Quantitative Aptitude PYQs

General solution:

Given:

 

Calculations:

Sum (Mar–May 2023) = 7,000 + 6,000 + 9,000 = 22,000

Sum (Mar–May 2022) = 5,000 + 7,000 + 8,000 = 20,000

Increase = 22,000 − 20,000 = 2,000

Percentage increase = (2,000 / 20,000) × 100 = 10.00%

Hence, the correct answer is 10.00%.

Shortcut Trick

Equating total work: (P + Q) × 10d = (1.5P + 0.75Q) × 8d

⇒ 10P + 10Q = 12P + 6Q ⇒ 2P = 4Q ⇒ P : Q = 2 : 1

Since P takes 20 hr, Q takes 20 × (2/1) = 40 hr. Then R takes 40 − 10 = 30 hr.

P + R time = (20 × 30) ÷ (20 + 30) = 600 ÷ 50 = 12 hr.

∴ The correct answer is 12 hr.

Alternate Method

Given: P takes 20 hr. Work done in Case 1 = (P + Q) × 10d. Case 2 = (1.5P + 0.75Q) × 8d.

⇒ 10dP + 10dQ = 12dP + 6dQ

⇒ 4dQ = 2dP ⇒ P = 2Q

⇒ Since Time ∝ 1/Efficiency, Time taken by Q = 2 × Time taken by P = 2 × 20 = 40 hr.

⇒ Time taken by R = 40 − 10 = 30 hr.

⇒ Total Work = LCM(20, 30) = 60 units.

⇒ Efficiency of P = 60 ÷ 20 = 3 units/hr.

⇒ Efficiency of R = 60 ÷ 30 = 2 units/hr.

⇒ Combined Efficiency (P + R) = 3 + 2 = 5 units/hr.

⇒ Time taken by P and R together = 60 ÷ 5 = 12 hr.

∴ The correct answer is 12 hr.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a task. If the ratio of efficiencies of two people is x : y, the ratio of time taken by them is y :

x. Combined Work Formula

If person A takes 'a' hours and person B takes 'b' hours, they together complete the work in (a × b) ÷ (a + b) hours.

Work Invariance Principle

Total Work = Efficiency × Time. When the same work is done with different efficiencies, (E₁ × T₁) = (E₂ × T₂).

I: 14x2 + 21x - 10x - 15 = 0

⇒ 7x(2x + 3) - 5(2x + 3) = 0

⇒ (2x + 3) × (7x - 5) = 0

⇒ x = -3/2, 5/7

II: 20y2 - 31y + 12 = 0

⇒ 20y2 - 15y - 16y + 12 = 0

⇒ 5y(4y - 3) - 4(4y - 3) = 0

⇒ (5y - 4) × (4y - 3) = 0

⇒ y = 4/5, 3/4

∴ x < y

Shortcut Trick

I. x2 + 74x – 480= 0

There are three terms = a, b, and c

Where 'a' = value of x2, 'b' = Value of x, 'c' = 480

'c' is negative here.

II. y2 + 64y – 1280 = 0

There are also three terms = a, b, and c

Where 'a' = value of y2, 'b' = Value of y, 'c' = 1280

'c' is also negative here.

If the value of 'c' is negative in both statements, then the answer always is "x = y or the relation between x and y can’t be established."

The basic and conventional method is given below:

Given:

I. x2 + 74x – 480= 0       

II. y2 + 64y – 1280 = 0

Calculation:

I. x2 + 74x – 480= 0       

⇒ x² + 80x – 6x – 480 = 0

⇒ x(x + 80) – 6(x + 80) = 0

⇒ (x + 80)(x – 6) = 0

⇒ x = (-80), 6    

II. y2 + 64y – 1280 = 0

⇒ y² + 80y – 16y – 1280 = 0

⇒ y(y + 80) – 16(y + 80) = 0

⇒ (y + 80)(y – 16) = 0

⇒ y = (-80), 16

When we compared the values of X and Y in the table above, we found that there are THREE relations between X and Y i.e. >, = and <. So, a relation cannot be defined.

