Quantitative Aptitude PYQs

Shortcut Trick

Apply BODMAS directly by adding like terms first, then dividing:

⇒ (3x² + 4x²) ÷ 7x

⇒ 7x² ÷ 7x

⇒ x

∴ The correct answer is x.

Alternate Method

Given:

Expression: (3x² + 4x²) ÷ 7x

Formula Used:

Addition of like terms: axⁿ + bxⁿ = (a + b)xⁿ

Law of exponents for division: a^m ÷ aⁿ = a^m−n

Calculations:

⇒ According to the BODMAS rule, solve the bracket first.

⇒ 3x² + 4x² = 7x²

⇒ Now, divide the result by 7x.

⇒ 7x² ÷ 7x = (7 ÷ 7) × (x² ÷ x¹)

⇒ 1 × x²⁻¹

⇒ x

∴ The correct answer is x.

Additional Information

BODMAS Rule

It stands for Bracket, Of, Division, Multiplication, Addition, and Subtraction. Always solve expressions inside parentheses first.

Law of Exponents (Multiplication)

When multiplying expressions with the same base, add the exponents: a^m × aⁿ = a^m+n

Law of Exponents (Division)

When dividing expressions with the same base, subtract the exponents: a^m ÷ aⁿ = a^m−n

Shortcut Trick

Net discount multiplier = (100 − 20)% × (100 − 11)% = 0.80 × 0.89 = 0.712.

Marked Price (MP) = Selling Price ÷ 0.712.

MP = 8,900 ÷ 0.712 = 12,500.

∴ The correct answer is ₹12,500.

Alternate Method

Given → Selling Price (SP) = ₹8,900 | Discount₁ = 20% | Discount₂ = 11%.
Formulas: SP = MP × (100 − D₁)% × (100 − D₂)% | Successive = a + b − ab ÷ 100.

⇒ Let the Marked Price be x.
⇒ First discount of 20%: Price becomes x × (80/100).
⇒ Second discount of 11%: Price becomes x × (80/100) × (89/100).
⇒ Set equation: x × 0.8 × 0.89 = 8,900.
⇒ x × 0.712 = 8,900.
⇒ x = 8,900 ÷ 0.712 = 12,500.

∴ The correct answer is ₹12,500.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price listed on the label. Discount is always calculated on the MP unless stated otherwise. SP = MP − Discount.

Successive Discount Formula

To find a single equivalent discount for two discounts a% and b%, use: Equivalent Discount = (a + b − ab/100)%.

Shortcut Trick

Ratio of Father : Daughter

Present → 4 : 1 (Difference = 3)

5 years ago → 7 : 1 (Difference = 6)

Make differences equal: Present ratio × 2 → 8 : 2

Change in units: 8 − 7 = 1 unit or 2 − 1 = 1 unit

1 unit = 5 years ∴ Daughter's present age = 2 units = 2 × 5 = 10 years

∴ The correct answer is 10 years.

Alternate Method

Given:

Present age ratio Father : Daughter = 4 : 1

Age ratio 5 years ago = 7 : 1

Let daughter's present age be x years.

⇒ Father's present age = 4x years.

⇒ 5 years ago, daughter's age = x − 5

⇒ 5 years ago, father's age = 4x − 5

⇒ According to question: (4x − 5) = 7 × (x − 5)

⇒ 4x − 5 = 7x − 35

⇒ 3x = 30

⇒ x = 10

∴ The correct answer is 10 years.

Additional Information

Age Difference Property

The difference between the ages of two people remains constant regardless of the number of years passed.

Ratio Balancing

To solve age problems using ratios, multiply the ratios by the difference of the other ratio to balance the units representing time intervals.

Shortcut Trick

When two objects meet and then travel towards their destinations, the ratio of their speeds is the inverse square root of the ratio of time taken to reach the destinations.

Speed Ratio S_A : S_B = √T_B : √T_A

Given: S_B = 1.5 × S_A ⇒ S_A/S_B = 1/1.5 = 2/3

2/3 = √(p / 95.4) ⇒ 4/9 = p / 95.4

p = (95.4 × 4) ÷ 9 = 10.6 × 4 = 42.4 hours

∴ The correct answer is 42.4 hours.

