Quantitative Aptitude PYQs

Shortcut Trick

Multiply each ratio part by its respective growth multiplier (1 + increment%).

New Ratio = (10 × 1.11) : (8 × 1.20) : (2 × 1.35)

⇒ 11.1 : 9.6 : 2.7

⇒ 111 : 96 : 27 (dividing by 3) → 37 : 32 : 9

∴ The correct answer is 37:32:9.

Alternate Method

Given: Ratio of salaries P : Q : R = 10 : 8 : 2. Increments are 11%, 20% and 35%.

Let the initial salaries be 1000, 800, and 200 for easier calculation.

⇒ New Salary of P = 1000 + (11% of 1000) = 1110

⇒ New Salary of Q = 800 + (20% of 800) = 800 + 160 = 960

⇒ New Salary of R = 200 + (35% of 200) = 200 + 70 = 270

⇒ New Ratio = 1110 : 960 : 270

⇒ Simplify by dividing by 30: 37 : 32 : 9

∴ The correct answer is 37:32:9.

Additional Information

Successive Percentage Change

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

Ratio Scaling

If a ratio a:b is increased by x% and y% respectively, the new ratio becomes a(1 + x/100) : b(1 + y/100).

Compounded Ratio

The compounded ratio of (a : b) and (c : d) is (ac : bd).

Shortcut Trick

Use the Difference of Two Squares algebraic identity: (x + y)(x − y) = x² − y².

Substituting the values x = 7a and y = 4b into the identity:

(7a)² − (4b)² = 49a² − 16b²

∴ The correct answer is 49a² − 16b².

Alternate Method

Given Expression: (7a + 4b)(7a − 4b)

Formula Used: Distributive Property (FOIL Method)

⇒ (7a × 7a) + (7a × (−4b)) + (4b × 7a) + (4b × (−4b)) ⇒ 49a² − 28ab + 28ab − 16b² ⇒ 49a² + 0 − 16b² ⇒ 49a² − 16b²

∴ The correct answer is 49a² − 16b².

Additional Information

Difference of Two Squares

The identity (a + b)(a − b) = a² − b² is used to multiply the sum and difference of the same two terms quickly.

Square of a Binomial (Sum)

The identity for the square of a sum is (a + b)² = a² + 2ab + b².

Square of a Binomial (Difference)

The identity for the square of a difference is (a − b)² = a² − 2ab + b².

Shortcut Trick

Total area of Regular Hexagon = 3√3/2 × side² = 3√3/2 × 36 = 54√3 cm².

Interior angle of a Regular Hexagon is 120°.

Area of Overlapping Sector = (120/360) × π × radius² = (1/3) × π × 3² = 3π cm².

Required non-overlapping area = Total Hexagon Area − Overlapping Sector Area = 54√3 − 3π cm².

∴ The correct answer is 54√3 − 3π cm².

Alternate Method

Given:

Side of Regular Hexagon (a) = 6 cm

Radius of Circle (r) = 3 cm

Formula Used:

Area of Regular Hexagon = (3√3 / 2) × a²

Interior angle of a regular polygon (θ) = ((n − 2) × 180) / n

Area of Circular Sector = (θ / 360) × πr²

Calculations:

⇒ Area of Regular Hexagon = (3√3 / 2) × 6²

⇒ Area of Regular Hexagon = (3√3 / 2) × 36 = 54√3 cm²

⇒ Interior angle (θ) = ((6 − 2) × 180) / 6 = 120°

⇒ Area of the overlapping Circular Sector = (120 / 360) × π × 3²

⇒ Sector Area = (1/3) × 9π = 3π cm²

⇒ Non-overlapping area = Total Area − Sector Area

⇒ Non-overlapping area = 54√3 − 3π cm²

∴ The correct answer is 54√3 − 3π cm².

Additional Information

Regular Hexagon Properties

A Regular Hexagon can be divided into 6 equilateral triangles.

Each interior angle is 120° and each exterior angle is 60°.

Area of a Sector

The area of a sector of a circle with radius 'r' and central angle 'θ' is given by A = (θ/360) × πr².

Sum of Interior Angles

The sum of all interior angles of a polygon with 'n' sides is (n − 2) × 180°.

Shortcut Trick

Total Distance = 100 + 150 + 50 + (60 × 2) = 420 km

Total Time = 1 + 1 + 1 + 2 = 5 hours

Average Speed = 420 ÷ 5 = 84 km/hr

∴ The correct answer is 84 km/hr.

