Quantitative Aptitude PYQs

Shortcut Trick

For decimal square roots, halve the number of decimal places directly.

⇒ √144 = 12

⇒ √0.0225 = 0.15 (4 decimal places → 2 decimal places)

⇒ √6.25 = 2.5 (2 decimal places → 1 decimal place)

⇒ 12 + 0.15 − 2.5 = 12.15 − 2.5 = 9.65

∴ The correct answer is 9.65.

Alternate Method

Given:

√144 + √0.0225 − √6.25

Calculation:

⇒ Convert the decimals to fractions to avoid errors while finding square roots:

⇒ √144 + √(225/10000) − √(625/100)

⇒ 12 + (15/100) − (25/10)

⇒ 12 + 0.15 − 2.5

⇒ 12.15 − 2.5

⇒ 9.65

∴ The correct answer is 9.65.

Additional Information

Square Roots of Decimals

When finding the square root of a decimal number, the number of decimal places in the square root is exactly half the number of decimal places in the original perfect square number (e.g., √0.0001 = 0.01).

BODMAS Rule

Always follow the correct order of operations: Brackets, Orders (roots/powers), Division/Multiplication, and Addition/Subtraction when simplifying basic mathematical expressions.

Common Squares

Memorizing common squares up to 30 helps in faster calculation: 12² = 144, 15² = 225, 25² = 625.

Shortcut Trick

Let the two consecutive even numbers be (x − 1) and (x + 1), where x is the odd number exactly between them.

Sum of their squares: (x − 1)² + (x + 1)² = 1684

⇒ 2(x² + 1) = 1684

⇒ x² + 1 = 842 ⇒ x² = 841 ⇒ x = 29

The numbers are 29 − 1 = 28 and 29 + 1 = 30. Their sum is 28 + 30 = 58.

∴ The correct answer is 58.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 1684.

Method Used:

Let the two consecutive even natural numbers be x and (x + 2).

Calculations:

⇒ x² + (x + 2)² = 1684

⇒ x² + x² + 4x + 4 = 1684

⇒ 2x² + 4x − 1680 = 0

⇒ x² + 2x − 840 = 0

⇒ x² + 30x − 28x − 840 = 0

⇒ x(x + 30) − 28(x + 30) = 0

⇒ (x − 28)(x + 30) = 0

⇒ x = 28 (ignoring the negative value −30 as x must be a natural number)

⇒ The second consecutive even number = 28 + 2 = 30

⇒ Sum of the numbers = 28 + 30 = 58

∴ The correct answer is 58.

Additional Information

Consecutive Numbers

Consecutive even or odd numbers always have a difference of 2. They are commonly represented algebraically as x, x + 2, x + 4, etc.

Quadratic Equations

For a standard quadratic equation ax² + bx + c = 0, the nature of roots is determined by the discriminant (b² − 4ac). The sum of roots is −b ÷ a and the product of roots is c ÷ a.

Algebraic Identities

A highly useful identity for solving consecutive number differences symmetrically is: (a − b)² + (a + b)² = 2(a² + b²).

Shortcut Trick

When Simple Interest and Time are equal for two investments, the principal is inversely proportional to the rate of interest.

Ratio of investments = P₁ : P₂ = R₂ : R₁ = 11 : 14

Total ratio parts = 11 + 14 = 25 parts

25 parts = ₹21,300 ⇒ 1 part = 21300 ÷ 25 = ₹852

Sum invested at 14% (P₁) = 11 parts = 11 × 852 = 9372

∴ The correct answer is ₹9,372.

Alternate Method

Given:

Total investment = ₹21,300

Rate 1 (R₁) = 14% p.a., Time 1 (T₁) = 3 years

Rate 2 (R₂) = 11% p.a., Time 2 (T₂) = 3 years

Simple Interest 1 (SI₁) = Simple Interest 2 (SI₂)

Formula Used:

SI = P × R × T ÷ 100

Total = ₹21,300

Part 1 (x)
14% p.a. | 3 Years

Part 2 (21300 − x)
11% p.a. | 3 Years

SI₁ = SI₂

Calculations:

Let the sum invested at 14% p.a. be x.

