75)² = 8² = 64.

∴ The correct answer is 64.

Alternate Method

Given:

Expression: 4.25 × 4.25 + 3.75 × 3.75 + 31.875

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ Let a = 4.25 and b = 3.75.

⇒ Calculating a² = 4.25 × 4.25 = 18.0625

⇒ Calculating b² = 3.75 × 3.75 = 14.0625

⇒ Checking the third term: 2 × a × b = 2 × 4.25 × 3.75 = 31.875

⇒ Therefore, the expression is in the form a² + b² + 2ab.

⇒ Expression = (4.25 + 3.75)²

⇒ Expression = (8.0)²

⇒ Result = 64

∴ The correct answer is 64.

Additional Information

Square of Difference Identity

The formula (a − b)² = a² + b² − 2ab is used when the middle term is subtracted instead of added.

Difference of Two Squares

The formula a² − b² = (a − b)(a + b) is a fundamental identity used to factorize the difference between two perfect squares.

Square of a Trinomial

For three terms, the identity is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

BODMAS Rule

Always follow the order of operations: Brackets, Orders (powers), Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

Apply the BODMAS rule: first evaluate the square, then multiplication, and finally addition/subtraction.

Evaluate: 15² = 225; 5 × 7 = 35; 4 × 18 = 72.

Combine terms: √(225 + 35 − 72 + 8) = √196.

The square root of 196 is 14.

∴ The correct answer is 14.

Alternate Method

Given: √(15² + 5 × 7 − 4 × 18 + 8)

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

⇒ √(225 + 5 × 7 − 4 × 18 + 8)

⇒ √(225 + 35 − 72 + 8)

⇒ √(260 − 72 + 8)

⇒ √(188 + 8)

⇒ √196

⇒ 14

∴ The correct answer is 14.

Additional Information

BODMAS Rule

It provides the standard order of operations to solve mathematical expressions: Brackets, Orders (Powers and Roots), Division, Multiplication, Addition, and Subtraction.

Perfect Squares

A perfect square is a number that is the result of multiplying an integer by itself. For example, 14 × 14 = 196, making 196 a perfect square.

Square Root Calculation

Finding the square root is the inverse operation of squaring a number. Memorizing squares from 1 to 30 is helpful for rapid simplification in exams.

Shortcut Trick

The diagonal of a square with side 'a' is always a√2.

Given the side = 12 cm.

Diagonal = 12 × √2 = 12√2 cm.

∴ The correct answer is 12√2 cm.

Alternate Method

Given:

Side of the square (a) = 12 cm

Formula Used:

In a right-angled triangle formed by two sides and the diagonal, by Pythagoras Theorem:

Diagonal (d)² = Side² + Side²

Calculations:

⇒ d² = 12² + 12²

⇒ d² = 144 + 144

⇒ d² = 288

⇒ d = √(144 × 2)

⇒ d = 12√2 cm

∴ The correct answer is 12√2 cm.

Additional Information

Area of a Square

The Area can be calculated as side² or as (diagonal² / 2).

Perimeter of a Square

The Perimeter is the total length of the boundary, calculated as 4 × side.

Inradius and Circumradius

The Inradius is side/2 and the Circumradius is diagonal/2 or (side × √2)/2.

Shortcut Trick

In a rhombus, the relation between sides and diagonals is: 4 × Side² = d₁² + d₂²

Side = Perimeter ÷ 4 = 58 ÷ 4 = 14.5 cm

4 × (14.5)² = 20² + d₂² ⇒ 4 × 210.25 = 400 + d₂²

841 − 400 = d₂² ⇒ d₂ = √441 = 21 cm

Sum of diagonals = 20 + 21 = 41 cm

∴ The correct answer is 41 cm.

Alternate Method

Given: Perimeter of rhombus = 58 cm and one diagonal (d₁) = 20 cm.

Formula Used: Side = Perimeter ÷ 4; In a right triangle formed by diagonals, Side² = (d₁/2)² + (d₂/2)².

