Quantitative Aptitude PYQs

Shortcut Trick

Let the total voters be 100%.

Voters who cast their votes = 60%.

Candidate B received 25% (100% − 75%) of the casted votes.

Fraction of B's votes = 60% × 25% = 0.60 × 0.25 = 0.15 (or 15%).

15% → 7500 ⇒ 1% → 500 ⇒ 100% → 50,000.

∴ The correct answer is 50,000 voters.

Alternate Method

Given:

Voter turnout = 60% of total registered voters.

Votes for Candidate A = 75% of polled votes.

Votes for Candidate B = 7500.

Calculations:

Let the total number of registered voters be x.

⇒ Total votes cast = 60% of x = 0.6x

⇒ Percentage of votes for Candidate B = 100% − 75% = 25%

⇒ Votes for B = 25% of (0.6x)

⇒ 7500 = (25/100) × (60/100) × x

⇒ 7500 = (1/4) × (3/5) × x

⇒ 7500 = (3/20) × x

⇒ x = (7500 × 20) ÷ 3

⇒ x = 2500 × 20

⇒ x = 50,000

∴ The correct answer is 50,000 voters.

Additional Information

Percentage Breakdown

In election problems, always distinguish between Total Voters, Votes Cast, and Valid Votes. Use the tree method for clarity.

Majority Rule

The Majority or Winning Margin is the difference between the votes of the winner and the runner-up: Margin = |Votes_A − Votes_B|.

Invalid Votes

If invalid votes are mentioned, subtract them from the total votes cast before distributing percentages to candidates.

Shortcut Trick

Total Salary is always considered as 100%.

If Savings = 30%, then Expenditure = 100% − 30% = 70%.

Expenditure = 70% of ₹1,890 = 0.7 × 1890.

Expenditure = ₹1,323.

∴ The correct answer is ₹1,323.

Alternate Method

Given:

Total Salary = ₹1,890

Savings percentage = 30%

Formula Used:

Expenditure = Total Salary − Savings

Calculations:

⇒ Amount of Savings = 30% of ₹1,890

⇒ Savings = (30 ÷ 100) × 1890

⇒ Savings = 0.3 × 1890 = ₹567

⇒ Expenditure = Total Salary − Savings

⇒ Expenditure = ₹1,890 − ₹567

⇒ Expenditure = ₹1,323

∴ The correct answer is ₹1,323.

Additional Information

Percentage Change

Percentage increase or decrease is calculated as: (Change ÷ Original Value) × 100.

Income and Expenditure Relation

Income = Expenditure + Savings. If any two are known, the third can be found by simple subtraction.

Successive Percentage

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

Shortcut Trick

Check the divisibility of numbers between 469 and 474.

Numbers to check: 470, 471, 472, 473.

471: 4 + 7 + 1 = 12 (divisible by 3). 473: 473 ÷ 11 = 43 (divisible by 11).

All numbers are composite.

∴ The correct answer is 0.

Alternate Method

Given: Range of numbers between 469 and 474.

Formula Used: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

⇒ 470: Ends in 0, so divisible by 2, 5, 10. (Composite)

⇒ 471: Sum of digits = 4 + 7 + 1 = 12. 12 is divisible by 3, so 471 is divisible by 3. (Composite)

⇒ 472: Even number, divisible by 2. (Composite)

⇒ 473: Sum of alternate digits = (4 + 3) − 7 = 0. Divisible by 11. (Composite)

⇒ Total prime numbers = 0.

∴ The correct answer is 0.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility Rule of 11

Subtract the sum of digits in even positions from the sum of digits in odd positions. If the result is 0 or divisible by 11, the number is divisible by 11.

Prime Numbers

To check if N is prime, test divisibility by primes up to √N. For 473, √473 ≈ 21.7. Primes to check: 2, 3, 5, 7, 11, 13, 17, 19.

Shortcut Trick

Substitute a standard angle A = 45° to verify the expression value.

⇒ Numerator: 2√(2−1) × √(1−1/2) = 2 × 1 × 1/√2 = √2

⇒ Denominator: √(1+1) × 3√(1−1/2) = √2 × 3 × 1/√2 = 3

⇒ Expression = √2 / 3

Checking options for A = 45°:

(1) (2 × 1/√2)/3 = √2 / 3. Matches perfectly.

