Quantitative Aptitude PYQs

Shortcut Trick

Numbers from 303 to 311: 303, 304, 305, 306, 307, 308, 309, 310, 311.

Eliminate even numbers and numbers ending in 5: 303, 307, 309, 311.

Check divisibility by 3 (sum of digits): 303 (sum 6) and 309 (sum 12) are multiples of 3.

Remaining numbers are 307 and 311, which are not divisible by primes up to √311 (7, 11, 13, 17).

Thus, 307 and 311 are the only prime numbers.

∴ The correct answer is 2.

Alternate Method

Given:

Range of numbers: 303 to 311 (both included)

Concept Used:

A prime number has only two distinct divisors: 1 and itself.

To check if a number N is prime, we divide it by all prime numbers ≤ √N.

Calculations:

⇒ List of numbers to check: 303, 304, 305, 306, 307, 308, 309, 310, 311.

⇒ Even numbers (304, 306, 308, 310) are divisible by 2.

⇒ Number ending in 5 (305) is divisible by 5.

⇒ Remaining odd numbers: 303, 307, 309, 311.

⇒ Checking divisibility by 3 (sum of digits must be a multiple of 3):

⇒ 303 → 3 + 0 + 3 = 6 (Divisible by 3)

⇒ 309 → 3 + 0 + 9 = 12 (Divisible by 3)

⇒ Remaining numbers to check for primality: 307 and 311.

⇒ Approximate square root of 311 ≈ 17.6.

⇒ Prime numbers ≤ 17 are 2, 3, 5, 7, 11, 13, 17.

⇒ Checking 307: Not divisible by 7, 11, 13, 17. Hence, 307 is a prime number.

⇒ Checking 311: Not divisible by 7, 11, 13, 17. Hence, 311 is a prime number.

⇒ Total count of prime numbers = 2.

∴ The correct answer is 2.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It has exactly two divisors: 1 and itself.

Primality Test

To check if a given number N is prime, find the approximate square root of N. Divide N by all prime numbers less than or equal to √N. If N is not divisible by any of these primes, it is a prime number.

Co-Prime Numbers

Two numbers are said to be co-prime (or mutually prime) if the only positive integer that divides both of them is 1. Their Highest Common Factor (HCF) is always 1.

Composite Numbers

Numbers that have more than two factors are called composite numbers. The number 1 is neither prime nor composite.

Shortcut Trick

When the radius and height of a cylinder are equal (r = h), the volume simplifies to πh³.

⇒ πh³ = 8624

⇒ (22/7) × h³ = 8624

⇒ h³ = 8624 × 7 ÷ 22 = 2744

⇒ h = ³√2744 = 14 cm

∴ The correct answer is 14 cm.

Alternate Method

Given:

Volume of the right circular cylinder = 8624 cm³

Radius of the cylinder (r) = Height of the cylinder (h)

Formula Used:

Volume of a cylinder = πr²h

h
r = h

Calculations:

⇒ Volume = π × r² × h

Substitute r = h in the formula:

⇒ Volume = π × h² × h

⇒ 8624 = πh³

⇒ 8624 = (22/7) × h³

⇒ h³ = 8624 × (7 ÷ 22)

⇒ h³ = 392 × 7

⇒ h³ = 2744

⇒ h = ³√2744

⇒ h = 14 cm

∴ The correct answer is 14 cm.

Additional Information

Curved Surface Area of a Cylinder

The area of the curved surface (excluding bases) is given by the formula 2πrh.

Total Surface Area of a Cylinder

The total area including the two circular bases is given by the formula 2πr(r + h).

Volume of a Cone

A cone with the same base radius and height as a cylinder has exactly one-third of its volume: (1/3)πr²h.

Shortcut Trick

Days between 7 Feb 2024 and 8 Apr 2024 = 22 (Feb) + 31 (Mar) + 8 (Apr) = 61 days.

Since 2024 is a leap year (366 days), Time (T) = 61 ÷ 366 = 1/6 years.

SI = (Principal × Rate × Time) ÷ 100

SI = (4000 × 7.5 × 1/6) ÷ 100

SI = 30000 ÷ 600 = 50

∴ The correct answer is ₹50.

Alternate Method

Given:

Principal (P) = ₹ 4,000

Rate of Interest (R) = 7.5% per annum

Date of deposit = 7 February 2024

Date of withdrawal = 8 April 2024

Formula Used:

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

7 Feb
29 Feb
31 Mar
8 Apr

22 Days

31 Days

8 Days

Principal: ₹ 4000 | Rate: 7.5% p.a.

