Quantitative Aptitude PYQs

Shortcut Trick

As per BODMAS rule, division and multiplication are evaluated from left to right.

Combine terms directly using laws of exponents: a^m ÷ aⁿ = a^{m − n} and a^m × aⁿ = a^{m + n}

Expression = 62^{3 − 2} × 62⁻¹³

Expression = 62¹ × 62⁻¹³ = 62^{1 − 13}

∴ The correct answer is 62⁻¹².

Alternate Method

Given:

62³ ÷ 62² × 62⁻¹³

Formula Used:

a^m ÷ aⁿ = a^{m − n}

a^m × aⁿ = a^{m + n}

Calculations:

⇒ According to the order of operations, evaluate division first (from left to right):

⇒ 62³ ÷ 62² = 62^{3 − 2}

⇒ 62^{3 − 2} = 62¹

⇒ Now multiply the obtained result by the remaining term:

⇒ 62¹ × 62⁻¹³ = 62^{1 + (−13)}

⇒ 62^{1 − 13} = 62⁻¹²

∴ The correct answer is 62⁻¹².

Additional Information

Product Rule

When multiplying two expressions with the same base, the exponents are added: a^m × aⁿ = a^{m + n}.

Quotient Rule

When dividing two expressions with the same base, the exponents are subtracted: a^m ÷ aⁿ = a^{m − n}.

Negative Exponent

A negative exponent represents the reciprocal of the base raised to a positive exponent: a⁻ⁿ = 1 ÷ aⁿ.

Order of Operations (BODMAS)

In the absence of brackets, Division and Multiplication have the same level of precedence and are always solved sequentially from left to right.

Shortcut Trick

Successive discount formula: D = x + y − (x × y) ÷ 100

Effective Discount = 32 + 22 − (32 × 22) ÷ 100 = 54 − 7.04 = 46.96%

Final Price = Marked Price × (100 − Effective Discount) ÷ 100

Final Price = 4000 × (100 − 46.96) ÷ 100 = 4000 × 0.5304 = ₹2121.6

∴ The correct answer is ₹2121.6 ; 46.96%.

Alternate Method

Given:

Marked Price (MP) = ₹4,000

First discount (d₁) = 32%

Second discount (d₂) = 22%

Formula Used:

Effective Discount % = d₁ + d₂ − (d₁ × d₂) ÷ 100

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

₹4000
Marked Price

₹2720

₹2121.6
Selling Price

−32%

−22%

Effective Discount = 46.96%

Calculations:

⇒ Effective Discount = 32 + 22 − (32 × 22) ÷ 100

⇒ Effective Discount = 54 − 7.04 = 46.96%

⇒ Selling Price = 4000 × (100 − 46.96) ÷ 100

⇒ Selling Price = 4000 × 0.5304

⇒ Selling Price = ₹2121.6

∴ The correct answer is ₹2121.6 ; 46.96%.

Additional Information

Successive Discounts Formula

For two successive discounts of x% and y%, the single equivalent discount is calculated as: D = x + y − (x × y) ÷ 100.

Marked Price vs Cost Price Relation

The relationship between Cost Price (CP) and Marked Price (MP) for a given Profit (P%) and Discount (D%) is: MP ÷ CP = (100 + P) ÷ (100 − D).

Discount Percentage

A discount is always calculated over the Marked Price. Formula: Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

Using the Net Deviation Method and assuming the base average as 41:

Deviation for the next 10 matches = 16 − 41 = −25

Total deviation for 10 matches = 10 × (−25) = −250

Net change in average = −250 ÷ 33 ≈ −7.575

Combined average = 41 − 7.575 = 33.425 ≈ 33.42 runs

∴ The correct answer is 33.42 runs.

Alternate Method

23 MatchesAverage: 41 runs10 MatchesAvg: 16 runsTotal 33 MatchesCombined Average = ?

Given:

Average runs scored in the first 23 matches = 41

Average runs scored in the next 10 matches = 16

Formula Used:

Combined Average = (Total Value of Group 1 + Total Value of Group 2) ÷ (Total Number of Matches)

Calculations:

⇒ Total runs in the first 23 matches = 23 × 41 = 943 runs

⇒ Total runs in the next 10 matches = 10 × 16 = 160 runs

⇒ Total runs in all 33 matches = 943 + 160 = 1103 runs

⇒ Combined Average = 1103 ÷ 33

⇒ Combined Average = 33.4242... runs

⇒ Rounding off to two decimal places, we get 33.42 runs.

