Quantitative Aptitude PYQs

Shortcut Trick

Solve the nested square roots from the innermost part to the outermost part.

√81 = 9

√(55 + 9) = √64 = 8

√(41 + 8) = √49 = 7

√(29 + 7) = √36 = 6

∴ The correct answer is 6.

Alternate Method

Given: \( \sqrt{29+\sqrt{41+\sqrt{55+\sqrt{81}}}} \)

Step-by-step simplification:

⇒ \( \sqrt{29+\sqrt{41+\sqrt{55+9}}} \) (Since √81 = 9)

⇒ \( \sqrt{29+\sqrt{41+\sqrt{64}}} \)

⇒ \( \sqrt{29+\sqrt{41+8}} \) (Since √64 = 8)

⇒ \( \sqrt{29+\sqrt{49}} \)

⇒ \( \sqrt{29+7} \) (Since √49 = 7)

⇒ \( \sqrt{36} \) = 6

∴ The correct answer is 6.

Additional Information

Perfect Squares

A perfect square is a number that is the square of an integer. Identifying them (like 36, 49, 64, 81) helps in solving nested roots quickly.

BODMAS in Surds

Always simplify the inner-most bracket or radical first, then move outwards following arithmetic operations.

Shortcut Trick

Since the Principal (P) and Time (T) are identical for both banks, calculate the difference in Interest Rates.

Difference in Rate of Interest = 7.5% − 3.5% = 4% per annum.

Total Interest Difference for 3 years = 4% × 3 = 12% of Principal.

Required Positive Difference = 12% of ₹2,60,000 = (12/100) × 2,60,000 = 12 × 2,600 = ₹31,200.

∴ The correct answer is ₹31,200.

Alternate Method

Given: Principal (P) = ₹2,60,000, Time (T) = 3 years, Rate A = 3.5%, Rate B = 7.5%

Formula Used: Simple Interest (SI) = (P × R × T) ÷ 100

⇒ Interest from Bank A = (2,60,000 × 3.5 × 3) ÷ 100

⇒ Interest from Bank A = 2,600 × 10.5 = ₹27,300

⇒ Interest from Bank B = (2,60,000 × 7.5 × 3) ÷ 100

⇒ Interest from Bank B = 2,600 × 22.5 = ₹58,500

⇒ Positive Difference = 58,500 − 27,300

⇒ Positive Difference = ₹31,200

∴ The correct answer is ₹31,200.

Additional Information

Simple Interest (SI) Properties

Simple interest is calculated only on the Principal amount. If the Principal and Time are constant, the interest is directly proportional to the Rate (SI ∝ R).

SI Difference Formula

The difference between SI at two different rates R₁ and R₂ for the same principal and time is given by: Difference = [P × T × (R₂ − R₁)] ÷ 100.

Amount Formula

The total Amount (A) at the end of the period is the sum of the Principal and the Simple Interest: A = P + SI.

Shortcut Trick

Use the direct relationship formula: MP/CP = (100 + Profit%)/(100 − Discount%).

MP/CP = (100 + 36)/(100 − 27) = 136/73.

Mark-up = MP − CP = 136 − 73 = 63 units.

Mark-up percentage = (63/73) × 100 ≈ 86.30%.

∴ The correct answer is 86.30%.

Alternate Method

Given: Discount = 27%, Profit = 36%.

Formulas: SP = CP × (100 + P%)/100 | SP = MP × (100 − D%)/100 | Mark-up% = (MP − CP)/CP × 100.

⇒ Let Cost Price (CP) be 100.

⇒ Selling Price (SP) = 100 + 36% of 100 = 136.

⇒ Marked Price (MP) = SP × 100 / (100 − Discount%) = 136 × 100 / 73 ≈ 186.3013.

⇒ Mark-up Percentage = (186.3013 − 100) = 86.3013%.

∴ The correct answer is 86.30%.

Additional Information

Marked Price and Mark-up

Marked price is the price displayed on an article. Mark-up percentage is the percentage increase from Cost Price to Marked Price.

The CP-MP Relationship

CP × (1 + Mark-up%) × (1 − Discount%) = CP × (1 + Profit%). This simplifies to the ratio MP/CP used in the shortcut.