Alternate Method

Given:

Time taken by A after meeting (T_A) = 95.4 hours

Time taken by B after meeting (T_B) = p hours

Speed of B (S_B) = Speed of A (S_A) + 50% of S_A = 1.5 S_A

Formula Used:

S_A / S_B = √(T_B / T_A)

Calculations:

⇒ S_A / 1.5S_A = √(p / 95.4)

⇒ 1 / 1.5 = √(p / 95.4)

⇒ 10 / 15 = √(p / 95.4)

⇒ 2 / 3 = √(p / 95.4)

Squaring both sides:

⇒ 4 / 9 = p / 95.4

⇒ p = (4 × 95.4) / 9

⇒ p = 4 × 10.6

⇒ p = 42.4

∴ The correct answer is 42.4 hours.

Additional Information

Post-Meeting Time Theorem

If two bodies start at the same time and meet at a point, then take T₁ and T₂ time to reach their destinations, their speed ratio is S₁/S₂ = √(T₂/T₁).

Meeting Time Formula

The time they took to meet each other from the start is given by T_meet = √(T₁ × T₂).

Relative Speed

When two objects move towards each other, their relative speed is the sum of their individual speeds: S_rel = S_A + S_B.

Shortcut Trick

Mean proportion M = √(a × b)

Given M = 66, a = x, and b = 99

66² = x × 99

x = (66 × 66) ÷ 99 = (2 × 33 × 2 × 33) ÷ (3 × 33) = (2 × 2 × 33) ÷ 3 = 4 × 11 = 44

∴ The correct answer is 44.

Alternate Method

Given:

Mean Proportion = 66

First and Third terms are x and 99 respectively.

Formula Used:

In a continued proportion a, b, c, the mean proportion b² = ac.

Calculations:

⇒ x : 66 = 66 : 99

⇒ x ÷ 66 = 66 ÷ 99

⇒ x = (66 × 66) ÷ 99

⇒ x = 4356 ÷ 99

⇒ x = 44

∴ The correct answer is 44.

Additional Information

Third Proportional

If a, b, c are in continued proportion, then c is called the third proportional to a and b, where c = b² ÷ a.

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional to a, b, and c, calculated as d = (b × c) ÷ a.

Duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Shortcut Trick

Total votes = 100%

Votes for N = 100% − (42% + 34%) = 24%

Votes for N = 24% of 37,700 = 24 × 377

Votes for N = 9048

∴ The correct answer is 9048 votes.

Alternate Method

Given: Total votes = 37,700 | Votes for M = 42% | Votes for R = 34%

⇒ Total percentage = Percentage(M) + Percentage(N) + Percentage(R)

⇒ 100% = 42% + Percentage(N) + 34%

⇒ Percentage(N) = 100% − 76% = 24%

⇒ Number of votes for N = (24 ÷ 100) × 37,700

⇒ Number of votes for N = 24 × 377 = 9048

∴ The correct answer is 9048 votes.

Additional Information

Total vs Valid Votes

In election problems, Total Votes = Valid Votes + Invalid Votes. If all votes are valid, then total polled equals valid.

Winner Margin

The margin of victory is the difference between the votes of the first and second-place candidates.

Percentage Calculation

To find the absolute value from a percentage: Value = (Percentage ÷ 100) × Total.

Shortcut Trick

Efficiency Ratio (Rishi : Shan) = 100 : 120 = 5 : 6.

Time Ratio is the inverse of Efficiency Ratio = 6 : 5.

If 6 units = 12 days (Rishi), then 1 unit = 2 days.

Days for Shan = 5 units = 5 × 2 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given: Rishi takes 12 days. Shan is 20% more efficient.

Formula Used: Total Work = Efficiency × Time; Days = Total Work ÷ Efficiency

⇒ Let Efficiency of Rishi = 5 units/day.