∴ The correct answer is (2 cos A)/3.

Alternate Method

Given: \( \frac{2\sqrt{\csc^{2}A-1} \times \sqrt{1-\cos^{2}A}}{\sqrt{1+\tan^{2}A} \times 3\sqrt{1-\sin^{2}A}} \)

Formula Used:

1. √(csc²A − 1) = √cot²A = cot A
2. √(1 − cos²A) = √sin²A = sin A
3. √(1 + tan²A) = √sec²A = sec A
4. √(1 − sin²A) = √cos²A = cos A

Calculations:

⇒ Numerator = 2 × cot A × sin A

⇒ Numerator = 2 × (cos A / sin A) × sin A = 2 cos A

⇒ Denominator = sec A × 3 × cos A

⇒ Denominator = (1 / cos A) × 3 × cos A = 3

⇒ Final Expression = (2 cos A) / 3

∴ The correct answer is (2 cos A)/3.

Additional Information

Pythagorean Identities

sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ | 1 + cot²θ = csc²θ.

Reciprocal Relations

sin θ = 1/csc θ | cos θ = 1/sec θ | tan θ = 1/cot θ.

Quotient Relations

tan θ = sin θ / cos θ | cot θ = cos θ / sin θ.

Shortcut Trick

If a + 1/a = k, then the direct formula for a² + 1/a² is k² − 2.

Given: k = 6.

Calculation: 6² − 2 = 36 − 2 = 34.

∴ The correct answer is 34.

Alternate Method

Given:

a + 1/a = 6

Formula Used:

The Algebraic Identity for the square of a binomial: (x + y)² = x² + y² + 2xy

Calculation:

⇒ Taking the square on both sides of the given equation:

⇒ (a + 1/a)² = 6²

⇒ a² + (1/a)² + 2 × a × (1/a) = 36

⇒ a² + 1/a² + 2 = 36

⇒ a² + 1/a² = 36 − 2

⇒ a² + 1/a² = 34

∴ The correct answer is 34.

Additional Information

Square of a Difference

If a − 1/a = k, then a² + 1/a² = k² + 2.

Cubic Identities

If a + 1/a = k, then a³ + 1/a³ = k³ − 3k. If a − 1/a = k, then a³ − 1/a³ = k³ + 3k.

Higher Power Identities

If a + 1/a = k, then a⁴ + 1/a⁴ = (k² − 2)² − 2.

Shortcut Trick

Identify the frequency of each observation in the dataset.

Observations: 45 (4 times), 42 (2 times), 48 (2 times), 52 (2 times), Others (1 time).

The Mode is the value that occurs most frequently.

Maximum frequency is 4, which corresponds to the value 45.

∴ The correct answer is 45.

Alternate Method

Given Data: 55, 45, 44, 48, 42, 45, 52, 53, 42, 50, 54, 49, 52, 45, 47, 43, 45, 48

Formula Used: Mode = Value with the highest frequency in the distribution.

Arrange the data points and count their occurrences:

⇒ Frequency of 42 = 2

⇒ Frequency of 45 = 4

⇒ Frequency of 48 = 2

⇒ Frequency of 52 = 2

⇒ Frequency of 43, 44, 47, 49, 50, 53, 54, 55 = 1 each

⇒ Comparing frequencies: 4 > 2 > 1

∴ The observation with the maximum frequency is 45.

∴ The correct answer is 45.

Additional Information

Empirical Relationship

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

Types of Mode

A data set can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes).

Median of Data

The Median is the middle value when data is arranged in ascending order. For 18 terms, it is the average of 9^th and 10^th terms.

Shortcut Trick

When two equal distances are covered at speeds x and y, Average Speed = (2 × x × y) ÷ (x + y).

Here, x = 60 km/h and y = 30 km/h.

Average Speed = (2 × 60 × 30) ÷ (60 + 30) = 3600 ÷ 90 = 40 km/h.

∴ The correct answer is 40 km/h.

Alternate Method

Given: Speed 1 (v₁) = 60 km/h, Speed 2 (v₂) = 30 km/h. Let total distance be 2d.