∴ The correct answer is 33.42 runs.

Additional Information

Weighted Average

Used to find the combined average of multiple groups directly: A_w = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Net Deviation Method

Instead of calculating large totals, assume a base average and calculate the net positive or negative deviation for each group, then divide it evenly over the total count to adjust the base.

Average Speed

If two equal distances are covered at speeds a and b, the Average Speed is given by (2ab) ÷ (a + b).

Shortcut Trick

Pranay's speed ratio (Original : New) = 1 : 2 ⇒ Time ratio = 2 :

1. Difference in time ratio = 2 − 1 = 1 unit.

Actual time difference = (Aman's time + 2) − (Aman's time − 3) = 5 hours.

⇒ 1 unit = 5 hours ⇒ Original time = 10 hours ⇒ Speed = 150 ÷ 10 = 15 km/hr.

∴ The correct answer is 15 km/hr.

Alternate Method

Given:

Total distance = 150 km

Pranay's original time = Aman's time + 2 hours

Pranay's time at double speed = Aman's time − 3 hours

Formula Used:

Speed = Distance ÷ Time

When Distance is constant, Speed ∝ 1 / Time

Start
End
Distance = 150 km

Speed = P

Time = t

Speed = 2P

Time = t − 5

Calculations:

⇒ Let original time taken by Pranay be T hours.

⇒ When Pranay's speed is doubled, his time becomes half = T / 2 hours.

⇒ Difference in Pranay's time = T − T / 2 = T / 2 hours.

⇒ From the given conditions, the actual difference in his time is also the difference between his times relative to Aman.

⇒ Time difference = (Aman's time + 2) − (Aman's time − 3) = 5 hours.

⇒ Therefore, T / 2 = 5 ⇒ T = 10 hours.

⇒ Pranay's original speed = Distance ÷ T

⇒ Pranay's original speed = 150 ÷ 10 = 15 km/hr.

∴ The correct answer is 15 km/hr.

Additional Information

Inverse Proportionality

When distance is constant, speed and time are inversely proportional. S₁ × T₁ = S₂ × T₂.

Average Speed

If equal distances are covered at speeds S₁ and S₂, the average speed is (2 × S₁ × S₂) ÷ (S₁ + S₂).

Relative Speed

For two objects moving in the same direction, Relative Speed = |S₁ − S₂|. For opposite directions, Relative Speed = S₁ + S₂.

Shortcut Trick

√169 = 13

√0.0144 = 0.12 (since 12² = 144, 4 decimal places inside the root become 2 outside)

√5.76 = 2.4 (since 24² = 576, 2 decimal places inside the root become 1 outside)

⇒ 13 + 0.12 − 2.4 = 13.12 − 2.4 = 10.72

∴ The correct answer is 10.72.

Alternate Method

Given:

Expression: √169 + √0.0144 − √5.76

Calculation:

Convert the decimals into fractions to easily calculate their square roots:

⇒ √0.0144 = √(144 ÷ 10000) = 12 ÷ 100 = 0.12

⇒ √5.76 = √(576 ÷ 100) = 24 ÷ 10 = 2.4

⇒ √169 = 13

Now, substitute the exact values back into the given expression:

⇒ 13 + 0.12 − 2.4

⇒ 13.12 − 2.4

⇒ 10.72

∴ The correct answer is 10.72.

Additional Information

Square Root of Decimals

When calculating the square root of a decimal number, count the number of decimal places. The resulting square root will have exactly half that number of decimal places.

BODMAS Rule

Follow the order of operations: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction. Always calculate roots before proceeding with addition or subtraction.

Common Perfect Squares

Memorizing perfect squares up to 30 (such as 12² = 144, 24² = 576) significantly reduces calculation time in simplification questions.

Shortcut Trick

Total days between 25 Feb and 26 Apr 2024 = 4 (Feb) + 31 (Mar) + 26 (Apr) = 61 days.

Since 2024 is a leap year, time in years = 61 ÷ 366 = 1/6 year.

SI = (P × R × T) ÷ 100

SI = (4000 × 6.75 × 1) ÷ (100 × 6) = 270 ÷ 6 = 45.

