Quantitative Aptitude PYQs

Shortcut Trick

Total balls = 5 + 4 + 3 = 12

P(Different colors) = 1 − P(Same color)

P(Same color) = [⁵C₂ + ⁴C₂ + ³C₂] ÷ ¹²C₂

P(Same color) = [10 + 6 + 3] ÷ 66 = 19/66

P(Different colors) = 1 − 19/66 = 47/66

∴ The correct answer is 47/66.

Alternate Method

Given: Red balls = 5, Blue balls = 4, Green balls = 3. Total balls = 12. Two balls drawn without replacement.

Formula Used: Probability = (Favorable Outcomes) ÷ (Total Outcomes)

⇒ Total ways to draw 2 balls from 12 = ¹²C₂ = (12 × 11) ÷ (2 × 1) = 66

⇒ Ways to draw 2 balls of same color = (2 Red) OR (2 Blue) OR (2 Green)

⇒ Ways = ⁵C₂ + ⁴C₂ + ³C₂

⇒ Ways = [(5×4)/2] + [(4×3)/2] + [(3×2)/2] = 10 + 6 + 3 = 19

⇒ Probability (Same color) = 19/66

⇒ Probability (Different colors) = 1 − 19/66 = 47/66

∴ The correct answer is 47/66.

Additional Information

Complementary Events

The sum of the probability of an event happening and not happening is always 1. P(A) + P(A') = 1.

Combinations Formula

Used for selection where order doesn't matter: ⁿC_r = n! ÷ [r! × (n−r)!].

Without Replacement Rule

When drawing items without replacement, the total number of available outcomes decreases by 1 after each draw.

Calculation:

The logic followed here is:

×2 + 2, ×3 + 3, ×4 + 4, ×5 + 5, ×6 + 6

⇒ 3 × 2 + 2 = 8

⇒ 8 × 3 + 3 = 27

⇒ 27 × 4 + 4 = 112

⇒ 112 × 5 + 5 = 565

⇒ 565 × 6 + 6 = 3396

Answer = 3396

Calculation:

The logic followed here is:

(3.2)² = 10.24

(4.2)² = 17.64

(5.2)² = 27.04

(6.2)² = 38.44

(7.2)² = 51.84

Answer = 51.84

Calculation:

The logic followed here is:

⇒ 152 × 4 - 15 = 593

⇒ 593 × 3 − 15 = 1764

⇒ 1764 × 2 - 15 = 3513

⇒ 3513 × 1 − 15 = 3498

Hence the correct answer is 1764.

Calculation:

The logic followed here is:

×3 + 1, +2, +3, +4…

⇒ 7 × 3 + 1 = 22

⇒ 22 × 3 + 2 = 68

⇒ 68 × 3 + 3 = 207

⇒ 207 × 3 + 4 = 625

⇒ 625 × 3 + 5 = 1880

Answer = 1880

Calculation:

The logic followed here is:

Check differences

⇒ 69 − 46 = 23

⇒ 103 − 69 = 34

⇒ 148 − 103 = 45

⇒ 204 − 148 = 56

Pattern:

23, 34, 45, 56 → increasing by +11

Next:

⇒ 204 + 67 = 271

Answer = 271

Shortcut Trick

Use the Alligation Method on the cost prices.

Selling Price (SP) = ₹36/kg and Profit = 20%.

Cost Price of Mixture (Mean CP) = SP × 100 ÷ (100 + P%) = 36 × 100 ÷ 120 = ₹30/kg.

Ratio (Type A : Type B) = (35 − 30) : (30 − 25) = 5 : 5 = 1 : 1.

∴ The correct answer is 1:1.

Alternate Method

Given → CP_A = ₹25/kg, CP_B = ₹35/kg, SP_mix = ₹36/kg, Profit% = 20%.
Formulas: CP = SP × 100 ÷ (100 + P%) | (Quantity 1 ÷ Quantity 2) = (CP₂ − Mean CP) ÷ (Mean CP − CP₁)

⇒ Calculate the Cost Price of the mixture: CP_mix = 36 × (100 ÷ 120) = 30.
⇒ Apply the alligation rule: Ratio = (Higher CP − Mean CP) : (Mean CP − Lower CP).
⇒ Ratio = (35 − 30) : (30 − 25) = 5 : 5.
⇒ Simplified Ratio = 1 :

1. ∴ The correct answer is 1:1.

Additional Information

Rule of Alligation

It is used to find the ratio in which two ingredients at given prices must be mixed to produce a mixture at a Mean Price.