Formula Used: Average Speed = Total Distance ÷ Total Time

⇒ Time for first half (t₁) = d ÷ 60

⇒ Time for second half (t₂) = d ÷ 30

⇒ Total Time (T) = (d ÷ 60) + (d ÷ 30) = (d + 2d) ÷ 60 = 3d ÷ 60 = d ÷ 20

⇒ Average Speed = 2d ÷ (d ÷ 20) = 2d × (20 ÷ d)

⇒ Average Speed = 40 km/h

∴ The correct answer is 40 km/h.

Additional Information

Harmonic Mean

When distance is constant, Average Speed is the Harmonic Mean of the individual speeds.

Equal Time Intervals

If a person travels at speed v₁ for time T and speed v₂ for the same time T, Average Speed = (v₁ + v₂) ÷ 2 (Arithmetic Mean).

Relative Speed

For two objects moving in opposite directions, relative speed = s₁ + s₂. In same direction, relative speed = |s₁ − s₂|.

Shortcut Trick

By using the standard trigonometric identity:

sec²θ − tan²θ = 1

Here, the angle θ = 50°.

sec² 50° − tan² 50° = 1

∴ The correct answer is 1.

Alternate Method

Given:

Expression: sec² 50° − tan² 50°

Formula Used:

Fundamental Trigonometric Identity: sec²θ − tan²θ = 1

Calculations:

⇒ Let θ = 50°

⇒ sec² 50° − tan² 50° = 1

∴ The correct answer is 1.

Additional Information

Pythagorean Trigonometric Identities

There are three fundamental Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

The functions secθ and tanθ can be written as secθ = 1/cosθ and tanθ = sinθ/cosθ.

Shortcut Trick
To find the least beneficial scheme for a customer, we look for the lowest overall discount percentage.
Scheme i: 34%.
Scheme ii: 37 + 8 − (37 × 8 ÷ 100) = 45 − 2.96 = 42.04%.
Scheme iii: 7 + 27 − (7 × 27 ÷ 100) = 34 − 1.89 = 32.11%.
Comparing the values: 32.11% < 34% < 42.04%.
The lowest discount is 32.11%, which is Scheme iii.
∴ The correct answer is Scheme iii.

Alternate Method
Given: Three different discount schemes. Goal: Identify the scheme with the minimum effective discount.
Formulas: Effective Discount for successive discounts a% and b% = (a + b − ab ÷ 100)%.

⇒ Scheme i: Net Discount = 34%.
⇒ Scheme ii: Net Discount = 37 + 8 − (37 × 8 / 100) = 45 − 2.96 = 42.04%.
⇒ Scheme iii: Net Discount = 7 + 27 − (7 × 27 / 100) = 34 − 1.89 = 32.11%.
⇒ Comparing all schemes, 32.11% is the smallest value.
∴ The correct answer is Scheme iii.

Additional Information
Effective Discount
When multiple discounts are applied, the total discount is always less than the sum of individual percentages because subsequent discounts are applied to already reduced prices.
Customer's Perspective
A customer always benefits from the highest discount percentage, while the least beneficial scheme is the one providing the lowest effective discount.

Shortcut Trick

Savings = 50% ⇒ Expenditure = 100% − 50% = 50%.

Since 50% (Expenditure) = ₹360,

Then 100% (Income) = ₹360 × 2 = ₹720.

∴ The correct answer is ₹720.

Alternate Method

Given:

Savings Percentage = 50%

Expenditure Value = ₹360

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let total Income be ₹X.

⇒ Expenditure % = 100% − Savings % = 100% − 50% = 50%.

⇒ 50% of X = 360

⇒ (50/100) × X = 360

⇒ X/2 = 360

⇒ X = 360 × 2 = 720

∴ The correct answer is ₹720.

Additional Information

Percentage Change

Percentage change = (Change / Original Value) × 100. This is used to calculate growth or reduction in income.

Income-Expenditure Relation

The fundamental relation is Income = Expenditure + Savings. If any two variables are known, the third can be found.

Successive Percentage

If income increases by a% and then b%, effective increase = (a + b + ab/100)%.

Shortcut Trick

Let the total capacity of the cistern be 12 units (LCM of 6 and 12).

Combined efficiency of P and Q = 12 ÷ 6 = 2 units/hour.