⇒ Side of the rhombus = 58 ÷ 4 = 14.5 cm

⇒ The diagonals of a rhombus bisect each other at 90°.

⇒ Half of the first diagonal (d₁/2) = 20 ÷ 2 = 10 cm

⇒ In ΔAOD, OA² + OD² = AD²

⇒ 10² + (d₂/2)² = 14.5²

⇒ 100 + (d₂/2)² = 210.25

⇒ (d₂/2)² = 210.25 − 100 = 110.25

⇒ Half of the second diagonal (d₂/2) = √110.25 = 10.5 cm

⇒ Second diagonal (d₂) = 10.5 × 2 = 21 cm

⇒ Sum of diagonals = 20 + 21 = 41 cm

∴ The correct answer is 41 cm.

Additional Information

Area of Rhombus

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

Diagonal Properties

The diagonals of a rhombus are not necessarily equal, but they always bisect each other at right angles (90°) and bisect the internal angles.

Perimeter and Side Relation

For any rhombus with side 'a', the perimeter is 4a and the relation with diagonals is given by 4a² = d₁² + d₂².

Shortcut Trick

Use the direct relation: (Sum of diagonals)² = 4 × (Side² + Area)

(d₁ + d₂)² = 4 × (37² + 840)

(d₁ + d₂)² = 4 × (1369 + 840) = 4 × 2209

(d₁ + d₂)² = 8836

d₁ + d₂ = √8836 = 94 cm

∴ The correct answer is 94 cm.

Alternate Method

Given: Side of the rhombus (s) = 37 cm, Area of the rhombus (A) = 840 cm².

Formula Used: Area = (d₁ × d₂)/2 and 4s² = d₁² + d₂².

⇒ Area = (d₁ × d₂)/2 = 840

⇒ d₁ × d₂ = 1680

⇒ 4 × (Side)² = d₁² + d₂²

⇒ 4 × (37)² = d₁² + d₂²

⇒ 4 × 1369 = d₁² + d₂² = 5476

⇒ (d₁ + d₂)² = d₁² + d₂² + 2d₁d₂

⇒ (d₁ + d₂)² = 5476 + 2(1680)

⇒ (d₁ + d₂)² = 5476 + 3360 = 8836

⇒ d₁ + d₂ = √8836 = 94 cm

∴ The correct answer is 94 cm.

Additional Information

Properties of Rhombus Diagonals

The diagonals of a rhombus bisect each other at right angles (90°), forming four congruent right-angled triangles.

Perimeter of a Rhombus

Since all sides of a rhombus are equal, the perimeter is calculated as 4 × side.

Relation between Side and Diagonals

Applying Pythagoras Theorem in one of the small triangles: (d₁/2)² + (d₂/2)² = side².

Shortcut Trick

The Highest Common Factor (HCF) of three numbers A, B, and C is determined by HCF(HCF(A, B), C).

Given HCF(A, B) = 36 and C = 18, we calculate HCF(36, 18).

Since 18 is a direct factor of 36 (18 × 2 = 36), the HCF must be 18.

∴ The correct answer is 18.

Alternate Method

Given:

Highest Common Factor of A and B = 36

The value of number C = 18

Formula Used:

HCF(A, B, C) = HCF(HCF(A, B), C)

Calculations:

⇒ HCF(A, B, C) = HCF(36, 18)

⇒ Prime factors of 36 = 2 × 2 × 3 × 3

⇒ Prime factors of 18 = 2 × 3 × 3

⇒ Common prime factors = 2 × 3 × 3 = 18

⇒ Therefore, HCF(36, 18) = 18

∴ The correct answer is 18.

Additional Information

HCF Definition

The Highest Common Factor (HCF) or Greatest Common Divisor (GCD) is the largest positive integer that divides each of the numbers without leaving a remainder.

Associative Property of HCF

For any set of numbers, the HCF can be found by taking the HCF of any two numbers first, then taking the HCF of that result and the third number: HCF(x, y, z) = HCF(HCF(x, y), z).