⇒ Efficiency of Shan = 5 + (20% of 5) = 5 + 1 = 6 units/day.

⇒ Total Work = Efficiency of Rishi × Days = 5 × 12 = 60 units.

⇒ Days taken by Shan = Total Work ÷ Efficiency of Shan = 60 ÷ 6 = 10 days.

∴ The correct answer is 10 days.

Additional Information

Inverse Proportionality

For a constant work, Time is inversely proportional to Efficiency (T ∝ 1/E). If efficiency increases, time taken decreases.

Combined Work

If two people work together, their combined efficiency is the sum of their individual efficiencies (E_total = E₁ + E₂).

Percentage to Fraction

20% more efficiency means a ratio of 120/100 or 6/5. This fractional approach simplifies ratio calculations significantly.

Shortcut Trick

Area of wet surface = Area of base + Lateral surface area

Area of wet surface = (length × width) + 2 × depth × (length + width)

Area of wet surface = (5 × 3) + 2 × 1.5 × (5 + 3)

Area of wet surface = 15 + 3 × 8 = 15 + 24 = 39 m²

∴ The correct answer is 39 m².

Alternate Method

Given:

Length (l) = 5 m

Width (w) = 3 m

Depth (h) = 1.50 m

Formula: Wet surface area = Area of bottom + Area of 4 vertical sides

⇒ Area of bottom = l × w = 5 × 3 = 15 m²

⇒ Area of 2 long sides = 2 × (l × h) = 2 × (5 × 1.5) = 2 × 7.5 = 15 m²

⇒ Area of 2 wide sides = 2 × (w × h) = 2 × (3 × 1.5) = 2 × 4.5 = 9 m²

⇒ Total wet area = 15 + 15 + 9 = 39 m²

∴ The correct answer is 39 m².

Additional Information

Total Surface Area of Cuboid

The total area of all six faces is given by 2(lb + bh + hl).

Lateral Surface Area (Area of 4 walls)

The area of the four vertical faces is 2h(l + b).

Volume of Rectangular Tank

The volume of water or capacity is calculated as Length × Width × Height.

Shortcut Trick

The Cost Price (CP) represents 100%.

Since there is a loss of 6%, the Selling Price (SP) represents 100% − 6% = 94%.

SP = 1,400 × (94 ÷ 100)

SP = 14 × 94 = 1,316.

∴ The correct answer is ₹1,316.

Alternate Method

Given: Cost Price (CP) = ₹1,400 and Loss Percentage = 6%.

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

⇒ Loss Amount = (6 × 1,400) ÷ 100

⇒ Loss Amount = 6 × 14 = ₹84

⇒ Selling Price (SP) = Cost Price − Loss

⇒ SP = 1,400 − 84 = ₹1,316

∴ The correct answer is ₹1,316.

Additional Information

Loss and Profit

Profit occurs when SP > CP. Loss occurs when CP > SP. Both are always calculated on the Cost Price unless stated otherwise.

Percentage Change

To find the new value after a percentage decrease: New Value = Original Value × (100 − decrease%) ÷ 100.

Shortcut Trick

Combine ratios A:B (2:3), B:C (4:5), and C:D (6:7) by making common terms equal.

Ratio A:B:C:D = (2×4×6) : (3×4×6) : (3×5×6) : (3×5×7) = 48 : 72 : 90 : 105.

Simplified Ratio A:B:C:D = 16 : 24 : 30 : 35 (Dividing by 3).

Total units = 16 + 24 + 30 + 35 = 105 units → ₹9,975. So, 1 unit = ₹95.

Share of A = 16 units = 16 × 95 = ₹1,520.

∴ The correct answer is ₹1,520.

Alternate Method

Given: Total Amount = ₹9,975. Ratios A:B = 2:3, B:C = 4:5, C:D = 6:7.

Formula: Joint Ratio calculation and Unitary method.