∴ After comparison all the values of x and y, the relation is “x = y or the relation between x and y can't be established.”

Given:

I. 4x² – 17x + 15 = 0

II. 3y² + 13y - 38 = 0

Calculation:

From I,

4x² – 17x + 15 = 0

⇒ 4x² – 12x - 5x + 15 = 0

⇒ 4x(x - 3) - 5(x - 3) = 0

⇒ (x - 3)(4x - 5) = 0

Taking,

⇒ (x - 3) = 0 or  (4x - 5) = 0

⇒ x = 3 or (5/4)

From II,

3y² + 13y - 38 = 0

⇒ 3y² + 19y - 6y - 38 = 0

⇒ y(3y + 19) - 2(3y + 19) = 0

⇒ (y - 2)(3y + 19) = 0

Taking,

⇒ (y - 2) = 0 or (3y + 19) = 0

⇒ y = 2 or (-19/3)

Comparison between x and y (via Tabulation):

 

∴ x = y or relation between x and y can not be established.

I. \((625)^\frac{{1}}{{4}} x + \sqrt{{1225}} = 155\)

⇒ \((5^4)^\frac{{1}}{{4}} x + 35 = 155\)

⇒ 5x = 155 – 35 ⇒ 5x = 120

⇒ \(x=\frac{{120}}{{5}} = 24\)

II. \(\sqrt{{196}} y + 13 = 279\)

⇒ 14y = 279 – 13 = 266

⇒ \(y = \frac{{266}}{{14}} = 19\)

Comparison between x and y (via Tabulation):

∴ Clearly x > y

I. 12x2 + 11x + 12 = 10x2 + 22x

⇒ 2x² – 11x + 12 = 0

⇒ 2x² – 8x – 3x + 12 = 0

⇒ 2x (x – 4) – 3 (x – 4) = 0

⇒ (x – 4) (2x – 3) = 0

⇒ x = 4 or 3/2

II. 13y2 – 18y + 3 = 9y2 – 10y

⇒ 4y² – 8y + 3 = 0

⇒ 4y² – 6y – 2y + 3 = 0

⇒ 2y (2y – 3) – 1 (2y – 3) = 0

⇒ y = 1/2 or 3/2 

Comparison between x and y (via Tabulation):

∴ Clearly x ≥ y

Shortcut Trick

Ratio of CSA to TSA = 2πrh : 2πr(h + r) = h : (h + r) = 5 : 8.

Given (h + r) = 24 cm, so 8 units = 24 ⇒ 1 unit = 3 cm.

Cylinder: h = 5 × 3 = 15 cm and r = 24 − 15 = 9 cm.

Cone: H = 15 − 3 = 12 cm and R = 9 + 1 = 10 cm.

Volume Ratio: (πr²h) : (1/3 πR²H) = (9² × 15) : (1/3 × 10² × 12) = 1215 : 400 = 243 : 80.

∴ The correct answer is 243:80.

Alternate Method

Given: CSA : TSA = 5 : 8, h + r = 24 cm.

Formula: Volume of Cylinder = πr²h, Volume of Cone = (1/3)πR²H.

⇒ CSA / TSA = (2πrh) / [2πr(h + r)] = h / (h + r) = 5 / 8

⇒ 8h = 5(h + r) ⇒ 3h = 5r ⇒ h/r = 5/3

⇒ Let h = 5x, r = 3x. Given h + r = 24 ⇒ 8x = 24 ⇒ x = 3

⇒ Cylinder: h = 15 cm, r = 9 cm

⇒ Cone: Height H = 15 − 3 = 12 cm, Radius R = 9 + 1 = 10 cm

⇒ Ratio = (π × 9 × 9 × 15) / [(1/3) × π × 10 × 10 × 12]

⇒ Ratio = (81 × 15) / (100 × 4) = 1215 / 400 = 243 / 80

∴ The correct answer is 243:80.