Then the sum invested at 11% p.a. will be (21300 − x).

According to the condition that the interest from both investments is equal:

⇒ (x × 14 × 3) ÷ 100 = ((21300 − x) × 11 × 3) ÷ 100

Canceling out 3 and 100 from both sides:

⇒ 14x = 11 × (21300 − x)

⇒ 14x = 234300 − 11x

⇒ 14x + 11x = 234300

⇒ 25x = 234300

⇒ x = 234300 ÷ 25

⇒ x = 9372

∴ The correct answer is ₹9,372.

Additional Information

Equal Simple Interest Rule

When a total sum is divided into multiple parts such that their Simple Interests are strictly equal, the ratio of the invested parts is P₁ : P₂ : P₃ = (1 ÷ R₁T₁) : (1 ÷ R₂T₂) : (1 ÷ R₃T₃).

Equal Amount Rule

If the final amounts (Principal + SI) obtained from multiple parts are equal, the ratio of the invested parts is P₁ : P₂ = 1 ÷ (100 + R₁T₁) : 1 ÷ (100 + R₂T₂).

Difference between SI and CI

For 2 years, the absolute difference between CI and SI on a certain principal sum is given by D = P(R ÷ 100)². For 3 years, the difference formula extends to D = P(R ÷ 100)² × ((300 + R) ÷ 100).

Shortcut Trick

The average of the first 'n' even numbers is directly given by the formula (n + 1).

Here, the number of terms n = 161.

⇒ Average = 161 + 1 = 162

∴ The correct answer is 162.

Alternate Method

Given:

Number of terms (n) = 161

First even number (a) = 2

The 161^st even number (l) = 161 × 2 = 322

First 161 Even Numbers in Arithmetic Progression

2

4

6

. . .

162

. . .

318

320

322

Average = Central Value = 162

Formula Used:

In an Arithmetic Progression, Average = (First term + Last term) ÷ 2

Calculations:

⇒ Average = (2 + 322) ÷ 2

⇒ Average = 324 ÷ 2

⇒ Average = 162

∴ The correct answer is 162.

Additional Information

Average of First 'n' Natural Numbers

The average of the first n natural numbers is given by (n + 1) ÷ 2.

Average of First 'n' Odd Numbers

The average of the first n consecutive odd numbers is strictly equal to n.

Average in an Arithmetic Progression

If the given terms are in A.P., their average is strictly the central value, which equals the average of the first and the last term.

Shortcut Trick

Profit of M = (6915 + 461) ÷ 2 = ₹3688 and Profit of N = 6915 − 3688 = ₹3227.

Profit ratio (M : N) = 3688 : 3227 = 8 : 7.

Capital ratio = (Profit of M ÷ Time of M) : (Profit of N ÷ Time of N) = (8 ÷ 2) : (7 ÷ 3) = 12 : 7.

Difference in capital ratio = 12 − 7 = 5 units = ₹38,000 ⇒ 1 unit = ₹7,600.

Capital contributed by M = 12 units = 12 × 7600 = ₹91,200.

∴ The correct answer is ₹91,200.

Alternate Method

Given:

Total profit = ₹6,915

M's profit − N's profit = ₹461

Time period of M = 2 months, Time period of N = 3 months

M's capital is ₹38,000 more than N's capital.

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Partner M
Time = 2 months
Capital = x + 38000
Profit = ₹3688

Partner N
Time = 3 months
Capital = x
Profit = ₹3227

Capital × Time ∝ Profit

Calculation:

⇒ Let M's profit be P_m and N's profit be P_n.

⇒ P_m + P_n = 6915

⇒ P_m − P_n = 461

⇒ Adding both equations: 2P_m = 7376 ⇒ P_m = 3688

⇒ P_n = 6915 − 3688 = 3227

⇒ Ratio of their profits = 3688 : 3227

⇒ Dividing by their common factor (461): Profit ratio = 8 : 7

⇒ Let N's capital be x. Then M's capital will be (x + 38000).

