Quantitative Aptitude PYQs

Shortcut Trick

Ratio of investments of A, B, and C is 2800 : 3600 : 4000 = 7 : 9 : 10.

Profit ratio = Investment × Time. Let C stay for x months.

Ratio of Profit (A : B : C) = (7 × 12) : (9 × 12) : (10 × x) = 84 : 108 : 10x.

Total Profit = 84 + 108 + 10x = 192 + 10x.

Given: C's share / Total share = 10x / (192 + 10x) = 5 / 17.

Solving: 170x = 5(192 + 10x) ⇒ 170x = 960 + 50x ⇒ 120x = 960 ⇒ x = 8.

∴ The correct answer is 8 months.

Alternate Method

Given → Investment of A = Rs. 2800, B = Rs. 3600, C = Rs. 4000. Time for A and B = 12 months. Time for C = x months.
Formulas: Profit Ratio = (I₁ × T₁) : (I₂ × T₂) : (I₃ × T₃) | Profit Share% = (Individual Profit / Total Profit) × 100

⇒ Profit A = 2800 × 12 = 33600
⇒ Profit B = 3600 × 12 = 43200
⇒ Profit C = 4000 × x = 4000x
⇒ Total Profit = 33600 + 43200 + 4000x = 76800 + 4000x
⇒ According to question: 4000x / (76800 + 4000x) = 5 / 17
⇒ 800x / (15360 + 800x) = 5 / 17 (Dividing numerator and denominator by 5)
⇒ 13600x = 5 × 15360 + 4000x
⇒ 9600x = 76800
⇒ x = 768 / 96 = 8.

∴ The correct answer is 8.

Additional Information

Partnership Basics

When time periods are different, the profit is distributed in the ratio of the product of capital and time (Investment × Time).

Ratio of Profit

If P is profit, I is investment, and T is time, then P₁ : P₂ = (I₁ × T₁) : (I₂ × T₂).

Shortcut Trick

Combined efficiency (A+B) = 1/12 and (A+B+C) = 1/6.

Efficiency of C = (A+B+C) − (A+B) = 1/6 − 1/12 = 1/12.

Time taken by C alone = 12 days. Given C = R + 8 ⇒ R = 4.

Efficiency of A = (3/4) × Efficiency of (A+B) = (3/4) × (1/12) = 1/16.

Time taken by A = 16 days = 4 + 12 = R + 12 days.

∴ The correct answer is R + 12 days.

Alternate Method

Given: Efficiency ratio A : B = 3 :

1. ⇒ Efficiency of C = (A+B+C) − (A+B) = 2 − 1 = 1 unit/day.

⇒ Time taken by C alone = Total Work ÷ Efficiency = 12 ÷ 1 = 12 days.

⇒ According to question, C takes R + 8 days ∴ R + 8 = 12 ⇒ R = 4.

⇒ Total efficiency of (A+B) = 1 unit/day. Dividing in ratio 3 : 1:

⇒ Efficiency of A = 1 × (3 ÷ (3+1)) = 3/4 = 0.75 units/day.

⇒ Time taken by A alone = 12 ÷ 0.75 = 16 days.

⇒ Expressing 16 in terms of R: 16 = 4 + 12 = R + 12.

∴ The correct answer is R + 12 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is x:y, the ratio of time taken is y:x.

Work and Wages

Wages are always distributed in the ratio of the work done or in the ratio of efficiencies if all persons work for the same duration.

Pipes and Cisterns

This concept follows the same logic as Time and Work, where an inlet pipe has positive efficiency and an outlet pipe has negative efficiency.

Difference between number of male teachers in school A and B is 10.

Ratio of number of male teachers in school A and B is 4:5.

So, we can say number of male teachers in school A is 4 × 10 = 40

Number of female teachers in school B is 5 ×10 = 50

Number of female teachers in school A is 40 + 4 = 44

Number of female teacher in school B is 50 × 13 /10 = 65

Number of male teachers in school C is 44 + 4 = 48

Number of female teachers in school C is 50 + 2 = 52

 

Calculation

Total number of teacher in school A and B together is 40 + 44 + 50 +65 = 199 = m +35

So, m = 199 -35 = 164

Difference between number of male teachers in school A and B is 10.

