Quantitative Aptitude PYQs

Shortcut Trick

Rectangle breadth : length = 3 : 4. Area = 3x × 4x = 12x² = 147.

⇒ x² = 12.25 ⇒ x = 3.5. Length (Side of square) = 4 × 3.5 = 14 cm.

Area of square = 14² = 196 cm².

Area of circle = (11/14) × 196 = 154 cm².

Since πr² = 154 and for r = 7, πr² is always 154, the radius is 7 cm.

∴ The correct answer is 7 cm.

Alternate Method

Given: Ratio of breadth to length of rectangle = 3 : 4, Area of rectangle = 147 cm², Side of square = Length of rectangle.

⇒ Let breadth (b) = 3x and length (l) = 4x.

⇒ Area = l × b = 4x × 3x = 12x² = 147

⇒ x² = 147 ÷ 12 = 12.25 ⇒ x = 3.5

⇒ Length of rectangle (l) = 4 × 3.5 = 14 cm.

⇒ Side of square (s) = 14 cm. Area of square = s² = 14 × 14 = 196 cm².

⇒ Area of circle : Area of square = 11 : 14

⇒ Area of circle = (11 ÷ 14) × 196 = 11 × 14 = 154 cm².

⇒ πr² = 154 ⇒ (22 ÷ 7) × r² = 154

⇒ r² = (154 × 7) ÷ 22 = 7 × 7 = 49

⇒ r = √49 = 7 cm.

∴ The correct answer is 7 cm.

Additional Information

Circle Mensuration

Area of circle = πr² and Circumference = 2πr. For r = 7, Area is always 154 and Circumference is 44.

Square and Rectangle

Area of Square = side². Area of Rectangle = length × breadth.

Common Ratios

If the side of a square is 2r, the ratio of the area of the inscribed circle to the square is π : 4 ≈ 11 : 14.

Shortcut Trick

Ratio of Speed (S₁ : S₂) = 60 : (60 − 20) = 60 : 40 = 3 : 2.

Since Distance is constant, Time Ratio is inverse of Speed Ratio = 2 : 3.

Difference in time ratio parts = 3 − 2 = 1 unit → 2 hours.

Original time (2 units) = 2 × 2 = 4 hours.

Distance = Original Speed × Original Time = 60 kmph × 4 hours = 240 km.

∴ The correct answer is 240 km.

Alternate Method

Given:

Initial speed (S₁) = 60 kmph

Reduced speed (S₂) = 60 − 20 = 40 kmph

Time difference (Δt) = 2 hours

Formula Used:

Distance (D) = [S₁ × S₂ ÷ (S₁ − S₂)] × Δt

Calculations:

⇒ D = [60 × 40 ÷ (60 − 40)] × 2

⇒ D = [2400 ÷ 20] × 2

⇒ D = 120 × 2

⇒ D = 240 km

∴ The correct answer is 240 km.

Additional Information

Inversely Proportional Relation

When distance is constant, speed is inversely proportional to time (S ∝ 1/T). This means S₁/S₂ = T₂/T₁.

Average Speed

If a certain distance is covered at speed 'a' and the same distance at speed 'b', the average speed is (2 × a × b) ÷ (a + b).

Relative Speed

When two objects move in the same direction, relative speed = S₁ − S₂. When they move in opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Work done by A and B in 20 days = 20 ÷ 30 = 2/3 part.

Remaining work = 1 − 2/3 = 1/3 part.

A finishes 1/3 part in 15 days.

A alone can finish full work in = 15 × 3 = 45 days.

∴ The correct answer is 45 days.

Alternate Method

⇒ Total Work = 30 units (assumed LCM based on given combined days).

⇒ Combined efficiency of (A + B) = 1 unit/day.

⇒ Work done by (A + B) in 20 days = 1 × 20 = 20 units.

⇒ Remaining Work = 30 − 20 = 10 units.

⇒ Efficiency of A = Remaining Work ÷ Days taken = 10 ÷ 15 = 2/3 unit/day.