∴ The correct answer is ₹45.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate of Interest (R) = 6.75% per annum

Date of deposit = 25 February 2024

Date of withdrawal = 26 April 2024

Formula Used:

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

25 Feb 2024
P = ₹4000

26 Apr 2024
SI = ?

T = 61 days
R = 6.75% p.a.

SI = ₹45

Calculations:

⇒ Year 2024 is a leap year (divisible by 4), so February has 29 days.

⇒ Number of days in February (excluding deposit date) = 29 − 25 = 4 days

⇒ Number of days in March = 31 days

⇒ Number of days in April (including withdrawal date) = 26 days

⇒ Total time (T) in days = 4 + 31 + 26 = 61 days

⇒ Total days in leap year = 366

⇒ Time in years (T) = 61 ÷ 366 = 1/6 year

⇒ SI = (4000 × 6.75 × 1/6) ÷ 100

⇒ SI = (40 × 6.75) ÷ 6

⇒ SI = 270 ÷ 6

⇒ SI = 45

∴ The correct answer is ₹45.

Additional Information

Leap Year Calculation

A year is a leap year if it is perfectly divisible by 4. Century years must be divisible by 400. In leap years, February has 29 days and the total number of days in the year is 366.

Counting Days for Interest

When calculating interest between two dates, typically the day of deposit is excluded but the day of withdrawal is included. This ensures the exact number of days is counted accurately.

Simple vs Compound Interest

Simple Interest is calculated uniformly on the initial principal amount throughout the period, whereas Compound Interest calculates interest on both the principal and the accumulated interest from previous periods.

Shortcut Trick

Profit ratio = Ratio of investments (since the time period is constant).

Profit Ratio = 1050 : 1320 : 1730 = 105 : 132 : 173

Total ratio parts = 105 + 132 + 173 = 410 parts

Deepa's share = (173 ÷ 410) × 1230 = 173 × 3

∴ The correct answer is ₹519.

Alternate Method

Given:

Investment by Sanjana = ₹1,050

Investment by Kartik = ₹1,320

Investment by Deepa = ₹1,730

Total profit at the end of the year = ₹1,230

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂) : (Capital₃ × Time₃)

Partnership Investment & Profit Ratio

₹1050
105
Sanjana

₹1320
132
Kartik

₹1730
173
Deepa

Total Profit = ₹1230 → Total Ratio Parts = 410

Calculation:

⇒ Since the time period is equal (1 year) for all partners, the profit is distributed in the ratio of their capitals.

⇒ Profit Ratio (Sanjana : Kartik : Deepa) = 1050 : 1320 : 1730

⇒ Simplifying the ratio by dividing by 10, we get: 105 : 132 : 173

⇒ Sum of all ratio parts = 105 + 132 + 173 = 410 units

⇒ Total profit = 410 units = ₹1,230

⇒ 1 unit = 1230 ÷ 410 = ₹3

⇒ Deepa's share = 173 units = 173 × 3 = ₹519

∴ The correct answer is ₹519.

Additional Information

Simple Partnership

When all the partners invest their capital for the same period of time, the profit or loss is distributed purely in the ratio of their invested capital.

Compound Partnership

When the partners invest their capital for different time periods, the final profit or loss ratio is calculated as the product of their capital and the respective time period (C₁T₁ : C₂T₂ : C₃T₃).

Working vs Sleeping Partner

A working partner manages the business and receives a specific salary or a fixed percentage of the profit before the remaining profit is divided. A sleeping partner only invests capital and shares the profit strictly based on their investment ratio.

Shortcut Trick

Total Work = LCM(22, 11, 5) = 110 units.

Efficiency of A = 110 ÷ 22 = 5, B = 110 ÷ 11 = 10.

Efficiency of (A + B + C) = 110 ÷ 5 = 22.

Efficiency of C = 22 − (5 + 10) = 7 units/day.

Time taken by C alone = 110 ÷ 7 = 15⁵/₇ days.

∴ The correct answer is 15⁵/₇ days.

Alternate Method

Given:

⇒ Time taken by A = 22 days

⇒ Time taken by B = 11 days

⇒ Time taken by A, B, and C together = 5 days

Formula Used:

⇒ Total Work = Time × Efficiency

Calculations:

⇒ Let Total Work be LCM(22, 11, 5) = 110 units

⇒ Combined efficiency of A, B, and C = 22 units/day

⇒ Efficiency of C = (Efficiency of A + B + C) − (Efficiency of A + Efficiency of B)

⇒ Efficiency of C = 22 − (5 + 10) = 7 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time taken by C alone = 110 ÷ 7 = 15⁵/₇ days

∴ The correct answer is 15⁵/₇ days.

