Quantitative Aptitude PYQs

Given:

(R + 3) : (H + 3) = 7 : 8

(R − 3) : (H − 9) = 5 : 4

Formula used:

Ratio ⇒ Cross multiplication

Calculations:

⇒ (R + 3)/(H + 3) = 7/8

⇒ 8(R + 3) = 7(H + 3)

⇒ 8R + 24 = 7H + 21

⇒ 8R − 7H = -3 ........(1)

⇒ (R − 3)/(H − 9) = 5/4

⇒ 4(R − 3) = 5(H − 9)

⇒ 4R − 12 = 5H − 45

⇒ 4R − 5H = -33 ........(2)

Now solve (1) and (2):

⇒ Multiply (2) by 2:

⇒ 8R − 10H = -66

Subtract (1):

⇒ (8R − 10H) − (8R − 7H) = -66 − (-3)

⇒ -3H = -63

⇒ H = 21

⇒ Age of H before 5 years = 21 − 5 = 16

∴ Final answer = 16 years

Shortcut Trick

Approximate Length (L) = 24 m and Side of Square (a) = (5/4) × 24 = 30 m.

Area of Square = 30² = 900 m².

Area of Rectangle = Area of Square − 444 = 900 − 444 = 456 m².

Breadth (B) = Area ÷ L = 456 ÷ 24 = 19 m ≈ 20 m.

∴ The correct answer is 20 m.

Alternate Method

Given: Length (L) ≈ 24 m, Length : Side = 4 : 5, Area Difference ≈ 444 m².

⇒ Let Length = 4x and Side of Square = 5x.

⇒ 4x = 24 ⇒ x = 6.

⇒ Side of Square (a) = 5 × 6 = 30 m.

⇒ Area of Square = a² = 30² = 900 m².

⇒ Area of Rectangle = Area of Square − 444.23 ≈ 900 − 444 = 456 m².

⇒ Area of Rectangle = Length × Breadth.

⇒ 24 × Breadth = 456.

⇒ Breadth = 456 ÷ 24 = 19 m.

⇒ Closest approximation among options is 20 m.

∴ The correct answer is 20 m.

Additional Information

Mensuration Formulas

Area of Rectangle = Length × Breadth; Area of Square = Side². Perimeter of Rectangle = 2 × (Length + Breadth).

Approximation Rule

When dealing with decimals like 24.23 or 444.23 in competitive exams, round to the nearest whole number to simplify calculations unless options are very close.

Given:

Total C (X + Y) = 84

Total X (A + B + C) = 95

Y(C) = Y(B) + 5

Y(A) = X(A) + 4

Y(C) = X(B) + n, where n = 32 = 9

Y(A) : Y(B) = 3 : 4

Let Y(A) = 3k, Y(B) = 4k

⇒ Y(C) = 4k + 5

Also,

⇒ Y(C) = X(B) + 9

⇒ X(B) = Y(C) − 9 = (4k + 5 − 9) = 4k − 4

Now,

Total C = X(C) + Y(C) = 84

⇒ X(C) = 84 − (4k + 5) = 79 − 4k

Now total X:

X(A) + X(B) + X(C) = 95

⇒ (3k − 4) + (4k − 4) + (79 − 4k) = 95

⇒ 3k − 4 + 4k − 4 + 79 − 4k = 95

⇒ 3k + 71 = 95

⇒ 3k = 24

⇒ k = 8

Final values:

X(A) = 20, X(B) = 28, X(C) = 47

Y(A) = 24, Y(B) = 32, Y(C) = 37

Calculation:

⇒ New X = 3/7 × 84 = 36

⇒ New Y = 4/7 × 84 = 48

⇒ X has 47 but needs 36

⇒ Shift = 47 − 36 = 11

∴ 11 books shifted from X to Y

Given:

Total C (X + Y) = 84

Total X (A + B + C) = 95

Y(C) = Y(B) + 5

Y(A) = X(A) + 4

Y(C) = X(B) + n, where n = 32 = 9

Y(A) : Y(B) = 3 : 4

Let Y(A) = 3k, Y(B) = 4k

⇒ Y(C) = 4k + 5

Also,

⇒ Y(C) = X(B) + 9

⇒ X(B) = Y(C) − 9 = (4k + 5 − 9) = 4k − 4

Now,

Total C = X(C) + Y(C) = 84

⇒ X(C) = 84 − (4k + 5) = 79 − 4k

Now total X:

X(A) + X(B) + X(C) = 95

⇒ (3k − 4) + (4k − 4) + (79 − 4k) = 95

⇒ 3k − 4 + 4k − 4 + 79 − 4k = 95

⇒ 3k + 71 = 95

⇒ 3k = 24

⇒ k = 8

Final values:

X(A) = 20, X(B) = 28, X(C) = 47

Y(A) = 24, Y(B) = 32, Y(C) = 37

Calculation:

⇒ X sells

B = 25% of 28 = 7 → unsold = 21

A = 20% of 20 = 4 → unsold = 16

⇒ X unsold = 21 + 16 = 37

⇒ Y sells

A = 25% of 24 = 6 → unsold = 18

B = 50% of 32 = 16 → unsold = 16

⇒ Y unsold = 18 + 16 = 34

⇒ Total unsold = 37 + 34 = 71

∴ Total unsold books = 71

Given:

Total C (X + Y) = 84

Total X (A + B + C) = 95

Y(C) = Y(B) + 5

Y(A) = X(A) + 4

Y(C) = X(B) + n, where n = 3² = 9

Y(A) : Y(B) = 3 : 4

Let Y(A) = 3k, Y(B) = 4k

⇒ Y(C) = 4k + 5

Also,

⇒ Y(C) = X(B) + 9

⇒ X(B) = Y(C) − 9 = (4k + 5 − 9) = 4k − 4

Now,

Total C = X(C) + Y(C) = 84

⇒ X(C) = 84 − (4k + 5) = 79 − 4k

Now total X:

X(A) + X(B) + X(C) = 95

⇒ (3k − 4) + (4k − 4) + (79 − 4k) = 95

⇒ 3k − 4 + 4k − 4 + 79 − 4k = 95

⇒ 3k + 71 = 95

⇒ 3k = 24

⇒ k = 8

Final values:

X(A) = 20, X(B) = 28, X(C) = 47

Y(A) = 24, Y(B) = 32, Y(C) = 37

Calculation:

Total A = 20 + 24 = 44

Total B = 28 + 32 = 60

⇒ Difference = 60 − 44 = 16

∴ Answer = 16

Given:

Total C (X + Y) = 84

Total X (A + B + C) = 95

Y(C) = Y(B) + 5

Y(A) = X(A) + 4

Y(C) = X(B) + n, where n = 32 = 9

Y(A) : Y(B) = 3 : 4

Let Y(A) = 3k, Y(B) = 4k

⇒ Y(C) = 4k + 5

Also,

⇒ Y(C) = X(B) + 9

⇒ X(B) = Y(C) − 9 = (4k + 5 − 9) = 4k − 4

Now,

Total C = X(C) + Y(C) = 84

⇒ X(C) = 84 − (4k + 5) = 79 − 4k

Now total X:

X(A) + X(B) + X(C) = 95

⇒ (3k − 4) + (4k − 4) + (79 − 4k) = 95

⇒ 3k − 4 + 4k − 4 + 79 − 4k = 95

⇒ 3k + 71 = 95

⇒ 3k = 24

⇒ k = 8

Final values:

X(A) = 20, X(B) = 28, X(C) = 47

Y(A) = 24, Y(B) = 32, Y(C) = 37

Calculation:

⇒ X(A) = 20 − 25% × 20 = 20 − 5 = 15

⇒ X(B) = 28 − 25% × 28 = 28 − 7 = 21

⇒ X(C) = 47

⇒ Total X = 15 + 21 + 47 = 83

⇒ Y(A) = 24 − 25% × 24 = 24 − 6 = 18

⇒ Y(B) = 32 − 25% × 32 = 32 − 8 = 24

⇒ Y(C) = 37

⇒ Total Y = 18 + 24 + 37 = 79

⇒ Ratio = 83 : 79

∴ Required Ratio = 83 : 79

Shortcut Trick

Let Cost Price (CP) = 100 units. Then Marked Price (MP) = 140 units.

Case 1: SP₁ = 140 × 90% = 126. Profit = 126 − 100 = 26 units. Given 26 units = Rs. 104.

Multiplier: 1 unit = 104 ÷ 26 = Rs. 4. Case 2: SP₂ = 140 × 80% = 112. Profit = 112 − 100 = 12 units.

New Profit = 12 × 4 = Rs. 48. ∴ The correct answer is Rs. 48.