Divisibility Rule for HCF

If one number is a factor of another number (i.e., it divides the other number completely), then the smaller number is the HCF of those two numbers.

Shortcut Trick

Find the LCM of the given intervals: 90, 120, and 150 seconds.

LCM (90, 120, 150) = 1800 seconds.

Convert seconds to minutes: 1800 ÷ 60 = 30 minutes.

Next simultaneous change = 5:20 p.m. + 30 minutes = 5:50 p.m.

∴ The correct answer is 5:50 p.m.

Alternate Method

Given:

Intervals of traffic lights = 90 sec, 120 sec, and 150 sec.

Initial simultaneous change time = 5:20 p.m.

Formula Used:

Next simultaneous event = Initial Time + LCM of individual intervals.

Calculations:

⇒ Prime factors of 90 = 2 × 3² × 5

⇒ Prime factors of 120 = 2³ × 3 × 5

⇒ Prime factors of 150 = 2 × 3 × 5²

⇒ LCM = 2³ × 3² × 5² = 8 × 9 × 25 = 1800 seconds.

⇒ Time in minutes = 1800 ÷ 60 = 30 minutes.

⇒ Required time = 5:20 p.m. + 30 minutes = 5:50 p.m.

∴ The correct answer is 5:50 p.m.

Additional Information

LCM for Recurring Events

The Least Common Multiple (LCM) is used to find the next time multiple independent events with different periodicities will occur at the same time.

Time Conversion

To convert seconds to minutes, divide the total seconds by 60 (1 minute = 60 seconds).

LCM of Fractions

If intervals are given in fractions, LCM = (LCM of Numerators) ÷ (HCF of Denominators).

Shortcut Trick

Time for 100% of the tank = 100 min

Time for 80% of the tank = 80% × 100 = 80 min

Convert minutes to seconds: 80 × 60 = 4800 sec

∴ The correct answer is 4800 sec.

Alternate Method

Given:

⇒ Total time to fill the tank = 100 minutes

⇒ Required filling level = 80%

Formula Used:

⇒ Time taken for part = Total Time × Percentage Part

⇒ 1 minute = 60 seconds

Calculations:

⇒ Time required for 80% capacity = 100 × (80 ÷ 100) min

⇒ Time in minutes = 80 min

⇒ Total time in seconds = 80 × 60

⇒ Required time = 4800 sec

∴ The correct answer is 4800 sec.

Additional Information

Rate and Time Relation

The amount of work done (or tank filled) is directly proportional to the time spent when the rate of flow is constant.

Unit Conversions

Standard time conversions: 1 hour = 60 minutes = 3600 seconds; 1 minute = 60 seconds.

Pipe Capacity

If a pipe fills a tank in 'x' minutes, the part filled in 1 minute is 1/x.

Shortcut Trick

Ratio of efficiencies: (Q + R) : P = 3 : 1 (Sum = 4) and (P + R) : Q = 5 : 2 (Sum = 7).

To equalize total efficiency (P + Q + R), multiply first ratio by 7 and second by 4.

⇒ (Q + R) : P = 21 : 7 and (P + R) : Q = 20 : 8.

Efficiency of P = 7 units/day and efficiency of (Q + R) = 21 units/day.

⇒ Total Work = 21 × 60 days. Time for P = (21 × 60) ÷ 7 = 180 days.

∴ The correct answer is 180 days.

Alternate Method

Given: Efficiency of (Q + R) = 3 × P; Efficiency of (P + R) = 2.5 × Q; Time for (Q + R) = 60 days.

Step 1: Determine efficiency ratios.

⇒ (Q + R) / P = 3 / 1 ⇒ (P + Q + R) = 4 units.

⇒ (P + R) / Q = 2.5 / 1 = 5 / 2 ⇒ (P + Q + R) = 7 units.

Step 2: Equalize total efficiency.

⇒ LCM of 4 and 7 = 28.