Ratio of number of male teachers in school A and B is 4:5.

So, we can say number of male teachers in school A is 4 × 10 = 40

Number of female teachers in school B is 5 ×10 = 50

Number of female teachers in school A is 40 + 4 = 44

Number of female teacher in school B is 50 × 13 /10 = 65

Number of male teachers in school C is 44 + 4 = 48

Number of female teachers in school C is 50 + 2 = 52

 

Calculation

Number of females married teacher in school B is 65 ×20/100 = 13

Number of females married teacher in school C is 52 ×25/100 = 13

So, required difference is 13 -13 = 0

Difference between number of male teachers in school A and B is 10.

Ratio of number of male teachers in school A and B is 4:5.

So, we can say number of male teachers in school A is 4 × 10 = 40

Number of female teachers in school B is 5 ×10 = 50

Number of female teachers in school A is 40 + 4 = 44

Number of female teacher in school B is 50 × 13 /10 = 65

Number of male teachers in school C is 44 + 4 = 48

Number of female teachers in school C is 50 + 2 = 52

 

Calculation

Total number of male teacher in school C and D is 41 × 2 = 82

Number of male teacher in school D is 82 – 48 = 34

So, total number male teacher in school B and D is 50 + 34  = 84

Difference between number of male teachers in school A and B is 10.

Ratio of number of male teachers in school A and B is 4:5.

So, we can say number of male teachers in school A is 4 × 10 = 40

Number of female teachers in school B is 5 ×10 = 50

Number of female teachers in school A is 40 + 4 = 44

Number of female teacher in school B is 50 × 13 /10 = 65

Number of male teachers in school C is 44 + 4 = 48

Number of female teachers in school C is 50 + 2 = 52

 

Calculation

Number of male sub staff in school B is 50 × 12/100 =6

Number of female sub staff in school B is  6 + 8 = 14

Difference between number of male teachers in school A and B is 10.

Ratio of number of male teachers in school A and B is 4:5.

So, we can say number of male teachers in school A is 4 × 10 = 40

Number of female teachers in school B is 5 ×10 = 50

Number of female teachers in school A is 40 + 4 = 44

Number of female teacher in school B is 50 × 13 /10 = 65

Number of male teachers in school C is 44 + 4 = 48

Number of female teachers in school C is 50 + 2 = 52

 

Calculation

Total age of all the teacher of school A is 48 × 40 + 44 × 34 = 3416

So, average age of teacher of school A is 3416 / 84 = 40.66

Shortcut Trick

Net change in total marks = (Sum of marks added) − (Sum of marks left)

Net change = 72.5 − (50.5 + 54.5) = 72.5 − 105 = −32.5

Let current number of students be n. Then initial students were n + 1 (as 2 left and 1 joined).

Total change = (New Avg × n) − (Old Avg × (n + 1)) ⇒ −32.5 = (65.5 × n) − (64 × (n + 1))

−32.5 = 1.5n − 64 ⇒ 1.5n = 31.5 ⇒ n = 21

∴ The correct answer is 21 students.

Alternate Method

Given: Initial Average = 64, Marks of leaving students = 50.5 and 54.5, Marks of joining student = 72.5, Average increase = 1.5.

Formula: Total = Average × Number of items

Let current number of students be x.

Number of students initially = x + 1 (Since 2 left and 1 joined, net change is −1).

⇒ Initial Total = 64 × (x + 1)

⇒ New Total = (64 + 1.5) × x = 65.5x

⇒ New Total = Initial Total − 50.5 − 54.5 + 72.5

⇒ 65.5x = 64(x + 1) − 105 + 72.5

⇒ 65.5x = 64x + 64 − 32.5

⇒ 65.5x − 64x = 31.5

⇒ 1.5x = 31.5

⇒ x = 31.5 ÷ 1.5 = 21

∴ The correct answer is 21 students.

Additional Information

Concept of Deviation

Net Change in Total = (Sum of added values − Sum of removed values). This net change is distributed among all current members to find the change in average.