⇒ To find A:B:C:D, multiply terms vertically and diagonally:

⇒ A = 2 × 4 × 6 = 48

⇒ B = 3 × 4 × 6 = 72

⇒ C = 3 × 5 × 6 = 90

⇒ D = 3 × 5 × 7 = 105

⇒ Ratio A:B:C:D = 48 : 72 : 90 : 105 = 16 : 24 : 30 : 35

⇒ Total components = 16 + 24 + 30 + 35 = 105

⇒ Value of 1 part = 9975 ÷ 105 = ₹95

⇒ Share of A = 16 × 95 = ₹1,520

∴ The correct answer is ₹1,520.

Additional Information

Combined Ratio Rule

If A:B = a:b, B:C = c:d, and C:D = e:f, then A:B:C:D = (a×c×e) : (b×c×e) : (b×d×e) : (b×d×f).

Compound Ratio

The compound ratio of (a:b), (c:d) and (e:f) is simply (a×c×e) : (b×d×f).

Proportional Division

To divide an amount S in the ratio a:b:c, the shares are [a/(a+b+c)]×S, [b/(a+b+c)]×S, and [c/(a+b+c)]×S.

Shortcut Trick

In any data set, the Mode is the observation with the maximum frequency (the value that repeats the most).

By observation: 86 appears 4 times, while 65, 85, 71, and 90 appear only 2 times each, and others appear once.

Since 86 has the highest repetition, it is the mode.

∴ The correct answer is 86.

Alternate Method

Given: Data set = 65, 86, 86, 85, 69, 90, 71, 60, 86, 71, 86, 65, 81, 82, 67, 72, 84, 66, 75, 85, 68, 78, 61, 64, 74, 90.

Formula Used: Mode = Value with the Highest Frequency.

Calculating the frequency of each observation from the given data:

⇒ Frequency of 60 = 1

⇒ Frequency of 65 = 2

⇒ Frequency of 71 = 2

⇒ Frequency of 85 = 2

⇒ Frequency of 90 = 2

⇒ Frequency of 86 = 4 (appears at positions 2, 3, 9, and 11)

⇒ All other values (61, 64, 66, 67, 68, 69, 72, 74, 75, 78, 81, 82, 84) appear only 1 time.

⇒ Maximum frequency = 4, which corresponds to the value 86.

∴ The correct answer is 86.

Additional Information

Mean (Average)

It is the sum of all observations divided by the total number of observations: Mean = (Σx_i) ÷ n.

Median

It is the middle-most value of the data when arranged in ascending or descending order.

Empirical Relationship

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

Types of Mode

Data can be Unimodal (one mode), Bimodal (two modes), or Multimodal (more than two modes).

Shortcut Trick

Interior angle of regular polygon with n sides = [(n − 2) × 180°] ÷ n

For n = 8, A = (6 × 180°) ÷ 8 = 135° ⇒ A/3 = 45°

For n = 10, B = (8 × 180°) ÷ 10 = 144° ⇒ 5B/6 = 120°

Value = sin²(45°) + cos³(120°) = (1/√2)² + (−1/2)³ = 1/2 − 1/8 = 3/8

∴ The correct answer is 3/8.

Alternate Method

Given: Sides of first polygon n₁ = 8, Sides of second polygon n₂ = 10.

Formula Used: Interior angle = 180° − (360°/n); sin 45° = 1/√2; cos 120° = −1/2.

A (n=8)A = 135°B (n=10)B = 144°Expression: sin²(A/3) + cos³(5B/6)

⇒ Angle A = 180° − (360° ÷ 8) = 180° − 45° = 135°

⇒ Angle B = 180° − (360° ÷ 10) = 180° − 36° = 144°

⇒ A/3 = 135° ÷ 3 = 45°

⇒ 5B/6 = (5 × 144°) ÷ 6 = 5 × 24° = 120°

⇒ sin²(45°) + cos³(120°) = (1/√2)² + (cos(180°−60°))³

⇒ 1/2 + (−cos 60°)³ = 1/2 + (−1/2)³

⇒ 1/2 − 1/8 = 4/8 − 1/8 = 3/8

∴ The correct answer is 3/8.

