Quantitative Aptitude PYQs

Given:

9² + (√4)² − 7√16 − 7

Formula used:

Square: a² = a × a

Square Root identity: (√a)² = a

Calculations:

To simplify the expression, solve each term separately:

9² + (√4)² − 7√16 − 7

⇒ 81 + 4 − (7 × 4) − 7

⇒ 81 + 4 − 28 − 7

⇒ 85 − 35

⇒ 50

∴ The correct answer is option (1).

Shortcut Trick

Cost Price (CP) of 1 notebook = 1000 ÷ 50 = ₹20.

Profit percentage = 20%.

Selling Price (SP) = CP × (100 + Profit)% = 20 × 120% = 20 × 1.2 = ₹24.

∴ The correct answer is ₹24.

Alternate Method

Given: Total quantity = 50 notebooks, Total CP = ₹1000, Profit = 20%.

Formula Used: CP per unit = Total CP ÷ Quantity; SP = CP + (Profit% of CP).

⇒ Cost Price of 1 notebook = 1000 ÷ 50

⇒ CP = ₹20

⇒ Profit amount = 20% of 20 = (20 ÷ 100) × 20

⇒ Profit = 0.2 × 20 = ₹4

⇒ Selling Price = CP + Profit = 20 + 4

⇒ SP = ₹24

∴ The correct answer is ₹24.

Additional Information

Profit and Loss Formulas

Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100. Always calculate percentage on the Cost Price unless stated otherwise.

Selling Price Calculation

SP = CP × (100 + Gain%) ÷ 100 when there is a profit, and SP = CP × (100 − Loss%) ÷ 100 when there is a loss.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

Ratio of Cost Price (CP) to Marked Price (MP) = (100 − Discount%) : (100 + Gain%)

⇒ CP : MP = (100 − 10) : (100 + 8) = 90 : 108 = 5 : 6

If no discount is allowed, Selling Price (SP) = Marked Price (MP) = 6 units.

⇒ Profit = SP − CP = 6 − 5 = 1 unit.

⇒ Gain percentage = (Profit / CP) × 100 = (1 / 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: Marked Price of refrigerator = ₹4,884, Discount = 10%, Gain = 8%.

Formula Used: CP = [MP × (100 − D%)] / (100 + G%)

⇒ Cost Price (CP) = [4884 × (100 − 10)] / (100 + 8)

⇒ CP = (4884 × 90) / 108

⇒ CP = 4884 × (5 / 6) = ₹4,070

⇒ If no discount is allowed, new Selling Price (SP) = Marked Price = ₹4,884

⇒ Profit = SP − CP = 4884 − 4070 = ₹814

⇒ Gain percentage = (Profit / CP) × 100

⇒ Gain percentage = (814 / 4070) × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Relationship between CP and MP

The direct formula connecting Cost Price and Marked Price is CP/MP = (100 − D%)/(100 + G%). This eliminates the need to calculate SP.

Marked Price (MP)

Also known as the list price or tag price. Discounts are always calculated on the Marked Price.

Successive Discounts

If two discounts d₁ and d₂ are given, the effective discount is d₁ + d₂ − (d₁ × d₂)/100.

Shortcut Trick

Sum 5 years ago = 50 − (2 × 5) = 40 years.

Ratio 5 years ago (Father : Son) = 3 : 1.

Total units = 3 + 1 = 4 units ⇒ 4 units = 40 years ⇒ 1 unit = 10 years.

Age Difference = 3 − 1 = 2 units = 2 × 10 = 20 years.

Age difference always remains constant over time.

∴ The correct answer is 20 years.

Alternate Method

Given:

Present sum of Father and Son = 50 years.

Age ratio 5 years ago = 3 : 1.

Let the Son's age 5 years ago be x years.

⇒ Father's age 5 years ago = 3x years.

⇒ Present Age of Father = 3x + 5

⇒ Present Age of Son = x + 5

⇒ (3x + 5) + (x + 5) = 50

⇒ 4x + 10 = 50

⇒ 4x = 40 ⇒ x = 10

⇒ Father's present age = 3(10) + 5 = 35 years.

⇒ Son's present age = 10 + 5 = 15 years.

⇒ Age difference = 35 − 15 = 20 years.

∴ The correct answer is 20 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals never changes with time, regardless of how many years pass or go back.