Additional Information

Cylinder Surface Area Formulas

Curved Surface Area (CSA) = 2πrh; Total Surface Area (TSA) = 2πr(h + r).

Volume Formulas

Volume of Cylinder = πr²h; Volume of Cone = 1/3 πr²h. If height and radius are same, Cylinder Volume = 3 × Cone Volume.

Cone Slant Height

Slant height (l) = √(r² + h²). CSA of cone = πrl and TSA = πr(l + r).

General Solution:

Given:

Total coupon users (A + B) across Sat, Sun, Mon = 8,000

Coupon users on Saturday = 4,200

Coupon users on Sunday = 2,400

Saturday: Coupon A : Coupon B = 5 : 2

Sunday: Coupon A users = 20% less than Saturday's A users

Monday: Coupon B users = 25% of Saturday's B users

Sunday coupon users (2,400) = 60% of total Sunday transactions

Total transactions on Saturday = total on Sunday + 500

Transactions on Monday with no coupon = 1,600

Total Monday coupon users = 8,000 − 4,200 − 2,400 = 1,400

Saturday (split A/B):

A : B = 5 : 2 ⇒ parts = 7

Coupon A (Sat) = (5/7) × 4200 = 3,000

Coupon B (Sat) = (2/7) × 4200 = 1,200

Sunday:

Coupon A (Sun) = 20% less than A(Sat) = 0.8 × 3000 = 2,400

Since total coupon users on Sunday = 2,400 ⇒ Coupon B (Sun) = 0

Monday:

Coupon B (Mon) = 25% of B(Sat) = 0.25 × 1200 = 300

Total coupon users Mon = 1,400 ⇒ Coupon A (Mon) = 1,400 − 300 = 1,100

Calculations:

Ratio: total Coupon B users (all days) : Coupon A users on Monday

Total Coupon B = B(Sat) + B(Sun) + B(Mon) = 1200 + 0 + 300 = 1,500

Coupon A on Monday = 1,100

Ratio = 1500 : 1100 = 15 : 11

Hence, the correct answer is 15 : 11.

General Solution:

Given:

Total coupon users (A + B) across Sat, Sun, Mon = 8,000

Coupon users on Saturday = 4,200

Coupon users on Sunday = 2,400

Saturday: Coupon A : Coupon B = 5 : 2

Sunday: Coupon A users = 20% less than Saturday's A users

Monday: Coupon B users = 25% of Saturday's B users

Sunday coupon users (2,400) = 60% of total Sunday transactions

Total transactions on Saturday = total on Sunday + 500

Transactions on Monday with no coupon = 1,600

Total Monday coupon users = 8,000 − 4,200 − 2,400 = 1,400

Saturday (split A/B):

A : B = 5 : 2 ⇒ parts = 7

Coupon A (Sat) = (5/7) × 4200 = 3,000

Coupon B (Sat) = (2/7) × 4200 = 1,200

Sunday:

Coupon A (Sun) = 20% less than A(Sat) = 0.8 × 3000 = 2,400

Since total coupon users on Sunday = 2,400 ⇒ Coupon B (Sun) = 0

Monday:

Coupon B (Mon) = 25% of B(Sat) = 0.25 × 1200 = 300

Total coupon users Mon = 1,400 ⇒ Coupon A (Mon) = 1,400 − 300 = 1,100​​​

Calculations:

Monday total = 3,000 ⇒ Tuesday total = 1.25 × 3000 = 3,750

Coupon% on Saturday = 4200 / 4500 = 14/15 = 93.333...%

Coupon users on Tuesday = (14/15) × 3750 = 3500

No-coupon Tuesday = 3750 − 3500 = 250

Hence, the correct answer is 250.