Additional Information

Interior and Exterior Angles

For any regular polygon, Interior Angle + Exterior Angle = 180°. Sum of all exterior angles is always 360°.

Trigonometric Ratios of Obtuse Angles

cos(180° − θ) = −cos θ. Since 120° is in the second quadrant, cosine is negative.

Sum of Interior Angles

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

Shortcut Trick

Total distance = Length of Train + Length of Platform = 200 + 600 = 800 m

Speed in m/sec = 50 × (5/18) = 125/9 m/sec

Time = Distance ÷ Speed = 800 ÷ (125/9) = (800 × 9) ÷ 125

Time = 6.4 × 9 = 57.6 sec

∴ The correct answer is 57.6 sec.

Alternate Method

Given:

Length of Train (L₁) = 200 m

Length of Platform (L₂) = 600 m

Speed (S) = 50 km/hr

Formula Used:

Time = Total Distance ÷ Speed

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

Calculations:

⇒ Total Distance = 200 + 600 = 800 m

⇒ Speed in m/s = 50 × (5/18) = 250/18 = 125/9 m/s

⇒ Time = 800 ÷ (125/9)

⇒ Time = (800 × 9) ÷ 125

⇒ Time = 7200 ÷ 125 = 57.6 sec

∴ The correct answer is 57.6 sec.

Additional Information

Unit Conversion

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

Relative Speed (Same Direction)

When two objects move in the same direction, the relative speed is the difference of their speeds (S₁ − S₂).

Relative Speed (Opposite Direction)

When two objects move in opposite directions, the relative speed is the sum of their speeds (S₁ + S₂).

Shortcut Trick

Total discount on First Article = a + b − (ab ÷ 100) = 30 + 18 − (30 × 18 ÷ 100) = 48% − 5.4% = 42.6%.

Total discount on Second Article = 48%.

The shopkeeper gets more benefit where the discount is lower (First Article).

Difference in discount % = 48% − 42.6% = 5.4%.

More benefit amount = 5.4% of 2,40,000 = 0.054 × 2,40,000 = 12,960.

∴ The correct answer is First article; ₹12,960.

Alternate Method

Given → Marked Price (MP) = ₹2,40,000. Case 1: Successive discounts 30% and 18%. Case 2: Single discount 48%.
Formulas: SP = MP × (100 − D₁)% × (100 − D₂)% | Profit is higher when SP is higher.

⇒ SP of First Article = 2,40,000 × (70 ÷ 100) × (82 ÷ 100) = 2,400 × 0.7 × 82 = ₹1,37,760.
⇒ SP of Second Article = 2,40,000 × (52 ÷ 100) = 2,400 × 52 = ₹1,24,800.
⇒ Shopkeeper benefit is higher when selling price is higher ∴ First Article.
⇒ Difference in benefit = 1,37,760 − 1,24,800 = ₹12,960.

∴ The correct answer is First article; ₹12,960.

Additional Information

Successive Discounts

Two successive discounts of x% and y% are equivalent to a single discount of (x + y − xy/100)%.

Shopkeeper Perspective

For a seller, a smaller total discount results in a higher Selling Price, thus yielding more benefit.

Shortcut Trick

The relationship between Mode, Median, and Mean is expressed as: Mode − Median = 2/3 × (Mode − Mean).

Given: Mode − Mean = 34.8

Applying the trick: Mode − Median = 2/3 × 34.8

Calculation: 2 × 11.6 = 23.2

∴ The correct answer is 23.2.

Alternate Method

Given: Difference between Mode and Mean = 34.8

Formula used: Empirical Relationship → Mode = 3 × Median − 2 × Mean

⇒ Let Mean = x

⇒ Mode = x + 34.8

⇒ Substituting in formula: x + 34.8 = 3 × Median − 2x

⇒ 3x + 34.8 = 3 × Median

⇒ Median = (3x + 34.8) ÷ 3

⇒ Median = x + 11.6

⇒ Difference (Mode − Median) = (x + 34.8) − (x + 11.6)

⇒ Mode − Median = 23.2

∴ The correct answer is 23.2.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Symmetric Distribution

In a perfectly symmetrical distribution, the values of Mean, Median, and Mode are identical (Mean = Median = Mode).