⇒ [2 × (x + 38000)] ÷ (3 × x) = 8 ÷ 7

⇒ 14 × (x + 38000) = 24x

⇒ 14x + 532000 = 24x

⇒ 10x = 532000

⇒ x = 53200

⇒ M's capital = x + 38000 = 53200 + 38000 = 91200

∴ The correct answer is ₹91,200.

Additional Information

Simple vs Compound Partnership

In simple partnership, capitals are invested for the same time period, so profit is distributed strictly in the ratio of their investments. In compound partnership, investments are made for different time periods, so profit ratio = (C₁ × T₁) : (C₂ × T₂).

Working vs Sleeping Partner

A working partner manages the business and usually receives a fixed salary or percentage of the profit before the remaining profit is distributed. A sleeping partner only invests capital and receives a share from the distributable profit based on their investment ratio.

Finding Capital from Profit and Time

If the profit ratio and time ratio of partners are given, the ratio of their invested capitals can be found by directly dividing the profit ratio by the time ratio: Capital Ratio = (P₁ ÷ T₁) : (P₂ ÷ T₂).

Shortcut Trick

Directly multiply the terms and combine them mentally.

For x² terms: x(2x) + 5(x²) = 2x² + 5x² = 7x²

For x terms: x(−8) = −8x

For constant terms: 5(−3) + 12 = −15 + 12 = −3

Combine all parts to get the final expression: 7x² − 8x − 3.

∴ The correct answer is 7x² − 8x − 3.

Alternate Method

Given:

x(2x − 8) + 5(x² − 3) + 12

Calculations:

⇒ Expand the first term using the distributive property:

⇒ x × 2x − x × 8 = 2x² − 8x

⇒ Expand the second term:

⇒ 5 × x² − 5 × 3 = 5x² − 15

⇒ Substitute these expanded forms back into the given expression:

⇒ (2x² − 8x) + (5x² − 15) + 12

⇒ Group the like terms together:

⇒ (2x² + 5x²) − 8x + (−15 + 12)

⇒ Add the coefficients of the grouped terms to simplify:

⇒ 7x² − 8x − 3

∴ The correct answer is 7x² − 8x − 3.

Additional Information

Distributive Property

Multiplying a sum by a number is exactly the same as multiplying each addend by the number and then adding the products: a(b + c) = ab + ac.

Combining Like Terms

In algebraic simplification, terms with the same variables and powers are grouped, and their coefficients are added or subtracted together.

BODMAS Rule

This rule determines the correct order of operations in calculations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Shortcut Trick

Total time in days = 3 (Feb) + 31 (Mar) + 27 (Apr) = 61 days.

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

SI = (P × R × T) ÷ 100

SI = (4000 × 6.75 × 1/6) ÷ 100

SI = 270 ÷ 6 = 45

∴ The correct answer is ₹45.

Alternate Method

Given:

Principal (P) = ₹4,000

Rate of Interest (R) = 6.75% per annum

Deposited on = 26 February 2024

Withdrawn on = 27 April 2024

Formula Used:

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

26 Feb 2024
Deposit
₹4,000

27 Apr 2024
Withdrawal
₹4,045

T = 61 Days

Rate = 6.75% p.a.
Simple Interest = ₹45

Calculations:

⇒ The year 2024 is divisible by 4, meaning it is a leap year, so February has 29 days.

⇒ While calculating interest duration, the day of deposit is generally excluded, but the day of withdrawal is included.

⇒ Number of days in February = 29 − 26 = 3 days

⇒ Number of days in March = 31 days

⇒ Number of days in April = 27 days

⇒ Total Time (T) = 3 + 31 + 27 = 61 days

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

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

⇒ SI = (4000 × 6.75 × (1/6)) ÷ 100

⇒ SI = 40 × 6.75 × (1/6)

⇒ SI = 270 ÷ 6

⇒ SI = 45

∴ The correct answer is ₹45.

