Quantitative Aptitude PYQs

Shortcut Trick

The Syrup is 20% of the mixture, which means the Milk (x) constitutes the remaining 80%.

Ratio of Syrup : Milk = 20% : 80% = 1 : 4

Given that 1 unit (Syrup) = 6 litres.

Therefore, 4 units (Milk) = 4 × 6 = 24 litres.

∴ The correct answer is 24 litres.

Alternate Method

Given:

Quantity of Syrup = 6 litres

Quantity of Milk = x litres

Syrup Percentage in the Mixture = 20%

Formula Used:

Percentage of a component = (Component Quantity ÷ Total Mixture Quantity) × 100

Calculations:

⇒ Total Mixture = (6 + x) litres

⇒ 20 = [6 ÷ (6 + x)] × 100

⇒ 20 ÷ 100 = 6 ÷ (6 + x)

⇒ 1 ÷ 5 = 6 ÷ (6 + x)

⇒ 6 + x = 6 × 5

⇒ 6 + x = 30

⇒ x = 30 − 6

⇒ x = 24 litres

∴ The correct answer is 24 litres.

Additional Information

Percentage Composition

The percentage of any ingredient in a mixture is calculated as (Part ÷ Whole) × 100. If one part is P%, the other part in a two-component mixture must be (100 − P)%.

Ratio and Proportion in Mixtures

The amount of substances in a mixture can be expressed as a ratio. If Syrup is 20% and Milk is 80%, the ratio is 20:80 or 1:4, which simplifies many algebraic steps.

Dilution Principle

When more solvent (like milk) is added to a solute (like syrup), the quantity of the solute remains constant while its percentage concentration in the total volume decreases.

Given:

Amit time = 15 days

Baban time = 25 days

Together work time = 5 days

Remaining work done by Amit + Chetan in 4 days

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Calculation:

⇒ LCM of 15, 25 = 75

⇒ Amit efficiency = 75 ÷ 15 = 5

⇒ Baban efficiency = 75 ÷ 25 = 3

⇒ (Amit + Baban) efficiency = 5 + 3 = 8

⇒ Work done in 5 days = 8 × 5 = 40

⇒ Remaining work = 75 - 40 = 35

⇒ (Amit + Chetan) efficiency = 35 ÷ 4

⇒ (Amit + Chetan) efficiency = 8.75

⇒ Chetan efficiency = 8.75 - 5

⇒ Chetan efficiency = 3.75

⇒ Time taken by Chetan = 75 ÷ 3.75

⇒ Time = 20 days

∴ The correct answer is 20 days.

Given:

(A + B) time = 10 days

C time = 40 days

A = B + C

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Calculation:

⇒ LCM of 10, 40 = 40

⇒ (A + B) efficiency = 40 ÷ 10 = 4

⇒ C efficiency = 40 ÷ 40 = 1

⇒ A efficiency = B + C

⇒ A = B + 1

⇒ A + B = 4

⇒ (B + 1) + B = 4

⇒ 2B + 1 = 4

⇒ 2B = 3

⇒ B = 1.5

⇒ A = 1.5 + 1 = 2.5

⇒ Time taken by A = 40 ÷ 2.5

⇒ Time = 16 days

∴ The correct answer is 16 days.

Given:

A time = 16 days

B time = 12 days

B worked = 8 days

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Calculation:

⇒ LCM of 16, 12 = 48

⇒ A efficiency = 48 ÷ 16 = 3

⇒ B efficiency = 48 ÷ 12 = 4

⇒ Work done by B in 8 days = 4 × 8 = 32

⇒ Remaining work = 48 - 32 = 16

⇒ Time taken by A = 16 ÷ 3

⇒ Time = 5 days 8 hours 

∴ The correct answer is 5 days 8 hours.

Given:

Perimeter of tile = 224 cm

Length : Breadth = 4 : 3

Floor length = 25.6 m = 2560 cm

Floor breadth = 14.4 m = 1440 cm

Formula Used:

Perimeter = 2 × (l + b)

Area = Length × Breadth

Calculation:

⇒ 2 × (l + b) = 224

⇒ l + b = 112

⇒ Let l = 4x and b = 3x

⇒ 4x + 3x = 112

⇒ 7x = 112

⇒ x = 16

⇒ l = 64 cm and b = 48 cm

⇒ Area of tile = 64 × 48 = 3072 cm²

⇒ Area of floor = 2560 × 1440 = 3686400 cm²

⇒ Number of tiles = 3686400 ÷ 3072

⇒ Number of tiles = 1200

∴ The correct answer is 1200.