Mean - Mode Relation

Another variation of the formula is: Mean − Mode = 3 × (Mean − Median).

Shortcut Trick

Total Work (LCM of 33, 22, 55) = 330 units.

Efficiency of A = 10, B = 15, Combined (A + B − C) = 6.

Efficiency of C = (10 + 15) − 6 = 19.

Time taken by C = 330 ÷ 19 = 17(7/19) minutes.

∴ The correct answer is 17(7/19) minutes.

Alternate Method

Given: Time for Pipe A = 33 min, Pipe B = 22 min, Combined (A+B-C) = 55 min.

⇒ Efficiency of Pipe A + Efficiency of Pipe B − Efficiency of Pipe C = Combined Efficiency

⇒ 10 + 15 − Efficiency of Pipe C = 6

⇒ 25 − Efficiency of Pipe C = 6

⇒ Efficiency of Pipe C = 25 − 6 = 19

⇒ Time taken by Pipe C to empty the tank = Total Work ÷ Efficiency of C

⇒ Time = 330 ÷ 19

⇒ Time = 17(7/19) minutes

∴ The correct answer is 17(7/19) minutes.

Additional Information

Negative Work Concept

In Pipes and Cisterns, an outlet pipe (emptying pipe) performs negative work, so its efficiency is subtracted from the total efficiency of inlet pipes.

Efficiency and Time Relationship

Efficiency is inversely proportional to Time when the total work is constant. Efficiency = Total Work ÷ Time.

Formula Method

For three pipes where C is an outlet: (1/T_A) + (1/T_B) − (1/T_C) = (1/T_Total). Solve for T_C directly.

Shortcut Trick

We can separate the square roots directly using the property √(a × b) = √a × √b:

⇒ (11 × 7) + (8 × 6) − (10 × 4)

⇒ 77 + 48 − 40

⇒ 125 − 40 = 85

∴ The correct answer is 85.

Alternate Method

Given:

Expression: √(121 × 49) + √(64 × 36) − √(100 × 16)

Formula Used:

√(x × y) = √x × √y

Calculation:

⇒ √(121 × 49) = √121 × √49 = 11 × 7 = 77

⇒ √(64 × 36) = √64 × √36 = 8 × 6 = 48

⇒ √(100 × 16) = √100 × √16 = 10 × 4 = 40

Substituting the calculated values back into the main expression:

⇒ 77 + 48 − 40

⇒ 125 − 40

⇒ 85

∴ The correct answer is 85.

Additional Information

BODMAS Rule

Always follow the strict sequence of Brackets, Order (Powers/Roots), Division/Multiplication, and Addition/Subtraction from left to right to ensure an accurate result.

Properties of Radicals

The product of two square roots is equivalent to the square root of their product: √a × √b = √(a × b). This property is universally valid for all non-negative real numbers.

Common Perfect Squares

Memorizing squares up to 30 heavily aids in faster calculation during simplification questions (e.g., 11² = 121, 15² = 225, 25² = 625).

Shortcut Trick

Find LCM of 28, 24, 56, and 29.

LCM (28, 24, 56, 29) = LCM(56, 24, 29) = 168 × 29 = 4872.

Multiples of 4872 near 9054: 4872 × 2 = 9744.

Required number = (Next Multiple + Remainder) − Given Number = (9744 + 6) − 9054 = 9750 − 9054 = 696.

∴ The correct answer is 696.

Alternate Method

Given: Numbers 28, 24, 56, 29; Remainder = 6; Target number > 9054.

