Quantitative Aptitude PYQs

Shortcut Trick

Curved Surface Area (4πr²) = 5544 ⇒ r² = 5544 ÷ (4 × 22/7) = 441 ⇒ r = 21 cm.

Cube side (a) = (64/336) × 21 = (4/21) × 21 = 4 cm.

Number of cubes (n) = (1/2 × Volume of Sphere) ÷ (Volume of Cube).

n = [1/2 × 4/3 × 22/7 × 21 × 21 × 21] ÷ [4 × 4 × 4] = 19404 ÷ 64 ≈ 303.18.

∴ The correct answer is 303 cubes.

Alternate Method

Given: Curved Surface Area of sphere = 5544 cm², Side of cube = (64/336) × r.

Formula Used: CSA = 4πr²; Volume of Sphere = (4/3)πr³; Volume of Cube = a³.

⇒ 4 × (22/7) × r² = 5544

⇒ r² = (5544 × 7) ÷ 88 = 63 × 7 = 441 ⇒ r = 21 cm.

⇒ Side of cube (a) = (64/336) × 21 = (8/42) × 21 = 4 cm.

⇒ Volume of 50% sphere = 1/2 × 4/3 × 22/7 × 21 × 21 × 21 = 2 × 22 × 21 × 21 = 19404 cm³.

⇒ Volume of one cube = 4³ = 64 cm³.

⇒ Max cubes = 19404 ÷ 64 = 303.1875.

∴ The correct answer is 303 cubes.

Additional Information

Volume Conservation

When a solid is melted to form new shapes, the total volume remains constant: n × V_new = V_old.

Sphere Properties

For a sphere of radius r, Surface Area = 4πr² and Volume = (4/3)πr³. For a hemisphere, CSA = 2πr² and TSA = 3πr².

Cube Mensuration

For a cube of side a, Volume = a³, Total Surface Area = 6a², and diagonal = a√3.

Common Calculation:

Given Data

The total number of boys and girls who played hockey, football, and volleyball are related as follows:

1. Boys (Hockey) : Girls (Football) = 5 : 3

2. Boys (Volleyball) = 2 × Girls (Football)

3. Girls (Volleyball) = Boys (Volleyball) - 30

4. Boys + Girls (Hockey) = 160

5. Boys (Football) = 25% more than Girls (Hockey)

6. Total Boys who played all three sports = 295

Let:

B_H = Boys who played Hockey

G_H = Girls who played Hockey

B_F = Boys who played Football

G_F = Girls who played Football

B_V = Boys who played Volleyball

G_V = Girls who played Volleyball

Using given relations:

G_F = (3/5) × B_H

B_V = 2 × G_F = (6/5) × B_H

G_V = B_V - 30 = (6/5)B_H - 30

B_H + G_H = 160 ⇒ G_H = 160 - B_H

B_F = (5/4) × G_H = (5/4)(160 - B_H)

Total boys = B_H + B_F + B_V = 295

⇒ B_H + [200 - (5/4)B_H] + (6/5)B_H = 295

Combine coefficients: 1 - 5/4 + 6/5 = 19/20

(19/20)B_H + 200 = 295 ⇒ (19/20)B_H = 95 ⇒ B_H = 100

Hence,

B_H = 100

G_H = 60

G_F = 60

B_F = 75

B_V = 120

G_V = 90

Check: B_H + B_F + B_V = 295 ✅

Calculation:

Required ratio = [120 + 90] : [100 + 75] = 210 : 175 = 42:35=6:5

Common Calculation:

Given Data

The total number of boys and girls who played hockey, football, and volleyball are related as follows:

1. Boys (Hockey) : Girls (Football) = 5 : 3

2. Boys (Volleyball) = 2 × Girls (Football)

3. Girls (Volleyball) = Boys (Volleyball) - 30

4. Boys + Girls (Hockey) = 160

5. Boys (Football) = 25% more than Girls (Hockey)

6. Total Boys who played all three sports = 295

Let:

B_H = Boys who played Hockey

G_H = Girls who played Hockey

B_F = Boys who played Football

G_F = Girls who played Football

B_V = Boys who played Volleyball

G_V = Girls who played Volleyball

Using given relations:

G_F = (3/5) × B_H

B_V = 2 × G_F = (6/5) × B_H

G_V = B_V - 30 = (6/5)B_H - 30

B_H + G_H = 160 ⇒ G_H = 160 - B_H

B_F = (5/4) × G_H = (5/4)(160 - B_H)

Total boys = B_H + B_F + B_V = 295

⇒ B_H + [200 - (5/4)B_H] + (6/5)B_H = 295

Combine coefficients: 1 - 5/4 + 6/5 = 19/20

(19/20)B_H + 200 = 295 ⇒ (19/20)B_H = 95 ⇒ B_H = 100

Hence,

B_H = 100

G_H = 60

G_F = 60

B_F = 75

B_V = 120

G_V = 90

Check: B_H + B_F + B_V = 295 ✅

Calculation:

Required difference = [60 +60+ 90] – [60  + 75 ]= 210 – 135 = 75

Common Calculation:

Given Data

The total number of boys and girls who played hockey, football, and volleyball are related as follows:

1. Boys (Hockey) : Girls (Football) = 5 : 3

2. Boys (Volleyball) = 2 × Girls (Football)

3. Girls (Volleyball) = Boys (Volleyball) - 30

4. Boys + Girls (Hockey) = 160

5. Boys (Football) = 25% more than Girls (Hockey)

6. Total Boys who played all three sports = 295

Let:

B_H = Boys who played Hockey

G_H = Girls who played Hockey

B_F = Boys who played Football

G_F = Girls who played Football

B_V = Boys who played Volleyball

G_V = Girls who played Volleyball

Using given relations:

G_F = (3/5) × B_H

B_V = 2 × G_F = (6/5) × B_H

G_V = B_V - 30 = (6/5)B_H - 30

B_H + G_H = 160 ⇒ G_H = 160 - B_H

B_F = (5/4) × G_H = (5/4)(160 - B_H)

Total boys = B_H + B_F + B_V = 295

⇒ B_H + [200 - (5/4)B_H] + (6/5)B_H = 295

Combine coefficients: 1 - 5/4 + 6/5 = 19/20

(19/20)B_H + 200 = 295 ⇒ (19/20)B_H = 95 ⇒ B_H = 100

Hence,

B_H = 100

G_H = 60

G_F = 60

B_F = 75

B_V = 120

G_V = 90

Check: B_H + B_F + B_V = 295 ✅

Calculation:

G_V = 90, B_F = 75

% Difference = ((G_V - B_F) / B_F) × 100 = (15/75) × 100 = 20%

Answer: 20% more

Common Calculation:

Given Data

The total number of boys and girls who played hockey, football, and volleyball are related as follows:

1. Boys (Hockey) : Girls (Football) = 5 : 3

2. Boys (Volleyball) = 2 × Girls (Football)

3. Girls (Volleyball) = Boys (Volleyball) - 30

4. Boys + Girls (Hockey) = 160

5. Boys (Football) = 25% more than Girls (Hockey)

6. Total Boys who played all three sports = 295

Let:

B_H = Boys who played Hockey

G_H = Girls who played Hockey

B_F = Boys who played Football

G_F = Girls who played Football

B_V = Boys who played Volleyball

G_V = Girls who played Volleyball

Using given relations:

G_F = (3/5) × B_H

B_V = 2 × G_F = (6/5) × B_H

G_V = B_V - 30 = (6/5)B_H - 30

B_H + G_H = 160 ⇒ G_H = 160 - B_H

B_F = (5/4) × G_H = (5/4)(160 - B_H)

