Quantitative Aptitude PYQs

Shortcut Trick
Initial Ratio (M:W) = 4 : 3 (Difference = 1 unit).
Final Ratio after adding 3L water = 6 : 5.
Equate Milk (multiply 4:3 by 3 and 6:5 by 2) → 12:9 and 12:10.
Increase in water = 10 − 9 = 1 unit → 1 unit = 3 liters.
Initial Difference = 12 − 9 = 3 units = 3 × 3 = 9 liters.
Given Difference = x + 5 → 9 = x + 5 → x = 4.
∴ The correct answer is 4.
Alternate Method

Given: Milk = 4a, Water = 3a. Difference = 4a − 3a = a.
Also given, Difference = x + 5 ⇒ a = x + 5.
According to the question, 3 liters of water is added:
⇒ 4a ÷ (3a + 3) = 6 ÷ 5
⇒ 20a = 18a + 18
⇒ 2a = 18
⇒ a = 9
Equating the values of a:
⇒ 9 = x + 5
⇒ x = 4
∴ The correct answer is 4.
Additional Information
Ratio and Proportion
When adding only one component to a mixture, keep the other component's ratio constant to solve quickly.
Mixture Difference Rule
If Quantity 1 and Quantity 2 are in ratio m:n, their difference is (m−n) units, provided m > n.

Given:

54.867 − 35.975 − 11.1491 − 1.743

Calculation:

⇒ 55 − 36 − 11 − 2

⇒ 55 − 49

⇒ 6

Approximation value is 6

Given:

(71.89 − 37.12) ÷ 5.23 + 16.92

Calculation:

⇒ (72 − 37) ÷ 5 + 17

⇒ 35 ÷ 5 + 17

⇒ 7 + 17

⇒ 24

Approximation value is 24

Given:

√323.89 − 11.91 + 28.91 × 2.901

Calculation:

⇒ √324 − 12 + 29 × 3

⇒ 18 − 12 + 87

⇒ 93

Approximation value is 93

Given:

323.11 ÷ 16.98 + 5⁴ ÷ 3.01² = ?

Calculation:

⇒ 323 ÷ 17 + 625 ÷ 9

⇒ 19 + 69

⇒ 88

Approximation value is 88

Given:

225.12 ÷ 2.9 + 31.8 × 9.9 = ? + 74.24

Calculation:

⇒ 225 ÷ 3 + 32 × 10 = ? + 74

⇒ 75 + 320 = ? + 74

⇒ 395 = ? + 74

⇒ ? = 321

Approximation value is 321

Shortcut Trick

Ratio of Pink : White = 4 : 7.

Ratio difference = 7 − 4 = 3 units.

3 units = 9 balls → 1 unit = 3 balls.

Pink balls = 4 × 3 = 12; White balls = 7 × 3 = 21; Total balls = 33.

Probability = (12/33) × (11/32) = (4/11) × (11/32) = 4/32 = 1/8.

∴ The correct answer is 1/8.

Alternate Method

Given: Ratio of pink balls to white balls = 4 : 7. Difference = 9.

Formula Used: Probability P(E) = n(E) / n(S); Combination ⁿC_r = n! / [r!(n−r)!]

⇒ Let pink balls be 4x and white balls be 7x.

⇒ 7x − 4x = 9 ⇒ 3x = 9 ⇒ x = 3.

⇒ Number of pink balls = 12, Number of white balls = 21.

⇒ Total balls = 12 + 21 = 33.

⇒ n(S) = ³³C₂ = (33 × 32) / (2 × 1) = 33 × 16 = 528.

⇒ n(E) = ¹²C₂ = (12 × 11) / (2 × 1) = 6 × 11 = 66.

⇒ Probability = 66 / 528 = 1/8.

∴ The correct answer is 1/8.

Additional Information

Probability of Successive Events

When picking items without replacement, the denominator decreases by 1 for each subsequent draw.

Combination Property

ⁿC₂ is a shortcut for selecting 2 items from n, calculated as n × (n−1) ÷ 2.

Shortcut Trick

Ratio of profit = Ratio of (Investment × Time).

A : B : C = (8000 × 6 + 13000 × 6) : (6000 × 6 + 8000 × 6) : (10000 × 6 + 15000 × 6).