Formula Used: Required number = [n × LCM(divisors) + Remainder] − 9054

⇒ Step 1: Prime factorization of divisors:

⇒ 28 = 2² × 7

⇒ 24 = 2³ × 3

⇒ 56 = 2³ × 7

⇒ 29 = 29 (Prime number)

⇒ LCM = 2³ × 3 × 7 × 29 = 8 × 3 × 7 × 29 = 168 × 29 = 4872

⇒ To find a number just greater than 9054, we take the 2^nd multiple:

⇒ 4872 × 2 = 9744

⇒ To leave a remainder of 6: 9744 + 6 = 9750

⇒ Smallest number to be added = 9750 − 9054 = 696

∴ The correct answer is 696.

Additional Information

LCM and Remainder Concept

The general form of a number that leaves a remainder 'r' when divided by x, y, and z is [k × LCM(x, y, z) + r].

Divisibility Rules

For large numbers, check divisibility of the final sum (9054 + x − 6) by simpler factors like 3 or 8 from the LCM set.

Prime Numbers

If one of the divisors is a prime number (like 29), it must be a direct factor of the LCM.

Shortcut Trick

Ratio of Expenditure to Savings = 200 : 100 = 2 : 1

Using Alligation/Weighted Average method:

Net Income change = (Expenditure % × E-ratio + Savings % × S-ratio) ÷ (E-ratio + S-ratio)

Income increase = (−4% × 2 + 35% × 1) ÷ (2 + 1) = (−8 + 35) ÷ 3 = 27 ÷ 3 = 9%

∴ The correct answer is 9%.

Alternate Method

Given:Expenditure is 100% more than Savings.

Formula Used:Income = Expenditure + Savings

⇒ Let Savings = 100

⇒ Expenditure = 100 + 100% of 100 = 200

⇒ Initial Income = 200 + 100 = 300

⇒ New Expenditure = 200 − 4% of 200 = 200 − 8 = 192

⇒ New Savings = 100 + 35% of 100 = 135

⇒ New Income = 192 + 135 = 327

⇒ Increase in Income = 327 − 300 = 27

⇒ % Increase = (27 ÷ 300) × 100 = 9%

∴ The correct answer is 9%.

Additional Information

Income, Expenditure and Savings Relationship

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

Weighted Average Concept

If components A and B change by x% and y%, the total change is (A×x + B×y) ÷ (A+B). This is the fastest way to solve income-expenditure problems.

Percentage Base Change

When a value is x% more than another, the ratio is (100+x) : 100. Always identify the base correctly before calculating increments or decrements.

Shortcut Trick

Let Length = L and Breadth = B.

From conditions: (L − 7)(B + 5) = LB − 21 ⇒ 5L − 7B = 14

From second condition: (L + 4)(B + 5) = LB + 67 ⇒ 5L + 4B = 47

Subtracting both: 11B = 33 ⇒ B = 3 m. Putting B in first: 5L − 21 = 14 ⇒ 5L = 35 ⇒ L = 7 m.

Original Area = L × B = 7 × 3 = 21 m².

∴ The correct answer is 21 m².

Alternate Method

Given: Reduction in area = 21 m², Increase in area = 67 m².

Formula: Area of Rectangle = Length (L) × Breadth (B).

⇒ Case 1: (L − 7)(B + 5) = LB − 21

⇒ LB + 5L − 7B − 35 = LB − 21

⇒ 5L − 7B = 14 ----(i)

⇒ Case 2: (L + 4)(B + 5) = LB + 67

⇒ LB + 5L + 4B + 20 = LB + 67

⇒ 5L + 4B = 47 ----(ii)

⇒ Subtracting (i) from (ii): (5L + 4B) − (5L − 7B) = 47 − 14

⇒ 11B = 33 ⇒ B = 3 m

⇒ Substituting B in (i): 5L − 7(3) = 14 ⇒ 5L − 21 = 14 ⇒ 5L = 35 ⇒ L = 7 m

⇒ Area = 7 × 3 = 21 m²

∴ The correct answer is 21 m².

Additional Information

Perimeter of Rectangle

The total boundary length is given by P = 2 × (Length + Breadth).

Diagonal of Rectangle

The length of the diagonal is calculated using Pythagoras theorem: d = √(L² + B²).

Percentage Change in Area

If length changes by x% and breadth by y%, net area change % = x + y + (xy/100).