Sum of Ages Rule

If the sum of ages of 'n' people is S at present, then 't' years ago, the sum was S − (n × t).

Ratio of Ages

If the ratio of ages is a : b and the difference is D, then the ages are [a/(a−b)] × D and [b/(a−b)] × D respectively.

Shortcut Trick

Difference in Time Period = 5 years − 3 years = 2 years.

Effective Interest Percentage difference = 2 years × 4% = 8%.

Given that 8% of the Principal corresponds to ₹784.

Required Simple Interest for 8 years = 8 years × 4% = 32%.

By unitary method: 32% = (784 ÷ 8) × 32 = 98 × 32 = 3136.

∴ The correct answer is ₹3136.

Alternate Method

Given:

Rate of Interest (R) = 4% per annum

Difference in Simple Interest = ₹784

Time Durations: T₁ = 5 years, T₂ = 3 years, T₃ = 8 years

Formula: Simple Interest (SI) = (P × R × T) ÷ 100

⇒ Let the Principal amount be P.

⇒ SI for 5 years − SI for 3 years = 784

⇒ (P × 4 × 5 ÷ 100) − (P × 4 × 3 ÷ 100) = 784

⇒ 20P ÷ 100 − 12P ÷ 100 = 784

⇒ 8P ÷ 100 = 784

⇒ P = (784 × 100) ÷ 8

⇒ P = ₹9800

⇒ SI for 8 years = (9800 × 4 × 8) ÷ 100

⇒ SI = 98 × 32

⇒ SI = ₹3136

∴ The correct answer is ₹3136.

Additional Information

Simple Interest Properties

For a fixed Principal and Rate, Simple Interest is directly proportional to Time. SI₁ ÷ T₁ = SI₂ ÷ T₂.

Amount Formula

The total Amount (A) is the sum of Principal (P) and Simple Interest (SI): A = P + (P × R × T ÷ 100) = P(1 + RT ÷ 100).

Rate Change Impact

If the Rate increases by r%, the additional interest earned for T years is (P × r × T) ÷ 100.

Shortcut Trick

When a sphere is melted and recast into a cone with the same radius (R = r):

Volume relationship: 4 × R = h

⇒ 4 × R = 84 cm ⇒ R = 21 cm

⇒ Volume of sphere = (4/3) × (22/7) × 21 × 21 × 21 = 88 × 441 = 38808 cm³.

∴ The correct answer is 38808 cm³.

Alternate Method

Given:

Radius of Sphere (R) = Radius of Cone (r)

Height of Cone (h) = 84 cm

Formula Used:

Volume of Sphere = (4/3) π R³

Volume of Cone = (1/3) π r² h

Calculations:

Since the sphere is melted and recast into a cone:

⇒ Volume of Sphere = Volume of Cone

⇒ (4/3) × π × R³ = (1/3) × π × r² × h

⇒ (4/3) × R³ = (1/3) × R² × 84 (Since R = r)

⇒ 4 × R = 84

⇒ R = 84 ÷ 4 = 21 cm

Now, calculate the volume of the original sphere:

⇒ Volume = (4/3) × (22/7) × 21 × 21 × 21

⇒ Volume = (4 × 22 × 21 × 21 × 21) ÷ 21

⇒ Volume = 88 × 441

⇒ Volume = 38808 cm³

∴ The correct answer is 38808 cm³.

Additional Information

Conservation of Volume

When a solid is melted and reshaped into another form, the total amount of material or volume remains constant (V₁ = V₂).

Total Surface Area of Sphere

The total surface area (TSA) of a sphere is given by the formula 4πR².

Slant Height of a Cone

The slant height (l) of a cone can be calculated using the Pythagorean theorem: l = √(r² + h²).

Shortcut Trick

Mass = Density × Volume

Weight (kg) = [4/3 × 22/7 × (13³ − 12³) × 39] ÷ 1000

⇒ Weight = [4/3 × 22/7 × 469 × 39] ÷ 1000

⇒ Weight = (4 × 22 × 67 × 13) ÷ 1000

⇒ Weight = 76648 ÷ 1000 = 76.648 kg

∴ The correct answer is 76.648 kg.