Additional Information

Leap Year Condition

A year divisible by 4 (and by 400 if it is a century year) is a leap year and consists of 366 days. February has 29 days in a leap year.

Counting Days for Interest

When computing simple interest between two dates, only one of the dates (either the deposit date or the withdrawal date) is counted to determine the duration.

Simple vs Compound Interest

Simple Interest (SI) is calculated uniformly only on the base principal amount, whereas Compound Interest (CI) is computed on both the principal and the continuously accumulated interest.

Shortcut Trick

LCM = HCF × a × b (where a, b are co-prime factors).

498 = 83 × a × b ⇒ a × b = 6.

First number = 166 = 83 × 2 ⇒ a = 2.

Thus, b = 6 ÷ 2 = 3.

Second number = HCF × b = 83 × 3 = 249.

∴ The correct answer is 249.

Alternate Method

Given:

HCF = 83

LCM = 498

One number (N₁) = 166

Formula Used:

Product of two numbers = HCF × LCM

N₁ × N₂ = HCF × LCM

Calculation:

⇒ 166 × N₂ = 83 × 498

⇒ N₂ = (83 × 498) ÷ 166

⇒ N₂ = 498 ÷ 2

⇒ N₂ = 249

∴ The correct answer is 249.

Additional Information

Product of Two Numbers Rule

For any two positive integers, the product of the numbers is strictly equal to the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM). This rule applies only to exactly two numbers, not three or more.

Co-prime Factors Property

If two numbers are represented as HCF × x and HCF × y, then x and y must be co-prime to each other. Their LCM is obtained directly using the formula: LCM = HCF × x × y.

HCF and LCM Dependency

The HCF of a set of numbers is always a perfect divisor of their LCM. If the given LCM is not perfectly divisible by the given HCF, no such numbers can mathematically exist.

Shortcut Trick

Average speed is calculated as the ratio of total distance to total time.

Total Distance = 200 + 210 + 204 = 614 km

Total Time = (200 ÷ 25) + (210 ÷ 21) + (204 ÷ 51) = 8 + 10 + 4 = 22 hours

Average Speed = 614 ÷ 22 = 27.909... ≈ 27.91 km/hr

∴ The correct answer is 27.91 km/hr.

Alternate Method

Given:

D₁ = 200 km, S₁ = 25 km/hr

D₂ = 210 km, S₂ = 21 km/hr

D₃ = 204 km, S₃ = 51 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

200 km
210 km
204 km

25 km/hr
21 km/hr
51 km/hr

8 hr

10 hr

4 hr

Total Distance = 614 km

Calculations:

⇒ Total Distance = 200 + 210 + 204 = 614 km

⇒ T₁ = 200 ÷ 25 = 8 hours

⇒ T₂ = 210 ÷ 21 = 10 hours

⇒ T₃ = 204 ÷ 51 = 4 hours

⇒ Total Time = 8 + 10 + 4 = 22 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ Average Speed = 614 ÷ 22

⇒ Average Speed = 307 ÷ 11

⇒ Average Speed = 27.909... ≈ 27.91 km/hr

∴ The correct answer is 27.91 km/hr.

Additional Information

Average Speed for Two Equal Distances

If a person travels two equal distances at speeds x and y, the average speed is directly given by (2 × x × y) ÷ (x + y).

Average Speed for Three Equal Distances

If a person travels three equal distances at speeds x, y, and z, the average speed is directly given by (3 × x × y × z) ÷ (x×y + y×z + z×x).

Unit Conversion

To convert km/hr to m/s, multiply the given speed by 5 ÷ 18. To convert m/s to km/hr, multiply by 18 ÷ 5.

Shortcut Trick

Between 468 and 476, ignore even numbers and multiples of 5.

Possible prime candidates are 469, 471, and 473.

469 = 7 × 67, 471 is divisible by 3 (sum of digits = 12), and 473 is divisible by 11 (4 − 7 + 3 = 0).