⇒ Efficiency of P = (1 ÷ 4) × 28 = 7 units/day.

⇒ Efficiency of Q = (2 ÷ 7) × 28 = 8 units/day.

⇒ Efficiency of R = 28 − (7 + 8) = 13 units/day.

Step 3: Calculate total work.

⇒ Total Work = (Efficiency of Q + Efficiency of R) × Time

⇒ Total Work = (8 + 13) × 60 = 21 × 60 = 1260 units.

Step 4: Calculate time taken by P.

⇒ Time for P = Total Work ÷ Efficiency of P

⇒ Time = 1260 ÷ 7 = 180 days.

∴ The correct answer is 180 days.

Additional Information

Efficiency and Time Relationship

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

Combined Work Formula

Total Work = (Efficiency₁ + Efficiency₂ + ...) × Time taken by all together. Efficiency is always additive for people working together.

Ratio Balancing

To compare ratios where the sum of parts represents the same total quantity, equalize the sum by multiplying each ratio by the appropriate factor (LCM method).

Shortcut Trick

To find time in minutes: Time = [Distance (m) ÷ Speed (km/hr)] × (60/1000)

Time = [450 ÷ 20] × (6/100)

Time = 22.5 × 0.06 = 1.35 minutes

In fraction: 135/100 = 27/20 min

∴ The correct answer is 27/20 min.

Alternate Method

Given:

Speed of the train = 20 km/hr

Length of the train (Distance) = 450 m

Formula: Time = Distance ÷ Speed

⇒ First, convert speed into m/s:

⇒ Speed = 20 × (5/18) = 50/9 m/s

⇒ Time taken to cross the man (in seconds) = 450 ÷ (50/9)

⇒ Time = (450 × 9) ÷ 50 = 9 × 9 = 81 seconds

⇒ Convert seconds to minutes: Time = 81 ÷ 60 minutes

⇒ Time = 27/20 min

∴ The correct answer is 27/20 min.

Additional Information

Crossing Objects without Length

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

Crossing Objects with Length

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

Relative Speed

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

Shortcut Trick

Distance to be covered = Length of train = 500 m = 0.5 km.

Speed of the train = 40 km/hr.

Time (in hours) = Distance ÷ Speed = 0.5 ÷ 40 = 1/80 hr.

Time (in minutes) = (1/80) × 60 = 60/80 = 3/4 min.

∴ The correct answer is 3/4 min.

Alternate Method

Given:

Speed of the train = 40 km/hr

Length of the train = 500 m

Formula Used:

Time = Distance ÷ Speed

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

Calculations:

⇒ Total Distance = 500 m

⇒ 1 km = 1000 m ∴ 500 m = 500/1000 = 0.5 km

⇒ Speed = 40 km/hr

⇒ Time (in hours) = 0.5 ÷ 40

⇒ Time (in minutes) = (0.5 ÷ 40) × 60

⇒ Time = (1/2 × 1/40) × 60

⇒ Time = 60 ÷ 80

⇒ Time = 3/4 min

∴ The correct answer is 3/4 min.

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 an Object with Length

If a train of length L₁ crosses a bridge or platform of length L₂, the total distance covered is L₁ + L₂.

Relative Speed

When two objects move in the same direction, relative speed = S₁ − S₂. In opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Observe the ratio parts: 3 + 4 = 7.

This implies (p − q) + (q − r) = (r − s).

⇒ p − r = r − s

⇒ p + s = 2r

⇒ (p + s) : r = 2 :

1. ∴ The correct answer is 2 : 1.

Alternate Method

Given: The ratio of differences is (p − q) : (q − r) : (r − s) = 3 : 4 : 7.

Let the common proportionality constant be k.