Alternate Method

Given:

Internal radius (r) = 12 cm

External radius (R) = 13 cm

Density of metal = 39 g/cm³

Formula Used:

Volume of hollow sphere = 4/3 × π × (R³ − r³)

Mass = Density × Volume

Calculations:

⇒ Volume = 4/3 × 22/7 × (13³ − 12³)

⇒ Volume = 4/3 × 22/7 × (2197 − 1728)

⇒ Volume = 4/3 × 22/7 × 469

⇒ Volume = 4/3 × 22 × 67 = 5896/3 cm³

⇒ Mass = 39 × (5896/3)

⇒ Mass = 13 × 5896 = 76648 g

⇒ Weight in kg = 76648 ÷ 1000

∴ The correct answer is 76.648 kg.

Additional Information

Volume of a Solid Sphere

The total space occupied by a solid sphere is given by the formula V = 4/3 πr³.

Surface Area of a Sphere

The total outer surface area of a sphere is calculated as A = 4πr².

Density-Mass-Volume Triangle

Mass = Density × Volume | Density = Mass ÷ Volume | Volume = Mass ÷ Density.

Unit Conversion

To convert grams to kilograms, divide the value by 1000 (1 kg = 1000 g).

Shortcut Trick

Equalize decimal places for all numbers: 5.00, 3.20, and 0.03.

Remove the decimal to treat them as integers: 500, 320, and 3.

Calculate LCM(500, 320, 3) = 24000.

Adjust the decimal back by dividing by 100: 24000 ÷ 100 = 240.

∴ The correct answer is 240.

Alternate Method

Given: Numbers are 5, 3.2, and 0.03.

Formula: LCM of Fractions = LCM of Numerators ÷ HCF of Denominators.

⇒ Convert decimals to simplest fractions:

⇒ 5 = 5/1

⇒ 3.2 = 32/10 = 16/5

⇒ 0.03 = 3/100

⇒ LCM of Numerators (5, 16, 3):

⇒ Prime factors: 5 = 5¹, 16 = 2⁴, 3 = 3¹

⇒ LCM = 2⁴ × 5¹ × 3¹ = 16 × 5 × 3 = 240

⇒ HCF of Denominators (1, 5, 100) = 1

⇒ Required LCM = 240 ÷ 1 = 240

∴ The correct answer is 240.

Additional Information

LCM of Fractions

LCM = LCM of numerators ÷ HCF of denominators. Ensure fractions are in their simplest form before applying the formula.

HCF of Fractions

HCF = HCF of numerators ÷ LCM of denominators. This is the inverse logic used for finding the LCM of fractions.

Product Rule for Two Numbers

For any two numbers 'a' and 'b', the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Shortcut Trick

Ratio of Initial Revenue to Final Revenue = 2,50,000 : 2,12,500

Divide both by 12,500 ⇒ 20 : 17

Decrease in units = 20 − 17 = 3 units

Percentage decrease = (3 ÷ 20) × 100 = 15%

∴ The correct answer is 15%.

Alternate Method

Given:

Initial Revenue = ₹2,50,000

Final Revenue = ₹2,12,500

Formula Used:

Percentage Decrease = [(Initial Value − Final Value) ÷ Initial Value] × 100

Calculations:

⇒ Amount of decrease = 2,50,000 − 2,12,500

⇒ Amount of decrease = ₹37,500

⇒ Percentage decrease = (37,500 ÷ 2,50,000) × 100

⇒ Percentage decrease = 3,75,00,00 ÷ 2,50,000

⇒ Percentage decrease = 375 ÷ 25

⇒ Percentage decrease = 15%

∴ The correct answer is 15%.

Additional Information

Percentage Increase

It is calculated as [(New Value − Old Value) ÷ Old Value] × 100 when the value grows over time.

Base Change Rule

In percentage calculations, the denominator is always the original (initial) value unless specified otherwise.

Fraction to Percentage Conversion

Multiplying a fraction by 100 converts it into a percentage. For example, 3/20 × 100 = 15%.

Shortcut Trick

Under the Buy 5, Get 2 Free offer, a customer pays for 5 markers but receives a total of 7 markers.

Total amount paid = 5 × ₹43 = ₹215

Effective price per marker = ₹215 ÷ 7 ≈ ₹30.71

∴ The correct answer is ₹30.71.