Efficiency of P alone = 12 ÷ 12 = 1 unit/hour ⇒ Efficiency of Q = 2 − 1 = 1 unit/hour.

One-fourth of the cistern = 12 ÷ 4 = 3 units.

Time taken by Q alone to fill one-fourth of the cistern = 3 ÷ 1 = 3 hours.

∴ The correct answer is 3 hours.

Alternate Method

Given:

Time taken by P and Q together = 6 hours

Time taken by P alone = 12 hours

Calculations:

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

⇒ Efficiency of Q = 2 − 1 = 1 unit/hour

⇒ One-fourth of the total work = 12 × (1 ÷ 4) = 3 units

⇒ Time taken by Q to complete 3 units of work = 3 ÷ 1 = 3 hours

∴ The correct answer is 3 hours.

Additional Information

Efficiency and Time Relationship

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

Pipes and Cisterns Negative Work

In inlet-outlet pipe systems, the outlet pipe performs negative work. Net efficiency is calculated by subtracting the outlet pipe efficiency from the inlet pipe efficiency.

Shortcut Trick

The given expression is in the form of (a³ + b³) ÷ (a² − ab + b²).

By Algebraic Identity, this always simplifies to (a + b).

Here, a = 9.73 and b = 7.27.

Result = 9.73 + 7.27 = 17.

∴ The correct answer is 17.

Alternate Method

Given:

The expression is (9.73 × 9.73 × 9.73 + 7.27 × 7.27 × 7.27) ÷ (9.73 × 9.73 − 9.73 × 7.27 + 7.27 × 7.27)

Formula Used:

We use the Sum of Cubes identity: a³ + b³ = (a + b)(a² − ab + b²)

Calculations:

Let a = 9.73 and b = 7.27.

⇒ The expression becomes: (a³ + b³) ÷ (a² − ab + b²)

⇒ Substituting the identity: [(a + b)(a² − ab + b²)] ÷ (a² − ab + b²)

⇒ The common term (a² − ab + b²) in the numerator and denominator cancels out.

⇒ Remaining expression = a + b

⇒ Required value = 9.73 + 7.27

⇒ Required value = 17.00

∴ The correct answer is 17.

Additional Information

Algebraic Identities for Cubes

Standard identities like a³ + b³ and a³ − b³ are essential for simplifying complex decimal or fraction calculations in Quantitative Aptitude.

Difference of Cubes

The formula a³ − b³ = (a − b)(a² + ab + b²) is the counterpart to the sum of cubes identity and follows similar simplification steps.

BODMAS Rule

While identities provide shortcuts, always follow the BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) priority for general simplification tasks.

Shortcut Trick

Subtract the remainders from the numbers: (1005 − 5), (244 − 4), (1343 − 3).

Resulting numbers: 1000, 240, 1340.

Find the HCF of 1000, 240, and 1340.

Since all numbers end in 0, 10 is a factor; check the remaining parts (100, 24, 134), their common factor is 2. ∴ HCF = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Numbers: 1005, 244, 1343

Remainders: 5, 4, 3

Calculations:

⇒ Number 1 = 1005 − 5 = 1000

⇒ Number 2 = 244 − 4 = 240

⇒ Number 3 = 1343 − 3 = 1340

⇒ To find the greatest number, calculate HCF(1000, 240, 1340).

⇒ Prime factorization:

⇒ 1000 = 2³ × 5³

⇒ 240 = 2⁴ × 3 × 5

⇒ 1340 = 2² × 5 × 67

⇒ HCF = 2² × 5 = 4 × 5 = 20

∴ The correct answer is 20.

Additional Information

HCF and Remainder Concept

To find the greatest number that divides x, y, and z leaving remainders a, b, and c, we calculate HCF((x−a), (y−b), (z−c)).

LCM and Remainder Concept

To find the least number which when divided by x, y, and z leaves a remainder r in each case, the answer is LCM(x, y, z) + r.

Difference Method for HCF

The HCF of a set of numbers is always a factor of the difference between any two numbers in that set.

Shortcut Trick

In problems where Sum (S) and Difference (D) are given:

Larger Quantity (Mangoes) = (Sum + Difference) ÷ 2

⇒ Mangoes = (96 + 48) ÷ 2 = 144 ÷ 2 = 72

∴ The correct answer is 72.