Total Time = 61 Days

Calculations:

⇒ Total days in February 2024 (Leap Year) = 29 days

⇒ Days the money was in the bank in February = 29 − 7 = 22 days

⇒ Days in March = 31 days

⇒ Days in April = 8 days

⇒ Total Number of days (T) = 22 + 31 + 8 = 61 days

⇒ As 2024 is a leap year, total days in the year = 366

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

⇒ SI = (4000 × 7.5 × 1) ÷ (100 × 6)

⇒ SI = (40 × 7.5) ÷ 6

⇒ SI = 300 ÷ 6 = 50

∴ The correct answer is ₹50.

Additional Information

Rule for Counting Days

While computing interest, either the day of deposit is excluded and the day of withdrawal is included, or vice versa. They are never both counted.

Leap Year Identification

A year is a leap year (366 days) if it is divisible by 4. If it is a century year (like 1900, 2000), it must be divisible by 400. February in a leap year has 29 days.

Simple Interest Amount

The total final amount at the end of the given time period is calculated as: Amount = Principal + Simple Interest.

Shortcut Trick

Let Mother's age be M and Daughter's be D. Given: 2D = M − 2 ⇒ M = 2D + 2.

Substitute M into the first equation: 3(2D + 2) = 5D + 26.

6D + 6 = 5D + 26 ⇒ D = 20 years.

M = 2(20) + 2 = 42 years. Difference = 42 − 20 = 22 years.

∴ The correct answer is 22 years.

Alternate Method

Given:

Let the present age of the mother be M years and the daughter be D years.

First condition: 3M = 5D + 26

Second condition: 2D = M − 2

Age Difference Visualization

Mother's Age: 42 years

Daughter's Age: 20 yrs

Diff = 22 years

Calculations:

⇒ From the second condition, M = 2D + 2

⇒ Substitute this value of M in the first equation:

⇒ 3(2D + 2) = 5D + 26

⇒ 6D + 6 = 5D + 26

⇒ 6D − 5D = 26 − 6

⇒ D = 20

⇒ The daughter's age is 20 years.

⇒ Substitute D = 20 in M = 2D + 2:

⇒ M = 2(20) + 2

⇒ M = 40 + 2 = 42

⇒ The mother's age is 42 years.

⇒ Difference between their ages = M − D

⇒ Difference = 42 − 20 = 22 years

∴ The correct answer is 22 years.

Additional Information

Linear Equations in Two Variables

A system of two linear equations can be solved rapidly using the substitution method. By isolating one variable in terms of the other, we can reduce the problem to a single-variable equation.

Problems on Ages

In age-related problems, always define variables for the present age first. Use standard linear equations to represent the conditions given like 'n times older' or 'x years less'.

Difference in Ages

The difference between the ages of any two individuals is an absolute constant. It remains exactly the same throughout their lifetimes, regardless of whether you refer to the past or the future.

Shortcut Trick

Radius of the sphere (r) = Diameter ÷ 2 = 22 ÷ 2 = 11 cm

Volume = (4/3) × π × r³

Volume = (4/3) × (22/7) × 11³ = (88 × 1331) ÷ 21

Volume = 117128 ÷ 21 = 5577.5238...

Rounding off to two decimal places, we get 5577.52 cm³

∴ The correct answer is 5577.52 cm³.

Alternate Method

Given:

Diameter of the sphere = 22 cm

π = 22/7

O

r = 11 cm

Diameter = 22 cm

Formula Used:

Volume of a sphere = (4/3) × π × r³

Where r is the radius of the sphere.

Calculations:

⇒ Radius (r) = Diameter ÷ 2 = 22 ÷ 2 = 11 cm

⇒ Volume = (4/3) × (22/7) × (11)³

⇒ Volume = (4/3) × (22/7) × 1331

⇒ Volume = (88 × 1331) ÷ 21

⇒ Volume = 117128 ÷ 21

⇒ Volume ≈ 5577.5238 cm³

⇒ Volume correct to two decimal places = 5577.52 cm³

∴ The correct answer is 5577.52 cm³.

Additional Information

Surface Area of a Sphere

The total surface area of a sphere is given by the formula 4 × π × r².

Volume of a Hemisphere

A hemisphere is exactly half of a sphere. Its volume is given by (2/3) × π × r³.

