Quantitative Aptitude PYQs

Given:

Number of students = 24

Initial average = 15

New average after first teacher = 16

New average after second teacher = 17

Formula Used:

Sum = Average × Number of terms

Calculation:

⇒ Initial sum = 24 × 15 = 360

⇒ New sum (with 1 teacher) = 25 × 16 = 400

⇒ First teacher age = 400 - 360 = 40

⇒ New sum (with 2 teachers) = 26 × 17 = 442

⇒ Second teacher age = 442 - 400 = 42

∴ The correct answer is 42.

Shortcut Trick

The Speed Ratio is 80 : 60 = 4 : 3.

Since distance is constant, the Time Ratio is the inverse = 3 : 4.

The difference in time units is 4 − 3 = 1 unit, which corresponds to 2 hours (1 hr early to 1 hr late).

Actual time taken at 80 km/h = 3 units × 2 = 6 hours.

Distance = Speed × Time = 80 km/h × 6 hours = 480 km.

∴ The correct answer is 480 km.

Alternate Method

Given:

Speed 1 (S₁) = 80 km/h (reaches 1 hr early)

Speed 2 (S₂) = 60 km/h (reaches 1 hr late)

Total time difference (ΔT) = 1 − (−1) = 2 hours

Formula Used:

Distance = [(S₁ × S₂) / (S₁ − S₂)] × ΔT

Calculations:

⇒ Distance = (80 × 60) / (80 − 60) × 2

⇒ Distance = (4800 / 20) × 2

⇒ Distance = 240 × 2

⇒ Distance = 480 km

⇒ Therefore, the correct option is (3).

∴ The correct answer is 480 km.

Additional Information

Relationship between Speed and Time

When distance is constant, speed is inversely proportional to time (Speed ∝ 1/Time). If speed increases by a factor of x/y, time decreases by the reciprocal factor.

Average Speed Formula

For a round trip or two equal distances with speeds x and y, the average speed is given by (2 × x × y) / (x + y).

Time Difference Calculation

When comparing 'early' and 'late' timings, the total time difference is the sum of the early and late durations (Early + Late).

Shortcut Trick

Ratio of Final Salary to Original Salary = 156.25 : 100 = 1.5625.

For two equal successive increments, the Multiplying Factor for each step is √(Total Change Factor).

Step Factor = √1.5625 = 1.25.

Percentage increase = (1.25 − 1) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given: Final salary = 156.25% of original salary. Number of increments = 2. Both increments are equal.

Formula Used: Final Value = Initial Value × (1 + r/100)ⁿ

Let the Original Salary be 100 and Increment be x%.

⇒ Final Salary = 100 × (1 + x/100) × (1 + x/100)

⇒ 156.25 = 100 × (1 + x/100)²

⇒ 156.25 / 100 = (1 + x/100)²

⇒ 1.5625 = (1 + x/100)²

⇒ √1.5625 = 1 + x/100

⇒ 1.25 = 1 + x/100

⇒ x/100 = 1.25 − 1 = 0.25

⇒ x = 25%

∴ The correct answer is 25%.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the net percentage change is given by the formula [a + b + (ab/100)]%.

Reverse Square Root Method

When n equal successive changes result in a total factor F, each single change factor is found by taking the n^th root of F (i.e., F^1/n).

Compounding Relationship

Successive increments follow the same logic as Compound Interest, where the principal increases by a fixed rate over multiple periods.

Given:

Initial population = 240000

Birth rate = 12%

Death rate = 7%

Time = 3 years

Formula Used:

Net increase = (1 + increase/100) × (1 - decrease/100)

Population after t years = P × (net factor)^t

Calculation:

⇒ Net factor = 1.12 × 0.93

⇒ Net factor = 1.0416

⇒ Population after 3 years = 240000 × (1.0416)³

⇒ (1.0416)² = 1.085

⇒ (1.0416)³ = 1.130

⇒ Population = 240000 × 1.130

⇒ Population = 271200

∴ The correct answer is 271200.

Given:

A = 8 days

B = 16 days

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Total time = Total work ÷ Total efficiency

Calculation:

⇒ LCM of 8 and 16 = 16

⇒ Efficiency of A = 16 ÷ 8 = 2

⇒ Efficiency of B = 16 ÷ 16 = 1

⇒ Total efficiency = 2 + 1 = 3

⇒ Time = 16 ÷ 3

⇒ Time = \(5\frac{1}{3}\)

∴ The correct answer is \(5\frac{1}{3}\).

Given:

Marked price = 1200

First discount = 20%

Second discount = 10%

Formula Used:

Successive discount price = MP × (1 - d1/100) × (1 - d2/100)

Calculation:

⇒ Selling price = 1200 × (1 - 20/100) × (1 - 10/100)

⇒ Selling price = 1200 × 0.8 × 0.9

⇒ Selling price = 1200 × 0.72

⇒ Selling price = 864

∴ The correct answer is 864.