Additional Information

Work and Wages Relation

Wages are always distributed in direct proportion to the amount of work done by each individual. If they work for the same total duration, wages are distributed in the direct ratio of their efficiencies.

Pipes and Cisterns

The core mathematical concept is identical to Time and Work. A pipe filling a tank functions with a positive efficiency, while a leak or an emptying pipe operates with a negative efficiency.

Alternate Day Work

When individuals perform work on alternate days, the work is calculated in cycles (e.g., a 2-day cycle for 2 people) and the total time is determined by evaluating the total number of full cycles plus the days required to complete the remaining work.

Shortcut Trick

Observe the x² coefficients: 3 × (7/3) = 7 and −7 × 1 = −7. They cancel each other out.

Now, compute the remaining terms directly:

x coefficient: 3 × (−27) − 7 × 8 = −81 − 56 = −137

Constant term: 3 × 14 − 7 × (−11) = 42 + 77 = 119

Combining them, we get −137x + 119.

∴ The correct answer is −137x + 119.

Alternate Method

Given:

3((7/3)x² − 27x + 14) − 7(x² + 8x − 11)

Calculations:

⇒ Expand the brackets by multiplying the outer constants with each term inside.

⇒ [3 × (7/3)x² − 3 × 27x + 3 × 14] − [7 × x² + 7 × 8x − 7 × 11]

⇒ [7x² − 81x + 42] − [7x² + 56x − 77]

⇒ 7x² − 81x + 42 − 7x² − 56x + 77

⇒ Combine the like terms (x², x, and constants):

⇒ (7x² − 7x²) + (−81x − 56x) + (42 + 77)

⇒ 0 − 137x + 119

⇒ −137x + 119

∴ The correct answer is −137x + 119.

Additional Information

BODMAS Rule

When simplifying mathematical expressions, always follow the BODMAS sequence: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Combining Like Terms

In algebra, like terms have the same variables raised to the same powers. They can be combined by adding or subtracting their numerical coefficients.

Distributive Property

This property states that multiplying a single term by two or more terms inside a set of brackets yields the same result as multiplying the single term by each of the terms inside separately: a(b + c) = ab + ac.

Shortcut Trick

Arrange the given data numbers in ascending order.

Sorted data: 17, 20, 21, 27, 35, 41, 42, 49, 68, 76, 95

For an odd number of terms (n = 11), the median is exactly the middle term.

Middle term = (11 + 1) ÷ 2 = 6^th term, which is 41.

∴ The correct answer is 41.

Alternate Method

Given:

Data set: 76, 35, 42, 49, 95, 21, 68, 41, 20, 27, 17

Total number of observations (n) = 11

Formula Used:

If n is odd, Median = Value of the [(n + 1) ÷ 2]^th observation

Sorted Data Array

17

20

21

27

35

41

42

49

68

76

95

Median = 6^th Term

Calculations:

⇒ First, arrange the given data in ascending order.

⇒ 17, 20, 21, 27, 35, 41, 42, 49, 68, 76, 95

⇒ Here, n = 11 (which is an odd number)

⇒ Median = [(11 + 1) ÷ 2]^th term

⇒ Median = [12 ÷ 2]^th term

⇒ Median = 6^th term

⇒ The 6^th term in the arranged data is 41.

∴ The correct answer is 41.

Additional Information

Median for Even Number of Terms

If the number of observations (n) is even, the median is the average of the (n ÷ 2)^th and [(n ÷ 2) + 1]^th terms.

Empirical Relationship

The relationship between Mean, Median, and Mode is given by: Mode = 3 × Median − 2 × Mean.

Mean

Mean is the average of all observations, calculated as the sum of all observations divided by the total number of observations.

Shortcut Trick

Teacher's age = Old Average + (Total Members × Increase in Average)

Number of people after including the teacher = 24 + 1 = 25

Increase in average = 47 − 46 = 1 year

Teacher's age = 46 + (25 × 1) = 46 + 25 = 71 years

∴ The correct answer is 71 years.