Since all candidates are composite, there are no prime numbers.

∴ The correct answer is 0.

Alternate Method

Given:

Range of numbers: 468 to 476 (both included).

Method Used:

List the numbers and eliminate composite numbers using basic divisibility rules.

Calculations:

⇒ The numbers are: 468, 469, 470, 471, 472, 473, 474, 475, 476.

⇒ Eliminate all even numbers (divisible by 2): 468, 470, 472, 474, 476.

⇒ Eliminate numbers ending in 5 (divisible by 5): 475.

⇒ The remaining numbers to check are 469, 471, and 473.

⇒ For 469: It is divisible by 7 (469 = 7 × 67).

⇒ For 471: The sum of digits is 4 + 7 + 1 = 12. Since 12 is divisible by 3, 471 is divisible by 3.

⇒ For 473: The difference of alternate digits sum is (4 + 3) − 7 = 0. So, it is divisible by 11 (473 = 11 × 43).

⇒ Thus, none of the numbers between 468 and 476 are prime.

∴ The correct answer is 0.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Divisibility Rule of 3

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

Divisibility Rule of 11

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

Checking Primality up to N

To check if a number N is prime, you only need to test divisibility by prime numbers up to √N.

Shortcut Trick

Relative Speed = Total distance ÷ Time taken

Total distance = 282 + 248 = 530 m

Relative Speed (in m/s) = 530 ÷ 12 = 265/6 m/s

Relative Speed (in km/hr) = (265/6) × (18/5) = 53 × 3 = 159 km/hr

Speed of second train = 159 − 68 = 91 km/hr

∴ The correct answer is 91 km/hr.

Alternate Method

Given:

Length of first train (L₁) = 282 m

Length of second train (L₂) = 248 m

Time taken (t) = 12 seconds

Speed of first train (S₁) = 68 km/hr

Formula Used:

Relative Speed = (L₁ + L₂) / Time

Conversion: km/hr to m/s = Multiply by 5/18

Calculations:

⇒ Let the speed of the second train be S₂ km/hr.

⇒ Relative speed = (68 + S₂) km/hr

⇒ (68 + S₂) × (5/18) = (282 + 248) / 12

⇒ (68 + S₂) × (5/18) = 530 / 12

⇒ 68 + S₂ = (530 / 12) × (18 / 5)

⇒ 68 + S₂ = (106) × (1.5)

⇒ 68 + S₂ = 159

⇒ S₂ = 159 − 68

⇒ S₂ = 91 km/hr

∴ The correct answer is 91 km/hr.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: S_rel = S₁ + S₂.

Same Direction Relative Speed

When two objects move in the same direction, their relative speed is the difference between their speeds: S_rel = |S₁ − S₂|.

Train Crossing Objects

To cross a stationary pole or man, a train covers its own length. To cross another train or platform, it covers the sum of both lengths.

Shortcut Trick

Total days from 5 February 2023 to 19 April 2023 = 23 (Feb) + 31 (Mar) + 19 (Apr) = 73 days.

Time (T) in years = 73 ÷ 365 = 1/5 year.

Simple Interest = (Principal × Rate × Time) ÷ 100

SI = (2000 × 7 × 1/5) ÷ 100 = 4 × 7 = 28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate of Interest (R) = 7% per annum

Time (T) = 5 February 2023 to 19 April 2023

Formula Used:

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

Calculations:

⇒ Calculation of days: Feb (28 − 5) = 23, March = 31, April = 19

⇒ Total days = 23 + 31 + 19 = 73 days

⇒ T (in years) = 73 ÷ 365 = 1/5 year

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (2000 × 7) ÷ 500

⇒ SI = 4 × 7 = ₹28

∴ The correct answer is ₹28.

Additional Information

Simple Interest Formula

The basic formula for SI is (P × R × T) ÷ 100, where P is principal, R is rate per annum, and T is time in years.