Alternate Method

Given:

Distance₁ = 100 km, Time₁ = 1 hr

Distance₂ = 150 km, Time₂ = 1 hr

Distance₃ = 50 km, Time₃ = 1 hr

Speed₄ = 60 km/hr, Time₄ = 2 hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Distance in last 2 hours = 60 × 2 = 120 km

⇒ Total Distance = 100 + 150 + 50 + 120 = 420 km

⇒ Total Time = 1 + 1 + 1 + 2 = 5 hours

⇒ Average Speed = 420 ÷ 5 = 84 km/hr

∴ The correct answer is 84 km/hr.

Additional Information

Average Speed (Equal Distances)

When two equal distances are covered at speeds u and v, Average Speed = 2uv ÷ (u + v).

Relative Speed

Same direction = S₁ − S₂ | Opposite direction = S₁ + S₂.

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

For a number to be divisible by 7, the difference between twice the unit digit and the remaining part of the number must be divisible by 7.

Number: 34X1 → Calculation: 34X − (2 × 1) = 34X − 2.

Trying options for X: If X = 5, then 345 − 2 = 343.

Since 343 is 7³, it is divisible by 7, so X = 5.

Value of X² = 5² = 25.

∴ The correct answer is 25.

Alternate Method

Given: Four-digit number 34X1 is divisible by 7.

Formula Used: Standard division method or place value expansion.

⇒ 3401 + 10X must be a multiple of 7.

⇒ Dividing 3401 by 7: 3401 = 485 × 7 + 6.

⇒ Remainder is 6. So, (6 + 10X) must be divisible by 7.

⇒ For X = 1: 6 + 10 = 16 (No)

⇒ For X = 5: 6 + 50 = 56 (Yes, 56 = 8 × 7)

⇒ X = 5

⇒ X² = 5² = 25.

∴ The correct answer is 25.

Additional Information

Divisibility Rule of 7

Subtract twice the last digit from the number formed by the remaining digits. If the result is 0 or a multiple of 7, the original number is divisible by 7.

Divisibility Rule of 11

The difference between the sum of digits at odd places and the sum of digits at even places must be 0 or a multiple of 11.

Divisibility Rule of 13

Add four times the last digit to the remaining number. If the sum is divisible by 13, the number is divisible by 13.

Shortcut Trick

Convert distance to km and time to hours directly.

Speed = (800 / 1000) ÷ (2 / 60) km/hr

Speed = 0.8 × 30 = 24 km/hr

∴ The correct answer is 24 km/hr.

Alternate Method

Given:

Distance = 800 m

Time = 2 minutes = 2 × 60 = 120 seconds

Formula Used:

Speed = Distance ÷ Time

Conversion: 1 m/s = (18/5) km/hr

Calculations:

⇒ Speed in m/s = 800 ÷ 120

⇒ Speed in m/s = 20/3 m/s

⇒ Speed in km/hr = (20/3) × (18/5)

⇒ Speed = 4 × 6

⇒ Speed = 24 km/hr

∴ The correct answer is 24 km/hr.

Additional Information

Unit Conversion for Speed

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

Average Speed

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

Relative Speed

Same direction: Subtract speeds (S₁ − S₂). Opposite direction: Add speeds (S₁ + S₂).

Shortcut Trick

In any quadrilateral PQRS, the angle formed by the bisectors of two adjacent angles (say ∠R and ∠S) is equal to half the sum of the other two angles (∠P and ∠Q):

⇒ ∠STR = (∠P + ∠Q) ÷ 2

⇒ 48° = (∠P + ∠Q) ÷ 2

⇒ ∠P + ∠Q = 96°

Given the ratio of ∠P : ∠Q = 3 : 5, the sum of ratio parts is 3 + 5 = 8 units.

⇒ 8 units = 96° ⇒ 1 unit = 12°

The difference between ∠Q and ∠P is 5 − 3 = 2 units:

⇒ Difference = 2 × 12° = 24°

∴ The correct answer is 24°.