⇒ p − q = 3k — (Eq. 1)

⇒ q − r = 4k — (Eq. 2)

⇒ r − s = 7k — (Eq. 3)

⇒ Adding Eq. 1 and Eq. 2:

⇒ (p − q) + (q − r) = 3k + 4k

⇒ p − r = 7k

⇒ From Eq. 3, we have r − s = 7k

⇒ Therefore, p − r = r − s

⇒ p + s = 2r

⇒ (p + s) / r = 2 / 1

∴ The correct answer is 2 : 1.

Additional Information

Arithmetic Progression (AP) Property

If three numbers a, b, and c are in Arithmetic Progression, then the common difference is constant: b − a = c − b, which simplifies to 2b = a + c.

Ratio and Proportion

When multiple ratios are given in a chain like a:b:c = x:y:z, we can assume a = xk, b = yk, and c = zk to solve for individual variables.

Midpoint Relation

In any sequence where the gap between the first and second terms equals the gap between the second and third terms, the middle term is the Arithmetic Mean of the outer terms.

Shortcut Trick

Initial ratio A : B = 3 : 1 (Difference = 2 parts)

New ratio A : B = 5 : 3 (Difference = 2 parts)

Since the difference is the same, compare the increase: 3 → 5 or 1 → 3 is 2 units.

Given increase = 2, so 2 units = 2 ⇒ 1 unit = 1.

Initial sum = 3 + 1 = 4 units ⇒ 4 × 1 = 4.

∴ The correct answer is 4.

Alternate Method

Given: Initial ratio of A and B = 3 : 1, Increment in both = 2, Final ratio = 5 : 3.

Formula Used: New Value = Initial Value + Increase

⇒ Let initial values be 3x and x.

⇒ (3x + 2) ÷ (x + 2) = 5 ÷ 3

⇒ 3 × (3x + 2) = 5 × (x + 2)

⇒ 9x + 6 = 5x + 10

⇒ 9x − 5x = 10 − 6

⇒ 4x = 4

⇒ x = 1

⇒ Sum of initial values = 3x + x = 4x

⇒ Sum = 4 × 1 = 4

∴ The correct answer is 4.

Additional Information

Ratio Constant Difference Concept

If the same value is added to or subtracted from two numbers, the difference between them remains unchanged. In ratios, we multiply to make the term differences equal before comparing parts.

Proportion Rule

If a : b = c : d, then the product of extremes (a × d) is equal to the product of means (b × c).

Scaling Ratios

To compare different ratios, always bring them to a common scale by equating the shared component or the difference between components.

Shortcut Trick

Ratio of Cost Price (CP) to Marked Price (MP) = 5 : 7

Ratio of Marked Price (MP) to Selling Price (SP) = 7 : 6

Combining the ratios, CP : MP : SP = 5 : 7 : 6

Profit Percentage = [(6 − 5) ÷ 5] × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given:

MP = (7/5) × CP

SP = (6/7) × MP

Calculations:

⇒ Let Cost Price (CP) = ₹100

⇒ Marked Price (MP) = (7/5) × 100 = ₹140

⇒ Selling Price (SP) = (6/7) × 140 = 6 × 20 = ₹120

⇒ Profit = Selling Price − Cost Price = 120 − 100 = ₹20

⇒ Profit Percentage = (Profit ÷ CP) × 100

⇒ Profit % = (20 ÷ 100) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Relationship between CP, SP, and MP

Selling Price (SP) can be derived from either Cost Price (SP = CP + Profit) or Marked Price (SP = MP − Discount).

Markup Percentage

Markup is the extra price added to the CP to reach the MP. Markup % = [(MP − CP) ÷ CP] × 100.

Discount Percentage

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

Profit/Loss Base

Profit and Loss are always calculated on the Cost Price unless specified otherwise in the question.

Shortcut Trick

To find the highest selling price, compare cases with the highest Marked Price (MP) and lowest Discount.

Between Option 1 (MP 8000, 25% discount) and Option 4 (MP 8000, 30% discount), Option 1 results in a higher price because the discount is lower.

Comparing 8000 at 25% discount with MP 6000 or 5000 is unnecessary as the base value (8000) is significantly larger.