Given:

Let side of square = x

Length of rectangle = x + 4

Breadth of rectangle = x - 2

Area difference = 48

Formula Used:

Area of square = x × x

Area of rectangle = Length × Breadth

Perimeter = 2 × (l + b)

Calculation:

⇒ (x + 4)(x - 2) - x × x = 48

⇒ (x² + 4x - 2x - 8) - x² = 48

⇒ 2x - 8 = 48

⇒ 2x = 56

⇒ x = 28

⇒ Perimeter of square = 4 × 28

⇒ Perimeter of square = 112 cm

⇒ Perimeter of rectangle = 2 × (28 + 4 + 28 - 2)

⇒ Perimeter of rectangle = 2 × (32 + 26)

⇒ Perimeter of rectangle = 2 × 58

⇒ Perimeter of rectangle = 116 cm

∴ The correct answer is 112 cm and 116 cm.

Given:

Radius of C₁ = 5

Radius of C₂ = 3

Distance between centers = 13

Formula used:

If distance between centers > sum of radii ⇒ 4 common tangents

Calculations:

Sum of radii

⇒ 5 + 3 = 8

Compare with distance

⇒ 13 > 8

Condition satisfied

⇒ Circles are separate (non-intersecting)

⇒ Number of common tangents = 4

∴ Total common tangents = 4

Given:

One angle = 84°

Remaining angles ratio = 4 : 3 : 5

Formula Used:

Sum of angles of quadrilateral = 360°

Calculation:

⇒ Sum of remaining angles = 360 - 84 = 276

⇒ 4x + 3x + 5x = 276

⇒ 12x = 276

⇒ x = 23

⇒ Angles = 4 × 23 = 92°, 3 × 23 = 69°, 5 × 23 = 115°

⇒ Difference = 115 - 69 = 46°

∴ The correct answer is 46°.

Given:

Slope = -3

Point = (0, -2)

Formula Used:

y = mx + c

Calculation:

⇒ -2 = -3 × 0 + c

⇒ c = -2

⇒ y = -3x - 2

∴ The correct answer is y = -3x - 2.

Given:

A time = 12 days

B time = 15 days

C time = 20 days

Formula Used:

Work = LCM of times

Efficiency = Work ÷ Time

Calculation:

⇒ LCM of 12, 15, 20 = 60

⇒ A efficiency = 60 ÷ 12 = 5

⇒ B efficiency = 60 ÷ 15 = 4

⇒ C efficiency = 60 ÷ 20 = 3

⇒ Work done in 3 days (A + B + C) = 5 + 4 + 3 = 12

⇒ Work done in 15 days = 12 × 5 = 60

⇒ Total work = 60

⇒ Time = 15 days

∴ The correct answer is 15 days.

Given:

Total cost = 4320

Cost per fan = 4320 ÷ 3 = 1440

Gain = 15%

Loss = 10%

Formula Used:

Selling Price = Cost Price × (1 ± Rate ÷ 100)

Profit% = [(Total SP - Total CP) ÷ Total CP] × 100

Calculation:

⇒ Total SP = 1440 × (115 ÷ 100) × 2 + 1440 × (90 ÷ 100)

⇒ Total SP = 1440 × 1.15 × 2 + 1440 × 0.9

⇒ Total SP = 3312 + 1296

⇒ Total SP = 4608

⇒ Profit = 4608 - 4320 = 288

⇒ Profit% = (288 ÷ 4320) × 100

⇒ Profit% = 6.67%

∴ The correct answer is 6.67% profit.

Given:

Marked Price = 10000

Discount₁ = 25%

Discount₂ = 2%

Formula Used:

Selling Price = Marked Price × (1 - d₁ ÷ 100) × (1 - d₂ ÷ 100)

Calculation:

⇒ Price after first discount = 10000 × (75 ÷ 100) = 7500

⇒ Final price = 7500 × (98 ÷ 100)

⇒ Final price = 7350

∴ The correct answer is 7350.

Given:

Selling Price = 1800

Discount = 4%

Formula Used:

Selling Price = Marked Price × (1 - Discount ÷ 100)

Calculation:

⇒ 1800 = Marked Price × (96 ÷ 100)

⇒ Marked Price = 1800 × (100 ÷ 96)

⇒ Marked Price = 180000 ÷ 96

⇒ Marked Price = 1875

∴ The correct answer is 1875.

Shortcut Trick

Using the Alligation Method to find the ratio of students:

Ratio (Old : New) = (Combined Avg − New Avg) : (Old Avg − Combined Avg)

⇒ Ratio = (19.5 − 18) : (20 − 19.5) = 1.5 : 0.5 = 3 : 1

Since 3 units correspond to 120 students, 1 unit = 120 ÷ 3 = 40.