⇒ Time taken by A alone = Total Work ÷ Efficiency of A = 30 ÷ (2/3) = 30 × (3/2).

⇒ Time taken by A = 45 days.

∴ The correct answer is 45 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to time. If efficiency ratio is x:y, then time ratio is y:x for the same work.

Work Equation

Total Work = Efficiency × Time. This core formula allows finding any unknown when two variables are known.

Man-Days Concept

M₁D₁ = M₂D₂ can be used when the same group of people works with constant efficiency.

Shortcut Trick

Ratio of final quantity to initial quantity of milk = (1 − x/60)ⁿ

48.6 ÷ 60 = (1 − x/60)²

0.81 = (1 − x/60)² ⇒ √0.81 = 1 − x/60

0.9 = 1 − x/60 ⇒ x/60 = 0.1

x = 6 litres

∴ The correct answer is 6 litres.

Alternate Method

Given:

Initial quantity (C) = 60 litres

Final quantity (F) = 48.6 litres

Number of operations (n) = 2

Formula Used:

F = C × (1 − x/C)ⁿ

Calculations:

⇒ 48.6 = 60 × (1 − x/60)²

⇒ 48.6 ÷ 60 = (1 − x/60)²

⇒ 486 ÷ 600 = (1 − x/60)²

⇒ 81 ÷ 100 = (1 − x/60)²

⇒ √(81/100) = 1 − x/60

⇒ 9/10 = 1 − x/60

⇒ x/60 = 1 − 0.9 = 0.1

⇒ x = 60 × 0.1 = 6 litres

∴ The correct answer is 6 litres.

Additional Information

Removal and Replacement Formula

The formula for final quantity is F = C(1 − x/C)ⁿ where C is initial capacity, x is quantity replaced, and n is iterations.

Ratio Method

If 1/k portion is removed, the remaining milk ratio after n times is (1 − 1/k)ⁿ. For n=2, if the ratio is 81/100, the single step ratio is 9/10.

Successive Percentage Change

This process is an application of successive percentage decrease where the value decreases by (x/C × 100)% each time.

Shortcut Trick

Given ratio of Nitin and Mayuri ages = 5 : 6.

Checking the second condition: (1/3 × 5x) : (1/2 × 6x) = (5x/3) : 3x = 5 : 9.

Since the second condition is derived directly from the first ratio, the equations are dependent.

We have two variables but only one independent equation.

∴ The correct answer is Can't be determined.

Alternate Method

Given: Ratio of ages of Nitin (N) and Mayuri (M) = 5 : 6

⇒ Let N = 5x and M = 6x

Condition: (1/3 × N) : (1/2 × M) = 5 : 9

⇒ (5x/3) / (6x/2) = 5 / 9

⇒ (5x/3) / 3x = 5 / 9

⇒ 5x / 9x = 5 / 9

⇒ 5/9 = 5/9

The variable x cancels out, meaning no specific value for x can be found.

∴ The correct answer is Can't be determined.

Additional Information

Consistency of Equations

To solve for n variables, we require n independent linear equations. If equations are dependent, infinite solutions exist.

Ratio and Proportion

A ratio alone does not provide absolute values unless at least one absolute quantity or a sum/difference is provided.

Let the number of 'international' videos downloaded from each website 'A', 'B', and 'E' = x 

Total number of 'regional' videos download from websites 'A', 'B', and 'E' together = 8 + 17 + 10 = 35

Total number of 'international' videos download from websites 'A', 'B', and 'E' together = 3x

Total number of videos 'downloaded' from websites 'A', 'B', and 'E' together = 335

Therefore, 35 + 3x = 335

⇒ 3x = 300

⇒ x = 100

The average number of 'international' videos downloaded from each of the given websites = x = 100

Hence, the correct answer is 100.