Alternate Method

Given: Markup% = 40%, Discount₁ = 10%, Profit₁ = Rs. 104, Discount₂ = 20%.
Formulas: SP = MP × (100 − D%) ÷ 100 | Profit = SP − CP.

⇒ Let CP = x. MP = 1.4x.
⇒ SP₁ = 1.4x × 0.9 = 1.26x.
⇒ Profit₁ = 1.26x − x = 0.26x.
⇒ 0.26x = 104 ⇒ x = 104 ÷ 0.26 = 400.
⇒ Case 2: SP₂ = MP × 0.8 = (1.4 × 400) × 0.8 = 560 × 0.8 = 448.
⇒ New Profit = SP₂ − CP = 448 − 400 = 48.

∴ The correct answer is Rs. 48.

Additional Information

Marked Price & Discount

MP is the price printed on the label. Discount is always calculated on MP: SP = MP − Discount.

Effective Profit Percentage

Effective Profit% = Markup% − Discount% − (Markup × Discount)/100.

Shortcut Trick

Assume Total CP = 1000 (10 units at 100 each). MP per unit = 150.

Net discount = 20 + 10 − (20 × 10 ÷ 100) = 28%.

Discounted SP per unit = 150 × 72 ÷ 100 = 108.

1 unit sold at 20% loss = 80. 9 units sold at 108 = 972.

Total SP = 80 + 972 = 1052. Profit% = (1052 − 1000) ÷ 1000 × 100 = 5.2%.

∴ The correct answer is 5.2%.

Alternate Method

Given → Markup% = 50%, Successive Discounts = 20% and 10%, Defective Goods = 10% sold at 20% loss.
Formulas: SP = CP × (100 + P%) ÷ 100 | Successive Discount = a + b − ab ÷ 100 | Net Profit% = (Total SP − Total CP) ÷ Total CP × 100.

⇒ Let total goods be 100 units at Rs. 100 each. Total CP = 10000.
⇒ MP per unit = 100 + 50% of 100 = 150.
⇒ Discounted SP per unit = 150 × (80/100) × (90/100) = Rs. 108.
⇒ Case 1: 10% goods (10 units) at 20% loss. SP₁ = 10 × (100 − 20) = 800.
⇒ Case 2: Remaining 90% goods (90 units) at discounted price. SP₂ = 90 × 108 = 9720.
⇒ Total SP = 800 + 9720 = 10520.
⇒ Effective Profit% = [(10520 − 10000) ÷ 10000] × 100 = 5.2%.

∴ The correct answer is 5.2%.

Additional Information

Markup and Discount Relationship

The relationship between CP and MP is given by CP/MP = (100 − D%)/(100 + P%).

Successive Discounts

Two discounts of x% and y% result in a single equivalent discount of (x + y − xy/100)%.

Shortcut Trick

Initial Ratio Boys : Girls = 7 : 5. Let them be 700 units and 500 units.

New Boys = 700 − 20% = 560 units. New Girls = 500 − 10% = 450 units.

Difference = 560 − 450 = 110 units → 99 students. So, 1 unit = 0.9.

Initial Total Students = (700 + 500) × 0.9 = 1200 × 0.9 = 1080.

∴ The correct answer is 1080 students.

Alternate Method

Given: Ratio Boys : Girls = 7 : 5. Let Boys = 7x and Girls = 5x.

⇒ Remaining Boys = 7x × (100 − 20)% = 7x × 0.8 = 5.6x

⇒ Remaining Girls = 5x × (100 − 10)% = 5x × 0.9 = 4.5x

⇒ Difference = 5.6x − 4.5x = 1.1x

⇒ 1.1x = 99

⇒ x = 99 ÷ 1.1 = 90

⇒ Initial total students = 7x + 5x = 12x

⇒ Total students = 12 × 90 = 1080

∴ The correct answer is 1080 students.

Additional Information

Percentage Change in Ratio

When components of a ratio increase or decrease, the new values are New = Old × (1 ± r/100).

Base Value Concept

Always assume base values like 100 or multiples of the ratio to avoid complex decimal calculations in competitive exams.

Ratio and Proportion

If A : B = x : y, then total units = (x + y). Individual values are found by multiplying the common factor k.

