Quantitative Aptitude PYQs

Common Calculation

Class A

Ratio B:G = 6:5, Girls = 45

⇒ 5 parts = 45

⇒ 1 part = 9

⇒ Boys = 6 × 9 = 54

⇒ Total = 54 + 45 = 99

Class B

Ratio B:G = 3:4, Girls = 64

⇒ 4 parts = 64

⇒ 1 part = 16

⇒ Boys = 3 × 16 = 48

⇒ Total = 48 + 64 = 112

Class C

Ratio B:G = 5:6, Girls = 48

⇒ 6 parts = 48

⇒ 1 part = 8

⇒ Boys = 5 × 8 = 40

⇒ Total = 40 + 48 = 88

Class D

Ratio B:G = 7:4, Girls = 32

⇒ 4 parts = 32

⇒ 1 part = 8

⇒ Boys = 7 × 8 = 56

⇒ Total = 56 + 32 = 88

Class E

Ratio B:G = 9:7, Girls = 42

⇒ 7 parts = 42

⇒ 1 part = 6

⇒ Boys = 9 × 6 = 54

⇒ Total = 54 + 42 = 96

Given:

A boys = 54, A girls = 45

D boys = 56, D girls = 32

Calculation:

⇒ 55.55% of 54 = 30

⇒ 25% of 32 = 8

⇒ New A = (54 − 30) + (45 + 8) = 24 + 53 = 77

⇒ New D = (56 + 30) + (32 − 8) = 86 + 24 = 110

⇒ Ratio = 77 : 110

⇒ Divide by 11 ⇒ 7 : 10

∴ Answer = 7 : 10

Common Calculation

Class A

Ratio B:G = 6:5, Girls = 45

⇒ 5 parts = 45

⇒ 1 part = 9

⇒ Boys = 6 × 9 = 54

⇒ Total = 54 + 45 = 99

Class B

Ratio B:G = 3:4, Girls = 64

⇒ 4 parts = 64

⇒ 1 part = 16

⇒ Boys = 3 × 16 = 48

⇒ Total = 48 + 64 = 112

Class C

Ratio B:G = 5:6, Girls = 48

⇒ 6 parts = 48

⇒ 1 part = 8

⇒ Boys = 5 × 8 = 40

⇒ Total = 40 + 48 = 88

Class D

Ratio B:G = 7:4, Girls = 32

⇒ 4 parts = 32

⇒ 1 part = 8

⇒ Boys = 7 × 8 = 56

⇒ Total = 56 + 32 = 88

Class E

Ratio B:G = 9:7, Girls = 42

⇒ 7 parts = 42

⇒ 1 part = 6

⇒ Boys = 9 × 6 = 54

⇒ Total = 54 + 42 = 96

Given:

B = 112, E = 96

A boys = 54, C boys = 40, D boys = 56

Calculation:

⇒ (B + E) = 112 + 96 = 208

⇒ Boys = 54 + 40 + 56 = 150

⇒ Ratio = 208 : 150

⇒ Divide by 2 ⇒ 104 : 75

∴ Answer = 104 : 75

Shortcut Trick

Length of Train A = (6/5) × 250 = 300 m.

Relative Speed = Distance ÷ Time = 300 ÷ 10 = 30 m/s.

Relative Speed in km/hr = 30 × (18/5) = 108 km/hr.

Speed of Train A = Relative Speed − Speed of Bird = 108 − 18 = 90 km/hr.

∴ The correct answer is 90 km/hr.

Alternate Method

Given:

Length ratio A : B = 6 : 5

Length of Train B = 250 m

Speed of Bird = 18 km/hr

Time to cross = 10 seconds

Formula Used:

Distance = Relative Speed × Time

Relative Speed (Opposite) = S₁ + S₂

Calculations:

⇒ 5 units = 250 m

⇒ 1 unit = 50 m

⇒ Length of Train A = 6 × 50 = 300 m

⇒ Speed of bird in m/s = 18 × (5/18) = 5 m/s

⇒ Let speed of Train A be 'v' m/s

⇒ Relative Speed = (v + 5) m/s

⇒ 300 = (v + 5) × 10

⇒ 30 = v + 5

⇒ v = 25 m/s

⇒ Speed of Train A in km/hr = 25 × (18/5) = 90 km/hr

∴ The correct answer is 90 km/hr.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, relative speed is the sum of their speeds (S₁ + S₂). 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.

Crossing Stationary/Moving Objects

When a train crosses a point object (bird, pole, man), the distance covered is equal to the length of the train itself.

Shortcut Trick

Total balls = 5 + 4 + 3 = 12. Total ways to draw 2 balls = ¹²C₂ = 66.

Probability(Different Colors) = 1 − Probability(Same Colors).

Ways for Same Color = (⁵C₂ + ⁴C₂ + ³C₂) = (10 + 6 + 3) = 19.

P(Different) = 1 − (19/66) = 47/66.

∴ The correct answer is 47/66.

Alternate Method

Given: Red = 5, Blue = 4, Green = 3. Total = 12.

Formula Used: Probability = Favorable Outcomes ÷ Total Outcomes.