Given:

First discount = 25%

Second discount = 10%

Formula Used:

Effective discount % = a + b - (a × b ÷ 100)

Calculation:

⇒ Effective discount = 25 + 10 - (25 × 10 ÷ 100)

⇒ Effective discount = 35 - (250 ÷ 100)

⇒ Effective discount = 35 - 2.5

⇒ Effective discount = 32.5%

∴ The correct answer is 32.5%.

Given:

Side of equilateral triangle = 75 m

Cost per meter = 15

Formula Used:

Perimeter of equilateral triangle = 3 × side

Total cost = Perimeter × cost per meter

Calculation:

⇒ Perimeter = 3 × 75 = 225

⇒ Total cost = 225 × 15 = 3375

∴ The correct answer is 3375.

Given:

Profit = 25%

Formula Used:

Profit % = (Water ÷ Milk) × 100

Calculation:

⇒ 25 = (Water ÷ Milk) × 100

⇒ Water ÷ Milk = 25 ÷ 100

⇒ Water ÷ Milk = 1 ÷ 4

∴ The correct answer is 1 : 4.

Given:

Values = ∛10 , √6 , 6^th root of 120

Formula Used:

a^1/n form for comparison

Calculation:

⇒ ∛10 = 10^1/3 ≈ 2.15

⇒ √6 = 6^1/2 ≈ 2.45

⇒ 120^1/6 ≈ 2.22

⇒ Descending order: 2.45 > 2.22 > 2.15

∴ The correct answer is \(\sqrt{6},\ \sqrt[6]{120},\ \sqrt[3]{10}\).

Given:

O is incentre

∠POR = 140°

Formula Used:

∠POR = 90° + (1 ÷ 2) × ∠PQR

Calculation:

⇒ 140 = 90 + (1 ÷ 2) × ∠PQR

⇒ 140 − 90 = (1 ÷ 2) × ∠PQR

⇒ 50 = (1 ÷ 2) × ∠PQR

⇒ ∠PQR = 100°

∴ The correct answer is 100°.

Shortcut Trick

Ratio sum 2 × (14 + 11) = 50 units → 150 cm ∴ 1 unit = 3 cm.

Area of Rectangle = (14 × 3) × (11 × 3) = 1386 cm².

Top surface area πr² = 1386 ⇒ r² = 1386 × 7/22 = 441 ⇒ radius (r) = 21 cm.

Height (h) = 21 ÷ 1.05 = 20 cm.

Curved Surface Area = 2 × 22/7 × 21 × 20 = 2640 cm².

∴ The correct answer is 2640 cm².

Alternate Method

Given:

Ratio of length (L) and breadth (B) = 14 : 11

Perimeter of rectangle = 150 cm

Radius (r) of cylinder = 105% of height (h)

Formula Used:

Perimeter = 2(L + B)

Area of circle = πr²

Curved Surface Area (CSA) = 2πrh

Calculations:

⇒ 2(14x + 11x) = 150

⇒ 50x = 150 ⇒ x = 3

⇒ Length = 14 × 3 = 42 cm, Breadth = 11 × 3 = 33 cm

⇒ Area of rectangle = 42 × 33 = 1386 cm²

⇒ πr² = 1386

⇒ (22/7) × r² = 1386

⇒ r² = (1386 × 7) ÷ 22 = 63 × 7 = 441

⇒ Radius (r) = √441 = 21 cm

⇒ Height (h) = 21 ÷ 1.05 = 20 cm

⇒ CSA = 2 × (22/7) × 21 × 20

⇒ CSA = 2 × 22 × 3 × 20 = 2640 cm²

∴ The correct answer is 2640 cm².

Additional Information

Total Surface Area of Cylinder

The Total Surface Area (TSA) of a cylinder includes the curved surface and two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Cylinder

The volume represents the capacity of the cylinder and is calculated as: V = πr²h.

Ratio and Percentage Relation

If a value A is x% of B, then A = (x/100) × B. For x > 100, A is greater than B; for x < 100, A is smaller than B.

Given:

Ratio of diagonals = 7 : 24

Side = 25 cm

Formula Used:

(d₁ ÷ 2)² + (d₂ ÷ 2)² = side²

Area of rhombus = (1 ÷ 2) × d₁ × d₂

Calculation:

⇒ Let d₁ = 7x, d₂ = 24x

⇒ (7x ÷ 2)² + (24x ÷ 2)² = 25²

⇒ (49x² ÷ 4) + (576x² ÷ 4) = 625

⇒ (625x² ÷ 4) = 625

⇒ x² = 4

⇒ x = 2

⇒ d₁ = 14 cm, d₂ = 48 cm

⇒ Area = (1 ÷ 2) × 14 × 48 = 336 cm²

∴ The correct answer is diagonals = 14 cm and 48 cm, area = 336 cm².