General Solution:

Given:

Total coupon users (A + B) across Sat, Sun, Mon = 8,000

Coupon users on Saturday = 4,200

Coupon users on Sunday = 2,400

Saturday: Coupon A : Coupon B = 5 : 2

Sunday: Coupon A users = 20% less than Saturday's A users

Monday: Coupon B users = 25% of Saturday's B users

Sunday coupon users (2,400) = 60% of total Sunday transactions

Total transactions on Saturday = total on Sunday + 500

Transactions on Monday with no coupon = 1,600

Total Monday coupon users = 8,000 − 4,200 − 2,400 = 1,400

Saturday (split A/B):

A : B = 5 : 2 ⇒ parts = 7

Coupon A (Sat) = (5/7) × 4200 = 3,000

Coupon B (Sat) = (2/7) × 4200 = 1,200

Sunday:

Coupon A (Sun) = 20% less than A(Sat) = 0.8 × 3000 = 2,400

Since total coupon users on Sunday = 2,400 ⇒ Coupon B (Sun) = 0

Monday:

Coupon B (Mon) = 25% of B(Sat) = 0.25 × 1200 = 300

Total coupon users Mon = 1,400 ⇒ Coupon A (Mon) = 1,400 − 300 = 1,100

Calculations:

Monday no-coupon = 1,600 (given) and coupon users Mon = 1,400 ⇒ total Mon = 1600 + 1400 = 3,000

Saturday total = 4,500

Reduction = (4500 − 3000) / 4500 = 1500 / 4500 = 1/3 = 33.33%

Hence, the correct answer is 33.33%

General Solution:

Given:

Total coupon users (A + B) across Sat, Sun, Mon = 8,000

Coupon users on Saturday = 4,200

Coupon users on Sunday = 2,400

Saturday: Coupon A : Coupon B = 5 : 2

Sunday: Coupon A users = 20% less than Saturday's A users

Monday: Coupon B users = 25% of Saturday's B users

Sunday coupon users (2,400) = 60% of total Sunday transactions

Total transactions on Saturday = total on Sunday + 500

Transactions on Monday with no coupon = 1,600

Total Monday coupon users = 8,000 − 4,200 − 2,400 = 1,400

Saturday (split A/B):

A : B = 5 : 2 ⇒ parts = 7

Coupon A (Sat) = (5/7) × 4200 = 3,000

Coupon B (Sat) = (2/7) × 4200 = 1,200

Sunday:

Coupon A (Sun) = 20% less than A(Sat) = 0.8 × 3000 = 2,400

Since total coupon users on Sunday = 2,400 ⇒ Coupon B (Sun) = 0

Monday:

Coupon B (Mon) = 25% of B(Sat) = 0.25 × 1200 = 300

Total coupon users Mon = 1,400 ⇒ Coupon A (Mon) = 1,400 − 300 = 1,100

Calculations:

No-coupon transactions: Saturday as % of Sunday

Sunday total transactions = 2400 / 0.60 = 4,000

No-coupon Sunday = 4000 − 2400 = 1,600

Saturday total transactions = Sunday + 500 = 4000 + 500 = 4,500

No-coupon Saturday = 4500 − 4200 = 300

Percentage = (300 ÷ 1600) × 100 = 18.75%

Hence, the correct answer is 18.75%

General Solution:

Given:

Total coupon users (A + B) across Sat, Sun, Mon = 8,000

Coupon users on Saturday = 4,200

Coupon users on Sunday = 2,400

Saturday: Coupon A : Coupon B = 5 : 2

Sunday: Coupon A users = 20% less than Saturday's A users

Monday: Coupon B users = 25% of Saturday's B users

Sunday coupon users (2,400) = 60% of total Sunday transactions

Total transactions on Saturday = total on Sunday + 500

Transactions on Monday with no coupon = 1,600

Total Monday coupon users = 8,000 − 4,200 − 2,400 = 1,400

Saturday (split A/B):

A : B = 5 : 2 ⇒ parts = 7

Coupon A (Sat) = (5/7) × 4200 = 3,000

Coupon B (Sat) = (2/7) × 4200 = 1,200

Sunday:

Coupon A (Sun) = 20% less than A(Sat) = 0.8 × 3000 = 2,400

Since total coupon users on Sunday = 2,400 ⇒ Coupon B (Sun) = 0

Monday:

Coupon B (Mon) = 25% of B(Sat) = 0.25 × 1200 = 300

Total coupon users Mon = 1,400 ⇒ Coupon A (Mon) = 1,400 − 300 = 1,100

Calculations:

Total number of Coupon A users across all three days:

Coupon A total = A(Sat) + A(Sun) + A(Mon) = 3000 + 2400 + 1100 = 6,500

Hence, the correct answer is 6,500

General solution:

Series I:

7

(7 × 2) = 14

(14 × 3) = 42

(42 × 4) = 168

(168 × 5) = 840

(840 × 6) = 5040

Thus the wrong number in series I = 8

Correct number in series I = 7

 

Series II:

(20 + 2³ )= 28

(28 + 3³ )= 55

(55 + 4³ )= 119

(119 + 5³ )= 244

(244 + 6³ )= 460

Thus, the wrong number in series I is 118

Correct number in series II is 119

Solution:

Correct number in series I = 7

Correct number in series II = 119

Require average = (7 + 119)/2 = 63

General solution:

Series I:

10

(10 × 11) = 110

(110 × 7) = 770

(770 × 5) = 3850

(3850 × 3) = 11550 

(11550 × 2) = 23100

Wrong number in series I is 8

Correct number in series I is 10

So, Wrong number II is 30

Correct number in series II is 35

Solution:

Wrong number in series I = 8

Wrong number in series II = 30

Required sum = (30 + 8) = 38

Calculation:

The series follows the following pattern

⇒ 89.5 + 20.5 = 110

⇒ 110 + 22.5 = 132.5

⇒ 132.5 + 24.5 = 157

⇒ 157 + 26.5 = 183.5

⇒ 183.5 + 28.5 = 212

∴ The wrong number is 193.5

Solution:

Series I: 

28

(28 ×  2) - 1 = 55

(55 ×  3) - 1 = 164

(164 ×  4) - 1 = 655

(655 ×  5 - 1) = 3274 

(3274 ×  6 - 1) = 19643

P = 55

(P - 8) = 47

Series II: 

47

47 + (3² + 3) = 59

59 + (5² + 5) = 89

89 + (7² + 7) = 145

145 + (11² + 11) = 277

277 +  (13² + 13) = 459

So the value of Q is 59 

From options

Option a)

(P - Q) = 39

(55 - 59) = - 4

Option a) is wrong 

Option b)

2P > (3Q - 10)

(2 ×  55) > (59 × 3 - 10)

(110) ⊁ (117 ) 

Option b) is wrong 

Option c) 

(3P + 4Q) < (2P + 5Q)

(3 ×  55 + 4 ×  59) < (2×  55 + 5 ×  59) 

(401) < (405)

Option c) is correct

Option d) 

 (2Q - P) = 34

(2 ×  59 - 55) = 63

So option d is wrong

Solution:

Series I:

100  + (4 ×  5) = 120

120 + (5 ×  6) = 150

150 + ( 6 ×  7) = 192

192 + (7 ×  8) = 248

248 + (8 × 9) = 320

Thus the value of P = 320

Series II: 

28    

(28 ×  1 - 3) = 25

(25 ×  2 - 3) = 47

(47 ×  3 - 3) = 138

(138 ×  4 - 3) = 549

(549 ×  5 - 3) = 2742    

Thus the value of Q = 28

From the options

Option a) 

(5Q - P) = 34

(5 ×  28 - 320) = (140 - 320) = (- 180)

Option a) is wrong 

Option b)

(3Q + P) > (2P - 4Q)

(3 ×  28 + 320) > (2 ×  320 - 4 ×  28) 

(404) > (528)

Option b is wrong

Option c)

(4P + Q) < (10Q - P)

(4 ×  320 + 28) < (28 ×  10 - 320)

(1308) < (- 40) 

Option c is wrong

Option d)

(3P - 10) > (8Q + 30)

(3 ×  320 - 10) > (28 ×  8 + 30) 

(960 - 10) > (254)

950 > 254 

Option d is correct

Shortcut Trick

Ratio of Speed of Boat (u) : Speed of Stream (v) = 4 : 1

Downstream speed (D) = 4 + 1 = 5 units; Upstream speed (U) = 4 − 1 = 3 units.