⇒ Total outcomes for drawing 2 balls = ¹²C₂ = (12 × 11) ÷ (2 × 1) = 66.

⇒ Favorable outcomes (Different colors) = (Red × Blue) + (Blue × Green) + (Red × Green).

⇒ Favorable outcomes = (5 × 4) + (4 × 3) + (5 × 3) = 20 + 12 + 15 = 47.

⇒ Probability = 47/66.

∴ The correct answer is 47/66.

Additional Information

Complementary Probability

P(A) + P(A') = 1. Often easier to calculate 1 − P(Opposite Event) for complex conditions like "at least" or "different".

Combination Formula (nCr)

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

Without Replacement Rule

When drawing balls one by one without replacement, the total count and category count decrease by 1 after the first draw.

Shortcut Trick

Work done by C = 7/18 of total work.

Remaining work done by A and B in 5 days = 1 − 7/18 = 11/18.

Combined efficiency of A and B = (11/18) ÷ 5 = 11/90.

Using 1/(x+5) + 1/(x+8) = 11/90 → By Option Verification: if x = 10, then 1/15 + 1/18 = (6+5)/90 = 11/90.

∴ The correct answer is 10.

Alternate Method

Given: Time for A = (x + 5) days, B = (x + 8) days, C = 18 days.

Formula Used: Work = Efficiency × Time

Calculations:

⇒ Work done by C in 7 days = 7/18

⇒ Work done by A and B together in 5 days = 1 − 7/18 = 11/18

⇒ (1-day work of A + 1-day work of B) × 5 = 11/18

⇒ [1/(x + 5) + 1/(x + 8)] × 5 = 11/18

⇒ [(x + 8 + x + 5) / (x + 5)(x + 8)] = 11/90

⇒ (2x + 13) / (x² + 13x + 40) = 11/90

⇒ 180x + 1170 = 11x² + 143x + 440

⇒ 11x² − 37x − 730 = 0

⇒ 11x² − 110x + 73x − 730 = 0

⇒ 11x(x − 10) + 73(x − 10) = 0

⇒ (x − 10)(11x + 73) = 0

⇒ x = 10 (as time cannot be negative)

∴ The correct answer is 10.

Additional Information

Total Work Concept

The total work is always considered as 1 unit or the LCM of the time taken by individuals.

Efficiency Formula

Efficiency is inversely proportional to time taken: Efficiency = 1 / Time.

Combined Work

If A and B work together, their combined 1-day work is (1/A + 1/B).

Given:

14.69% of 300.25 + 14.89 × 23.33 – 345.68

Calculation:

⇒ 15% of 300 + 15 × 23 – 346

⇒ 45 + 345 – 346

⇒ 44

∴ Answer ≈ 44

Given:

24.89% of  639.87 – 147.89 + 212.32 % of 49.95 = ?

Calculation:

24.89% of  639.87 – 147.89 + 212.32 % of 49.95 = ?

Applying approximation concept,

⇒ 25% of 640 – 148 + 212% of 50 = ?

⇒ 160 – 148 + 106 = ?

⇒ 12 + 106 = ?

⇒ ? = 118

Hence, the correct answer is option (4).

Given

(499.89 × 19.8 ÷ 9.9) + (50.23% of 799.75) − (39.01² ÷ 3) + 7  ≈ ?

Calculation:

⇒ (500 × 20 ÷ 10) + (50% of 800) − (39² ÷ 3) + 7

⇒ 1000 + 400 – 507 + 7

⇒ 900

∴ Approximation value is 900

Given

(804.89 × 296.44 ÷ 49.89 × 6.01) + (49.87% of 400.25) − (24.22²) ≈ ?

Calculation:

⇒ (800 × 300 ÷ 50 × 6) + (50% of 400) − (24²)

⇒ 28800 + 200 – 576

⇒ 28424

∴ Approximation value is 28400

Given

(19.8% of 1500) + (29.7% of 1000) + (63.01² − 36.87² ÷ 26) - 42 ≈ ?

Calculation:

⇒ (20% of 1500) + (30% of 1000) + (63² − 37² ÷ 26 ) - 42

⇒ 300 + 300 + (3969 − 53) - 16

⇒ 300 + 300 + 3916 -16

⇒ 4500

∴ Approximation value is 4500

Shortcut Trick

Net rate of filling = Pipe A (Inlet) − Pipe B (Outlet)

1/10 = 1/x − 1/40

1/x = 1/10 + 1/40 = 5/40 = 1/8

∴ The correct answer is 8 minutes.

Alternate Method

⇒ Net efficiency = Efficiency of Pipe A + Efficiency of Pipe B

⇒ 4 = Efficiency of Pipe A + (−1)

⇒ Efficiency of Pipe A = 4 + 1 = 5 units/min

⇒ Time taken by Pipe A (x) = Total Work ÷ Efficiency of A

⇒ x = 40 ÷ 5 = 8 minutes

∴ The correct answer is 8 minutes.

Additional Information

Inlet and Outlet Concept

Inlet pipes add water (positive work), while outlet pipes remove water (negative work). Net work = Σ (Inlet rates) − Σ (Outlet rates).

