Quantitative Aptitude PYQs

Shortcut Trick

Let the initial average weight of 20 contestants be A.

Weight of contestant who left (L) = 20 × A − 19 × (A + 5.75) = A − 109.25

Weight of contestant who joined (J) = 20 × (A + 5.75 − 2.75) − 19 × (A + 5.75) = A − 49.25

Difference (J − L) = (A − 49.25) − (A − 109.25) = 109.25 − 49.25 = 60 kg.

∴ The correct answer is 60 kg.

Alternate Method

Given: Initial count = 20, Change when 1 left = +5.75 kg (on 19 people), Change when 1 joined = −2.75 kg (on 20 people).

⇒ Total initial weight = 20 × A

⇒ Total weight after 1 leaves = 19 × (A + 5.75) = 19A + 109.25

⇒ Weight of contestant who left (L) = 20A − (19A + 109.25) = A − 109.25

⇒ Total weight after new one joins = 20 × (A + 5.75 − 2.75) = 20 × (A + 3) = 20A + 60

⇒ Weight of contestant who joined (J) = (20A + 60) − (19A + 109.25) = A − 49.25

⇒ Difference = J − L = (A − 49.25) − (A − 109.25) = 109.25 − 49.25

⇒ Difference = 60 kg.

∴ The correct answer is 60 kg.

Additional Information

Average weight Change

When a person leaves, New Avg = (Total Sum − Left Weight) ÷ (n − 1). If the average increases, the person who left was lighter than the average.

Joiner Weight Formula

New Weight = Old Avg + (New count × Change in Avg). This helps in calculating the weight of a person entering a group.

Replacement Logic

If one leaves and one joins, New Weight − Old Weight = Total Strength × Total change in average.

Shortcut Trick Given average speed = 6.4 km/hr and speed a is an integer. Possible values for a must satisfy (a − 1) < 6.4 < a, so a = 7. Speeds are 7 km/hr (snowing) and 6 km/hr (not snowing). Let x be the fraction of distance covered while snowing: 6.4 = 1 / [x/7 + (1−x)/6]. Solving for x gives 7/16. ∴ The correct answer is 7/16. Alternate Method Given: Average speed = 6.4 km/hr, speeds are a and a − 1, where a is an integer. Formula Used: Average Speed = Total Distance ÷ Total Time ⇒ Since average speed (6.4) must lie between the two speeds: a − 1 < 6.4 < a ⇒ a = 7 km/hr and a − 1 = 6 km/hr ⇒ Let total distance be 1 unit and x be the fraction covered at 7 km/hr. ⇒ 6.4 = 1 ÷ [x/7 + (1 − x)/6] ⇒ 6.4 = 1 ÷ [(6x + 7 − 7x)/42] ⇒ 6.4 = 42 ÷ (7 − x) ⇒ 7 − x = 42 ÷ 6.4 ⇒ 7 − x = 420 ÷ 64 = 105 ÷ 16 ⇒ x = 7 − 105/16 = (112 − 105)/16 ⇒ x = 7/16 ∴ The correct answer is 7/16. Additional Information Average Speed Concept Average speed is the total distance divided by the total time. It always lies between the minimum and maximum speeds of the journey segments. Harmonic Mean If equal distances are covered, average speed is the harmonic mean of speeds: 2v1v2/(v1+v2). Integer Constraint When a value is specified as an integer, use the range of the average to narrow down possible values to the nearest whole numbers.

Calculation:

The series follows the following pattern:

⇒ 12 × 0.5 – 0.5 = 5.5

⇒ 5.5 × 1 + 1 = 6.5

⇒ 6.5 × 2 – 2 = 11

⇒ 11 × 4 + 4 = 48

⇒ 48 × 8 – 8 = 376

∴ The value of ? is 48

GIVEN:

930, 812, 506, ?, 272, 156

Calculation: 

⇒ 930 = 31² - 31

⇒ 812 = 292 - 29

⇒ 506 = 232 - 23

⇒ ? = 342 = 192 - 19

⇒ 272 = 172 - 17

⇒ 156 = 132 - 13

∴ The required value of "?" is 342.