Total boys = B_H + B_F + B_V = 295

⇒ B_H + [200 - (5/4)B_H] + (6/5)B_H = 295

Combine coefficients: 1 - 5/4 + 6/5 = 19/20

(19/20)B_H + 200 = 295 ⇒ (19/20)B_H = 95 ⇒ B_H = 100

Hence,

B_H = 100

G_H = 60

G_F = 60

B_F = 75

B_V = 120

G_V = 90

Check: B_H + B_F + B_V = 295 ✅

Calculation:

Boys who played chess = 20% more than girls who played hockey = 1.2 × 60 = 72

Girls who played chess = twice the number of boys who played football = 2 × 75 = 150

Difference = |150 - 72| = 78

Answer: 78

Common Calculation:

Given Data

The total number of boys and girls who played hockey, football, and volleyball are related as follows:

1. Boys (Hockey) : Girls (Football) = 5 : 3

2. Boys (Volleyball) = 2 × Girls (Football)

3. Girls (Volleyball) = Boys (Volleyball) - 30

4. Boys + Girls (Hockey) = 160

5. Boys (Football) = 25% more than Girls (Hockey)

6. Total Boys who played all three sports = 295

Let:

B_H = Boys who played Hockey

G_H = Girls who played Hockey

B_F = Boys who played Football

G_F = Girls who played Football

B_V = Boys who played Volleyball

G_V = Girls who played Volleyball

Using given relations:

G_F = (3/5) × B_H

B_V = 2 × G_F = (6/5) × B_H

G_V = B_V - 30 = (6/5)B_H - 30

B_H + G_H = 160 ⇒ G_H = 160 - B_H

B_F = (5/4) × G_H = (5/4)(160 - B_H)

Total boys = B_H + B_F + B_V = 295

⇒ B_H + [200 - (5/4)B_H] + (6/5)B_H = 295

Combine coefficients: 1 - 5/4 + 6/5 = 19/20

(19/20)B_H + 200 = 295 ⇒ (19/20)B_H = 95 ⇒ B_H = 100

Hence,

B_H = 100

G_H = 60

G_F = 60

B_F = 75

B_V = 120

G_V = 90

Check: B_H + B_F + B_V = 295 ✅

Calculation:

Ratio (Girls Cricket : Boys Football) = 11 : 15

⇒ G_C = (11/15) × 75 = 55

Total Cricket Players = 60% of Volleyball Players = 0.6 × 210 = 126

Hence Boys Cricket = 126 - 55 = 71

Answer: 71

Shortcut Trick

Let Train Length = L and Platform Length = P.

Case 2: (L + 3L) / S = 24 ⇒ 4L/S = 24 ⇒ L/S = 6.

Case 1: (L + 0.5P) / 0.4S = 30 ⇒ L + 0.5P = 12S.

Substitute L = 6S: 6S + 0.5P = 12S ⇒ 0.5P = 6S ⇒ P/S = 12/1.

∴ The correct answer is 12:1.

Alternate Method

Given:

Let Length of Train = L m and Original Speed = S m/s.

Let Length of Platform = P m.

Calculations:

⇒ According to Case 2: (L + 3L) / S = 24

⇒ 4L = 24S ⇒ L = 6S

⇒ According to Case 1: (L + 0.5P) / (0.4S) = 30

⇒ L + 0.5P = 30 × 0.4S

⇒ L + 0.5P = 12S

⇒ Substitute L = 6S: 6S + 0.5P = 12S

⇒ 0.5P = 6S

⇒ P / S = 6 / 0.5 = 12 / 1

⇒ P : S = 12 : 1

∴ The correct answer is 12:1.

Additional Information

Relative Speed

When two objects move in the same direction, relative speed = S₁ − S₂. In opposite directions, relative speed = S₁ + S₂.

Train Crossing Concepts

Total distance covered by a train crossing a platform = Length of Train + Length of Platform.