Weighted Average

When two groups are combined, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Inclusion and Exclusion

When a person joins, New Average × New Total Count = Old Average × Old Total Count + New Member Value.

Shortcut Trick

Perimeter equality: 4 × (a + 10) = 2 × (3a + a)

4a + 40 = 8a ⇒ 4a = 40 ⇒ a = 10

Square side = 20; Rectangle = 30 × 10

Difference = |20² − (30 × 10)| = |400 − 300| = 100

∴ The correct answer is 100 sq. units.

Alternate Method

Given: Side of square = a + 10, Length of rectangle (L) = 3a, Breadth (B) = a.

Formula: Perimeter of square = 4 × side; Perimeter of rectangle = 2(L + B).

⇒ 4(a + 10) = 2(3a + a)

⇒ 4a + 40 = 2(4a)

⇒ 4a + 40 = 8a

⇒ 4a = 40 ⇒ a = 10

⇒ Side of square = 10 + 10 = 20 units

⇒ Dimensions of rectangle = 30 units and 10 units

⇒ Area of square = 20 × 20 = 400 sq. units

⇒ Area of rectangle = 30 × 10 = 300 sq. units

⇒ Difference = 400 − 300 = 100 sq. units

∴ The correct answer is 100 sq. units.

Additional Information

Square Properties

Area = side² and Perimeter = 4 × side.

All sides are equal and angles are 90°.

Rectangle Properties

Area = length × breadth and Perimeter = 2(length + breadth).

Isoperimetric Theorem

For a given perimeter, a square has a larger area than any other rectangle.

Shortcut Trick

Let Upstream speed be x km/hr. Then Downstream speed = x + 2 × Stream speed = x + 6 km/hr.

Time difference = 72/x − 72/(x + 6) = 2 hours.

Test options: If x = 12, then 72/12 − 72/18 = 6 − 4 = 2 hours. Condition satisfied.

∴ The correct answer is 12 km/hr.

Alternate Method

Given: Distance = 72 km, Speed of stream (v) = 3 km/hr, Time difference = 2 hours.

Formula: Time = Distance ÷ Speed

⇒ Let Speed of boat in still water = u km/hr.

⇒ Upstream speed = (u − 3) km/hr and Downstream speed = (u + 3) km/hr.

⇒ 72/(u − 3) − 72/(u + 3) = 2

⇒ 36 × [(u + 3) − (u − 3)] / [(u − 3)(u + 3)] = 1

⇒ 36 × 6 = u² − 9

⇒ 216 + 9 = u² ⇒ u² = 225 ⇒ u = 15 km/hr.

⇒ Upstream speed = u − 3 = 15 − 3 = 12 km/hr.

∴ The correct answer is 12 km/hr.

Additional Information

Downstream and Upstream Speed

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

Speed of Boat in Still Water

Boat speed (u) = (D + U) ÷ 2.

Speed of Current

Stream speed (v) = (D − U) ÷ 2.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

Building C:

Private sector employees in Building C = 135

40% of private sector employees move to Building D:

0.40 × 135 = 54

New private sector employees in Building C:

135 - 54 = 81

Building D:

Private sector employees in Building D = 48

54 private sector employees move from Building C to Building D:

48 + 54 = 102

New private sector employees in Building D: 102

Building A:

Public sector employees in Building A = 120

25% of public sector employees move to Building B:

0.25 × 120 = 30

New public sector employees in Building A:

120 - 30 = 90

Building B:

Public sector employees in Building B = 30

30 public sector employees move from Building A to Building B:

30 + 30 = 60

New public sector employees in Building B: 60

Building C:

Public sector employees in Building C = 90 (unchanged)

Private sector employees in Building C = 81 (updated)

Vacant flats in Building C = 75 (unchanged)

Total occupied flats in Building C = Public + Private

90 + 81 = 171

Building D:

Public sector employees in Building D = 60 (unchanged)

Private sector employees in Building D = 102 (updated)

Vacant flats in Building D = 132 (unchanged)

Total occupied flats in Building D = Public + Private

60 + 102 = 162

Ratio = Total occupied flats in Building C / Total occupied flats in Building D = 171 / 162