Shortcut Trick

Mean Proportional (MP) = √(a × b)

88 = √(P × 242)

Square both sides: 88 × 88 = P × 242

P = (88 × 88) ÷ 242 = 7744 ÷ 242 = 32

∴ The correct answer is 32.

Alternate Method

Given: First number = P, Second number = 242, Mean Proportional = 88.

Formula Used: If b is the mean proportional between a and c, then b² = a × c.

⇒ 88² = P × 242

⇒ 7744 = P × 242

⇒ P = 7744 ÷ 242

⇒ P = 32

∴ The correct answer is 32.

Additional Information

Third Proportional

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

Fourth Proportional

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

Continued Proportion

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

Shortcut Trick

Identify the Highest Common Factor (HCF) by checking the options from largest to smallest.

Check if 180 divides all numbers: 5880 ÷ 180 is not an integer.

Check if 120 divides all numbers: 2520/120 = 21, 3600/120 = 30, 5880/120 = 49.

Since 120 divides all three numbers and is the largest such value among options, it is the HCF.

∴ The correct answer is 120.

Alternate Method

Given: Numbers are 2520, 3600, and 5880.

Method: Prime Factorization Method.

⇒ Prime factors of 2520 = 2³ × 3² × 5¹ × 7¹

⇒ Prime factors of 3600 = 2⁴ × 3² × 5²

⇒ Prime factors of 5880 = 2³ × 3¹ × 5¹ × 7²

⇒ HCF = Product of lowest powers of common prime factors

⇒ HCF = 2³ × 3¹ × 5¹

⇒ HCF = 8 × 3 × 5 = 120

∴ The correct answer is 120.

Additional Information

Definition of HCF

The Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD), is the largest positive integer that divides each of the integers without a remainder.

Relationship between HCF and LCM

For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. This property does not directly apply to three or more numbers in the same simple product form.

Division Method for HCF

Divide the larger number by the smaller one, then divide the previous divisor by the remainder. Continue until the remainder is zero; the last divisor is the HCF.

Shortcut Trick

Calculate the semi-perimeter (s): (13 + 14 + 15) ÷ 2 = 21 cm.

Apply Heron's Formula: Area = √[s(s−a)(s−b)(s−c)]

Area = √[21 × (21−13) × (21−14) × (21−15)] = √(21 × 8 × 7 × 6) = 84 cm².

∴ The correct answer is 84 cm².

Alternate Method

Given: The lengths of the three sides of the triangle are a = 13 cm, b = 14 cm, and c = 15 cm.

Formula Used:

To find the area of a scalene triangle, we use Heron's Formula:

1. Semi-perimeter (s) = (a + b + c) ÷ 2
2. Area = √[s × (s − a) × (s − b) × (s − c)]

Calculations:

⇒ s = (13 + 14 + 15) ÷ 2

⇒ s = 42 ÷ 2 = 21 cm

⇒ Area = √[21 × (21 − 13) × (21 − 14) × (21 − 15)]

⇒ Area = √[21 × 8 × 7 × 6]

⇒ Area = √[7056]

⇒ Area = 84 cm²

∴ The correct answer is 84 cm².

Additional Information

Heron's Formula Application

This formula is particularly useful for finding the area of a scalene triangle when the vertical height is unknown but all three side lengths are available.

Perimeter vs. Semi-perimeter

The Perimeter is the sum of all sides (a+b+c), while the Semi-perimeter (s) is exactly half of that value.

Right-Angled Triangle Area

For a triangle where base (b) and height (h) are perpendicular, the area simplifies to: Area = 1/2 × b × h.

Shortcut Trick

Combined Efficiency method: Total Work (LCM) ÷ Combined Efficiency.

LCM of 3, 18, and 9 = 18 units (Original Work).

Efficiency of A + B + C = (18÷3) + (18÷18) + (18÷9) = 6 + 1 + 2 = 9 units/day.

Time for twice the work = (2 × 18) ÷ 9 = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Time taken by A = 3 days, B = 18 days, C = 9 days.

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

⇒ Total combined efficiency of A + B + C = 6 + 1 + 2 = 9 units/day.

⇒ Original Work = 18 units.

⇒ Twice the work = 2 × 18 = 36 units.

⇒ Time required = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Pipes & Cisterns

This logic is identical to filling a tank where efficiency represents the rate of flow per hour/minute.

Work & Wages

Wages are always distributed in the ratio of efficiency (or work done) of the individuals, not the days taken.