A : B : C = (48 + 78) : (36 + 48) : (60 + 90) = 126 : 84 : 150.

Simplified Ratio A : B : C = 21 : 14 : 25.

B's share 14 units = Rs.4200 ⇒ 1 unit = 300.

Total units = 21 + 14 + 25 = 60 units ⇒ 60 × 300 = Rs.18000.

∴ The correct answer is Rs.18000.

Alternate Method

Given → Investments of A, B, C for two 6-month periods.
Formula: Profit Share = ∑(Investment × Time Period).

⇒ Profit of A = (8000 × 6) + (13000 × 6) = 126000.
⇒ Profit of B = (6000 × 6) + (8000 × 6) = 84000.
⇒ Profit of C = (10000 × 6) + (15000 × 6) = 150000.
⇒ Ratio A:B:C = 126:84:150 = 21:14:25.
⇒ B's Profit = (14 ÷ 60) × Total Profit = 4200.
⇒ Total Profit = (4200 × 60) ÷ 14 = 300 × 60 = 18000.

∴ The correct answer is Rs.18000.

Additional Information

Partnership Basics

Profit is directly proportional to both Capital and Time. Formula: P₁:P₂ = C₁T₁:C₂T₂.

Active vs Sleeping Partner

An Active Partner manages the business and may receive a salary or extra profit percentage before the remaining profit is divided.

Given:

Number of males:

A = 340

B = 320

C = 380

D = 360

E = 330

Condition:

For A, C, E → Female = Male − 130

For B, D → Female = Male + 130

Formula Used:

Female = Male ± 130

Calculation:

⇒ Office A

Female = 340 − 130 = 210

⇒ Office B

Female = 320 + 130 = 450

⇒ Office C

Female = 380 − 130 = 250

⇒ Office D

Female = 360 + 130 = 490

⇒ Office E

Female = 330 − 130 = 200

Final Table:

A → Male = 340, Female = 210

B → Male = 320, Female = 450

C → Male = 380, Female = 250

D → Male = 360, Female = 490

Calculation:

⇒ From B to C

B female = 450 → 30% = 135

⇒ New C (before leaving)

Male = 380

Female = 250 + 135 = 385

Total = 765

⇒ 20% leave

20% of 765 = 153

⇒ Remaining

= 765 − 153

= 612

Answer:

Total employees in C = 612

Given:

Number of males:

A = 340

B = 320

C = 380

D = 360

E = 330

Condition:

For A, C, E → Female = Male − 130

For B, D → Female = Male + 130

Formula Used:

Female = Male ± 130

Calculation:

⇒ Office A

Female = 340 − 130 = 210

⇒ Office B

Female = 320 + 130 = 450

⇒ Office C

Female = 380 − 130 = 250

⇒ Office D

Female = 360 + 130 = 490

⇒ Office E

Female = 330 − 130 = 200

Final Table:

A → Male = 340, Female = 210

B → Male = 320, Female = 450

C → Male = 380, Female = 250

D → Male = 360, Female = 490

Given:

E → Female = 200

Calculation:

⇒ C

Male = 380 − 200 = 180

⇒ D

Male = 360 + 200 = 560

Answer:

Ratio = 180 : 560

⇒ Divide by 20

= 9 : 28

Given:

Number of males:

A = 340

B = 320

C = 380

D = 360

E = 330

Condition:

For A, C, E → Female = Male − 130

For B, D → Female = Male + 130

Formula Used:

Female = Male ± 130

Calculation:

⇒ Office A

Female = 340 − 130 = 210

⇒ Office B

Female = 320 + 130 = 450

⇒ Office C

Female = 380 − 130 = 250

⇒ Office D

Female = 360 + 130 = 490

⇒ Office E

Female = 330 − 130 = 200

Final Table:

A → Male = 340, Female = 210

B → Male = 320, Female = 450

C → Male = 380, Female = 250

D → Male = 360, Female = 490

Calculation:

⇒ A + C

A = 340 + 210 = 550

C = 380 + 250 = 630

Total = 1180

⇒ B + E

B = 320 + 450 = 770

E = 330 + 200 = 530

Total = 1300

Answer:

Difference = 1300 − 1180 = 120

Percentage = (120 / 1300) × 100

= 9.23% less

Given:

Number of males:

A = 340

B = 320

C = 380

D = 360

E = 330

Condition:

For A, C, E → Female = Male − 130

For B, D → Female = Male + 130

Formula Used:

Female = Male ± 130

Calculation:

⇒ Office A

Female = 340 − 130 = 210

⇒ Office B

Female = 320 + 130 = 450

⇒ Office C

Female = 380 − 130 = 250

⇒ Office D

Female = 360 + 130 = 490

⇒ Office E

Female = 330 − 130 = 200

Final Table:

A → Male = 340, Female = 210

B → Male = 320, Female = 450

C → Male = 380, Female = 250

D → Male = 360, Female = 490

Given:

B → M = 320 → 25% = 80

D → F = 490 → 20% = 98

Calculation:

⇒ New B

Male = 320 − 80 = 240

Female = 450

Total B = 240 + 450 = 690

⇒ New D

Female = 490 − 98 = 392

Male = 360

Total D = 360 + 392 = 752

Answer:

Ratio = 690 : 752

⇒ Divide by 2

= 345 : 376

Given:

Number of males:

A = 340

B = 320

C = 380

D = 360

E = 330

Condition:

For A, C, E → Female = Male − 130

For B, D → Female = Male + 130

Formula Used:

Female = Male ± 130

Calculation:

⇒ Office A

Female = 340 − 130 = 210

⇒ Office B

Female = 320 + 130 = 450

⇒ Office C

Female = 380 − 130 = 250

⇒ Office D

Female = 360 + 130 = 490

⇒ Office E

Female = 330 − 130 = 200

Final Table:

A → Male = 340, Female = 210

B → Male = 320, Female = 450

C → Male = 380, Female = 250

D → Male = 360, Female = 490

Given:

E → F = 200 → 40% shift = 80

A → M = 340 → 20% shift = 68

Calculation:

⇒ New A

Male = 340 − 68 = 272

Female = 210 + 80 = 290

Total A = 272 + 290 = 562

⇒ New E

Male = 330 + 68 = 398

Female = 200 − 80 = 120

Total E = 398 + 120 = 518

Answer:

Difference = 562 − 518 = 44

Shortcut Trick

Efficiency Ratio A : B = 5 : 4

Time Ratio B : C = 3 : 4 → Efficiency Ratio B : C = 4 : 3

Combined Efficiency Ratio A : B : C = 5 : 4 : 3

Total Work = (5 + 4 + 3) × 10 = 120 units

Days taken by C = 120 ÷ 3 = 40 days

Given C = x + 20 ⇒ 40 = x + 20 ⇒ x = 20

∴ The correct answer is 20.

Alternate Method

Given: Efficiency of A : B = 5 : 4; Days of B : C = 3 : 4; Together = 10 days.

Formula Used: Total Work = Combined Efficiency × Time; Efficiency ∝ 1/Days

⇒ Efficiency Ratio A : B = 5 : 4

⇒ Time Ratio B : C = 3 : 4 ∴ Efficiency Ratio B : C = 4 : 3

⇒ Combined Efficiency A : B : C = 5 : 4 : 3

⇒ Total Efficiency = 5 + 4 + 3 = 12 units/day

⇒ Total Work = 12 units/day × 10 days = 120 units

⇒ Days taken by C = 120 ÷ 3 = 40 days

⇒ According to question: x + 20 = 40

⇒ x = 40 − 20 = 20

∴ The correct answer is 20.

Additional Information

Efficiency and Time Relation

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

Work and Wages

Wages are distributed in the ratio of the actual work done by each person, which is equivalent to the ratio of their efficiencies if they work for the same duration.

Combined Work Formula

If A takes 'a' days and B takes 'b' days, together they take (a × b) ÷ (a + b) days.

Shortcut Trick

Area = 3k × 1k = 192 ⇒ 3k² = 192 ⇒ k² = 64 ⇒ k = 8

Length = 24 m, Breadth = 8 m

Perimeter = 2(24 + 8) = 64 m

x + 20 = 64 ⇒ x = 44

∴ The correct answer is 44.