Time and Efficiency Relationship

Time is inversely proportional to efficiency. If a pipe fills a tank in 't' minutes, its efficiency is 1/t units per minute.

Pipes and Cisterns Formula

If Pipe A fills in x and Pipe B empties in y, and the tank is filled in T time, then 1/T = 1/x − 1/y.

Shortcut Trick

Total units for Cost Price = 5 + 3 = 8 units.

8 units = Rs. 320 ⇒ 1 unit = Rs. 40. CP₁ = Rs. 200, CP₂ = Rs. 120.

Profit on first book = 25% of 200 = Rs. 50. Overall profit = Rs. 46. ∴ Loss on second book = 50 − 46 = Rs. 4. Loss percentage (x) = (4 ÷ 120) × 100 = 100 ÷ 30 = 3.33%.

∴ The correct answer is 3.33%.

Alternate Method

Given → Total CP = Rs. 320, Ratio = 5:3, P₁% = 25%, Total Profit = Rs. 46. Formulas: Profit = SP − CP | Profit = CP × Profit% | Loss% = (Loss ÷ CP) × 100

⇒ Calculate individual CP: CP₁ = (5/8) × 320 = 200; CP₂ = (3/8) × 320 = 120.
⇒ Profit from Book 1 = 200 × 25% = Rs. 50. ⇒ Let loss on Book 2 be L. Total Profit = Profit₁ − Loss₂.
⇒ 46 = 50 − Loss₂ ⇒ Loss₂ = Rs. 4. ⇒ x = (4 ÷ 120) × 100 = 3.33%.

∴ The correct answer is 3.33%.

Additional Information

Profit and Loss on Multiple Items

Total Profit or Loss is the algebraic sum of individual profits and losses. Net P/L% = (Total P/L ÷ Total CP) × 100.

Ratio to Value Conversion

To find a value from a ratio a:b with total T: Value A = [a ÷ (a + b)] × T.

Shortcut Trick

In a meeting case where boats move towards each other, the speed of stream cancels out.

Relative Speed = Total Distance ÷ Time = 250 ÷ 5 = 50 km/hr.

Sum of speed ratios (3 + 2) = 5 units → 50 km/hr.

Difference in speed ratios (3 − 2) = 1 unit → 10 km/hr.

∴ The correct answer is 10 km/hr.

Alternate Method

Given: Ratio of speeds of boat A and B = 3:2, Total Distance = 250 km, Time = 5 hours, Speed of stream = 5 km/hr.

Formula: Relative Speed (opposite direction) = (u₁ + v) + (u₂ − v) = u₁ + u₂

⇒ Let the speed of Boat A be 3x and Boat B be 2x.

⇒ Relative Speed = Total Distance ÷ Time = 250 ÷ 5 = 50 km/hr.

⇒ Since they move towards each other, Relative Speed = (3x + 5) + (2x − 5) = 50.

⇒ 5x = 50 ⇒ x = 10.

⇒ Speed of Boat A = 3 × 10 = 30 km/hr.

⇒ Speed of Boat B = 2 × 10 = 20 km/hr.

⇒ Difference = 30 − 20 = 10 km/hr.

∴ The correct answer is 10 km/hr.

Additional Information

Relative Speed in Rivers

When two boats move towards each other, one is downstream and one is upstream. The stream speed (v) always cancels out: (u₁+v) + (u₂−v) = u₁+u₂.

Downstream & Upstream

Downstream Speed = Speed of boat + Speed of stream. Upstream Speed = Speed of boat − Speed of stream.

Meeting Point Distance

Distance from starting point of Boat A to meeting point = (Speed of Boat A + Stream) × Meeting Time.

Shortcut Trick

Net discount = a + b − (ab ÷ 100) = 10 + 20 − (10 × 20 ÷ 100) = 28%.

SP = CP × (100 + P%) ÷ 100 = 1440 × 120 ÷ 100 = 1728.

MP = SP ÷ (100 − D%) × 100 = 1728 ÷ 72 × 100 = 2400.

Given MP = x + 400 ⇒ 2400 = x + 400 ⇒ x = 2000.

∴ The correct answer is 2000.

Alternate Method

Given → Cost Price (CP) = 1440, Profit% = 20%, Successive Discounts = 10% and 20%, Marked Price (MP) = x + 400.
Formulas: SP = CP × (100 + Profit%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%)

⇒ Selling Price (SP) = 1440 × (120 ÷ 100) = 1440 × 1.2 = 1728.
⇒ Equivalent Single Discount = 10 + 20 − (10 × 20 ÷ 100) = 28%.
⇒ Marked Price (MP) = 1728 × 100 ÷ (100 − 28) = 1728 × 100 ÷ 72 = 2400.
⇒ x + 400 = 2400.
⇒ x = 2400 − 400 = 2000.

∴ The correct answer is 2000.

Additional Information

Successive Discounts

When two discounts d₁% and d₂% are given, the total discount is d₁ + d₂ − (d₁ × d₂ ÷ 100)%.

Relation between CP and MP

The ratio of MP to CP can be found using: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).