Given:

A investment = 40000

B investment = 60000

Time = 6 months

Total profit = 40000

Formula Used:

Profit share ∝ Investment × Time

Calculation:

⇒ A's share ∝ 40000 × 6 = 240000

⇒ B's share ∝ 60000 × 6 = 360000

⇒ Ratio = 240000 : 360000 = 2 : 3

⇒ B's share = (3 ÷ 5) × 40000

⇒ B's share = 24000

∴ The correct answer is 24000.

Given:

Number of students = 36

Initial average = 40

New average = 44

Total students = 44

Formula Used:

Sum = Average × Number of terms

Calculation:

⇒ Initial sum = 36 × 40 = 1440

⇒ New sum = 48 × 44 = 2112

⇒ Sum of score of new students = 2112 - 1440 = 672

So, 

 Average score of new students = 672 / 12 = 56

∴ The correct answer is 56.

Shortcut Trick

New Mean Average = 40,000 − 3,000 = 37,000.

Apply Alligation between Original Average (40,000) and New Joiners Average (25,000).

Ratio of (Original Employees) : (New Joiners) = (37,000 − 25,000) : (40,000 − 37,000).

Ratio = 12,000 : 3,000 = 4 :

1. Given 1 unit = 4 employees, then 4 units = 4 × 4 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Original Average Salary = ₹40,000

Number of New Employees = 4

Average Salary of new employees = ₹25,000

Reduction in Average = ₹3,000

Calculations:

⇒ Let the original number of employees be n.

⇒ New total average = 40,000 − 3,000 = 37,000.

⇒ Total salary of original employees = 40,000 × n.

⇒ Total salary of new employees = 4 × 25,000 = 100,000.

⇒ New Total Average = (40,000n + 100,000) ÷ (n + 4).

⇒ 37,000 = (40,000n + 100,000) ÷ (n + 4).

⇒ 37,000(n + 4) = 40,000n + 100,000.

⇒ 37,000n + 148,000 = 40,000n + 100,000.

⇒ 40,000n − 37,000n = 148,000 − 100,000.

⇒ 3,000n = 48,000.

⇒ n = 48,000 ÷ 3,000 = 16.

∴ The correct answer is 16.

Additional Information

Weighted Average Formula

When two groups are combined, the resulting average is given by: A_avg = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Alligation Rule

The ratio of the quantities of two components is inversely proportional to the differences of their individual averages from the mean average: (Quantity 1) ÷ (Quantity 2) = (Average 2 − Mean) ÷ (Mean − Average 1).

Basic Average Concept

Average = (Sum of all observations) ÷ (Total number of observations). ∴ Sum = Average × Count.

Given:

Total distance = 36 km

Time = 2.5 hours

Formula Used:

Distance = Speed × Time

Calculation:

⇒ Let covered distance = x

⇒ Remaining distance = 36 - x

⇒ x = 3/5 × (36 - x)

⇒ 5x = 3(36 - x)

⇒ 5x = 108 - 3x

⇒ 8x = 108

⇒ x = 108 ÷ 8 = 13.5 km

⇒ Speed = 13.5 ÷ 2.5

⇒ Speed = 5.4 km/h

∴ The correct answer is 5.4.

Given:

Total distance = 120 km

Time = 2 hours 40 minutes = 8/3 hours

Formula Used:

Distance = Speed × Time

Calculation:

⇒ Let covered distance = x

⇒ Remaining distance = 120 - x

⇒ x = 7/8 × (120 - x)

⇒ 8x = 7(120 - x)

⇒ 8x = 840 - 7x

⇒ 15x = 840

⇒ x = 840 ÷ 15 = 56 km

⇒ Speed = 56 ÷ (8/3)

⇒ Speed = 56 × 3 ÷ 8 = 21 km/h

⇒ Speed = 21 × 5 ÷ 18

⇒ Speed = 105 ÷ 18 = 5.83 m/s

∴ The correct answer is 5.83.

Given:

A + B = 15 days

A = 25 days

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Total efficiency = Sum of individual efficiencies

Calculation:

⇒ LCM of 15 and 25 = 75

⇒ Efficiency of A + B = 75 ÷ 15 = 5

⇒ Efficiency of A = 75 ÷ 25 = 3

⇒ Efficiency of B = 5 - 3 = 2

⇒ Time of B = 75 ÷ 2

⇒ Time of B = 37.5 days

∴ The correct answer is 37.5.

Given:

Selling price = 864

First discount = 20%

Second discount = 10%

Formula Used:

Selling price = Marked price × (1 - d1/100) × (1 - d2/100)

Calculation:

⇒ 864 = MP × 0.8 × 0.9

⇒ 864 = MP × 0.72

⇒ MP = 864 ÷ 0.72

⇒ MP = 1200

∴ The correct answer is 1200.