Total Surface Area of a Hemisphere

The total surface area of a hemisphere includes the curved surface and the flat circular base, given by 3 × π × r².

Shortcut Trick

Total Distance = Sum of lengths = 259 + 431 = 690 m

Relative Speed (Opposite) = 52 + 56 = 108 km/h

Convert to m/s: 108 × (5/18) = 30 m/s

Time = Distance ÷ Relative Speed = 690 ÷ 30 = 23 sec

∴ The correct answer is 23 sec.

Alternate Method

Given:

Length of first train (L₁) = 259 m

Length of second train (L₂) = 431 m

Speed of first train (S₁) = 52 km/h

Speed of second train (S₂) = 56 km/h

Formula Used:

Time = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 259 + 431 = 690 m

⇒ Relative Speed = 52 + 56 = 108 km/h

⇒ Relative Speed in m/s = 108 × (5/18) = 30 m/s

⇒ Time taken = 690 ÷ 30

⇒ Time = 23 sec

∴ The correct answer is 23 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). In the same direction, they are subtracted (S₁ − S₂).

Unit Conversion

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

Train Crossing Logic

When a train crosses another train, the total distance covered is always the sum of the lengths of both trains.

Shortcut Trick

Let the total votes be 100%.

Candidate 1 = 42% | Candidate 2 = 100% − 42% = 58%.

Margin = 58% − 42% = 16%.

Given 16% = 416 votes.

100% = (416 ÷ 16) × 100 = 26 × 100 = 2600.

∴ The correct answer is 2600 votes.

Alternate Method

Given: Votes of losing candidate = 42%, Defeat margin = 416 votes.

Formula: Total Votes = (Margin × 100) ÷ (Difference in % share).

⇒ Let Total Votes = V.

⇒ Winning Candidate Share = 100% − 42% = 58%.

⇒ Difference in share = 58% − 42% = 16%.

⇒ 16% of V = 416.

⇒ (16 ÷ 100) × V = 416.

⇒ V = (416 × 100) ÷ 16.

⇒ V = 26 × 100 = 2600.

∴ The correct answer is 2600 votes.

Additional Information

Percentage Change

To calculate the percentage change between two values: [(New Value − Old Value) ÷ Old Value] × 100.

Invalid Votes Concept

If invalid votes are present: Total Valid Votes = Total Cast Votes − Invalid Votes. The winner/loser percentages are usually calculated on valid votes.

Base Change Rule

If a candidate gets x% of the total votes, the other gets (100 − x)% provided there are only two candidates and no invalid votes.

Shortcut Trick

Using the Multiplying Factor method for successive changes:

Final Price = 100 × 0.60 × 0.75 × 0.55 × 1.55

Final Price = 100 × 0.383625 = 38.3625

Net change = 38.3625 − 100 = −61.6375%

∴ The correct answer is Decreases by 61.64%.

Alternate Method

Given: Original Price = 100 units. Changes: −40%, −25%, −45%, +55%.

⇒ Price after 1^st month = 100 − (40% of 100) = 60

⇒ Price after 2^nd month = 60 − (25% of 60) = 60 − 15 = 45

⇒ Price after 3^rd month = 45 − (45% of 45) = 45 − 20.25 = 24.75

⇒ Price after 4^th month = 24.75 + (55% of 24.75) = 24.75 + 13.6125 = 38.3625

⇒ Percentage Change = (38.3625 − 100) / 100 × 100 = −61.6375%

⇒ Approximately 61.64% decrease.

∴ The correct answer is Decreases by 61.64%.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, net change = (a + b + ab/100)%.

Multiplying Factor (MF)

For an x% increase, MF = (1 + x/100). For an x% decrease, MF = (1 − x/100).

Order of Operations

In successive percentage changes, the order in which increases or decreases are applied does not affect the final result.

Shortcut Trick

Check LCM of 6 and 9 → LCM(6, 9) = 18. All numbers must be multiples of 18.

Numbers divisible by 12 are also divisible by 4 (since 12 = 3 × 4).

For 126: Last two digits (26) are not divisible by 4, so it is not divisible by 12.

For 72, 108, 252: All are divisible by 12 (72/12=6, 108/12=9, 252/12=21).

∴ The correct answer is 126.

Alternate Method

Given: Numbers 126, 252, 72, 108.

Conditions: Divisible by 6 AND 9 (meaning divisible by 18) AND NOT divisible by 12.