Total time = 100/5k + 30/3k = 20/k + 10/k = 30/k = 10 hours ⇒ k = 3.

Downstream speed z = 5 × 3 = 15 km/hr; Still water speed = 4 × 3 = 12 km/hr.

Time for 216 km = 216 ÷ 12 = 18 hours. Check options: 2(15) − 12 = 18.

∴ The correct answer is (2z - 12) hr.

Alternate Method

Given: Ratio of boat speed (u) to stream speed (v) = 4 :

1. Total distance DS = 100 km, US = 30 km. Total time = 10 hr.

Formula Used: Time = Distance ÷ Speed; Downstream (D) = u + v; Upstream (U) = u − v

⇒ Let u = 4k and v = k. Then D = 5k and U = 3k.

⇒ 100/(5k) + 30/(3k) = 10

⇒ 20/k + 10/k = 10 ⇒ 30/k = 10 ⇒ k = 3 km/hr.

⇒ Downstream speed z = 5k = 5 × 3 = 15 km/hr.

⇒ Speed of boat in still water (u) = 4k = 4 × 3 = 12 km/hr.

⇒ Time to travel 216 km in still water = 216 ÷ 12 = 18 hours.

⇒ Comparing with z = 15: 2z − 12 = 2(15) − 12 = 30 − 12 = 18.

∴ The correct answer is (2z - 12) hr.

Additional Information

Downstream and Upstream Speeds

Downstream Speed (D) = u + v and Upstream Speed (U) = u − v, where u is boat speed and v is stream speed.

Still Water Speed Calculation

The speed of the boat in still water can also be expressed as u = (D + U) ÷ 2.

Stream Speed Calculation

The speed of the stream can be expressed as v = (D − U) ÷ 2.

General Solution:

Known totals: A = 800, B = 1200, C = 1000, E = 900 ⇒ sum = 3,900 ⇒ D total = 5,400 − 3,900 = 1,500.

Use ratios to get Managers (M) and Clerks (Ck):

A (1:3): Managers = 800 × 1/4 = 200; Clerks = 600

B (1:5): Managers = 1200 × 1/6 = 200; Clerks = 1000

C: Managers given = 300; Clerks = 700

D (1:2): Managers = 1500 × 1/3 = 500; Clerks = 1000

E (2:7): Managers = 900 × 2/9 = 200; Clerks = 700

Use given male totals / female totals:

Company A: Total male = 480 ⇒ male managers = 200 − 80 = 120; male clerks = 480 − 120 = 360; female clerks = 600 − 360 = 240; female total = 320.

Company B: Female total = 540 ⇒ male total = 1200 − 540 = 660. Male managers = 200 − 120 = 80; male clerks = 660 − 80 = 580; female clerks = 1000 − 580 = 420.

Company C: Total male = 500 ⇒ male managers = 300 − 100 = 200; male clerks = 500 − 200 = 300; female clerks = 700 − 300 = 400.

Company D: Total male = 1050 ⇒ male managers = 500 − 150 = 350; male clerks = 1050 − 350 = 700; female clerks = 1000 − 700 = 300; female total = 450.

Company E: Total male = 360 (given), male clerks = 220 (given) ⇒ male managers = 360 − 220 = 140; female managers = 200 − 140 = 60; female clerks = 700 − 220 = 480; female total = 540.

Calculation:

Female clerks: A = 240, B = 420, C = 400, D = 300, E = 480.

Total = 240 + 420 + 400 + 300 + 480 = 1,840

Hence, the correct answer is 1,840.