Alternate Method

Given: Average age of 24 students = 46 years. When the teacher is included, the total group size becomes 25 and the new average becomes 47 years.

Formula: Total Sum = Average × Number of items

⇒ Total age of 24 students = 24 × 46 = 1104 years

⇒ Total age including teacher = 25 × 47 = 1175 years

⇒ Teacher's age = Total sum with teacher − Total sum of students

⇒ Teacher's age = 1175 − 1104 = 71 years

∴ The correct answer is 71 years.

Additional Information

Average Properties

If every observation is increased or decreased by a constant value 'k', the new average also increases or decreases by 'k'.

Inclusion/Exclusion Rule

Value of New Member = (New Average × New Quantity) − (Old Average × Old Quantity).

Net Deviation

The sum of deviations of all terms from the arithmetic mean is always 0. This property is used to find missing values quickly.

Shortcut Trick

Use identity: a³ − b³ = (a − b)(a² + b² + ab)

Here a = 4.5, b = 0.5

Numerator has (a² + b² + ab), so cancel

Expression = 8 ÷ (a − b) = 8 ÷ 4

∴ Answer = 2

Alternate Method

Given:

8 ÷ (4.5³ − 0.5³) × (4.5 × 4.5 + 0.25 + 4.5 × 0.5)

Concept Used:

BODMAS rule

Calculation:

⇒ 8 ÷ [(4.5 − 0.5)(4.5² + 0.5² + 4.5 × 0.5)] × (4.5 × 4.5 + 0.25 + 4.5 × 0.5)

⇒ 8 ÷ [4 × (4.5² + 0.5² + 4.5 × 0.5)] × (4.5 × 4.5 + 0.25 + 4.5 × 0.5)

⇒ 8 ÷ (4 × K) × K

⇒ 8 ÷ 4

⇒ 2

∴ The correct answer is 2.

Additional Information

Identity Used

a³ − b³ = (a − b)(a² + b² + ab)

Cancellation Trick

Look for common terms in numerator and denominator.

BODMAS Rule

Solve brackets first, then division/multiplication.

Simplification Tip

Convert decimals to fractions if needed.

Shortcut Trick

Let C be the number of crows. Total animals = 32 + 41 + 7 + C = 80 + C.

Each animal has at least 2 feet. Extra feet come from 4-legged animals (sheep and horses).

Equation: Total feet − Total heads = 211.

Since (Feet − Heads) for a 2-legged animal is 1 and for a 4-legged animal is 3, we use the net surplus.

80 + C + (2 × 41 + 2 × 7) = 211 ⇒ 80 + C + 96 = 211 ⇒ 176 + C = 211 ⇒ C = 35.

∴ The correct answer is 35 crows.

Alternate Method

Given: Ducks = 32, Sheep = 41, Horses = 7, Crows = x.

Heads (H): 32 + 41 + 7 + x = 80 + x.

Feet (F): (32 × 2) + (41 × 4) + (7 × 4) + (x × 2).

⇒ F = 64 + 164 + 28 + 2x = 256 + 2x.

Condition: F − H = 211.

⇒ (256 + 2x) − (80 + x) = 211.

⇒ 256 − 80 + 2x − x = 211.

⇒ 176 + x = 211.

⇒ x = 211 − 176 = 35.

∴ The correct answer is 35 crows.

Additional Information

Head and Foot Count Logic

For any animal, Number of Heads is always 1. 2-legged animals (birds) contribute 2 feet, while 4-legged animals contribute 4 feet.

The Difference Formula

The difference (Feet − Heads) = (1 × 2-legged) + (3 × 4-legged). This is a fast way to check consistency in animal word problems.

Linear Equations in Two Variables

Often solved using the formula: 4-legged animals = (Total Feet − 2 × Total Heads) ÷ 2.

Shortcut Trick

The area of a triangle is directly related to its inradius and semi-perimeter.

Logic: Area = Inradius × Semi-perimeter

⇒ Semi-perimeter = Perimeter ÷ 2 = 60 ÷ 2 = 30 cm

⇒ Area = 8 × 30 = 240 cm²

∴ The correct answer is 240 cm².