∴ The correct answer is 40.

Alternate Method

Given:

Initial students (n₁) = 120 | Initial average (A₁) = 20 years

New students average (A₂) = 18 years | Combined average (A_m) = 19.5 years

Let the number of new students be x.

Total age of old students = 120 × 20 = 2400 years.

Total age of new students = x × 18 = 18x.

Total students in the school = 120 + x.

Total age of all students = (120 + x) × 19.5.

⇒ 2400 + 18x = 19.5(120 + x)

⇒ 2400 + 18x = 2340 + 19.5x

⇒ 2400 − 2340 = 19.5x − 18x

⇒ 60 = 1.5x

⇒ x = 60 ÷ 1.5 = 40.

∴ The correct answer is 40.

Additional Information

Weighted Average Formula

The combined average of two groups is given by A_m = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n is the quantity and A is the average.

Alligation Rule

The ratio of the weights is inversely proportional to the differences of their averages from the mean: n₁/n₂ = (A₂ − A_m) / (A_m − A₁).

Deviation Method

The sum of deviations of all observations from the combined average is always zero. This is used for faster calculation in complex average problems.

Given:

Total days = 180

Productive days = 60%

Formula Used:

Work ∝ Days × Daily output

Calculation:

⇒ Productive days = 180 × 60 ÷ 100 = 108

⇒ Let initial daily output = 1

⇒ Total work = 180 × 1 = 180

⇒ Required daily output = 180 ÷ 108

⇒ Required daily output = 1.6667

⇒ Increase = 1.6667 - 1 = 0.6667

⇒ Increase% = 0.6667 × 100

⇒ Increase% = 66.67%

∴ The correct answer is 66.67%.

Given:

Total students = 800

Math = 55%

Physics = 45%

Both = 30%

Remaining prefer CS = 25%

Formula Used:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Calculation:

⇒ Students liking Math or Physics = 55 + 45 - 30 = 70%

⇒ Students liking Math or Physics = 70 × 800 ÷ 100 = 560

⇒ Remaining students = 800 - 560 = 240

⇒ Students liking CS = 25 × 240 ÷ 100 = 60

⇒ Students with no preference = 240 - 60 = 180

∴ The correct answer is 180.

Shortcut Trick

A perfect square never ends with an odd number of zeros.

In the given options, 900 has two zeros (even), while 640 has only one zero (odd).

Therefore, 640 cannot be a perfect square without any further calculation.

∴ The correct answer is 640.

Alternate Method

Given: Numbers (1) 900, (2) 441, (3) 729, (4) 640.

We check the square roots of each number to identify the non-perfect square.

⇒ 900 = 30 × 30 = 30² (Perfect Square)

⇒ 441 = 21 × 21 = 21² (Perfect Square)

⇒ 729 = 27 × 27 = 27² (Perfect Square)

⇒ 640 = 64 × 10 = 8² × 10. Since 10 is not a perfect square, 640 is not a perfect square.

∴ The correct answer is 640.

Additional Information

Last Digit Rule

A perfect square always ends with 0, 1, 4, 5, 6, or 9 at the unit place. It never ends in 2, 3, 7, or 8.

Digital Sum Property

The digital sum (sum of digits until a single digit is reached) of a perfect square is always 1, 4, 7, or 9.

Trailing Zeros

If a number ends in zeros, it must have an even number of trailing zeros (2, 4, 6...) to be a perfect square.

Given:

Investment ratio of A and B = 4:5

Share of B in profit = Rs 1000

Formula Used:

Total profit = Share of B × (Sum of ratio ÷ Ratio of B)

Calculation:

⇒ Sum of ratio = 4 + 5 = 9

⇒ Total profit = 1000 × (9 ÷ 5)

⇒ Total profit = 1000 × 1.8

⇒ Total profit = 1800

∴ The correct answer is 1800.

Given:

A = 18000

B = 12000

C = 20000

Formula Used:

Profit ∝ Investment

Percentage share = (Individual ÷ Total) × 100

Calculation:

⇒ Total investment = 18000 + 12000 + 20000 = 50000

⇒ C’s share = (20000 ÷ 50000) × 100

⇒ C’s share = 40%

∴ The correct answer is 40%.

Given:

A investment = 20000

B investment = 30000

Difference in profit = 1000

Formula Used:

Profit ∝ Investment

Calculation:

⇒ Ratio A : B = 20000 : 30000 = 2 : 3

⇒ Difference in ratio = 3 - 2 = 1 part

⇒ 1 part = 1000

⇒ Total parts = 2 + 3 = 5

⇒ Total profit = 5 × 1000

⇒ Total profit = 5000

∴ The correct answer is 5000.