Time Conversion for Interest

When time is given in days, divide by 365 (ordinary year) or 366 (leap year). 2023 is not a leap year, so we use 365.

Counting Days Rule

Standard banking practice excludes the day of deposit but includes the day of withdrawal (or vice versa).

Amount Calculation

The final Amount (A) is the sum of the Principal and the Interest: A = P + SI.

Shortcut Trick

If (m − n) is even, then both m and n must be either both even or both odd.

Case 1: Let m = 3 and n = 1 (both odd) ⇒ m² − n² = 9 − 1 = 8 (divisible by 4).

Case 2: Let m = 4 and n = 2 (both even) ⇒ m² − n² = 16 − 4 = 12 (divisible by 4).

In both cases, the result is divisible by 4.

∴ The correct answer is 4.

Alternate Method

Given: m and n are positive integers and (m − n) is even.

Formula Used: m² − n² = (m − n)(m + n)

⇒ If (m − n) is even, then the difference between m and n is a multiple of 2.

⇒ This implies that m and n have the same parity (both even or both odd).

⇒ If both are even or both are odd, their sum (m + n) must also be even.

⇒ m² − n² = (m − n) × (m + n)

⇒ Since both factors are even, let (m − n) = 2a and (m + n) = 2b.

⇒ m² − n² = 2a × 2b = 4ab.

⇒ Therefore, the expression is always divisible by 4.

∴ The correct answer is 4.

Additional Information

Parity Rules

Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd.

Difference of Squares

For any integers, m² − n² is divisible by (m − n) and (m + n).

Divisibility of Consecutive Squares

The difference between squares of two consecutive odd integers is always divisible by 8.

Shortcut Trick

In a right-angled triangle, if tan A = 3/4, the sides represent a Pythagorean triplet (3, 4, 5).

Here, Perpendicular (P) = 3 and Base (B) = 4.

The Hypotenuse (H) is always 5 for this triplet.

Since cosec A = Hypotenuse / Perpendicular, we get 5/3.

∴ The correct answer is 5/3.

Alternate Method

Given: tan A = 3/4 in ΔABC right-angled at B.

Formula Used: tan A = Perpendicular / Base; cosec A = Hypotenuse / Perpendicular; H² = P² + B²

⇒ Let Perpendicular (BC) = 3k and Base (AB) = 4k.

⇒ AC² = AB² + BC²

⇒ AC² = (4k)² + (3k)² = 16k² + 9k² = 25k²

⇒ AC = √(25k²) = 5k

⇒ cosec A = AC / BC = 5k / 3k

⇒ cosec A = 5/3

∴ The correct answer is 5/3.

Additional Information

Trigonometric Ratios

Standard ratios: sin θ = P/H, cos θ = B/H, tan θ = P/B, cosec θ = H/P, sec θ = H/B, cot θ = B/P.

Pythagorean Triplets

Common sets of integers (a, b, c) such that a² + b² = c². Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

Reciprocal Identities

cosec A = 1 / sin A, sec A = 1 / cos A, cot A = 1 / tan A.

Shortcut Trick

The mode is the value that appears most frequently in a data set.

By scanning the given list: 1 appears 2 times, 3 appears 3 times, 5 appears 2 times, 9 appears 2 times, and 6 appears 5 times.

Since 6 has the highest frequency, it is the mode.

∴ The correct answer is 6.

Alternate Method

Given: Observations = {9, 6, 8, 1, 9, 6, 4, 5, 7, 1, 6, 5, 3, 6, 6, 3, 3}

Formula Used: Mode = Observation with the maximum frequency.

Frequency Distribution of the data:

⇒ 1 occurs 2 times

⇒ 3 occurs 3 times

⇒ 4 occurs 1 time

⇒ 5 occurs 2 times

⇒ 6 occurs 5 times

⇒ 7 occurs 1 time

⇒ 8 occurs 1 time

⇒ 9 occurs 2 times

⇒ Maximum Frequency = 5

⇒ The observation corresponding to the maximum frequency is 6.