Fraction Method

Combined 1-day work = (1/d₁ + 1/d₂ + 1/d₃). Total time = Total Work / Combined 1-day work.

Shortcut Trick

Height (h) = (2 × Area) ÷ Base = (2 × 48) ÷ 12 = 8 units.

The height divides the base into two equal parts of 6 units each.

Using Pythagorean triplet (6, 8, 10), the equal side (a) = 10 units.

Perimeter = (2 × equal side) + base = 20 + 12 = 32 units.

∴ The correct answer is 32 units.

Alternate Method

Given:

Area of Isosceles Triangle = 48 square units

Base (b) = 12 units

Formula Used:

Area = (1/2) × Base × Height

Perimeter = 2a + b (where a = equal sides)

Calculations:

⇒ 48 = (1/2) × 12 × Height

⇒ 48 = 6 × Height

⇒ Height = 8 units

In an Isosceles Triangle, the Height bisects the Base at 90°. Let equal side be 'a'.

Using Pythagoras Theorem in the right-angled half:

⇒ a² = 8² + (12 ÷ 2)²

⇒ a² = 64 + 36 = 100

⇒ a = √100 = 10 units

⇒ Perimeter = a + a + b

⇒ Perimeter = 10 + 10 + 12 = 32 units

∴ The correct answer is 32 units.

Additional Information

General Area Formula

The area of an isosceles triangle with equal sides 'a' and base 'b' can also be calculated as Area = (b/4) × √(4a² − b²).

Altitude Property

The altitude (height) drawn to the unequal side of an isosceles triangle acts as both the median and the angle bisector.

Pythagorean Triplets

Standard integer sets that satisfy a² + b² = c², such as (3, 4, 5), (6, 8, 10), and (5, 12, 13), help in solving geometry questions quickly.

Shortcut Trick

Net multiplier for successive discounts of 10% and 36%:

Multiplier = (100 − 10)/100 × (100 − 36)/100 = 0.9 × 0.64 = 0.576

Marked Price (MP) = SP ÷ 0.576 = 5,400 ÷ 0.576

MP = 5,400,000 ÷ 576 = 9,375

∴ The correct answer is ₹9,375.

Alternate Method

Given: Selling Price (SP) = ₹5,400, Discount₁ = 10%, Discount₂ = 36%.
Formulas: SP = MP × (100 − D₁%)/100 × (100 − D₂%)/100

⇒ Let Marked Price be x.
⇒ 5,400 = x × (90/100) × (64/100)
⇒ 5,400 = x × (9/10) × (16/25)
⇒ x = (5,400 × 10 × 25) ÷ (9 × 16)
⇒ x = (600 × 250) ÷ 16
⇒ x = 150,000 ÷ 16 = 9,375

∴ The correct answer is ₹9,375.

Additional Information

Successive Discounts

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

Marked Price

MP is the price printed on the label, and discounts are always calculated on this value to reach the SP.

Shortcut Trick

Check options: 5 years ago, Father's age must be 7 times Son's age.

Option (4): 40 − 5 = 35 and 10 − 5 = 5. Since 35 = 7 × 5, it satisfies the first condition.

After 20 years: 40 + 20 = 60 and 10 + 20 = 30. Since 60 = 2 × 30, it satisfies the second condition.

∴ The correct answer is 40, 10 years.

Alternate Method

Let Son's age 5 years ago be x and Father's age be 7x.

⇒ Present ages: Father = 7x + 5, Son = x + 5.

⇒ After 20 years: Father = 7x + 25, Son = x + 25.

⇒ According to question: 7x + 25 = 2 × (x + 25)

⇒ 7x + 25 = 2x + 50

⇒ 5x = 25 ⇒ x = 5

⇒ Father's present age = 7(5) + 5 = 40 years.

⇒ Son's present age = 5 + 5 = 10 years.

∴ The correct answer is 40, 10 years.

Additional Information

Ratio Difference Concept

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

Linear Equations in Ages

If 'n' years are added to the present age, the ratio changes. Always define variables based on the earliest time frame given to simplify calculations.

Verification Tip

In multiple-choice questions, back-substitution of options into the given conditions is often the fastest way to find the answer.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 150 + 90 = 240 km

Total Time = 5 + 3 = 8 hours

Average Speed = 240 ÷ 8 = 30 km/hr

∴ The correct answer is 30 km/hr.