Alternate Method

Given:

Cost of one marker = ₹43

Offer = Buy 5, Get 2 Free

Formula Used:

Effective price per marker = Total cost paid ÷ Total markers received

Calculations:

⇒ Total markers received = 5 + 2 = 7 markers

⇒ Total price paid = 5 × ₹43 = ₹215

⇒ Effective price per marker = ₹215 ÷ 7

⇒ Effective price per marker = ₹30.7142...

⇒ Effective price per marker ≈ ₹30.71

∴ The correct answer is ₹30.71.

Additional Information

Discount % in 'Buy X Get Y Free' Offer

The discount percentage is calculated as: Discount % = [Y ÷ (X + Y)] × 100. For 'Buy 5, Get 2 Free', the equivalent discount is [2 ÷ 7] × 100 ≈ 28.57%.

Effective Price Formula

Effective Price = Original Price × (1 − Discount Rate) = Original Price × [X ÷ (X + Y)].

Successive Discounts Rule

If two successive discounts of a% and b% are offered on an item, the equivalent single discount is given by: (a + b − ab/100)%.

Shortcut Trick

For a Rhombus, the area is always half the product of its diagonals.

Area = 1/2 × (d₁ × d₂)

Area = 1/2 × 19 × 29 = 551 ÷ 2 = 275.5

∴ The correct answer is 275.5 cm².

Alternate Method

Given:

Length of first diagonal (d₁) = 19 cm

Length of second diagonal (d₂) = 29 cm

Formula Used:

Area of Rhombus = (d₁ × d₂) ÷ 2

Calculations:

⇒ Area = (19 × 29) ÷ 2

⇒ Area = 551 ÷ 2

⇒ Area = 275.5 cm²

∴ The correct answer is 275.5 cm².

Additional Information

Perpendicular Bisectors

The diagonals of a rhombus always bisect each other at a 90° angle (perpendicularly).

Relation between Side and Diagonals

The side length s is related to the diagonals by: 4s² = d₁² + d₂² or s = 1/2 × √(d₁² + d₂²).

Perimeter of Rhombus

Since all four sides of a rhombus are equal, the Perimeter = 4 × side.

Area with Base and Height

The area can also be calculated as Base × Height, similar to a parallelogram.

Shortcut Trick

For any regular polygon, the sum of an interior angle and its corresponding exterior angle is always 180°.

Exterior Angle = 180° − 150° = 30°

The sum of all exterior angles of any polygon is always 360°.

Number of sides (n) = 360° ÷ Exterior Angle

n = 360° ÷ 30° = 12

∴ The correct answer is 12 sides.

Alternate Method

Given: Each interior angle of a regular polygon = 150°

Formula Used: Interior angle of a regular polygon with n sides = [(n − 2) × 180°] ÷ n

Calculations:

⇒ 150 = [(n − 2) × 180] ÷ n

⇒ 150n = (n − 2) × 180

⇒ 150n = 180n − 360

⇒ 180n − 150n = 360

⇒ 30n = 360

⇒ n = 360 ÷ 30

⇒ n = 12

∴ The correct answer is 12 sides.

Additional Information

Sum of Interior Angles

For any polygon with n sides, the sum of all interior angles is given by (n − 2) × 180°.

Exterior Angle Property

The sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Regular Polygons

In a regular polygon, all sides are equal and all interior angles are equal. A polygon with 12 sides is specifically called a dodecagon.

Shortcut Trick

Mean Price of the mixture = 1600 ÷ 45 = ₹320/9 per kg.

Apply Alligation between ₹32 and ₹40 with mean price ₹320/9.

Ratio = (40 − 320/9) : (320/9 − 32) = 40/9 : 32/9 = 40 : 32.

Simplifying the ratio: 40 : 32 → 5 : 4.

∴ The correct answer is 5:4.

Alternate Method

Given: Cost of Type 1 rice = ₹32/kg, Cost of Type 2 rice = ₹40/kg, Total Quantity = 45 kg, Total Cost = ₹1600.

Step 1: Calculate Mean Price (M) of the mixture.

⇒ M = Total Cost ÷ Total Quantity

⇒ M = 1600 ÷ 45 = 320/9

Step 2: Let Quantity 1 be Q₁ and Quantity 2 be Q₂.