∴ The correct answer is 6.

Additional Information

Types of Mode

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

Empirical Formula

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

Median for Discrete Data

To find the median, arrange the data in ascending order. If N is odd, Median = ((N+1)/2)^th term. If N is even, Median is the average of (N/2)^th and (N/2 + 1)^th terms.

Shortcut Trick

Total amount to be deducted = ₹(62 + 17 + 79) = ₹158.

Remaining amount = ₹10,415 − ₹158 = ₹10,257.

Sum of ratio parts = 15 + 10 + 14 = 39 units.

1 unit = ₹10,257 ÷ 39 = ₹263.

Difference between Q and R (in ratio) = 14 − 10 = 4 units.

Difference in original shares = (4 × 263) + (79 − 17) = 1052 + 62 = ₹1,114.

∴ The correct answer is ₹1,114.

Alternate Method

Given: Total Sum = ₹10,415; Ratio after deductions = 15 : 10 : 14.

Formula: Original Share = (Ratio part × common factor) + deducted amount.

⇒ Total reduction = 62 + 17 + 79 = 158

⇒ New total sum = 10415 − 158 = 10257

⇒ 15x + 10x + 14x = 10257

⇒ 39x = 10257

⇒ x = 10257 ÷ 39 = 263

⇒ Original share of Q = (10 × 263) + 17 = 2630 + 17 = 2647

⇒ Original share of R = (14 × 263) + 79 = 3682 + 79 = 3761

⇒ Difference = 3761 − 2647 = 1114

∴ The correct answer is ₹1,114.

Additional Information

Ratio & Proportion Basics

If a total sum S is divided into a:b:c, then share of A = (a / (a+b+c)) × S.

Inverse Variation

If x varies inversely as y, then x × y = k (constant). In ratios, if a:b is the ratio of quantities, the inverse ratio is 1/a : 1/b.

Compound Ratio

The compound ratio of (a : b) and (c : d) is (ac : bd).

Shortcut Trick

The product of the number of sides n and (n − 3) is always twice the number of diagonals.

n × (n − 3) = 2 × 665 = 1330

Now, find a value of n such that n × (n − 3) = 1330.

By checking the options, if n = 38, then 38 × (38 − 3) = 38 × 35 = 1330.

This satisfies the condition perfectly.

∴ The correct answer is 38.

Alternate Method

Given:

Number of diagonals of the regular polygon = 665

Formula Used:

Number of diagonals of an n-sided polygon = [n(n − 3)] / 2

⇒ 665 = [n(n − 3)] / 2

⇒ 1330 = n² − 3n

⇒ n² − 3n − 1330 = 0

⇒ n² − 38n + 35n − 1330 = 0

⇒ n(n − 38) + 35(n − 38) = 0

⇒ (n − 38)(n + 35) = 0

⇒ n = 38 or n = −35

Since the number of sides n must be a positive integer, we take n = 38.

∴ The correct answer is 38.

Additional Information

Sum of Interior Angles

The sum of all interior angles of an n-sided polygon is (n − 2) × 180°.

Exterior Angles Property

The sum of all exterior angles of any convex polygon is always 360°.

Regular Polygon Angles

For a regular polygon, each interior angle is [(n − 2) × 180°] / n and each exterior angle is 360° / n.

Shortcut Trick

Calculate the semi-perimeter (s) of the triangle: s = (22 + 18 + 24) ÷ 2 = 32 cm.

Triangle Area = √(32 × 10 × 14 × 8) = 32√35 cm².

Since Triangle Area = 1/3 × Parallelogram Area, then Parallelogram Area = 3 × 32√35 = 96√35 cm².

Height (h) = Area ÷ Base = (96 × √35) ÷ 24 = 4 × √35 ≈ 4 × 5.916 = 23.66 cm.

∴ The correct answer is 23.6 cm.

Alternate Method

Given: Sides of triangle (a, b, c) = 22 cm, 18 cm, 24 cm. Base of parallelogram (b_p) = 24 cm.