Given:

6000, ?, 3000, 4500, 9000, 22500

Calculation:

⇒ 6000 × 0.5 = 3000

⇒ 3000 × 1 = 3000

⇒ 3000 × 1.5 = 4500

⇒ 4500 × 2 = 9000

⇒ 9000 × 2.5 = 22500

∴ ? = 3000

Calculation:

1512 + 12 = 1524

1524 + 16 = 1540

1540 + 23 = 1563

1563 + 33 = 1596

1596 + 46 = 1642

∴ ? = 1596

Given:

290 , ? , 374 , 421 , 474 , 533 

Calculation:

Here the logic follows as,

The series follows the addition of prime numbers from 41

Then, The second term will be,

= 290 + 41 

= 331

\(\therefore\) The correct answer is 331.

Shortcut Trick

Effective Rate of Interest for 2 years (compounded annually) = 20 + 20 + (20 × 20)/100 = 44%

Effective Rate of Interest for 1 year (compounded half-yearly) = 10 + 10 + (10 × 10)/100 = 21%

Difference in interest rates = 44% − 21% = 23%

23% of Sum = ₹4,600 → 1% = ₹200 → 100% = ₹20,000

∴ The correct answer is Rs. 20,000.

Alternate Method

Given: Rate (R) = 20% p.a., Difference in Interest = ₹4,600

Formula Used: CI = P[(1 + R/100)ⁿ − 1]

⇒ Case 1 (Annually, 2 yrs): CI₁ = P[(1 + 20/100)² − 1] = P[(1.2)² − 1] = 0.44P

⇒ Case 2 (Half-yearly, 1 yr): R = 10%, n = 2; CI₂ = P[(1 + 10/100)² − 1] = P[(1.1)² − 1] = 0.21P

⇒ Difference = 0.44P − 0.21P = 0.23P

⇒ 0.23P = 4600

⇒ P = 4600 / 0.23 = 20,000

∴ The correct answer is Rs. 20,000.

Additional Information

Effective Rate for Compounding

For two successive cycles of rate r%, the effective rate is [r + r + (r × r)/100]%. This applies to 2 years annually or 1 year half-yearly.

Compounding Half-yearly

When interest is compounded half-yearly, the Annual Rate is halved (R/2) and the Time period is doubled (2n).

Relationship between CI and SI

Difference between CI and SI for 2 years is P(R/100)². For 1 year, CI and SI are equal if compounded annually.

Shortcut Trick

Equate the total parts of each bag to their LCM (6, 10, 7) = 210, then scale by volume ratios 3 : 5 : 7.

Bag 1 (3 × 210): Lime = 3 × 175 = 525, Silica = 3 × 35 = 105

Bag 2 (5 × 210): Lime = 5 × 147 = 735, Silica = 5 × 63 = 315

Bag 3 (7 × 210): Lime = 7 × 90 = 630, Silica = 7 × 120 = 840

Total Lime : Total Silica = (525 + 735 + 630) : (105 + 315 + 840) = 1890 : 1260 = 3 : 2.

∴ The correct answer is 3 : 2.

Alternate Method

Given: Volume Ratio = 3 : 5 : 7. Ratios (L:S) = 5:1, 7:3, 3:4.

Formula: Total Lime = ∑ (Volume_i × Fraction of Lime_i)

⇒ Total Lime = 3 × (5/6) + 5 × (7/10) + 7 × (3/7)

⇒ Total Lime = 15/6 + 35/10 + 3 = 2.5 + 3.5 + 3 = 9

⇒ Total Silica = 3 × (1/6) + 5 × (3/10) + 7 × (4/7)

⇒ Total Silica = 3/6 + 15/10 + 4 = 0.5 + 1.5 + 4 = 6

⇒ Final Ratio = 9 : 6 = 3 : 2

∴ The correct answer is 3 : 2.

Additional Information

Mixture Balancing

To mix quantities of different ratios, first equalize the total parts by taking the LCM of the sums of the ratio parts before applying the volume weights.

Alligation for Three Components

When three or more mixtures are combined, the final ratio is the sum of the weighted averages of each individual component's concentration.