Alternate Method

Given:

Distance₁ = 150 km, Time₁ = 5 hours

Distance₂ = 90 km, Time₂ = 3 hours

Formula Used:

Average Speed = (D₁ + D₂) ÷ (T₁ + T₂)

Calculations:

⇒ Total Distance = 150 km + 90 km = 240 km

⇒ Total Time = 5 hours + 3 hours = 8 hours

⇒ Average Speed = 240 ÷ 8

⇒ Average Speed = 30 km/hr

∴ The correct answer is 30 km/hr.

Additional Information

Average Speed for Equal Distances

When two equal distances are covered at speeds x and y, the average speed = 2xy ÷ (x + y).

Relative Speed

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

Profit on Selling Price = 6.25% = 1/16.

This means if SP = 16 units, then Profit = 1 unit.

Cost Price (CP) = SP − Profit = 16 − 1 = 15 units.

Actual Profit% (on CP) = (Profit ÷ CP) × 100.

Actual Profit% = (1 ÷ 15) × 100 = 20 ÷ 3 = 6(2/3)%.

∴ The correct answer is 6(2/3)%.

Alternate Method

Given: Profit on SP = 6.25%
Formulas: Profit = SP − CP | Profit% on SP = (Profit ÷ SP) × 100 | Actual Profit% = (Profit ÷ CP) × 100

⇒ Let the Selling Price (SP) be 100.
⇒ Profit = 6.25% of 100 = 6.25.
⇒ Cost Price (CP) = SP − Profit = 100 − 6.25 = 93.75.
⇒ Actual Profit% = (6.25 ÷ 93.75) × 100.
⇒ Actual Profit% = (625 ÷ 9375) × 100 = (1 ÷ 15) × 100 = 6.66% or 6(2/3)%.

∴ The correct answer is 6(2/3)%.

Additional Information

Profit Calculation Bases

Standard Profit Percentage is always calculated on the Cost Price unless specified otherwise. When calculated on SP, the denominator changes, affecting the final percentage value.

Fractional Equivalents

6.25% = 1/16 | 6.66% = 1/15 | 8.33% = 1/12. Knowing these helps in fast ratio conversions.

Shortcut Trick

Ratio of Profit = Ratio of Investment (when time is constant).

Total profit = ₹6,600 and Pushparaj's profit = ₹2,200.

Ratio (Pushparaj : Total) = 2,200 : 6,600 = 1 : 3.

Since the total investment is ₹48,600, Pushparaj's investment = 1/3 × 48,600.

Pushparaj's investment = ₹16,200.

∴ The correct answer is ₹16,200.

Alternate Method

Given: Total Investment = ₹48,600, Total Profit = ₹6,600, Pushparaj's Profit = ₹2,200.
Formulas: Profit Share = (Individual Investment × Time) / (Total Investment × Time) × Total Profit.

⇒ Let the investment of Pushparaj be P and Himani be H.
⇒ P + H = 48,600.
⇒ Profit ratio P : H = 2,200 : (6,600 − 2,200) = 2,200 : 4,400 = 1 : 2.
⇒ Investment ratio is also 1 : 2.
⇒ Pushparaj's investment = [1 / (1 + 2)] × 48,600.
⇒ Pushparaj's investment = 1/3 × 48,600 = 16,200.

∴ The correct answer is ₹16,200.

Additional Information

Partnership Basics

When the time period for investment is the same for all partners, the profit is distributed in the ratio of their initial investments.

Compound Partnership

If the time periods are different, the profit is distributed in the ratio of (Investment × Time Period) for each partner.

Shortcut Trick

For conjugate pairs x = a + b and y = a − b, the sum of their squares is x² + y² = 2(a² + b²).

Here, a = 3 and b = 2√2.

⇒ x² + y² = 2 × (3² + (2√2)²)

⇒ 2 × (9 + 8) = 2 × 17 = 34.

∴ The correct answer is 34.

Alternate Method

Given: x = 3 + 2√2 and y = 3 − 2√2

Formula Used: (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab

⇒ x² = (3 + 2√2)² = 3² + (2√2)² + 2 × 3 × 2√2

⇒ x² = 9 + 8 + 12√2 = 17 + 12√2

⇒ y² = (3 − 2√2)² = 3² + (2√2)² − 2 × 3 × 2√2

⇒ y² = 9 + 8 − 12√2 = 17 − 12√2

⇒ x² + y² = (17 + 12√2) + (17 − 12√2)

⇒ x² + y² = 17 + 17 = 34

∴ The correct answer is 34.