Alternate Method

Given:

Inradius (r) = 8 cm

Sum of sides (Perimeter) (P) = 60 cm

Formula Used:

Area of Triangle (A) = r × s

Where s = Semi-perimeter = P ÷ 2

Calculations:

⇒ Semi-perimeter (s) = Perimeter ÷ 2

⇒ s = 60 ÷ 2

⇒ s = 30 cm

⇒ Area of Triangle (A) = Inradius × Semi-perimeter

⇒ A = 8 × 30

⇒ A = 240 cm²

∴ The correct answer is 240 cm².

Additional Information

Inradius and Area Relation

Shortcut Trick

Total work = 4 Men × 8 days = 32 Man-days = 2 Women × 10 days = 20 Woman-days.

Efficiency ratio: 32M = 20W ⇒ 8M = 5W.

We need to find the time for 5 women and 8 men: (5W + 8M) ⇒ (8M + 8M) = 16M.

Since 4 men take 8 days, 16 men will take (4 × 8) ÷ 16 = 2 days.

∴ The correct answer is 2 days.

Alternate Method

Given:

Time taken by 4 men = 8 days

Time taken by 2 women = 10 days

Formula Used:

Total Work = Number of workers × Efficiency × Time

Calculations:

⇒ Total Work = 4M × 8 = 32M

⇒ Total Work = 2W × 10 = 20W

⇒ 32M = 20W

⇒ Efficiency of 1 Man (M) ÷ Efficiency of 1 Woman (W) = 20 ÷ 32 = 5 ÷ 8

⇒ Let efficiency of 1 Man = 5 units/day, and efficiency of 1 Woman = 8 units/day

⇒ Total Work = 32 × 5 = 160 units

⇒ Efficiency of 5 women and 8 men = (5 × 8) + (8 × 5) = 40 + 40 = 80 units/day

⇒ Number of days required = Total Work ÷ Combined Efficiency = 160 ÷ 80 = 2 days

∴ The correct answer is 2 days.

Additional Information

Work-Efficiency Relation

Efficiency is inversely proportional to the time taken when work is constant: E₁ × T₁ = E₂ × T₂.

Pipes & Cisterns and Wages

Wages are always distributed in the direct proportion of the total work done by individual workers or their efficiencies when they work for the same duration.

Shortcut Trick

Put θ = 45°

⇒ (1 + tan² 45°) ÷ (1 + cot² 45°)

⇒ (1 + 1²) ÷ (1 + 1²) = 2 ÷ 2 = 1

Check options: tan² 45° = 1² = 1

∴ The correct answer is tan² θ.

Alternate Method

Given: Expression \( \frac{1 + \tan^2 \theta}{1 + \cot^2 \theta} \)

Formula Used: 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ

⇒ Expression = sec² θ ÷ cosec² θ

⇒ Expression = (1/cos² θ) ÷ (1/sin² θ)

⇒ Expression = (1/cos² θ) × (sin² θ/1)

⇒ Expression = sin² θ ÷ cos² θ

⇒ Expression = (sin θ / cos θ)²

⇒ Expression = tan² θ

∴ The correct answer is tan² θ.

Additional Information

Trigonometric Identities

Fundamental identities include sin² θ + cos² θ = 1 and the relationship tan θ = 1 ÷ cot θ.

Reciprocal Relations

sec θ = 1 ÷ cos θ and cosec θ = 1 ÷ sin θ are essential for simplifying complex trigonometric fractions.

Shortcut Trick

Difference in Valid Votes percentage = 40% − 32% = 8%

Given that 8% of Valid Votes = 1,60,000

Total Valid Votes (100%) = (1,60,000 ÷ 8) × 100

Total Valid Votes = 20,00,000

∴ The correct answer is 20,00,000.

Alternate Method

Given:

Winner share = 40% of valid votes

Runner-up share = 32% of valid votes

Vote Difference = 1,60,000

Calculations:

Let the total Valid Votes be V.

⇒ Votes of Winner = 0.40V

⇒ Votes of Runner-up = 0.32V

⇒ Difference = 0.40V − 0.32V = 1,60,000

⇒ 0.08V = 1,60,000

⇒ V = 1,60,000 ÷ 0.08

⇒ V = 1,60,00,000 ÷ 8

⇒ V = 20,00,000

∴ The correct answer is 20,00,000.

Additional Information

Valid vs Invalid Votes

Total Votes = Valid Votes + Invalid Votes.

Only Valid Votes are considered for determining the winning margin.