Calculation:

((72.08 + 144.23 × 4.12) ÷ 18.16 + 28.23)

⇒ ((72 + 144 × 4) ÷ 18 + 28)

⇒ (72 + 576) ÷ 18 + 28

⇒ 648 ÷ 18 + 28

⇒ 36 + 28 = 64

Approximate value is 64

Calculation:

(40.23% of (125.28 ÷ 24.69 × 4.36 + 14.77))

⇒ 40% of (125 ÷ 25 × 4 + 15)

⇒ 40% of (5 × 4 + 15)

⇒ 40% of 35 = 14

Approximate value is 14

Calculation:

(7.14² + 162.012 ÷ 3.09² − 7.12 × 9.16)

⇒ (7² + 162 ÷ 3² − 7 × 9)

⇒ 49 + 162 ÷ 9 − 63

⇒ 49 + 18 − 63 = 4

Approximate value is 4

Calculation:

(80.12% of 600.12 + 191.96 ÷ 32.02)

⇒ (80% of 600 + 192 ÷ 32)

⇒ 480 + 6 = 486

⇒ √36 ≈ 6

⇒ 486 = 6 × ?

⇒ ? = 81

Approximate value is 81

Calculation:

(12.03 – [26.06 – {2.18 + 5.25 × (5.97 – 3.13)}])

⇒ (12 – [26 – {2 + 5 × (6 – 3)}])

⇒ 12 – [26 – (2 + 15)]

⇒ 12 – (26 – 17)

⇒ 12 – 9 = 3

Approximate value is 3

Shortcut Trick

Ratio of Boat Speed (u) to Stream Speed (v) = 7 : 2

Downstream ratio = 7 + 2 = 9 units | Upstream ratio = 7 − 2 = 5 units

Time difference = 180/5k − 180/9k = 8 ⇒ 36/k − 20/k = 8 ⇒ 16/k = 8 ⇒ k = 2

Downstream Speed = 9 × 2 = 18 km/hr

Given x + 8 = 18 ⇒ x = 10

∴ The correct answer is 10.

Alternate Method

Given:

⇒ Speed of boat in still water (u) : Speed of stream (v) = 7 : 2

⇒ Let u = 7k and v = 2k

⇒ Downstream Speed (D) = u + v = 9k

⇒ Upstream Speed (U) = u − v = 5k

Calculations:

⇒ Time taken Upstream − Time taken Downstream = 8 hours

⇒ 180 / 5k − 180 / 9k = 8

⇒ 36/k − 20/k = 8

⇒ 16/k = 8 ⇒ k = 2

⇒ Downstream Speed = 9 × 2 = 18 km/hr

⇒ Given Downstream speed = x + 8

⇒ x + 8 = 18 ⇒ x = 10

∴ The correct answer is 10.

Additional Information

Downstream and Upstream Formulas

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

Boat and Stream Speed from D and U

Boat speed in still water u = (D + U) / 2 and Stream speed v = (D − U) / 2.

Average Speed for Round Trip

Average Speed = (D × U) / u = (u² − v²) / u when the distance covered upstream and downstream is the same.

Shortcut Trick

Using the combined work formula for two people: 1/D_A + 1/D_B = 1/D_combined

We have: 1/(x + 6) + 1/(x + 18) = 1/8

Substitute options to find the value of x that satisfies the equation:

For x = 6: 1/(6 + 6) + 1/(6 + 18) = 1/12 + 1/24 = (2 + 1)/24 = 3/24 = 1/8

i.e. if x = 6, then work will be done in 8 days if they work together.

∴ The correct answer is 6.

Alternate Method

Given:

Time taken by A = (x + 6) days

Time taken by B = (x + 18) days

Time taken together = 8 days

Formula Used:

1/A + 1/B = 1/Total Time

Calculations:

⇒ 1/(x + 6) + 1/(x + 18) = 1/8

⇒ [(x + 18) + (x + 6)] / [(x + 6)(x + 18)] = 1/8

⇒ (2x + 24) / (x² + 24x + 108) = 1/8

⇒ 8 × (2x + 24) = x² + 24x + 108

⇒ 16x + 192 = x² + 24x + 108

⇒ x² + 8x − 84 = 0

⇒ x² + 14x − 6x − 84 = 0

⇒ x(x + 14) − 6(x + 14) = 0

⇒ (x − 6)(x + 14) = 0

⇒ x = 6 or x = −14 (Since days cannot be negative, x = 6)

∴ The correct answer is 6.

Additional Information

Combined Work Rate Formula

When two entities A and B complete work in 'a' and 'b' days respectively, their combined working rate is given by 1/a + 1/b = 1/T, where T is the total time taken working together.