⇒ 126 ÷ 6 = 21 (Yes)

⇒ 126 ÷ 9 = 14 (Yes)

⇒ 126 ÷ 12 = 10.5 (No - Not Divisible)

⇒ 252 ÷ 12 = 21 (Divisible)

⇒ 72 ÷ 12 = 6 (Divisible)

⇒ 108 ÷ 12 = 9 (Divisible)

∴ The correct answer is 126.

Additional Information

Divisibility Rule of 6

A number is divisible by 6 if it is divisible by both 2 (even number) and 3 (sum of digits is a multiple of 3).

Divisibility Rule of 9

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

Divisibility Rule of 12

A number is divisible by 12 if it is divisible by both 3 and 4 (last two digits divisible by 4).

Shortcut Trick

Combined efficiency of A, B, and C = (1/3 + 1/15 + 1/10) of work per day.

LCM of 3, 15, 10 is 30. Combined efficiency = (10 + 2 + 3)/30 = 15/30 = 1/2 work per day.

Time for single work = 2 days. For twice the work = 2 × 2 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Time taken by A = 3 days, B = 15 days, C = 10 days.

Formula Used: Efficiency = Total Work ÷ Time taken

⇒ Total Efficiency of (A + B + C) = 10 + 2 + 3 = 15 units/day.

⇒ Total Work to be done = 2 × 30 = 60 units.

⇒ Required Time = Total Work ÷ Combined Efficiency = 60 ÷ 15 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Total Work (LCM Method)

The most efficient way to solve Time & Work problems is to assume Total Work as the LCM of given days, making efficiency an integer.

Work & Time Relationship

Time taken is directly proportional to the amount of Work and inversely proportional to Efficiency (T = W/E).

Pipes & Cisterns

Same logic applies where Filling Pipe has positive efficiency and Leaking/Emptying Pipe has negative efficiency.

Shortcut Trick

Convert Rate 5% to fraction: 5/100 = 1/20.

For 2 years, the ratio of Principal to Amount is 20² : 21² = 400 : 441.

Given 400 units = ₹2,000 ⇒ 1 unit = ₹5.

Amount = 441 units × ₹5 = ₹2,205.

∴ The correct answer is ₹2,205.

Alternate Method

Given: Principal (P) = ₹2,000, Rate (R) = 5%, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 2000 × (1 + 5/100)²

⇒ A = 2000 × (1 + 1/20)²

⇒ A = 2000 × (21/20) × (21/20)

⇒ A = 2000 × (441/400)

⇒ A = 5 × 441

⇒ A = ₹2,205

∴ The correct answer is ₹2,205.

Additional Information

Effective Rate of Interest

For 2 years, the effective Compound Interest rate can be calculated as R + R + (R×R)/100.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is P(R/100)².

Compounding Half-Yearly

If interest is compounded half-yearly, the Rate is halved (R/2) and the Time is doubled (2n).

Shortcut Trick

Assume Emma's age is 1 unit.

Saloni's age = 2 × 1 unit + 7 = 2 units + 7.

Total age = 1 unit + (2 units + 7) = 3 units + 7 = 25 years.

3 units = 25 − 7 = 18 → 1 unit = 6 years.

∴ The correct answer is 6 years.

Alternate Method

Given:

Sum of Saloni and Emma's ages = 25 years.

Saloni's age = 2 × (Emma's age) + 7.

Let Emma's age be x years.

Then Saloni's age = 2x + 7.

⇒ x + (2x + 7) = 25

⇒ 3x + 7 = 25

⇒ 3x = 25 − 7

⇒ 3x = 18

⇒ x = 18 ÷ 3

⇒ x = 6

∴ The correct answer is 6 years.

Additional Information

Linear Equations in One Variable

An equation of the form ax + b = c, where x is the unknown variable to be solved by isolating it.

Age Relation Problems

Always define the independent age (usually the younger person) as 'x' to simplify the mathematical expression for the older person.

Verification Method

Plug the result back: if Emma is 6, Saloni is (2×6)+7 = 19. Sum = 6 + 19 = 25. The answer is consistent with the data.

Shortcut Trick

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

SP = 6,420 × (100 − 30) ÷ 100

SP = 6,420 × 0.7

SP = 4,494

∴ The correct answer is ₹4,494.

Alternate Method

Given: Marked Price (MP) = ₹6,420 and Discount = 30%.
Formula: Selling Price (SP) = MP − Discount Amount | Discount Amount = (Discount% ÷ 100) × MP.