Alternate Method

Given:

∠STR = 48°

∠P : ∠Q = 3 : 5

Formula Used:

Sum of angles in a triangle = 180°

Sum of angles in a quadrilateral = 360°

Calculations:

In ▵STR, the sum of angles is 180°:

⇒ ∠TSR + ∠TRS + ∠STR = 180°

⇒ ∠TSR + ∠TRS + 48° = 180°

⇒ ∠TSR + ∠TRS = 132°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2 and ∠TRS = ∠R ÷ 2

⇒ (∠S ÷ 2) + (∠R ÷ 2) = 132°

⇒ ∠S + ∠R = 264°

In quadrilateral PQRS, the sum of all interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 264° = 360°

⇒ ∠P + ∠Q = 96°

Given that the ratio of ∠P to ∠Q is 3 : 5:

⇒ Let ∠P = 3x and ∠Q = 5x

⇒ 3x + 5x = 96°

⇒ 8x = 96°

⇒ x = 12°

Now, find the difference between the measures of ∠Q and ∠P:

⇒ Difference = ∠Q − ∠P = 5x − 3x = 2x

⇒ Difference = 2 × 12° = 24°

∴ The correct answer is 24°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining two angles: ∠STR = (∠P + ∠Q) ÷ 2.

Sum of Interior Angles of a Polygon

The sum of the interior angles of any n-sided polygon is given by the formula (n − 2) × 180°. For a quadrilateral (n = 4), the sum is always 360°.

Opposite Angle Bisectors

If the bisectors of two opposite angles of a quadrilateral intersect, the angle between them is equal to half the difference of the other two opposite angles.

Shortcut Trick

When a larger object is melted into n smaller identical objects, the Radius Ratio is R/r = ³√n.

Given n = 27, R/r = ³√27 = 3.

Since the large radius R = 9 cm, the small radius r = 9 ÷ 3 = 3 cm.

Surface Area of small sphere = 4πr² = 4 × π × 3² = 36π cm².

∴ The correct answer is 36π cm².

Alternate Method

Given:

Radius of solid metallic sphere (R) = 9 cm

Number of smaller spheres (n) = 27

Formula Used:

Volume of Sphere = (4/3)πr³

Surface Area of Sphere = 4πr²

Calculations:

⇒ Volume of large sphere = 27 × Volume of one small sphere

⇒ (4/3) × π × (9)³ = 27 × (4/3) × π × (r)³

⇒ 9 × 9 × 9 = 27 × r³

⇒ 729 = 27 × r³

⇒ r³ = 729 ÷ 27 = 27

⇒ r = 3 cm

⇒ Surface Area of one small sphere = 4 × π × r²

⇒ Surface Area = 4 × π × (3)²

⇒ Surface Area = 4 × π × 9 = 36π cm²

∴ The correct answer is 36π cm².

Additional Information

Volume and Surface Area of Sphere

Volume = (4/3)πr³ and Surface Area = 4πr². The surface area is also equal to the area of four circles of the same radius.

Hemisphere Formulas

Volume = (2/3)πr³, Curved Surface Area = 2πr², and Total Surface Area = 3πr².

Recasting Property

When one solid is melted to form another, the total volume remains constant. If n identical small solids are formed, then V_large = n × V_small.

Shortcut Trick

Using the Successive Multiplier method.

Initial Cost Price = ₹15,000.

Net multiplier = (100+20)÷100 × (100+25)÷100 = 1.2 × 1.25 = 1.5.

Final Selling Price = 15,000 × 1.5 = ₹22,500.

∴ The correct answer is ₹22,500.

Alternate Method

Given: CP of Wholesaler = ₹15,000, Profit₁ = 20%, Profit₂ = 25%.

Formulas: SP = CP × (100 + P%) ÷ 100.

⇒ Selling Price for Wholesaler (CP for Retailer) = 15,000 × (120 ÷ 100) = ₹18,000.

⇒ Final Selling Price to Customer = 18,000 × (125 ÷ 100).

⇒ Final SP = 18,000 × 1.25 = ₹22,500.

∴ The correct answer is ₹22,500.

Additional Information

Successive Percentage Change

When a value is increased by a% and then b%, the net percentage increase is given by: a + b + (ab ÷ 100)%.

Effective Profit Percentage

In this case, net profit % = 20 + 25 + (20 × 25 ÷ 100) = 45 + 5 = 50%. Total SP = 15,000 × 150% = 22,500.

Shortcut Trick

Combined ratio X : Y : Z = 5 : 9 : 7.

Total units = 5 + 9 + 7 = 21 units.

21 units = ₹6,874.

Share of Y = (9 ÷ 21) × 6,874 = 2,946.

∴ The correct answer is ₹2,946.

Alternate Method

Given:

Total amount = ₹6,874

Ratio X : Y = 5 : 9

Ratio Y : Z = 9 : 7

Calculations:

⇒ Since Y is common and its value is 9 in both ratios, the combined ratio X : Y : Z = 5 : 9 : 7.