Alternate Method

Given: Ratio of Length (L) and Breadth (B) = 3 : 1

Formula Used: Area = L × B; Perimeter = 2(L + B)

⇒ Let L = 3k and B = k

⇒ Area = 3k × k = 192

⇒ 3k² = 192

⇒ k² = 64 ⇒ k = 8 m

⇒ L = 3 × 8 = 24 m; B = 1 × 8 = 8 m

⇒ Perimeter = 2(24 + 8) = 2(32) = 64 m

⇒ Given Perimeter = x + 20

⇒ x + 20 = 64

⇒ x = 64 − 20 = 44

∴ The correct answer is 44.

Additional Information

Properties of Rectangle

Opposite sides are equal and parallel.

Each internal angle is 90°. Diagonals are equal and bisect each other.

Diagonal Formula

The length of the diagonal (d) of a rectangle is given by d = √(L² + B²).

Area and Perimeter of Square

For a square with side 'a', Area = a² and Perimeter = 4a.

Formula used:

Herbivores = Total animals − (Carnivores + Omnivores)

Ratio of Herbivores : Omnivores = H : O

Omnivores = Herbivores × O-part ÷ H-part

Carnivores = (Carnivores + Omnivores) − Omnivores

Calculations:

 

Given:

H(C) = 84, H(D) = 90

C(D) = 96, O(D) = 60

Formula Used:

Average = Sum / 2

Increase 25% ⇒ ×1.25

Calculation:

⇒ H(F) = (84 + 90) / 2

⇒ 174 / 2 = 87

⇒ C(F) = 96 × 1.25

⇒ 120

⇒ O(F) = 60 × 1.25

⇒ 75

⇒ Total F = 87 + 120 + 75

⇒ = 282

∴ Total animals in Zoo F = 282

Formula used:

Herbivores = Total animals − (Carnivores + Omnivores)

Ratio of Herbivores : Omnivores = H : O

Omnivores = Herbivores × O-part ÷ H-part

Carnivores = (Carnivores + Omnivores) − Omnivores

Calculations:

 

Given:

D = 246, E = 264

B = 234, C = 248

Formula Used:

Difference = (Sum₁ − Sum₂)

Calculation:

⇒ (D + E) = 246 + 264 = 510

⇒ (B + C) = 234 + 248 = 482

⇒ Difference = 510 − 482 = 28

∴ Required difference = 28

Formula used:

Herbivores = Total animals − (Carnivores + Omnivores)

Ratio of Herbivores : Omnivores = H : O

Omnivores = Herbivores × O-part ÷ H-part

Carnivores = (Carnivores + Omnivores) − Omnivores

Calculations:

 

Given:

Total A = 214, Total C = 248

Formula Used:

% = (Difference / Base) × 100

Calculation:

⇒ Difference = 248 − 214 = 34

⇒ % = (34 / 214) × 100

⇒ ≈ 15.89% = 16%

∴ Zoo C is 16% more than Zoo A

Formula used:

Herbivores = Total animals − (Carnivores + Omnivores)

Ratio of Herbivores : Omnivores = H : O

Omnivores = Herbivores × O-part ÷ H-part

Carnivores = (Carnivores + Omnivores) − Omnivores

Calculations:

 

Given:

D → H = 90, C = 96, O = 60

E → H = 99, C = 88, O = 77

Formula Used:

Remaining = Old − Out + In

Calculation:

⇒ 30% H from D → E

⇒ 90 × 30/100 = 27

⇒ New H(D) = 90 − 27 = 63

⇒ New H(E) = 99 + 27 = 126

⇒ 25% C from E → D

⇒ 88 × 25/100 = 22

⇒ New C(E) = 88 − 22 = 66

⇒ New C(D) = 96 + 22 = 118

Now Zoo E total:

⇒ H + C + O = 126 + 66 + 77 = 269

∴ Total animals in Zoo E = 269

Formula used:

Herbivores = Total animals − (Carnivores + Omnivores)

Ratio of Herbivores : Omnivores = H : O

Omnivores = Herbivores × O-part ÷ H-part

Carnivores = (Carnivores + Omnivores) − Omnivores

Calculations:

 

Given:

C(A) = 84, C(B) = 86, C(C) = 84, C(D) = 96, C(E) = 88

Formula Used:

Average = Sum / 5

Calculation:

⇒ Sum = 84 + 86 + 84 + 96 + 88 = 438

⇒ Average = 438 / 5 = 87.6

∴ Average = 87.6