Quadratic Roots Method

For any equation of form ax² + bx + c = 0, the variable value represents real-life positive parameters, so negative solutions are discarded.

Shortcut Trick

Ratio of Investment × Time determines the profit share.

A profit unit = (4000 × 4) + (5000 × 8) = 16000 + 40000 = 56000.

B profit unit = 6000 × 12 = 72000.

Profit Ratio A : B = 56000 : 72000 = 7 : 9.

Total ratio parts = 7 + 9 = 16 units ⇒ 6080.

B share = (6080 ÷ 16) × 9 = 380 × 9 = 3420.

∴ The correct answer is Rs. 3420.

Alternate Method

Given → Initial A = 4000, Initial B = 6000. Increase for A = 25%. Time = 12 months. Total Profit = 6080.
Formulas: Profit Share = Investment × Time Period | Profit Ratio = (I₁ × T₁) : (I₂ × T₂)

⇒ A investment for first 4 months = 4000.
⇒ A new investment = 4000 + (25% of 4000) = 4000 + 1000 = 5000 for remaining 8 months.
⇒ A total equivalent investment = (4000 × 4) + (5000 × 8) = 16000 + 40000 = 56000.
⇒ B total equivalent investment = 6000 × 12 = 72000.
⇒ Ratio of Profit (A : B) = 56000 : 72000 = 56 : 72 = 7 : 9.
⇒ Sum of ratios = 7 + 9 = 16.
⇒ B profit share = (9 ÷ 16) × 6080 = 9 × 380 = 3420.

∴ The correct answer is Rs. 3420.

Additional Information

Partnership Basics

When investments are made for different time periods, we calculate the effective capital by multiplying Capital × Time.

Ratio Distribution

To find the share of one partner, use the formula: (Individual Ratio ÷ Total Ratio) × Total Profit.

Common Calculation

Class A

Ratio B:G = 6:5, Girls = 45

⇒ 5 parts = 45

⇒ 1 part = 9

⇒ Boys = 6 × 9 = 54

⇒ Total = 54 + 45 = 99

Class B

Ratio B:G = 3:4, Girls = 64

⇒ 4 parts = 64

⇒ 1 part = 16

⇒ Boys = 3 × 16 = 48

⇒ Total = 48 + 64 = 112

Class C

Ratio B:G = 5:6, Girls = 48

⇒ 6 parts = 48

⇒ 1 part = 8

⇒ Boys = 5 × 8 = 40

⇒ Total = 40 + 48 = 88

Class D

Ratio B:G = 7:4, Girls = 32

⇒ 4 parts = 32

⇒ 1 part = 8

⇒ Boys = 7 × 8 = 56

⇒ Total = 56 + 32 = 88

Class E

Ratio B:G = 9:7, Girls = 42

⇒ 7 parts = 42

⇒ 1 part = 6

⇒ Boys = 9 × 6 = 54

⇒ Total = 54 + 42 = 96

Given:

D = 88, B = 112

C = 88, E = 96

Calculation:

⇒ (D + B) = 88 + 112 = 200

⇒ (C + E) = 88 + 96 = 184

⇒ x = 200 − 184 = 16

⇒ 7x + 224 = 7 × 16 + 224

⇒ = 112 + 224 = 336

∴ Answer = 336

Common Calculation

Class A

Ratio B:G = 6:5, Girls = 45

⇒ 5 parts = 45

⇒ 1 part = 9

⇒ Boys = 6 × 9 = 54

⇒ Total = 54 + 45 = 99

Class B

Ratio B:G = 3:4, Girls = 64

⇒ 4 parts = 64

⇒ 1 part = 16

⇒ Boys = 3 × 16 = 48

⇒ Total = 48 + 64 = 112

Class C

Ratio B:G = 5:6, Girls = 48

⇒ 6 parts = 48

⇒ 1 part = 8

⇒ Boys = 5 × 8 = 40

⇒ Total = 40 + 48 = 88

Class D

Ratio B:G = 7:4, Girls = 32

⇒ 4 parts = 32

⇒ 1 part = 8

⇒ Boys = 7 × 8 = 56

⇒ Total = 56 + 32 = 88

Class E

Ratio B:G = 9:7, Girls = 42

⇒ 7 parts = 42

⇒ 1 part = 6

⇒ Boys = 9 × 6 = 54

⇒ Total = 54 + 42 = 96

Given:

A = 99, C = 88

Calculation:

⇒ Difference = 99 − 88 = 11

∴ Answer = 11