⇒ Let the shares be 5k, 9k, and 7k.

⇒ 5k + 9k + 7k = 6,874

⇒ 21k = 6,874

⇒ k = 6,874 ÷ 21

⇒ k = 327.333...

⇒ Share of Y = 9 × k = 9 × (6,874 ÷ 21)

⇒ Share of Y = 3 × (6,874 ÷ 7)

⇒ Share of Y = 3 × 982 = 2,946

∴ The correct answer is ₹2,946.

Additional Information

Combined Ratio

To combine A:B and B:C, make the common term B equal in both ratios by multiplying with suitable constants.

Proportional Division

If a quantity S is divided in the ratio a:b:c, the share of the second part is [b ÷ (a+b+c)] × S.

Ratio Scaling

Multiplying or dividing all terms of a ratio by the same non-zero number does not change the ratio value.

Shortcut Trick

Savings percentage = 100% − 70% = 30%

Given, 30% of Income = ₹18,000

100% of Income = (18,000 ÷ 30) × 100

100% of Income = 600 × 100 = ₹60,000

∴ The correct answer is ₹60,000.

Alternate Method

Given:

Expenditure = 70% of Income

Savings = ₹18,000

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let the total income be 'x'.

⇒ Savings percentage = 100% − 70% = 30%

⇒ (30 ÷ 100) × x = 18,000

⇒ 0.3x = 18,000

⇒ x = 18,000 ÷ 0.3

⇒ x = 1,80,000 ÷ 3

⇒ x = 60,000

∴ The correct answer is ₹60,000.

Additional Information

Income, Expenditure, and Savings Relationship

The basic formula is Income = Expenditure + Savings. If any two are known as percentages or values, the third can be found.

Percentage Change in Income

New Income = Old Income × (100 + % Increase) ÷ 100. This is used in questions where income and expenditure both change.

Base Value Concept

In income-based problems, the total income is always treated as the 100% base for calculations unless specified otherwise.

Shortcut Trick

Identify the observation with the maximum frequency (most repetitions) in the dataset.

Number 53 appears 4 times in the list.

Numbers 48, 49, and 50 appear only 2 times each.

All other numbers (40, 41, 45, 46, 47, 51, 54, 55) appear only once.

The Mode is the observation that occurs most frequently.

∴ The correct answer is 53.

Alternate Method

Given Data: 53, 47, 53, 48, 53, 53, 40, 41, 50, 50, 55, 48, 54, 49, 49, 51, 46, 45

Formula: Mode = Value with the highest frequency (f).

Let us count the frequency (repetition) of each unique number:

⇒ Frequency of 40 = 1

⇒ Frequency of 41, 45, 46, 47 = 1 each

⇒ Frequency of 48 = 2 (numbers at pos 4 and 12)

⇒ Frequency of 49 = 2 (numbers at pos 14 and 15)

⇒ Frequency of 50 = 2 (numbers at pos 9 and 10)

⇒ Frequency of 51, 54, 55 = 1 each

⇒ Frequency of 53 = 4 (numbers at pos 1, 3, 5, and 6)

⇒ Since 53 has the maximum frequency of 4, it is the mode.

∴ The correct answer is 53.

Additional Information

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Types of Mode

Data with one mode is Unimodal; two modes is Bimodal; more than two modes is Multimodal.

Mode for Grouped Data

Mode = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h, where L is lower limit of modal class.

Shortcut Trick

When distance is constant, Speed is inversely proportional to Time.

Speed Ratio (Usual : New) = 6 : 5 → Time Ratio (Usual : New) = 5 : 6.

Difference in time units = 6 − 5 = 1 unit.

Given 1 unit = 24 minutes, then Usual Time = 5 × 24 = 120 minutes.

∴ The correct answer is 120 min.

Alternate Method

Given: New speed = 5/6 of normal speed, Time delay = 24 minutes.

Formula: Distance = Speed × Time

⇒ Let usual speed be S and usual time be T.

⇒ New speed = 5S/6 and New time = T + 24.

⇒ S × T = (5S/6) × (T + 24)

⇒ T = 5/6 × (T + 24)

⇒ 6T = 5T + 120

⇒ T = 120 minutes.

∴ The correct answer is 120 min.

Additional Information

Inverse Proportion of Speed and Time

If distance is constant, Speed₁/Speed₂ = Time₂/Time₁. This is the most efficient way to solve late/early arrival problems.