General solution:

2400 + 5500 + 280X + 32000 ×  X/100 = 15400

2400 + 5500 + 280X + 320X = 15400

600X = 7500

X = 25/2

Village A:

Total population = 2400 

Percentage of females = (50 - 12.5) = 37.5%

Number of females = (2400×  37.5/100) = 900

Number of males = 2400 - 900 = 1500

Village B:

Total population = 5500 

Percentage of females= (19.5 + 12.5) = 32%

Number of females = 5500 ×  (32/100) = 1760

Number of males = (5500 - 1760) = 3740

Village C:

Total population = (280 × 12.5) = 3500

Percentage of females = 40.2%

Number of females = 3500 ×  (40.2/100) = 1407

Number of males = (3500 - 1407) = 2093

Village D: 

Total population = 32000 ×  (12.5/100) = 4000

Percentage of females = 30%

Number of females = 4000 ×  (30/100) = 1200

Number of males = (4000 - 1200) = 2800

Solution:

Total number of males in village D = 2800

Number of graduated males working = 2800 ×   2/7 = 800

Number of non graduated males working = 2800 - 800 = 2000

Number of male job workers = 800 ×  (40/100) = 320

Required difference = (2000 - 320) = 1680

General solution:

2400 + 5500 + 280X + 32000 ×  X/100 = 15400

2400 + 5500 + 280X + 320X = 15400

600X = 7500

X = 25/2

Village A:

Total population = 2400 

Percentage of females = (50 - 12.5) = 37.5%

Number of females = (2400×  37.5/100) = 900

Number of males = 2400 - 900 = 1500

Village B:

Total population = 5500 

Percentage of females= (19.5 + 12.5) = 32%

Number of females = 5500 ×  (32/100) = 1760

Number of males = (5500 - 1760) = 3740

Village C:

Total population = (280 × 12.5) = 3500

Percentage of females = 40.2%

Number of females = 3500 ×  (40.2/100) = 1407

Number of males = (3500 - 1407) = 2093

Village D: 

Total population = 32000 ×  (12.5/100) = 4000

Percentage of females = 30%

Number of females = 4000 ×  (30/100) = 1200

Number of males = (4000 - 1200) = 2800

Solution:

Number of females in village B = 1760

Number of females in village E = 1760 ×  9/8 = 1980

Average number of females in village E and A = (1980 + 900)/2 = 1440

General solution:

2400 + 5500 + 280X + 32000 ×  X/100 = 15400

2400 + 5500 + 280X + 320X = 15400

600X = 7500

X = 25/2

Village A:

Total population = 2400 

Percentage of females = (50 - 12.5) = 37.5%

Number of females = (2400×  37.5/100) = 900

Number of males = 2400 - 900 = 1500

Village B:

Total population = 5500 

Percentage of females= (19.5 + 12.5) = 32%

Number of females = 5500 ×  (32/100) = 1760

Number of males = (5500 - 1760) = 3740

Village C:

Total population = (280 × 12.5) = 3500

Percentage of females = 40.2%

Number of females = 3500 ×  (40.2/100) = 1407

Number of males = (3500 - 1407) = 2093

Village D: 

Total population = 32000 ×  (12.5/100) = 4000

Percentage of females = 30%

Number of females = 4000 ×  (30/100) = 1200

Number of males = (4000 - 1200) = 2800

Solution:

Number of  males in village B = 3740

Number of males in village C = 2093

Number of females in village A = 900

Number of females in village B = 1760

Required difference = (5833 - 2660) = 3173

General solution:

2400 + 5500 + 280X + 32000 ×  X/100 = 15400

2400 + 5500 + 280X + 320X = 15400

600X = 7500

X = 25/2

Village A:

Total population = 2400 

Percentage of females = (50 - 12.5) = 37.5%

Number of females = (2400×  37.5/100) = 900

Number of males = 2400 - 900 = 1500

Village B:

Total population = 5500 

Percentage of females= (19.5 + 12.5) = 32%

Number of females = 5500 ×  (32/100) = 1760

Number of males = (5500 - 1760) = 3740

Village C:

Total population = (280 × 12.5) = 3500

Percentage of females = 40.2%

Number of females = 3500 ×  (40.2/100) = 1407

Number of males = (3500 - 1407) = 2093

Village D: 