Additional Information

Algebraic Identity for Sum of Squares

The expression x² + y² can be efficiently calculated using the sum and product: x² + y² = (x + y)² − 2xy.

Product of Conjugates

For any conjugate pair a + √b and a − √b, their product is always a rational number: (a + √b)(a − √b) = a² − b.

Sum of Conjugates

The sum of (a + √b) and (a − √b) is 2a, which eliminates the irrational part of the expression.

Shortcut Trick

Required percentage = (Value ÷ Base) × 100

Required percentage = [(1/30) ÷ (1/80)] × 100

Required percentage = (80 ÷ 30) × 100 = 800 ÷ 3 = 266.666...

Rounding off to two decimal places, we get 266.67%.

∴ The correct answer is 266.67%.

Alternate Method

Given:

First fraction = 1/30

Second fraction (Base) = 1/80

Formula Used:

Percentage = (A / B) × 100

Calculations:

⇒ Required % = [(1/30) / (1/80)] × 100

⇒ Required % = (1/30) × (80/1) × 100

⇒ Required % = (80 / 30) × 100

⇒ Required % = (8/3) × 100

⇒ Required % = 800 / 3

⇒ Required % = 266.666...

⇒ Required % ≈ 266.67%

∴ The correct answer is 266.67%.

Additional Information

Percentage Change

Percentage change = [(New Value − Old Value) / Old Value] × 100.

Fraction to Percentage

To convert any fraction x/y into a percentage, multiply it by 100 and add the % symbol.

Reciprocal Percentages

1/3 = 33.33%, 2/3 = 66.67%, 4/3 = 133.33%, 8/3 = 266.67%.

Shortcut Trick

Assume a base mean of 40 and calculate the net deviation.

Deviations: (27−40), (69−40), (15−40), (88−40), (15−40), (54−40), (53−40), (20−40), (19−40)

⇒ −13 + 29 − 25 + 48 − 25 + 14 + 13 − 20 − 21 = 0

Correct Mean = Assumed Mean + (Net Deviation ÷ Total Count) = 40 + (0 ÷ 9) = 40.

∴ The correct answer is 40.

Alternate Method

Given: Observations are 27, 69, 15, 88, 15, 54, 53, 20, and 19.

Formula Used: Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

⇒ Total count of observations (n) = 9

⇒ Sum = 27 + 69 + 15 + 88 + 15 + 54 + 53 + 20 + 19

⇒ Sum = 360

⇒ Arithmetic Mean = 360 ÷ 9

⇒ Mean = 40

∴ The correct answer is 40.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of items from their arithmetic mean is always zero, i.e., ∑(x − x̄) = 0.

Weighted Arithmetic Mean

Used when different observations have different importance or weights: x̄_w = ∑(w × x) ÷ ∑w.

Median and Mode

Median is the middle value in a sorted list; Mode is the value that appears most frequently in the data set.

Shortcut Trick
The empirical relationship is defined as: (Mode − Mean) = 3 × (Median − Mean).
Given: Mode − Mean = 10.8.
Substitute the value: 10.8 = 3 × (Median − Mean) ⇒ (Median − Mean) = 3.6.
Required value: (Mode − Median) = (Mode − Mean) − (Median − Mean).
Calculated: 10.8 − 3.6 = 7.2.
∴ The correct answer is 7.2.
Alternate Method
Given:
Difference between Mode and Mean = 10.8
⇒ Mode − Mean = 10.8
⇒ Mean = Mode − 10.8
Formula Used:
Empirical formula: Mode = 3 × Median − 2 × Mean

Calculations:
⇒ Mode = 3 × Median − 2 × (Mode − 10.8)
⇒ Mode = 3 × Median − 2 × Mode + 21.6
⇒ Mode + 2 × Mode − 3 × Median = 21.6
⇒ 3 × Mode − 3 × Median = 21.6
⇒ 3 × (Mode − Median) = 21.6
⇒ Mode − Median = 21.6 ÷ 3
⇒ Mode − Median = 7.2
∴ The correct answer is 7.2.
Additional Information
Empirical Relationship
In a moderately skewed distribution, the relationship is approximately Mode = 3 Median − 2 Mean.
Symmetric Distribution
For a perfectly symmetrical distribution (like the Normal distribution), the Mean = Median = Mode, and the difference between any two is 0.
Skewness Property
If Mode > Median > Mean, the distribution is negatively skewed (skewed to the left). If Mean > Median > Mode, it is positively skewed (skewed to the right).