Average Speed

For a journey with two equal distances covered at speeds x and y, Average Speed = 2xy/(x + y).

Relative Speed

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

Shortcut Trick

Rate 10% = 1/10 → Principal : Amount ratio for 1 year = 10 : 11.

For 2 years, the ratio becomes 10² : 11² = 100 : 121.

Given, 100 units = ₹4,900 ⇒ 1 unit = 49.

Amount (121 units) = 121 × 49 = ₹5,929.

∴ The correct answer is ₹5,929.

Alternate Method

Given:

Principal (P) = ₹4,900

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4900 × (1 + 10/100)²

⇒ A = 4900 × (1 + 0.1)²

⇒ A = 4900 × (1.1)²

⇒ A = 4900 × 1.21

⇒ A = ₹5,929

∴ The correct answer is ₹5,929.

Additional Information

Effective Interest Rate

For a rate of R% for 2 years compounded annually, the effective rate is (2R + R²/100)%. Here, 10 + 10 + (100/100) = 21%.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is given by the formula D = P(R/100)².

Amount for n Compounding Periods

If interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2n) in the standard formula.

Shortcut Trick

Apply the Empirical Formula: Mode = 3 × Median − 2 × Mean

Substitute values: 21 = 3 × Median − 2 × 60

3 × Median = 21 + 120 = 141

Median = 141 ÷ 3 = 47

∴ The correct answer is 47.

Alternate Method

Given:

Mode = 21

Mean = 60

Formula Used:

In a moderately skewed distribution, the relationship is: Mode = 3(Median) − 2(Mean)

Calculations:

⇒ Mode = 3 × Median − 2 × Mean

⇒ 21 = 3 × Median − 2 × 60

⇒ 21 = 3 × Median − 120

⇒ 3 × Median = 21 + 120

⇒ 3 × Median = 141

⇒ Median = 141 ÷ 3

⇒ Median = 47

∴ The correct answer is 47.

Additional Information

Empirical Formula

For a moderately skewed distribution, Mode ≈ 3 × Median − 2 × Mean. This is also written as (Mean − Mode) = 3 × (Mean − Median).

Symmetric Distribution

In a perfectly symmetrical distribution (like the Normal Distribution), the Mean, Median, and Mode are all equal (Mean = Median = Mode).

Skewness Types

If Mean > Median > Mode, the distribution is positively skewed. If Mode > Median > Mean, it is negatively skewed.

Shortcut Trick

Apply the standard algebraic identity: (P + Q)² = P² + Q² + 2PQ.

To find P² + Q², rearrange the formula: P² + Q² = (P + Q)² − 2PQ.

Substitute the given values: P² + Q² = 17² − 2 × 56.

P² + Q² = 289 − 112 = 177.

∴ The correct answer is 177.

Alternate Method

Given:

Sum of numbers (P + Q) = 17

Product of numbers (PQ) = 56

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ (P + Q) = 17

Squaring both sides:

⇒ (P + Q)² = 17²

⇒ P² + Q² + 2PQ = 289

Substitute the value of PQ = 56:

⇒ P² + Q² + 2 × 56 = 289

⇒ P² + Q² + 112 = 289

⇒ P² + Q² = 289 − 112

⇒ P² + Q² = 177

∴ The correct answer is 177.

Additional Information

Difference of Squares and Identities

The square of the difference is (a − b)² = a² + b² − 2ab. This can be used to find P² + Q² if the difference and product are known.

Relation between Sum and Difference

The identities are linked by (a + b)² − (a − b)² = 4ab, which helps in finding the difference if the sum and product are given.

Sum of Squares Shortcut

For any two numbers, the sum of their squares is also equal to [(a + b)² + (a − b)²] ÷ 2.

Shortcut Trick

Total work (LCM of 7 and 56) = 56 units.

Efficiency of P + Q = 56 ÷ 7 = 8 units/hr.

Efficiency of P = 56 ÷ 56 = 1 unit/hr.

Efficiency of Q = 8 − 1 = 7 units/hr.

Time for Q to fill 1/4^th work = (56 × 1/4) ÷ 7 = 14 ÷ 7 = 2 hours.

∴ The correct answer is 2 hours.

Alternate Method

Given: Time taken by (P + Q) = 7 hours, Time taken by P = 56 hours.