Number of 'regional' videos downloaded from website 'F' = (17/16) × 16 = 17

Number of 'international' videos downloaded from website 'F' = 17 + 326 = 343

Let the total number of 'viewed' videos on website 'F' = 100x

Total number of 'videos' downloaded from the website 'F' = 30% × 100x = 30x

ATQ, 30x = 17 + 343

⇒ 100x = 1200

The total number of 'viewed' videos on website 'F' = 1200

Total data consumed by person 'X' if he views all the 'videos' of website 'F = 1200 × 250 = 300000 MB = 300 GB

Hence, the correct answer is 300 GB.

Let the total number of 'viewed' videos on website 'C' = 100x

Total number of 'downloaded' videos from website 'C' = 72% × 100x = 72x

Difference between the total number of 'viewed' videos on website 'C' and the number of 'international' videos downloaded from the same website is 187.

Therefore, the number of 'international' videos downloaded from website 'C' = 100x - 187

Number of 'regional' videos downloaded from website 'C' = 72x - (100x -187) = 187 - 28x 

ATQ, 187 - 28x = 19

⇒ x = 6

Total number of 'downloaded' videos from website 'C' = 72x = 432

Number of 'videos' from website 'C' which were stored in the hard disk = 432 × 75% = 324

Hence, the correct answer is 324.

Total number of 'international' videos downloaded from website 'B' = 153

Total number of 'videos' downloaded from website 'B' = 153 + 17 = 170

Let the total number of 'viewed' videos on website 'B' = 100x

ATQ, 40x = 170

⇒ 100x = 425

Total number of 'viewed' videos on website 'B' = 425

Similarly, total number of 'videos' downloaded from website 'D' = 164 + 16 = 180

Let the total number of 'viewed' videos on website 'D' = 100y

ATQ, 60y = 180

⇒ 100y = 300

Total number of 'viewed' videos on website 'D' = 300

Total number of 'viewed' videos on website 'B' and 'D' together = 425 + 300 = 725

Total number of 'regional' videos downloaded from websites 'A', 'B', 'C' and 'D' together = 8 + 17 + 19 + 16 = 60

Required ratio = 725 : 60 = 145 : 12

Hence, the correct answer is 145 : 12.

Let the total number of 'viewed' videos on the website 'E' = 100x

Total number of 'viewed' videos on website 'A' = 250% × 100x = 250x

Total number of downloaded videos from website 'E' = 50% × 100x = 50x

Total number of downloaded videos from website 'A' = 250x × 30% = 75x

Total number of videos downloaded from websites 'A' and 'E' together = 250

ATQ, 50x + 75x = 250

⇒ x = 2

Total number of 'international' videos downloaded from website 'E' = 50x - 10 = 90

Total number of 'international' videos downloaded from website 'A' = 75x - 8 = 142

Total number of 'international' videos downloaded from websites 'A' and 'E' together = 90 + 142 = 232

Hence, the correct answer is 232.

Considering the given series

12, 13, 28, 88, 352, 1765

The logic of the given series can be explained as

12 × 1 + 1 = 13

13 × 2 + 2 = 28

28 × 3 + 3 = 87

87 × 4 + 4= 352

352 × 5 + 5 = 1765

∴ The wrong term in the given number series is 88.

16 × 1 + 1² = 17

17 × 2 + 2² = 38

38 × 3 + 3² = 123

123 × 4 + 4² = 508

508 × 5 + 5² = 2565

∴ Wrong term in given number series is 510.

Calculation:

The given number series follows this pattern:

2 × 1 = 2

2 × 2 = 4

4 × 3 = 12

12 × 4 = 48

48 × 5 = 240

240 × 6 = 1440

∴ The wrong number in the given series is 200.