⇒ Discount Amount = (30 ÷ 100) × 6,420
⇒ Discount Amount = 0.3 × 6,420 = 1,926
⇒ Selling Price = 6,420 − 1,926
⇒ Selling Price = 4,494

∴ The correct answer is ₹4,494.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP to arrive at the Selling Price (SP).

Relationship Formula

SP = MP × (100 − D%) / 100. If multiple discounts are given, they are applied successively.

Shortcut Trick

Ratio of articles P : Q : R = 3 : 4 : 5

Cost of one set = (3 × 280) + (4 × 180) + (5 × 100) = 840 + 720 + 500 = ₹2060

Number of sets = Total expenditure ÷ Cost of one set = 26780 ÷ 2060 = 13

Articles of type Q = 4 × 13 = 52

∴ The correct answer is 52.

Alternate Method

Given: Prices of P, Q, R are ₹280, ₹180, ₹100. Ratio of quantity = 3 : 4 : 5. Total = ₹26,780.

Formula Used: Total Cost = ∑(Price × Quantity)

⇒ Let the quantities be 3x, 4x, and 5x respectively.

⇒ Total Cost = (280 × 3x) + (180 × 4x) + (100 × 5x) = 26780

⇒ 840x + 720x + 500x = 26780

⇒ 2060x = 26780

⇒ x = 26780 ÷ 2060 = 13

⇒ Number of articles of type Q = 4x = 4 × 13 = 52

∴ The correct answer is 52.

Additional Information

Ratio and Proportion

When items are bought in a ratio a : b : c, the actual quantities are ax, bx, and cx where x is the common multiplier.

Weighted Average Logic

Total value is the sum of products of individual values and their respective weights (quantities). Total = ∑(v_i × w_i).

Shortcut Trick

((3²)⁶)⁵ ⇒ multiply powers

= 3^2×6×5 = 3⁶⁰

Now 9 = 3²

⇒ 3⁶⁰ = (3²)³⁰ = 9³⁰

∴ Exponent = 30

Alternate Method

Given:

((3²)⁶)⁵

Concept Used:

Laws of exponents

Calculation:

⇒ (3²)⁶ = 3¹²

⇒ (3¹²)⁵ = 3⁶⁰

⇒ 9 = 3²

⇒ 3⁶⁰ = (3²)³⁰ = 9³⁰

∴ The correct answer is 30.

Additional Information

Power Rule

(a^m)ⁿ = a^mn

Base Conversion

Express numbers in same base to compare powers

Exponent Simplification

Multiply exponents when powers are raised again

Shortcut Trick

Use the options substitution method for rapid verification.

From Option (1): Price of mango = 3, Price of apple = 8.

Check first condition: (10 × 3) + (13 × 8) = 30 + 104 = 134. (True)

Check second condition: (15 × 3) − (4 × 8) = 45 − 32 = 13. (True)

Since both conditions are satisfied, Option (1) is the correct pair.

∴ The correct answer is 3, 8.

Alternate Method

Given → Let cost of one mango be m and one apple be a.
Formulas: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂)

⇒ Multiply Eq 1 by 4 and Eq 2 by 13 to eliminate 'a'.
⇒ (40m + 52a = 536) + (195m − 52a = 169)
⇒ 235m = 705
⇒ m = 705 ÷ 235 = 3
⇒ Substitute m = 3 in Eq 1: 10(3) + 13a = 134
⇒ 30 + 13a = 134 ⇒ 13a = 104 ⇒ a = 8

∴ The correct answer is 3, 8.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c can be solved using the Elimination Method or Substitution Method to find the unique values of variables.

Verification of Roots

Always plug the values back into the original constraints to ensure both logical conditions are satisfied simultaneously.

Shortcut Trick

Substitute standard trigonometric values directly.

sin 30° = 1/2, cos 60° = 1/2, cot 45° = 1.

Expression = (1/2 + 1/2) ÷ 1 = 1 ÷ 1 = 1.

∴ The correct answer is 1.

Alternate Method

Given: sin 30°, cos 60°, cot 45°

Formula Used: Values of trigonometric ratios at specific angles.

⇒ sin 30° = 1/2

⇒ cos 60° = 1/2

⇒ cot 45° = 1

⇒ Value = (sin 30° + cos 60°) ÷ cot 45°

⇒ Value = (1/2 + 1/2) ÷ 1

⇒ Value = 1 ÷ 1 = 1

∴ The correct answer is 1.