Formula Used: Efficiency = Total Work ÷ Time

⇒ Efficiency of Q = Efficiency of (P + Q) − Efficiency of P

⇒ Efficiency of Q = 8 − 1 = 7 units/hr

⇒ One-fourth of the cistern capacity = 56 ÷ 4 = 14 units

⇒ Time taken by Q to fill 14 units = 14 ÷ 7 = 2 hours

∴ The correct answer is 2 hours.

Additional Information

Pipes & Cisterns Efficiency

Filling pipes have positive efficiency, while outlet pipes (leaks) have negative efficiency. Total efficiency is the algebraic sum.

Work-Time Relationship

Time taken is inversely proportional to efficiency. If efficiency doubles, the time taken to complete the same work is halved.

Shortcut Trick

Ratio of Savings to Expenditure = 100 : 250 = 2 : 5

Using Alligation/Weighted Average for Income change:

% Change in Income = [(Savings × % Change) + (Expenditure × % Change)] ÷ Total Ratio

% Change = [(2 × 21) + (5 × −7)] ÷ (2 + 5) = [42 − 35] ÷ 7 = 7 ÷ 7 = 1%

∴ The correct answer is 1%.

Alternate Method

Given: Expenditure is 150% more than Savings.

Formula Used: Income = Expenditure + Savings

⇒ Let Savings = 100

⇒ Expenditure = 100 + 150% of 100 = 100 + 150 = 250

⇒ Initial Income = 100 + 250 = 350

⇒ New Savings = 100 + 21% of 100 = 121

⇒ New Expenditure = 250 − 7% of 250 = 250 − 17.5 = 232.5

⇒ New Income = 121 + 232.5 = 353.5

⇒ Increase in Income = 353.5 − 350 = 3.5

⇒ Percentage Increase = (3.5 ÷ 350) × 100 = 1%

∴ The correct answer is 1%.

Additional Information

Income, Expenditure, and Savings Relationship

The fundamental equation is Income = Expenditure + Savings. Any percentage change in two components directly determines the change in the third.

Weighted Average Concept

If Income (I) is composed of Expenditure (E) and Savings (S), then %ΔI = (%ΔE × E + %ΔS × S) ÷ (E + S).

Percentage Change Shortcut

When values are defined as "x% more than", the multiplier is (1 + x/100). Here, 150% more means 2.5 times the base value.

Shortcut Trick

Discount = Marked Price − Selling Price = 606 − 304 = 302.

Discount % = (Discount ÷ Marked Price) × 100.

Discount % = (302 ÷ 606) × 100.

Discount % = 30200 ÷ 606 ≈ 49.83%.

∴ The correct answer is 49.83%.

Alternate Method

Given: Marked Price (MP) = ₹606, Selling Price (SP) = ₹304.

Formulas: Discount = MP − SP | Discount % = (Discount ÷ MP) × 100

⇒ Step 1: Calculate the total discount: 606 − 304 = 302.

⇒ Step 2: Calculate discount percentage: (302 ÷ 606) × 100.

⇒ Step 3: 30200 ÷ 606 = 49.8349...

⇒ Step 4: Rounding off to two decimal places, we get 49.83%.

∴ The correct answer is 49.83%.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP unless stated otherwise.

Rounding Rules

When rounding to two decimal places, if the third decimal is 5 or greater, round up; otherwise, keep the second decimal as is.

Shortcut Trick

Convert all to base 0.2

0.04 = (0.2)², 0.008 = (0.2)³

(0.04)² = (0.2)⁴

Expression = (0.2)⁴ ÷ (0.2)³ × (0.2)⁶

= (0.2)^(4−3+6) = (0.2)⁷

∴ x = 7

Alternate Method

Given:

(0.04)² ÷ (0.008) × (0.2)⁶ = (0.2)^x

Concept Used:

Index laws: a^m ÷ aⁿ = a^m−n, a^m × aⁿ = a^m+n

Calculation:

⇒ 0.04 = (0.2)²

⇒ 0.008 = (0.2)³

⇒ (0.04)² = ((0.2)²)² = (0.2)⁴

⇒ Expression = (0.2)⁴ ÷ (0.2)³ × (0.2)⁶

⇒ = (0.2)⁽⁴⁻³⁾ × (0.2)⁶

⇒ = (0.2)¹ × (0.2)⁶

⇒ = (0.2)⁷

⇒ x = 7

∴ The correct answer is 7.

Additional Information

Laws of Exponents

a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n

Power of Power

(a^m)ⁿ = a^mn

Base Conversion

Convert decimals into powers of a common base for easy solving.

Quick Insight

Combining powers early simplifies expressions quickly.