171:162 = 19:18

Thus, the new ratio of total occupied flats in Building C to Building D is 19:18.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

Building B:

Total flats = 100

Vacant flats = 20

Occupied flats = Total flats Vacant flats

Occupied flats in B = 100 - 20 = 80

Building D:

Total flats = 240

Vacant flats = 132

Occupied flats = Total flats - Vacant flats

Occupied flats in D = 240 - 132 = 108

Building B:

Average rent per occupied flat = ₹12,000

Total rent = Occupied flats × Average rent

Total rent in B = 80 × 12,000 = ₹9,60,000

Building D:

Average rent per occupied flat = ₹18,000

Total rent = Occupied flats × Average rent

Total rent in D = 108 × 18,000 = ₹19,44,000

Difference = Total rent in D - Total rent in B

Difference = 19,44,000 - 9,60,000 = ₹9,84,000

Thus, the difference between the total rent collected from Building B and Building D is ₹9,84,000.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

Total public sector employees in Building A = 120

Ratio of male to female employees in Building A = 5 : 3

Let the number of male employees = 5x

Let the number of female employees = 3x

Total employees = 5x + 3x = 8x

8x = 120

x = 15

Male employees in Building A = 5x = 5 × 15 = 75

Female employees in Building A = 3x = 3 × 15 = 45

Total public sector employees in Building C = 90

Ratio of male to female employees in Building C = 4 : 5

Let the number of male employees = 4y

Let the number of female employees = 5y

Total employees =4y + 5y = 9y

9y = 90

y = 10

Male employees in Building C = 4y = 4 × 10 = 40

Female employees in Building C = 5y = 5 × 10 = 50

Total public sector employees in Building A = 120

Total public sector employees in Building C = 90

Total employees in A and C = 120 + 90 = 210

Female employees in Building A = 45

Female employees in Building C = 50

Total female employees = 45 + 50 = 95

Percentage of female employees = (Total female employees / Total employees in A and C) × 100

Percentage = 95 / 210 × 100 = 45.24%

Thus, the percentage of female public sector employees in Buildings A and C together is 45.24%.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

From the table:

Total flats in Building D = 240

Percentage of vacant flats = 55%

Number of vacant flats = 55%of 240

Vacant flats = 0.55 × 240 = 132

75% of 132 vacant flats are converted into community halls:

Flats converted to community halls = 0.75 × 132 = 99

Remaining vacant flats after conversion:

Remaining vacant flats = 132 - 99 = 33

33.33% of the remaining 33 vacant flats are converted into small offices:

Flats converted to small offices = 13 × 33 = 11

Remaining vacant flats after conversion:

Remaining vacant flats = 33 - 11 = 22

Thus, the number of flats that remain vacant in Building D is 22.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

Building B:

Private sector employees = 50

Building D:

Public sector employees = 60

Increase in private sector employees in Building B by 40%:

Increase = 50 × 0.40 = 20

New private sector employees = 50 + 20 = 70

Decrease in public sector employees in Building D by 25%:

Decrease = 60 × 0.25 = 15

New public sector employees = 60 - 15 = 45

The new ratio of private sector employees in Building B to public sector employees in Building D is:

Ratio = Private sector employees in Building B / Public sector employees in Building D = 70 / 45

70 / 45 = 14 / 9

Thus, the new ratio is 14:9.

General Solution:

From the table:

Public sector employees in Building A = 120

Public sector employees in Building B = a

Public sector employees in Building C = 90

Public sector employees in Building D = 2a

Total public sector employees =120 + a + 90 + 2a = 300

210 + 3a = 300

3a = 90

a = 30

Building A:

Public sector employees = 120

Ratio of public to private = 3 : 2

Private sector employees = 2/3 × 120 = 80

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

120 + 80 + 0.20F = F

200 = 0.80F

F = 250

Total flats in Building A = 250

Building B:

Public sector employees =a = 30

Ratio of public to private = 3 : 5

Private sector employees = 5/3 × 30 = 50

Vacant flats = 20% of total flats

Let total flats = F

Vacant flats = 0.20F

Total flats = Public + Private + Vacant

30 + 50 + 0.20F = F

80 = 0.80F

F = 100

Total flats in Building B = 100

Building C:

Public sector employees = 90

Ratio of public to private = 2 : 3

Private sector employees = 3/2 × 90 = 135

Vacant flats = 25% of total flats

Let total flats = F

Vacant flats = 0.25F

Total flats = Public + Private + Vacant

90 + 135 + 0.25F = F

225 = 0.75F

F = 300

Total flats in Building C = 300

Building D:

Public sector employees = 2a = 60

Ratio of public to private = 5 : 4

Private sector employees = 4/5 × 60 = 48

Vacant flats = 55% of total flats

Let total flats = F

Vacant flats = 0.55F

Total flats = Public + Private + Vacant

60 + 48 + 0.55F = F

108 = 0.45F

F = 240

Total flats in Building D = 240

Thus,

 

Calculations:

Building E has the same total number of flats as Buildings B and C combined.

Total flats in Building B = 100

Total flats in Building C = 300

Total flats in Building E = 100 + 300 = 400.

The percentage of vacant flats in Building E is the average of the percentages in Buildings B and D.

Percentage of vacant flats in Building B = 20%

Percentage of vacant flats in Building D = 55%

 

Average percentage = (20 + 55) / 2 = 37.5%.

So, the percentage of vacant flats in Building E is 37.5%.

Total flats in Building E = 400

Percentage of vacant flats = 37.5%

Number of vacant flats = 37.5% × 400 = 0.375 × 400 = 150.

Thus, the Building E will have 150 vacant flats.

Shortcut Trick

Ratio of Individual Salary (Previous : Current) = 3 : 4

Ratio of Number of Employees (Previous : Current) = 5 : 3

Ratio of Total Expenditure (Previous : Current) = (3 × 5) : (4 × 3) = 15 : 12 = 5 : 4

Profit (Reduction in Expenditure) = 5 − 4 = 1 unit → ₹90000

Previous Expenditure = 5 units = 5 × ₹90000 = ₹450000

∴ The correct answer is ₹450000.

Alternate Method

Given: Ratio of salaries = 3 : 4 | Ratio of employees = 5 : 3 | Profit = ₹90000

Formula: Total Expenditure = Individual Salary × Number of Employees

⇒ Previous Expenditure = 3 × 5 = 15 units

⇒ Current Expenditure = 4 × 3 = 12 units

⇒ Reduction in Expenditure (Profit) = 15 − 12 = 3 units

⇒ 3 units = ₹90000

⇒ 1 unit = ₹30000

⇒ Previous Expenditure = 15 × ₹30000 = ₹450000

∴ The correct answer is ₹450000.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (Term 1 × Term 1 : Term 2 × Term 2), it is called a compound ratio. Here, total expenditure is the compound ratio of salary and staff count.

Ratio & Proportion Relationship

If A : B = a : b and C : D = c : d, then the product (A×C) : (B×D) = (a×c) : (b×d). This property helps in calculating total costs or values directly.

Profit as Expenditure Reduction

In terms of corporate budget, any decrease in total expenditure while keeping output constant is treated as an increase in the company's profit margin.

Shortcut Trick

Initial Ratio of Milk : Water = 12 : 13 (Total 25 units)

Final Ratio of Milk : Water = 9 : 16 (Total 25 units)

Since the total quantity remains the same (replaced with equal water), compare Milk change: 12 → 9 (Reduction of 3 units out of 12).

Fraction of mixture replaced = 3/12 = 1/4

Value of x = 1/4 × 200 = 50 litres.

∴ The correct answer is 50 lit.

Alternate Method

Given: Total volume = 200 lit, Initial Ratio = 12:13, Final Ratio = 9:16.

Step 1: Calculate initial Milk and Water.

⇒ Milk = (12/25) × 200 = 96 lit

⇒ Water = (13/25) × 200 = 104 lit

Step 2: After removing x lit mixture, remaining Milk.