Conversion Formula

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

Common Calculation

Pattern in Series I:

Observing the pattern after the 3^rd term:

10 × 2 + 1 = 21

21 × 4 + 1 = 85

85 × 8 + 1 = 681

Multipliers are doubling each time: 2, 4, 8.

So before 10, the multiplier should be 1.

A × 1 + 1 = 10 → A = 9

✅ A = 9

Pattern in Series II:

Differences between consecutive terms:

52 - 44 = 8

64 - 52 = 12

80 - 64 = 16

Differences increase by +4 each time.

Next difference = 20 → B = 80 + 20 = 100

Next difference = 24 → 124 = 100 + 24 (Correct)

Calculation:

A = 9 → 3² (perfect square) → True

B = 10 × 9 = 90 So, it is false

B / (A + 1) = 100 / 10 = 10 → True

Only I and III true

Common Calculation

Pattern in Series I:

Observing the pattern after the 3^rd term:

10 × 2 + 1 = 21

21 × 4 + 1 = 85

85 × 8 + 1 = 681

Multipliers are doubling each time: 2, 4, 8.

So before 10, the multiplier should be 1.

A × 1 + 1 = 10 → A = 9

A = 9

Pattern in Series II:

Differences between consecutive terms:

52 - 44 = 8

64 - 52 = 12

80 - 64 = 16

Differences increase by +4 each time.

Next difference = 20 → B = 80 + 20 = 100

Next difference = 24 → 124 = 100 + 24 (Correct)

Calculation:

B + 69 = 169 → 13² → True

A = 9 → not prime → False

B + 25 = 125 → divisible by 25 → True

 

Statements

(i) and (iii) are true.

Common Calculation

[(386 / 2)] + 1 = 194, [(194 / 2) + 1] = 98 ………..

Rule: Each term = (previous term ÷ 2) + 1

834 → 834/2 = 417 → +1 = 418 (given)

418 → 418/2 = 209 → +1 = P = 210

210 → 210/2 = 105 → +1 = Q = 106

106 → 106/2 = 53 → +1 = R = 54

54 → 54/2 = 27 → +1 = S = 28

28 → 28/2 = 14 → +1 = 15 (matches final term)

Thus, P = 210, Q = 106, R = 54, S = 28.

Calculation

Find 50% of Q + R

Q + R = 106 + 54 = 160

50% of (Q + R) = 0.5 × 160 = 80

Answer: 80

Common Calculation

[(386 / 2)] + 1 = 194, [(194 / 2) + 1] = 98 ………..

Rule: Each term = (previous term ÷ 2) + 1

834 → 834/2 = 417 → +1 = 418 (given)

418 → 418/2 = 209 → +1 = P = 210

210 → 210/2 = 105 → +1 = Q = 106

106 → 106/2 = 53 → +1 = R = 54

54 → 54/2 = 27 → +1 = S = 28

28 → 28/2 = 14 → +1 = 15 (matches final term)

Thus, P = 210, Q = 106, R = 54, S = 28.

​Calculation

R + S = 54 + 28 = 82 → not 92

2Q = 212; P + (S/4) = 210 + 7 = 217 → not equal

(2P/15) = 420/15 = 28 → equals S

Only Statement III is True.

Common Calculation

[(386 / 2)] + 1 = 194, [(194 / 2) + 1] = 98 ………..

Rule: Each term = (previous term ÷ 2) + 1

834 → 834/2 = 417 → +1 = 418 (given)

418 → 418/2 = 209 → +1 = P = 210

210 → 210/2 = 105 → +1 = Q = 106

106 → 106/2 = 53 → +1 = R = 54

54 → 54/2 = 27 → +1 = S = 28

28 → 28/2 = 14 → +1 = 15 (matches final term)

Thus, P = 210, Q = 106, R = 54, S = 28.

Calculation:

P + R = 210 + 54 = 264 → even → not odd.