Total population = 32000 ×  (12.5/100) = 4000

Percentage of females = 30%

Number of females = 4000 ×  (30/100) = 1200

Number of males = (4000 - 1200) = 2800

Solution:

Number of females in village D = 1200

Number of females in village F is 1200/75 ×  100 = 1600

Total number of males and females in village F = 3400

Number of males in village F = (3400 - 1600) = 1800

General solution:

2400 + 5500 + 280X + 32000 ×  X/100 = 15400

2400 + 5500 + 280X + 320X = 15400

600X = 7500

X = 25/2

Village A:

Total population = 2400 

Percentage of females = (50 - 12.5) = 37.5%

Number of females = (2400×  37.5/100) = 900

Number of males = 2400 - 900 = 1500

Village B:

Total population = 5500 

Percentage of females= (19.5 + 12.5) = 32%

Number of females = 5500 ×  (32/100) = 1760

Number of males = (5500 - 1760) = 3740

Village C:

Total population = (280 × 12.5) = 3500

Percentage of females = 40.2%

Number of females = 3500 ×  (40.2/100) = 1407

Number of males = (3500 - 1407) = 2093

Village D: 

Total population = 32000 ×  (12.5/100) = 4000

Percentage of females = 30%

Number of females = 4000 ×  (30/100) = 1200

Number of males = (4000 - 1200) = 2800

Solution:

Number of males in village B = 3740

Number of married males in village B = 1/5 × 3740 = 748/33

Number of unmarried males in village B = 3740 - 748 = 2992

Number of females in village A = 900

Number of married females in village A = 900/3 = 300

Number of unmarried females in village A = 900 - 300 = 600

Required ratio = 600 : 2992 = 75 : 374

Shortcut Trick

Ratio of investments of A and B is 2 : (5 − 2) = 2 : 3.

Ratio of profits of A and B is (7 − 3) : 3 = 4 : 3.

Profit ratio = (Investment × Time) ratio.

4 : 3 = (2 × 8) : (3 × T_B) ⇒ 4 : 3 = 16 : 3T_B.

12T_B = 48 ⇒ T_B = 4 months.

∴ The correct answer is 4 months.

Alternate Method

Given → Investment of A = 2/5 of Total | Profit of B = 3/7 of Total | Time of A = 8 months.
Formulas: Profit_A : Profit_B = (Capital_A × Time_A) : (Capital_B × Time_B)

⇒ Let total investment be 5 units. A's investment = 2 units, B's investment = 5 − 2 = 3 units.
⇒ Let total profit be 7 units. B's profit = 3 units, A's profit = 7 − 3 = 4 units.
⇒ According to partnership rule: (Investment_A × Time_A) / (Investment_B × Time_B) = Profit_A / Profit_B
⇒ (2 × 8) / (3 × T_B) = 4 / 3
⇒ 16 / 3T_B = 4 / 3
⇒ 3T_B × 4 = 16 × 3
⇒ 12T_B = 48
⇒ T_B = 4 months.

∴ The correct answer is 4 months.

Additional Information

Partnership Basics

In a partnership, the profit sharing ratio is directly proportional to the product of capital invested and the time period of investment.

Variable Ratios

If profits are shared in ratio P₁:P₂ and investments are C₁:C₂, then the time ratio is (P₁/C₁) : (P₂/C₂).

Shortcut Trick

Work done by P in 5 hours = 5/15 = 1/3

Work done by Q in (5 + 3) = 8 hours = 8/20 = 2/5

Work done by R in (5 + 3) = 8 hours = 1 − (1/3 + 2/5) = 1 − 11/15 = 4/15

Efficiency of R = (4/15) ÷ 8 = 1/30

∴ The correct answer is 30 hours.

Alternate Method

Given: P = 15 hours, Q = 20 hours, R = N hours.

All worked 5 hrs; P left; Q and R worked 3 more hrs.

⇒ Let efficiency of R be x units/hr.

⇒ Total work completed = 5 × (4 + 3 + x) + 3 × (3 + x) = 60

⇒ 5 × (7 + x) + 3 × (3 + x) = 60

⇒ 35 + 5x + 9 + 3x = 60

⇒ 44 + 8x = 60

⇒ 8x = 16

⇒ x = 2 units/hr

⇒ Time taken by R alone (N) = Total Work ÷ Efficiency of R

⇒ N = 60 ÷ 2 = 30 hours

∴ The correct answer is 30 hours.