Alternate Method

Let the number of Oranges be x.

Then the number of Mangoes = x + 48

⇒ Total fruits = Mangoes + Oranges = 96

⇒ (x + 48) + x = 96

⇒ 2x + 48 = 96

⇒ 2x = 96 − 48 = 48

⇒ x = 24

⇒ Mangoes = 24 + 48 = 72

∴ The correct answer is 72.

Additional Information

Sum and Difference Formula

If A + B = S and A − B = D, then the larger value A = (S + D) ÷ 2 and the smaller value B = (S − D) ÷ 2.

Ratio and Proportion

Once values are found, the ratio of Mangoes to Oranges can be expressed as 72:24, which simplifies to 3:1.

Linear Equations

Problems involving two variables can always be solved by establishing two independent equations based on the given conditions.

Shortcut Trick

Surface area of a sphere is proportional to r² (radius squared).

When a value depends on the product of two variables (r × r), apply the successive percentage change formula: x + y + (x × y)/100.

Net percentage increase = 25 + 25 + (25 × 25)/100

⇒ 50 + 625/100 = 50 + 6.25 = 56.25%

∴ The correct answer is 56.25%.

Alternate Method

Given: Percentage increase in radius (r) = 25%

Formula Used: Surface Area (A) = 4πr²

⇒ Let the initial radius (r₁) be 100 units.

⇒ Initial Surface Area (A₁) = 4 × π × (100)² = 40000π

⇒ New radius (r₂) after 25% increase = 125 units.

⇒ New Surface Area (A₂) = 4 × π × (125)² = 4 × π × 15625 = 62500π

⇒ Increase in Surface Area = 62500π − 40000π = 22500π

⇒ Percentage Increase = (Increase ÷ Initial Area) × 100

⇒ Percentage Increase = (22500π ÷ 40000π) × 100 = 56.25%

∴ The correct answer is 56.25%.

Additional Information

Volume Change of Sphere

Since Volume (V) = 4/3 πr³, if the radius increases by x%, the volume increases by [(1 + x/100)³ − 1] × 100%.

Ratio of Areas

If the ratio of the radii of two spheres is a : b, then the ratio of their surface areas is a² : b² and the ratio of their volumes is a³ : b³.

Hemisphere Surface Area

For a hemisphere of radius r, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Shortcut Trick

Convert 2 days into hours: 2 × 24 = 48 hours.

Required Ratio = 38 : 48.

Divide both by the Highest Common Factor (HCF) which is 2.

Simplified Ratio = 19 : 24.

∴ The correct answer is 19:24.

Alternate Method

Given:

First value = 38 hours

Second value = 2 days

Conversion:

⇒ 1 day = 24 hours

⇒ 2 days = 2 × 24 = 48 hours

Calculation:

⇒ Ratio = 38 / 48

⇒ Ratio = (2 × 19) / (2 × 24)

⇒ Ratio = 19 : 24

∴ The correct answer is 19:24.

Additional Information

Unit Consistency in Ratio

A ratio must always be calculated between quantities of the same unit. Always convert the larger unit to the smaller unit to avoid decimals.

Simplest Form of Ratio

A ratio a:b is in its simplest form if the HCF of a and b is 1. In this case, HCF(38, 48) = 2, so we divide both by 2.

Time Conversion Constants

1 Day = 24 Hours, 1 Hour = 60 Minutes, 1 Minute = 60 Seconds.

Shortcut Trick

Net deviation of given 5 numbers from Mean (26) = (25−26) + (29−26) + (25−26) + (32−26) + (24−26) = −1 + 3 − 1 + 6 − 2 = +5.

To maintain the mean at 26, the deviation of x must be −5 ⇒ x = 26 − 5 = 21.

Arranging data in ascending order: 21, 24, 25, 25, 29, 32.

For n = 6 (even), Median = Average of 3^rd and 4^th terms = (25 + 25) ÷ 2 = 25.

∴ The correct answer is 25.

Alternate Method

Given: Observations = 25, 29, 25, 32, 24, x | Number of observations (n) = 6 | Mean = 26.