8000 × 0.75 = ₹6000, while 8000 × 0.70 = ₹5600, 6000 × 0.75 = ₹4500, and 5000 × 0.70 = ₹3500.

∴ The correct answer is Marked price = Rs. 8000 and Discount = 25 percent.

Alternate Method

Given: Four different combinations of Marked Price and Discount percentage.

Formula Used: Selling Price (SP) = MP × (100 − Discount%)/100

⇒ Option 1: SP = 8000 × (1 − 25/100) = 8000 × 0.75 = ₹6000

⇒ Option 2: SP = 6000 × (1 − 25/100) = 6000 × 0.75 = ₹4500

⇒ Option 3: SP = 5000 × (1 − 30/100) = 5000 × 0.70 = ₹3500

⇒ Option 4: SP = 8000 × (1 − 30/100) = 8000 × 0.70 = ₹5600

Comparing the calculated values: 6000 > 5600 > 4500 > 3500.

∴ The correct answer is Marked price = Rs. 8000 and Discount = 25 percent.

Additional Information

Marked Price (MP) and Discount

The Marked Price is the price printed on the label. Discount is always calculated as a percentage of the MP: Discount = MP × (D / 100).

Selling Price (SP) Relation

SP is the final price after discount. SP = MP − Discount. Alternatively, SP = MP × (100 − D)%.

Successive Discounts

If two discounts d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − d₁d₂/100)%.

Shortcut Trick

Use the Ratio Method for 25% discount: 25% = 1/4 ⇒ MP = 4 units, Discount = 1 unit.

Selling Price (SP) = 4 − 1 = 3 units.

Given SP = Rs. 3000 ⇒ 3 units = 3000 ⇒ 1 unit = 1000.

Marked Price (MP) = 4 units × 1000 = Rs. 4000. This condition directly satisfies the question requirement.

∴ The correct answer is SP = Rs. 3000 and Discount = 25%.

Alternate Method

Given:

Selling Price (SP) = Rs. 3000

Discount = 25%

Formula Used:

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

Calculations:

⇒ MP = [3000 × 100] ÷ [100 − 25]

⇒ MP = 300000 ÷ 75

⇒ MP = Rs. 4000

Similarly, checking other options:

⇒ Option 2: MP = 2500 ÷ 0.85 ≈ Rs. 2941.17

⇒ Option 3: MP = 3800 ÷ 0.88 ≈ Rs. 4318.18

⇒ Option 4: MP = 3200 ÷ 0.85 ≈ Rs. 3764.70

∴ The correct answer is SP = Rs. 3000 and Discount = 25%.

Additional Information

Relationship between MP, SP and Discount

Discount is always calculated on the Marked Price (MP). Formula: SP = MP − Discount.

Successive Discounts

Two successive discounts of a% and b% are equivalent to a single discount of [a + b − (a × b)/100]%.

Markup Percentage

It is the percentage by which the Marked Price is higher than the Cost Price (CP). Formula: Markup % = [(MP − CP) / CP] × 100.

Shortcut Trick

The fourth proportional to a, b, and c is calculated as (b × c) ÷ a.

Given values: a = 12, b = 4, c = 18.

Required value = (4 × 18) ÷ 12

Required value = 72 ÷ 12 = 6.

∴ The correct answer is 6.

Alternate Method

Given:

First term (a) = 12

Second term (b) = 4

Third term (c) = 18

Formula Used:

In a proportion a : b :: c : d, the Product of Extremes is equal to the Product of Means.

⇒ a × d = b × c

Calculations:

⇒ 12 × d = 4 × 18

⇒ 12d = 72

⇒ d = 72 ÷ 12

⇒ d = 6

∴ The correct answer is 6.

Additional Information

Mean Proportional

For two numbers a and b, the mean proportional x is calculated as x = √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional and c = b² ÷ a.

Properties of Proportion

Invertendo: If a/b = c/d, then b/a = d/c. Alternando: If a/b = c/d, then a/c = b/d.