The series follows the following pattern:

11

11 + 14 = 25

25 + 28 = 53 ≠ 63

53 + 56 = 109

109 + 112 = 221

221 + 224 = 445

∴ The wrong number is 63

11, 25, 53, 109, 221, 445

    14, 28, 56, 112, 224

      × 2   × 2   × 2   × 2

∴ The wrong number is 675

Shortcut Trick

Cumulative rate for 3 years = 7% + (7+2)% + (9+2)% = 7% + 9% + 11% = 27%

Total interest = 27% of Principal = ₹10,260

1% of Principal = 10,260 ÷ 27 = ₹380

Principal (100%) = 380 × 100 = ₹38,000

∴ The correct answer is ₹38,000.

Alternate Method

Given:

Rate for 1^st year (R₁) = 7%

Rate for 2^nd year (R₂) = 7 + 2 = 9%

Rate for 3^rd year (R₃) = 9 + 2 = 11%

Total Simple Interest (SI) = ₹10,260

Formula Used:

Total SI = P × (R₁ + R₂ + R₃) ÷ 100

Calculations:

⇒ 10,260 = P × (7 + 9 + 11) ÷ 100

⇒ 10,260 = P × (27 ÷ 100)

⇒ P = (10,260 × 100) ÷ 27

⇒ P = 380 × 100

⇒ P = 38,000

∴ The correct answer is ₹38,000.

Additional Information

Variable Simple Interest

When interest rates change annually, total SI is calculated by summing individual interests: SI = (P×R₁×T₁)/100 + (P×R₂×T₂)/100 + ...

Effective Rate of Simple Interest

The overall percentage interest is simply the arithmetic sum of rates if the principal remains constant over equal time periods.

Simple Interest vs Amount

Amount (A) = Principal (P) + Simple Interest (SI). In SI, interest is only calculated on the initial principal every period.

Shortcut Trick

Let current speed be v. Difference in time = [Distance ÷ (u − v)] − [Distance ÷ (u + v)] = 10.

200 ÷ (48 − v) − 200 ÷ (48 + v) = 10 ⇒ 20 ÷ (48 − v) − 20 ÷ (48 + v) = 1.

By Option Verification, if v = 32:

20 ÷ (48 − 32) − 20 ÷ (48 + 32) = 20 ÷ 16 − 20 ÷ 80 = 1.25 − 0.25 = 1. (Satisfied)

∴ The correct answer is 32 m/min.

Alternate Method

Given: Speed of man (u) = 48 m/min, Distance (D) = 200 m, Time difference (Δt) = 10 min.

Formula Used: Time = Distance ÷ Speed; Upstream speed = u − v; Downstream speed = u + v.

⇒ 200 ÷ (48 − v) − 200 ÷ (48 + v) = 10

⇒ 200 × [(48 + v − (48 − v)) ÷ (48² − v²)] = 10

⇒ 20 × [2v ÷ (2304 − v²)] = 1

⇒ 40v = 2304 − v²

⇒ v² + 40v − 2304 = 0

⇒ v² + 72v − 32v − 2304 = 0

⇒ v(v + 72) − 32(v + 72) = 0

⇒ (v − 32)(v + 72) = 0

⇒ v = 32 m/min (Speed cannot be negative)

∴ The correct answer is 32 m/min.

Additional Information

Relative Speed in Water

Downstream Speed = Speed of boat in still water + Speed of stream. Upstream Speed = Speed of boat in still water − Speed of stream.

Average Speed for Round Trip

Average speed = (u² − v²) ÷ u, where u is boat speed and v is stream speed.

Still Water Speed Calculation

Speed in still water = (Downstream speed + Upstream speed) ÷ 2.

Shortcut Trick

Total profit = 40000, Chittu's share = 15000 ⇒ Combined share of Chotu and Bittu = 25000.

Profit ratio (Chotu+Bittu : Chittu) = 25000 : 15000 = 5 : 3.

Investment × Time ratio: (50000×12 + 80000×12) : (X×8) = 5 : 3.

130000 × 12 : 8X = 5 : 3.

1560000 ÷ 8X = 5 ÷ 3 ⇒ 40X = 4680000 ⇒ X = 117000.

∴ The correct answer is Rs. 117000.