⇒ Remaining Milk = 96 − (12/25)x

Step 3: After adding x lit Water, total volume remains 200 lit. Final Milk quantity:

⇒ Final Milk = (9/25) × 200 = 72 lit

Step 4: Equating Milk quantities.

⇒ 96 − (12/25)x = 72

⇒ 24 = (12/25)x

⇒ x = (24 × 25) ÷ 12 = 50 lit

∴ The correct answer is 50 lit.

Additional Information

Replacement Formula

Final Quantity = Initial Quantity × (1 − x/V)ⁿ, where x is quantity replaced, V is total volume, and n is number of operations.

Mixture Concentration

When a part of mixture is replaced with another component, the concentration of the non-replaced component decreases proportionally.

Alligation Application

Replacement problems can be solved using alligation by considering the added liquid as 0% of the original component.

Shortcut Trick

Ratio of investments of A and C = 35 : 53.

Difference in ratio units = 53 − 35 = 18 units.

Value of 18 units = Rs. 3600 ⇒ 1 unit = Rs. 200. Investment difference between A and C in terms of x = (x + 6500) − (x − 2500) = 9000.

Investment of B relative to A = (x + 2000) − (x − 2500) = 4500 more than A.

Profit of B = Profit of A + (4500 ÷ 18 × 1 unit value) = 35 × 200 + (4500 ÷ 500) × 200 = 7000 + 1800 = Rs. 8800. ∴ The correct answer is Rs. 8800.

Alternate Method

Given → Investment of A = (x − 2500), B = (x + 2000), C = (x + 6500). Ratio A : C = 35 : 53.
Formulas: Profit Share Ratio = Investment Ratio × Time Ratio | Profit share difference ∝ Investment difference (when time is constant).

⇒ (x − 2500) ÷ (x + 6500) = 35 ÷ 53
⇒ 53x − 132500 = 35x + 227500
⇒ 18x = 360000 ⇒ x = 20000.
⇒ Investment of A = 17500, B = 22000, C = 26500.
⇒ Ratio of Profit A : B : C = 17500 : 22000 : 26500 = 35 : 44 : 53.
⇒ Given difference (C − A) = 53 − 35 = 18 units = Rs. 3600. ⇒ 1 unit = 200.
⇒ Profit share of B = 44 units × 200 = Rs. 8800. ∴ The correct answer is Rs. 8800.

Additional Information

Partnership Basics

Profit is distributed in the ratio of the product of Investment and Time. If time is constant, Profit Ratio = Investment Ratio.

Variable Handling

When investments are given in terms of a variable x, use the provided ratio of two partners to find the value of x first.

Shortcut Trick

Efficiency Ratio A : B = 5 : 4

Total Work = (Combined Efficiency of A & B) × Time = (5 + 4) × 20/3 = 60 units

Efficiency of C: (B + C) efficiency = 60 ÷ (60/7) = 7 ⇒ C = 7 − 4 = 3 units

Work done by (A + C) in 6 min = (5 + 3) × 6 = 48 units

Remaining work = 60 − 48 = 12 units → x% = (12/60) × 100 = 20%

∴ The correct answer is 20%.

Alternate Method

Given: Efficiency ratio A : B = 5 : 4

⇒ Total Efficiency of A + B = 5 + 4 = 9 (Matches table efficiency)

⇒ Efficiency of B = 4

⇒ Efficiency of C = (B + C) − B = 7 − 4 = 3

⇒ Efficiency of A + C = 5 + 3 = 8

⇒ Work done by A and C in 6 min = 8 × 6 = 48 units

⇒ Remaining Work = Total Work − Work Done = 60 − 48 = 12 units

⇒ Value of x (Remaining %) = (Remaining Work / Total Work) × 100

⇒ x = (12 / 60) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Total Work and Efficiency

Total work is the product of Efficiency and Time. Efficiency is inversely proportional to the time taken to complete a task.

Pipes and Cisterns Logic

Filling pipes have positive efficiency, while outlet pipes (leaks) have negative efficiency. The net work done is the sum of their individual efficiencies.

Percentage Work

To find the percentage of work remaining, use the formula: (Remaining Work ÷ Total Work) × 100.