Additional Information

Complementary Angles

For any angle θ, sin θ = cos(90° − θ). Therefore, sin 30° = cos(90° − 30°) = cos 60°.

Reciprocal Identities

cot θ is the reciprocal of tan θ. Since tan 45° = 1, cot 45° = 1/1 = 1.

Pythagorean Identity

sin²θ + cos²θ = 1 is the fundamental identity for all trigonometric calculations.

Shortcut Trick

Total sum of 8 numbers = 121 × 8 = 968.

Sum of 7 numbers = 35 + 35 + 49 + 83 + 102 + 80 + 20 = 404.

Eighth number = Total Sum − Sum of 7 numbers.

8^th number = 968 − 404 = 564.

∴ The correct answer is 564.

Alternate Method

Given:

Average of 8 numbers = 121

Value of 7 numbers = 35, 35, 49, 83, 102, 80, 20

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Sum of 8 numbers = 121 × 8 = 968

⇒ Sum of 7 numbers = 35 + 35 + 49 + 83 + 102 + 80 + 20

⇒ Sum of 7 numbers = 404

⇒ Let the eighth number be x.

⇒ Sum of 8 numbers = (Sum of 7 numbers) + x

⇒ 968 = 404 + x

⇒ x = 968 − 404

⇒ x = 564

∴ The correct answer is 564.

Additional Information

Arithmetic Mean

The average or arithmetic mean is calculated by adding all the values and dividing by the count of values: Average = Σx_i ÷ n.

Sum of Observations

Total sum can always be found if the average and total count are known: Sum = Average × Count.

Deviation Method

For larger numbers, the sum of deviations of all observations from the average is always zero.

Shortcut Trick

Multiply initial ratio parts by their respective percentage growth factors.

New Ratio = (4 × 190) : (18 × 190) : (21 × 140)

Simplify: (4 × 19) : (18 × 19) : (21 × 14) ⇒ 76 : 342 : 294

Divide by 2: 38 : 171 : 147.

∴ The correct answer is 38:171:147.

Alternate Method

Given: Initial ratio of salaries X : Y : Z = 4 : 18 : 21. Let salaries be 400, 1800, and 2100.

Formula: New Value = Original Value × (100 + Increment %)/100

⇒ New Salary of X = 400 + (90% of 400) = 400 + 360 = 760

⇒ New Salary of Y = 1800 + (90% of 1800) = 1800 + 1620 = 3420

⇒ New Salary of Z = 2100 + (40% of 2100) = 2100 + 840 = 2940

⇒ New Ratio = 760 : 3420 : 2940

⇒ New Ratio = 76 : 342 : 294 (Dividing by 10)

⇒ New Ratio = 38 : 171 : 147 (Dividing by 2)

∴ The correct answer is 38:171:147.

Additional Information

Compound Ratio

The ratio obtained by multiplying the corresponding terms of two or more ratios (a:b and c:d becomes ac:bd).

Percentage Increment

When a value increases by r%, the new value is (100 + r)/100 times the original value.

Ratio Simplification

A ratio remains unchanged if all its terms are multiplied or divided by the same non-zero constant.

Shortcut Trick

Assume a base mean of 50 and calculate deviations: (−15, +3, −12, +39, +21, +19, +36, −10, −9).

Net sum of deviations = 72.

Average deviation = 72 ÷ 9 = 8.

Actual Arithmetic Mean = 50 + 8 = 58.

∴ The correct answer is 58.

Alternate Method

Given: Observations are 35, 53, 38, 89, 71, 69, 86, 40, and 41.

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

Step 1: Calculate the Sum of Observations

⇒ Sum = 35 + 53 + 38 + 89 + 71 + 69 + 86 + 40 + 41

⇒ Sum = 522

Step 2: Identify Total Number of Observations

⇒ n = 9

Step 3: Calculate the Mean

⇒ Mean = 522 ÷ 9

⇒ Mean = 58

∴ The correct answer is 58.

Additional Information

Arithmetic Mean Property

The sum of the deviations of all observations from their arithmetic mean is always zero. If each observation is increased by a constant k, the mean also increases by k.

Sum of Observations Formula

Total Sum = Average × Total Number of Terms. This is used to find missing values in a data set when the mean is known.

Assumed Mean Method

Useful for large numbers: Actual Mean = A + (∑d ÷ n), where A is the assumed mean and d is the deviation (Value − A).