Formula Used: Mean = Sum of observations ÷ Total number of observations

⇒ Mean = (25 + 29 + 25 + 32 + 24 + x) ÷ 6 = 26

⇒ 135 + x = 26 × 6

⇒ 135 + x = 156

⇒ x = 156 − 135 = 21

⇒ Arranging observations in ascending order: 21, 24, 25, 25, 29, 32

⇒ Since n = 6 (even), Median = [(n/2)^th term + (n/2 + 1)^th term] ÷ 2

⇒ Median = [3^rd term + 4^th term] ÷ 2

⇒ Median = (25 + 25) ÷ 2 = 25

∴ The correct answer is 25.

Additional Information

Arithmetic Mean

The average of a set of numbers, calculated by dividing the sum of all values by the total count of values.

Median for Odd Observations

If n is odd, the median is the middle-most value, calculated as the ((n + 1)/2)^th observation.

Empirical Relationship

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

Shortcut Trick

Simplified Ratio = 4/6 : 1 : 1 = 2/3 : 1 : 1 = 2 : 3 : 3

Total units = 2 + 3 + 3 = 8 units

8 units = 896 → 1 unit = 112

Third monkey (3 units) = 3 × 112 = 336

∴ The correct answer is 336 bananas.

Alternate Method

Given: Total bananas = 896 | Ratio = 4/6 : 2/2 : 2/2

Step 1: Simplify the fractions in the ratio

⇒ 4/6 = 2/3

⇒ 2/2 = 1

⇒ Ratio = 2/3 : 1 : 1

Step 2: Convert to whole numbers by multiplying by 3

⇒ 2 : 3 : 3

Step 3: Calculate the share of the third monkey

⇒ Sum of ratio parts = 2 + 3 + 3 = 8

⇒ Third monkey's share = (3/8) × 896

⇒ 3 × 112 = 336

∴ The correct answer is 336 bananas.

Additional Information

Ratio Simplification

To simplify a ratio involving fractions, multiply each term by the Least Common Multiple (LCM) of the denominators.

Proportional Division

If a quantity 'A' is divided in ratio a:b:c, the third part is calculated as [c / (a + b + c)] × A.

Duplicate Ratio

The duplicate ratio of a:b is a² : b².

Shortcut Trick

For a square, the relationship between Area and Perimeter is:

Perimeter = 4 × √Area

Perimeter = 4 × √15.21 = 4 × 3.9 = 15.6

∴ The correct answer is 15.6 cm.

Alternate Method

Given:

Area of the square = 15.21 cm²

Formula Used:

1. Area of square = (side)²
2. Perimeter of square = 4 × side

Calculations:

⇒ Let the side length of the square be a cm.

⇒ a² = 15.21

⇒ a = √15.21

⇒ a = 3.9 cm

⇒ Perimeter = 4 × a

⇒ Perimeter = 4 × 3.9

⇒ Perimeter = 15.6 cm

∴ The correct answer is 15.6 cm.

Additional Information

Properties of a Square

A square is a regular quadrilateral with four equal sides and four equal angles (each 90°).

Diagonal of a Square

The length of the diagonal (d) is given by d = side × √2.

Relationship between Area and Diagonal

The Area can also be calculated using the diagonal: Area = (diagonal)² ÷ 2.

Shortcut Trick

For divisibility by 11, the difference between the sum of digits at odd positions and even positions must be 0 or a multiple of 11.

Sum of odd position digits (2 + 7 + Y) − Sum of even position digits (X + 3 + 5) = 0, 11, ...

⇒ (9 + Y) − (X + 8) = 0

⇒ 1 + Y − X = 0 ⇒ X − Y = 1.

∴ The correct answer is 1.

Alternate Method

Given: The number is 2X73Y5 and it is divisible by 11.

Step-by-step Calculation:

⇒ Sum of digits at odd places (S₁) = 2 + 7 + Y = 9 + Y

⇒ Sum of digits at even places (S₂) = X + 3 + 5 = X + 8

⇒ According to Divisibility Rule of 11: (S₁ − S₂) = 0 or 11k

⇒ (9 + Y) − (X + 8) = 0

⇒ 9 + Y − X − 8 = 0

⇒ 1 + Y − X = 0

⇒ X − Y = 1

∴ The correct answer is 1.

Additional Information

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or a multiple of 11.

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is divisible by 9.

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.