Alternate Method

Given: Chotu's investment = 50000, Bittu's investment = 80000, Chittu's investment = X, Time for Chotu/Bittu = 12 months, Time for Chittu = 8 months, Total Profit = 40000, Chittu's Profit = 15000.

Formulas: Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

⇒ Total investment-months for Chotu = 50000 × 12 = 6,00,000

⇒ Total investment-months for Bittu = 80000 × 12 = 9,60,000

⇒ Total investment-months for Chittu = X × 8 = 8X

⇒ Ratio of profits = 6,00,000 : 9,60,000 : 8X = 75,000 : 1,20,000 : X

⇒ Chittu's profit share = [8X ÷ (600000 + 960000 + 8X)] × 40000 = 15000

⇒ 8X ÷ (1560000 + 8X) = 15000 ÷ 40000 = 3/8

⇒ 64X = 3(1560000 + 8X) ⇒ 64X = 4680000 + 24X

⇒ 40X = 4680000 ⇒ X = 117000

∴ The correct answer is Rs. 117000.

Additional Information

Partnership Basics

In a partnership, the profit is distributed among partners in the ratio of the product of their investment and the time period for which the investment was made.

Effective Investment

When time periods differ, calculate the equivalent investment for 1 unit of time (usually months) to find the profit-sharing ratio.

Shortcut Trick

Bike Selling Price (SP) = 13200 × (100 − 18)% = 13200 × 0.82.

Since Bike SP = Mobile Cost Price (CP), Mobile CP = 10824.

Required Mobile Selling Price for 50% profit = 10824 × 1.5.

Final SP = 10824 + 5412 = 16236.

∴ The correct answer is Rs. 16236.

Alternate Method

Given: Bike CP = Rs. 13200, Loss% = 18%, Mobile Profit% = 50%.
Formulas: SP = CP × (100 − Loss%) ÷ 100 | SP = CP × (100 + Profit%) ÷ 100

⇒ Bike Selling Price = 13200 × (100 − 18) ÷ 100
⇒ Bike SP = 13200 × 82 ÷ 100 = 132 × 82 = Rs. 10824.
⇒ According to the question, Mobile Cost Price = Bike SP = Rs. 10824.
⇒ Required Mobile Selling Price = 10824 × (100 + 50) ÷ 100
⇒ Mobile SP = 10824 × 1.5 = Rs. 16236.

∴ The correct answer is Rs. 16236.

Additional Information

Cost Price and Selling Price Relationship

When a series of transactions occur where the SP of one item equals the CP of another, the values can be chained using multipliers.

Percentage Calculations

A 50% profit implies the final value is 1.5 times the original value (100% + 50%).

Shortcut Trick

By Alligation Method:

Ratio of Tech Leads to Freshers = (2000 − 1200) : (5500 − 2000) = 800 : 3500 = 8 : 35

Given, 8 units (Tech Leads) = 16

⇒ 1 unit = 2

⇒ 35 units (Freshers) = 35 × 2 = 70

∴ The correct answer is 70 freshers.

Alternate Method

Given:

Average salary of Tech Leads = Rs. 5500

Average salary of Freshers = Rs. 1200

Average salary of all staff = Rs. 2000

Number of Tech Leads = 16

Calculations:

Let the number of freshers be x.

⇒ Total salary = Average × Total number of staff

⇒ 5500 × 16 + 1200 × x = 2000 × (16 + x)

⇒ 88000 + 1200x = 32000 + 2000x

⇒ 88000 − 32000 = 2000x − 1200x

⇒ 56000 = 800x

⇒ x = 56000 ÷ 800

⇒ x = 70

∴ The correct answer is 70 freshers.

Additional Information

Alligation Rule

It is used to find the ratio in which two ingredients at given prices must be mixed to produce a mixture at a given price. Ratio = (Dearer Price − Mean Price) : (Mean Price − Cheaper Price).

Weighted Average

When groups with different sizes (n₁, n₂) and averages (a₁, a₂) are combined, the average A = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).