Cost Price vs Selling Price

Always ensure all values used in the Alligation diagram are of the same type (all Cost Prices or all Selling Prices with same profit/loss).

Shortcut Trick

Initial Average = 65 | Total Students = 40.

Net change from 8 students leaving: 8 × (65 − 70) = −40.

Net change from 12 students joining: 12 × (60 − 65) = −60.

Total Deviation = −40 − 60 = −100. New strength = 40 − 8 + 12 = 44.

New Average = 65 + (−100 ÷ 44) = 65 − 2.27 = 62.73.

∴ The correct answer is 62.73 marks.

Alternate Method

Given: Initial students = 40, Initial average = 65, Students left = 8 (avg 70), Students joined = 12 (avg 60).

Formula Used: Average = Total Sum ÷ Number of observations

⇒ Total initial marks = 40 × 65 = 2600

⇒ Marks of students who left = 8 × 70 = 560

⇒ Marks of students who joined = 12 × 60 = 720

⇒ New Total Sum = 2600 − 560 + 720 = 2760

⇒ New Number of students = 40 − 8 + 12 = 44

⇒ New Average = 2760 ÷ 44 = 62.7272...

∴ The correct answer is 62.73 marks.

Additional Information

Weighted Average

When multiple groups join or leave, the combined average is (n₁a₁ ± n₂a₂) ÷ (n₁ ± n₂).

Net Deviation Method

New Average = Old Average + (Total Deviation ÷ New Number of terms). This saves calculation time by avoiding large multiplications.

Shortcut Trick

Total Relative Speed = (90 + 72) × 5/18 = 162 × 5/18 = 45 m/s.

Total Distance = Speed × Time = 45 × 8 = 360 m.

Sum of lengths = (3x + 60) + (2x + 100) = 5x + 160.

5x + 160 = 360 ⇒ 5x = 200 ⇒ x = 40.

∴ The correct answer is 40.

Alternate Method

Given:

Length of Train A (L₁) = (3x + 60) m

Length of Train B (L₂) = (2x + 100) m

Speed of Train A (S₁) = 90 km/hr = 90 × 5/18 = 25 m/s

Speed of Train B (S₂) = 72 km/hr = 72 × 5/18 = 20 m/s

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Distance = Speed × Time

Calculations:

⇒ Total Distance = L₁ + L₂ = (3x + 60) + (2x + 100) = 5x + 160

⇒ Relative Speed = 25 + 20 = 45 m/s

⇒ Distance = Relative Speed × Time

⇒ 5x + 160 = 45 × 8

⇒ 5x + 160 = 360

⇒ 5x = 360 − 160

⇒ 5x = 200

⇒ x = 200 ÷ 5 = 40

∴ The correct answer is 40.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S₁ + S₂). When moving in the same direction, it is the difference |S₁ − S₂|.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Train Crossing Distance

When a train crosses another train or a platform/bridge, the total distance traveled is the sum of the lengths of the train and the other object.

Shortcut Trick

When a cylinder is melted into a cone of the same height, their volumes are equal.

Volume of Cylinder = Volume of Cone

π × R_cyl² × H = (1/3) × π × R_cone² × H

R_cone² = 3 × R_cyl² ⇒ R_cone = R_cyl × √3

Given R_cyl = 14 cm ⇒ R_cone = 14√3 cm.

∴ The correct answer is 14√3 cm.

Alternate Method

Given:

Radius of Cylinder (r₁) = 14 cm

Height of Cylinder (h₁) = 21 cm

Height of Cone (h₂) = 21 cm

Formula Used:

Volume of Cylinder = πr²h

Volume of Cone = (1/3)πr²h

Calculations:

⇒ Volume of Cylinder = Volume of Cone

⇒ π × (14)² × 21 = (1/3) × π × R² × 21

⇒ 14 × 14 × 21 = (1/3) × R² × 21

⇒ 196 = R² / 3

⇒ R² = 196 × 3

⇒ R = √(196 × 3)

⇒ R = 14√3 cm

∴ The correct answer is 14√3 cm.

Additional Information

Volume Invariance

When one 3D shape is melted and recast into another, the Total Volume remains constant unless wastage is mentioned.

Cylinder and Cone Relationship

If a cylinder and a cone have the same radius and height, the volume of the cylinder is exactly 3 times the volume of the cone.

Surface Area of Cone

Curved Surface Area (CSA) = πrl, where l is the slant height calculated as l = √(r² + h²).