Shortcut Trick

For a polygon with n sides, the number of triangles formed by joining its vertices is given by the combination formula ⁿC₃.

Here, n = 10.

Number of triangles = ¹⁰C₃ = (10 × 9 × 8) ÷ (3 × 2 × 1)

Number of triangles = 10 × 3 × 4 = 120.

∴ The correct answer is 120.

Alternate Method

Given:

Number of vertices of the polygon (n) = 10

Formula Used:

To form a triangle, we need to select 3 vertices out of n available vertices.

Number of ways = ⁿC_r = n! ÷ [r! × (n − r)!]

Calculations:

⇒ Total vertices = 10

⇒ Number of triangles = ¹⁰C₃

⇒ ¹⁰C₃ = (10 × 9 × 8) ÷ (3 × 2 × 1)

⇒ ¹⁰C₃ = 720 ÷ 6

⇒ ¹⁰C₃ = 120

∴ The correct answer is 120.

Additional Information

Number of Diagonals

The number of diagonals in a polygon of n sides is given by [n × (n − 3)] ÷ 2.

Sum of Interior Angles

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

Triangles with No Common Sides

Number of triangles that can be formed using vertices of a polygon such that no side of the triangle is a side of the polygon is ⁿC₃ − n − n(n − 4).

Shortcut Trick

Assume Total Votes = 100 units.

Valid Votes = 100 − 30% of 100 = 70 units.

Winner gets 60% of valid votes → Loser gets (100% − 60%) = 40% of valid votes.

Loser% of total = 40% of 70 = 28 units.

∴ The correct answer is 28%.

Alternate Method

Given:

Invalid votes = 30% of Total

Winner share = 60% of Valid votes

Calculations:

⇒ Let Total Votes = V

⇒ Valid Votes = V − 0.30V = 0.70V

⇒ Losing candidate share = (100% − 60%) of Valid Votes

⇒ Losing candidate share = 40% of 0.70V

⇒ Losing candidate share = 0.40 × 0.70V = 0.28V

⇒ Percentage = (0.28V / V) × 100 = 28%

∴ The correct answer is 28%.

Additional Information

Percentage Change Base

Always identify if the percentage is calculated on the Initial Total or the Remaining/Valid amount to avoid calculation errors.

Successive Percentage Rule

For two successive changes a% and b%, the net effect is [a + b + (ab/100)]%. In elections, multiplication of factors (Valid% × Share%) is often faster.

Valid vs Invalid Votes

Total Votes = Valid Votes + Invalid Votes.

Only Valid Votes are considered for determining the winning or losing margin.

Shortcut Trick

Compare the total quantities purchased by Arohi and Rhea.

Arohi: 3 apples + 5 bananas = ₹85

Rhea: 2 apples + 4 bananas = ₹60

Subtracting Rhea's total from Arohi's total: (3 − 2) apples + (5 − 4) bananas = ₹85 − ₹60

1 apple + 1 banana = ₹25

∴ The correct answer is ₹25.

Alternate Method

Given:

Cost of 3 apples and 5 bananas = ₹85

Cost of 2 apples and 4 bananas = ₹60

Let the cost of one apple be x and the cost of one banana be y.

⇒ 3x + 5y = 85 → (Equation 1)

⇒ 2x + 4y = 60 → (Equation 2)

Subtracting Equation 2 from Equation 1:

⇒ (3x + 5y) − (2x + 4y) = 85 − 60

⇒ (3x − 2x) + (5y − 4y) = 25

⇒ x + y = 25

∴ The correct answer is ₹25.

Additional Information

Elimination Method

A process of solving a system of equations by adding or subtracting them to eliminate one variable, making it easier to solve for the other.

Linear Equations in Two Variables

Equations of the form ax + by = c, where a, b, and c are constants and x, y are variables representing unknown quantities.

Substitution Method

Expressing one variable in terms of the other from one equation (e.g., x = 30 − 2y) and substituting it into the second equation.