Additional Information

Total Work Concept

Total Work is always assumed as the LCM of time taken by individual workers to simplify efficiency calculations.

Efficiency Formula

Efficiency = Total Work ÷ Time taken. Efficiency is inversely proportional to the time taken to complete the work.

Work Left/Remaining

When a person leaves, the remaining work must be divided by the combined efficiency of the remaining workers to find the extra time.

Shortcut Trick

Curved Surface Area of Hemisphere = 2πr² = 882π ⇒ r² = 441 ⇒ r = 21 cm.

Volume of milk in Hemisphere = (2/3)πr³.

Since this fills the Cylinder completely: (2/3)πr³ = πr²h.

Since radii are same: h = (2/3)r = (2/3) × 21 = 14 cm.

∴ The correct answer is 14 cm.

Alternate Method

Given:

Surface area of Hemispherical pot = 882π cm²

Radius of Cylindrical can = Radius of Hemisphere (r)

Formula Used:

1. Curved Surface Area of Hemisphere = 2πr²
2. Volume of Hemisphere = (2/3)πr³
3. Volume of Cylinder = πr²h

Calculations:

⇒ 2πr² = 882π

⇒ r² = 882 ÷ 2 = 441

⇒ r = √441 = 21 cm

As per the question, Volume of Hemisphere = Volume of Cylinder

⇒ (2/3)πr³ = πr²h

⇒ (2/3)r = h

⇒ h = (2/3) × 21

⇒ h = 14 cm

∴ The correct answer is 14 cm.

Additional Information

Total Surface Area of Hemisphere

The total surface area includes the curved part and the flat circular top, calculated as 3πr².

Cylinder Surface Area

Curved Surface Area = 2πrh and Total Surface Area = 2πr(h + r).

Volume Conservation

When a liquid is poured from one container to another, the volume remains constant unless stated otherwise.

Concept Used:

Follow the BODMAS rule according to the table given below:

Calculation:

14.972 × 4.99 ÷ 44.78 + 299.99 ÷ √5624.98 × 12.93 = ?

⇒ 15² × 5 ÷ 45 + 300 ÷ √5625 × 13 = ?

⇒ 225 × 1 ÷ 9 + 300 ÷ 75 × 13 = ?

⇒ 25 + 4 × 13 = ?

⇒ 25 + 52 = ?

⇒ 77 = ?

∴ Required value of ? is 77.

Correct answer  is: 1

Concept used:

1. If the digits after decimal in a given number is less than 5, then the digits after decimal are omitted and the rest number will be the approximate value.

2. If the digits after decimal in a given number is more than 5, then the digits after decimal are omitted and the rest of the number is increased by 1. The obtained number will be the approximate value.

3. The BODMAS rule should be followed to solve the question as per the given table:

Calculation:

{12960.31 ÷ 79.79 × (9.75)2} ÷ 89.89 + √195.79 - 10.23 = ?

⇒ {12960 ÷ 80 × 10²} ÷ 90 + √196 - 10 = ?

⇒ (162 × 100) ÷ 90 + 14 - 10 = ?

⇒ 16200 ÷ 90 + 14 - 10 = ?

⇒ 180 + 14 - 10 = ?

⇒ ? = 184

∴ The value of ? is 184

Correct  answer is: 4

Concept used:

1. If the digits after decimal in a given number is less than 5, then the digits after decimal are omitted and the rest number will be the approximate value.

2. If the digits after decimal in a given number is more than 5, then the digits after decimal are omitted and the rest of the number is increased by 1. The obtained number will be the approximate value.

3. The BODMAS rule should be followed to solve the question as per the given table:

Calculation:

14.66 × 22.28 - 344.81 ÷ 23.38 + (11.73)2 - 68.87 = ?

⇒ 15 × 22 - 345 ÷ 23 + 12² - 69 = ?

⇒ 15 × 22 - 15 + 144 - 69 =?

⇒ 330 - 15 + 144 -69 =?

⇒ 474 - 84 = ?

⇒ ? = 390

∴ The value of ? is 390