Percentage Base Change

If asked for Total Votes Cast: Total = Valid Votes ÷ (100% − Invalid%). In this case, 20,00,000 ÷ 0.80 = 25,00,000.

Winning Margin Rule

Margin = (Winner% − Runner-up%) × Total Valid Votes. Always ensure the percentages refer to the same base.

Shortcut Trick

Use the Successive Percentage formula to find the net change.

Net Profit% = a − b − (a × b) ÷ 100, where a = Markup% and b = Discount%.

Net Profit% = 35 − 15 − (35 × 15) ÷ 100.

Net Profit% = 20 − 5.25 = 14.75%.

∴ The correct answer is 14.75%.

Alternate Method

Given: Markup = 35% above CP, Discount = 15% on MP.
Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100.

⇒ Let the Cost Price (CP) be 100.

⇒ Marked Price (MP) = 100 + 35% of 100 = 135.

⇒ Selling Price (SP) = 135 − 15% of 135 = 135 − 20.25 = 114.75.

⇒ Profit = SP − CP = 114.75 − 100 = 14.75.

⇒ Profit Percentage = (14.75 ÷ 100) × 100 = 14.75%.

∴ The correct answer is 14.75%.

Additional Information

Marked Price & Discount

The price fixed by the seller on the label is the Marked Price. Discount is always calculated on the MP. SP = MP − Discount.

Successive Percentage Change

When a value is changed by x% and then y%, the net change is given by [x + y + (xy/100)]%. For discounts, use negative values.

Shortcut Trick

Speed Ratio of trucks = 5 : 4

Since distance is constant, Time Ratio = Inverse of Speed Ratio = 4 : 5

Difference in time units = 5 − 4 = 1 unit → 2 hours

Time taken by slower truck (5 units) = 5 × 2 = 10 hours

Speed of slower truck = Total Distance ÷ Time = 200 ÷ 10 = 20 km/hr

∴ The correct answer is 20 km/hr.

Alternate Method

Given:

Ratio of speeds = 5 : 4

Distance (D) = 200 km

Time difference = 2 hours

Formula Used:

Speed = Distance ÷ Time | Time = Distance ÷ Speed

Calculations:

⇒ Let the speed of the faster truck be 5x and slower truck be 4x.

⇒ Time taken by slower truck − Time taken by faster truck = 2

⇒ (200 ÷ 4x) − (200 ÷ 5x) = 2

⇒ (50 ÷ x) − (40 ÷ x) = 2

⇒ 10 ÷ x = 2

⇒ x = 5

⇒ Speed of slower truck = 4 × 5 = 20 km/hr

∴ The correct answer is 20 km/hr.

Additional Information

Inverse Proportion of Speed and Time

When the distance is constant, speed is inversely proportional to time (S ∝ 1/T). If speed ratio is a:b, the time ratio is b:a.

Average Speed Formula

For the same distance covered at two different speeds u and v, Average Speed = (2 × u × v) ÷ (u + v).

Relative Speed

In the same direction, relative speed = S₁ − S₂. In the opposite direction, relative speed = S₁ + S₂.

Shortcut Trick

Efficiency of (A + B) = Total Work ÷ Days = 6 ÷ 3 = 2 units/day.

Efficiency of B = Total Work ÷ Days = 6 ÷ 6 = 1 unit/day.

Efficiency of A = (A + B) − B = 2 − 1 = 1 unit/day.

Double work = 6 × 2 = 12 units.

Time for A = 12 ÷ 1 = 12 days.

∴ The correct answer is 12 days.

Alternate Method

Given:

Time taken by (A + B) = 3 days

Time taken by B = 6 days

Formula Used:

Total Work = LCM of time taken

Efficiency = Total Work ÷ Time

Calculations:

⇒ Efficiency of A = Efficiency of (A + B) − Efficiency of B

⇒ Efficiency of A = 2 − 1 = 1 unit/day

⇒ Work to be done = Double of original work = 2 × 6 = 12 units

⇒ Time taken by A = Double Work ÷ Efficiency of A

⇒ Time taken by A = 12 ÷ 1 = 12 days

∴ The correct answer is 12 days.

Additional Information

Combined Efficiency Rule

The efficiency of two people working together is the sum of their individual efficiencies: E_A+B = E_A + E_B.

Work-Time Relationship

Time is inversely proportional to efficiency for a constant amount of work. If work is doubled, time taken at the same efficiency also doubles.