⇒ Q₁ : Q₂ = (Price₂ − M) : (M − Price₁)

⇒ Q₁ : Q₂ = (40 − 320/9) : (320/9 − 32)

⇒ Q₁ : Q₂ = (360 − 320)/9 : (320 − 288)/9

⇒ Q₁ : Q₂ = 40/9 : 32/9

⇒ Q₁ : Q₂ = 40 : 32 = 5 : 4

∴ The correct answer is 5:4.

Additional Information

Alligation Rule

It states that (Quantity of Cheaper / Quantity of Dearer) = (Rate of Dearer − Mean Rate) / (Mean Rate − Rate of Cheaper).

Mean Price

The Mean Price is the cost price of a unit quantity of the mixture obtained by mixing two or more ingredients.

Mixture Calculations

If two ingredients of concentrations C₁ and C₂ are mixed in ratio n₁:n₂, the resulting concentration C_avg = (n₁C₁ + n₂C₂) ÷ (n₁ + n₂).

Shortcut Trick

Let Time₃ = x, then Time₁ = x + 9 and Time₂ = x + 4.

Apply the rule: If 1/A + 1/B = 1/C, then C = √(a × b) where A = C+a and B = C+b.

Time₃ = √(9 × 4) = 6 hr; therefore Time₂ = 6 + 4 = 10 hr.

Together (2 & 3) = (10 × 6) ÷ (10 + 6) = 60/16 = 3 3/4 hr.

∴ The correct answer is 3 3/4 hr.

Alternate Method

Given:

Let the time taken by Pipe 3 alone be t₃ hr.

⇒ Pipe 2 time (t₂) = t₃ + 4 hr

⇒ Pipe 1 time (t₁) = t₂ + 5 = (t₃ + 4) + 5 = t₃ + 9 hr

Formula Used:

Efficiency(1) + Efficiency(2) = Efficiency(3) ⇒ 1/t₁ + 1/t₂ = 1/t₃

Calculations:

⇒ 1/(t₃ + 9) + 1/(t₃ + 4) = 1/t₃

⇒ (t₃ + 4 + t₃ + 9) / ((t₃ + 9)(t₃ + 4)) = 1/t₃

⇒ (2t₃ + 13) / (t₃² + 13t₃ + 36) = 1/t₃

⇒ 2t₃² + 13t₃ = t₃² + 13t₃ + 36

⇒ t₃² = 36 ⇒ t₃ = 6 hr

⇒ t₂ = 6 + 4 = 10 hr

⇒ Total Efficiency (2 & 3) = 3 + 5 = 8 units/hr

⇒ Time taken together = Total Work ÷ Total Efficiency = 30 ÷ 8

⇒ Time = 15/4 = 3 3/4 hr

∴ The correct answer is 3 3/4 hr.

Additional Information

Root Rule for Combined Work

If A takes 'a' days more and B takes 'b' days more than (A+B) together, then the time taken by (A+B) is √(a × b).

Pipes and Cisterns Basics

If a pipe fills a tank in 'h' hours, the part filled in 1 hour is 1/h. For inlet pipes, efficiency is positive; for outlet pipes, it is negative.

Efficiency and Capacity

Capacity of the tank = Total Time × Combined Efficiency of the filling pipes.

Shortcut Trick

Let the Distance between City A and City B be D.

Forward time = D ÷ 60; Return speed = 60 + 20 = 80 km/h.

Return journey includes 15 km (30 km/h × 0.5 hr) at 30 km/h and (D − 15) km at 80 km/h.

Total time equation: (D/60) + 0.5 + (D − 15)/80 = 7 ⇒ (D/60) + (D − 15)/80 = 6.5.

Multiplying by LCM 240: 4D + 3D − 45 = 1560 ⇒ 7D = 1605 ⇒ D = 229 2/7 km.

∴ The correct answer is 229 2/7 km.

Alternate Method

Given:

Speed (A to B) = 60 km/h

Speed (B to A, normal) = 60 + 20 = 80 km/h

Repair Speed = 30 km/h | Repair Duration = 30 min = 0.5 hr

Total Round Trip Time = 7 hr

Calculations:

⇒ Let Distance between A and B be D km.

⇒ Distance covered during repair = Speed × Time = 30 × 0.5 = 15 km.

⇒ Remaining distance for return journey = D − 15 km.

⇒ Time for forward journey = D/60 hr.