Formula Used:

Heron's Formula for Triangle Area = √(s(s−a)(s−b)(s−c))

Area of Parallelogram = Base × Height

⇒ Semi-perimeter of triangle (s) = (22 + 18 + 24) ÷ 2 = 32 cm

⇒ Area of triangle = √(32 × (32 − 22) × (32 − 18) × (32 − 24))

⇒ Area of triangle = √(32 × 10 × 14 × 8) = √35840 = 32√35 cm²

⇒ Area of parallelogram = 3 × Area of triangle = 3 × 32√35 = 96√35 cm²

⇒ Base × Height = 96√35

⇒ 24 × h = 96√35

⇒ h = (96 ÷ 24) × √35 = 4√35

⇒ h ≈ 4 × 5.916 = 23.664 cm

∴ The correct answer is 23.6 cm.

Additional Information

Heron's Formula

Used to find the area of any triangle when all three sides are known: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter.

Parallelogram Properties

Area is calculated as Base × Perpendicular Height. The diagonal of a parallelogram divides it into two triangles of equal area.

Relationship between Triangle and Parallelogram

If a triangle and a parallelogram have the same base and height, the area of the triangle is exactly half the area of the parallelogram.

Shortcut Trick

Combined efficiency of P + Q = 1/12 units/hr.

Efficiency of P = 1/24 units/hr.

Efficiency of Q = 1/12 − 1/24 = 1/24 units/hr.

Time for Q to fill 1/4 cistern = (1/4) ÷ (1/24) = 6 hours.

∴ The correct answer is 6 hours.

Alternate Method

⇒ Total capacity of the cistern = 24 units (LCM of 12 and 24).

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

⇒ Efficiency of Q = 2 − 1 = 1 unit/hr.

⇒ Work to be done = 1/4 of total cistern = 24 × (1/4) = 6 units.

⇒ Time taken by Q = 6 ÷ 1 = 6 hours.

∴ The correct answer is 6 hours.

Additional Information

Time and Work Relation

Total Work = Efficiency × Time. Efficiency is inversely proportional to the time taken to complete a task.

Pipes and Cisterns

Filling pipes have positive efficiency, while outlet/leakage pipes have negative efficiency. Net work is the sum of their individual efficiencies.

LCM Method

Using the LCM of the given time durations as the total capacity of the tank simplifies calculations by avoiding fractions.

Shortcut Trick

Ratio 5 years ago (Meera : Kavya) = 3 : 1 (Difference = 2)

Ratio 7 years later (Meera : Kavya) = 2 : 1 (Difference = 1)

Equalize difference: 5 yrs ago = 3 : 1 | 7 yrs later = 4 : 2 (Multiplied by 2)

Ratio change 1 unit → 5 + 7 = 12 years

Meera 5 years ago = 3 × 12 = 36 years

Current age = 36 + 5 = 41 years

∴ The correct answer is 41 years.

Alternate Method

Let Meera and Kavya current ages be M and K respectively.

Given:

5 years ago: (M − 5) = 3(K − 5)

7 years later: (M + 7) = 2(K + 7)

⇒ M − 5 = 3K − 15 ⇒ M − 3K = −10 --- (i)

⇒ M + 7 = 2K + 14 ⇒ M − 2K = 7 --- (ii)

⇒ Subtracting (i) from (ii): (M − 2K) − (M − 3K) = 7 − (−10)

⇒ K = 17 years

⇒ Substitute K in (ii): M − 2(17) = 7

⇒ M − 34 = 7

⇒ M = 41 years

∴ The correct answer is 41 years.

Additional Information

Age Difference Constant

The difference between the ages of two people remains the same regardless of the number of years passed or before.

Ratio Method for Ages

When solving age problems using ratios, always ensure the numerical difference between the ratio parts of the two individuals is made equal across different time periods.

Linear Equation Method

Standard algebraic approach involves setting up equations for "n years ago" (x − n) and "n years later" (x + n).