264 ÷ 4 = 66 (exactly divisible by 4)

R + S = 54 + 28 = 82 → 82 < 50 → False

Only Statement II is True.

Common Calculation

100% = 8000

So, 1% = 80.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

Calculation:

C + D + E = 1760 + 1640 + 2240 = 5640

2/3 of 5640 = (2/3) × 5640 = 3760

Answer: 3760

Common Calculation

100% = 8000

So, 1% = 80%.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

Calculation:

Let females in E = f _E.

Then males in E = f _E + 740.

Total E = 2240.

So, f _E + (f _E + 740) = 2240

⇒ 2f_E + 740 = 2240

⇒ 2f_E = 1500

⇒ f _E = 750

Males in B = 20% more than females in E = 1.2 × 750 = 900

Total B = 1000

⇒ females in B = 1000 - 900 = 100

Answer: 100 females in B

Common Calculation

100% = 8000

So, 1% = 80%.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

Calculation:

Total number of MALES in A, D and F together = 2760.

The difference between female participants in A and D together and female participants in F is 640, i.e.

(female _A + female _D) – female _F = 640.

Also given: female _D > female _F.

Let fa = females in A, fd = females in D, ff = females in F, mF = males in F.

Then males _A = 1360 - fa, males _D = 1640 - fd.

Given: males _A + males _D + mF = 2760.

So (1360 - fa) + (1640 - fd) + mF = 2760

⇒ 3000 - (fa + fd) + mF = 2760

⇒ mF = (fa + fd) - 240 ......(1)

Given: (fa + fd) - ff = 640 ⇒ ff = (fa + fd) - 640  ......(2)

Compute mF - ff:

mF - ff = [(fa + fd) - 240] - [(fa + fd) - 640] = 400

Therefore |mF - ff| = 400

Answer: 400

Common Calculation

100% = 8000

So, 1% = 80%.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

Calculation:

B + D = 1000 + 1640 = 2640

Fraction of whole = 2640 / 8000 = 0.33

Central angle = 0.33 × 360° = 118.8°

Common Calculation

100% = 8000

So, 1% = 80%.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

Calculation:

12.5% of A = 0.125 × 1360 = 170

Difference = C - 170 = 1760 - 170 = 1590

Answer: 1590

Common Calculation

100% = 8000

So, 1% = 80%.

According to pie chart is

[17% + x% + 22% + (x +8) % + (2x + 3) %] = 100%

So, 4x = 50

Or, x = 50/4 = 12.5

Percentage distribution of number of players in A, B, C, D, and E is 17%, 12.5%, 22%, 20.5%, and 28% respectively.

Number participate in competition in A is 17 × 80 = 1360

Similarly, Number of players in others competition B, C, D and E is 1000, 1760, 1640 and 2240.

 

Calculation:

|B - C| = |1000 - 1760| = 760

Ratio = A : |B - C| = 1360 : 760.

⇒ 34 : 19

Answer: 34 : 19

1. 4x² – 71x + 177 = 0

4x² – 12x – 59x + 177 = 0

4x(x – 3) – 59(x – 3) = 0

(4x – 59) (x – 3) = 0

x = 59/4, 3

2. y² = (25y – 33)/2

2y² – 25y + 33 = 0

2y² – 22y – 3y + 33 = 0

2y(y – 11) – 3(y – 11) = 0

(2y – 3) (y – 11) = 0

y = 3/2, 11

 

∴ x = y or relation doesn’t exist

i. 50x² - 45x + 9 = 0

⇒ 50x² - 15x - 30x + 9 = 0

⇒ (5x - 3) (10x - 3) = 0

⇒ x = 3/5, 3/10

 

ii. 25y² - 35y + 12 = 0

⇒ 25y² - 15y - 20y + 12 = 0

⇒ (5y - 4) (5y - 3) = 0

⇒ y = 4/5, 3/5

 

 

Hence, x ≤ y.