⇒ Time for return journey = 0.5 + (D − 15)/80 hr.

⇒ Total Time = D/60 + 0.5 + (D − 15)/80 = 7

⇒ D/60 + (D − 15)/80 = 6.5

⇒ (4D + 3(D − 15)) ÷ 240 = 6.5

⇒ 4D + 3D − 45 = 6.5 × 240

⇒ 7D − 45 = 1560

⇒ 7D = 1605

⇒ D = 1605 ÷ 7 = 229 2/7 km.

∴ The correct answer is 229 2/7 km.

Additional Information

Average Speed

It is defined as the total distance traveled divided by the total time taken. Average Speed = Total Distance ÷ Total Time.

Speed-Time-Distance Relationship

Distance = Speed × Time. If distance is constant, speed is inversely proportional to time (S ∝ 1/T).

Unit Conversion

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

Shortcut Trick

Scheme A: Successive 50% and 16% ⇒ 50 + 16 − (50 × 16 ÷ 100) = 58%.

Scheme B: Successive 49%, 8%, and 13% ≈ 59.18% (Multiple successive drops).

Scheme C: 8 toys ⇒ (6 × 35% + 2 × 45%) ÷ 8 = (210% + 90%) ÷ 8 = 37.5%.

Scheme D: Buy 8, pay for 4 ⇒ 4 free ÷ 8 total × 100 = 50%.

The least beneficial scheme is the one with the minimum total discount (37.5%).

∴ The correct answer is Scheme C.

Alternate Method

Assume the Marked Price (MP) of each toy is ₹100. For 8 toys, total MP = 8 × 100 = ₹800.

Formula Used: Selling Price (SP) = MP × [ (100 − d₁)/100 ] × [ (100 − d₂)/100 ] ...

Calculation of Total Selling Price (SP) for 8 toys:

⇒ Scheme A: SP = 8 × [ 100 × 0.50 × 0.84 ] = 8 × 42 = ₹336.

⇒ Scheme B: SP = 8 × [ 100 × 0.51 × 0.92 × 0.87 ] ≈ 8 × 40.82 = ₹326.56.

⇒ Scheme C: SP = (6 × 65) + (2 × 55) = 390 + 110 = ₹500.

⇒ Scheme D: SP = Customer billed for 4 toys = 4 × 100 = ₹400.

Comparing the total Selling Prices: ₹326.56 (B) < ₹336 (A) < ₹400 (D) < ₹500 (C).

Since the customer pays the maximum amount in Scheme C, it is the least beneficial.

∴ The correct answer is Scheme C.

Additional Information

Successive Discount Formula

For two successive discounts x% and y%, the single equivalent discount is [x + y − (xy ÷ 100)]%.

Buy X Get Y Free

The effective discount percentage is calculated as: [Free items (Y) ÷ Total items (X + Y)] × 100.

Least vs Most Beneficial

From a customer's perspective, the least beneficial scheme is the one that offers the lowest discount or the highest final price.

Shortcut Trick

Use the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, a = 24, b = −5, and c = −19.

Checking the sum: 24 + (−5) + (−19) = 24 − 24 = 0.

Result = 3 × 24 × (−5) × (−19) = 72 × 95 = 6840.

∴ The correct answer is 6840.

Alternate Method

Given: The expression is 24³ + (−5)³ + (−19)³.

Formula Used: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

⇒ Let a = 24, b = −5, and c = −19.

⇒ Calculate the sum (a + b + c): 24 + (−5) + (−19) = 24 − 5 − 19 = 0.

⇒ Since (a + b + c) = 0, the expression (a + b + c)(a² + b² + c² − ab − bc − ca) becomes 0.

⇒ a³ + b³ + c³ − 3abc = 0

⇒ a³ + b³ + c³ = 3abc

⇒ Substituting the values: 3 × 24 × (−5) × (−19)

⇒ 72 × 95

⇒ 72 × (100 − 5) = 7200 − 360 = 6840.

∴ The correct answer is 6840.

Additional Information

General Cubic Identity

The expression a³ + b³ + c³ − 3abc is always equal to (a + b + c) × 0.5 × [(a−b)² + (b−c)² + (c−a)²].

Conditional Identity

If a³ + b³ + c³ = 3abc, it implies that either a + b + c = 0 or a = b = c.

Square of Trinomial

The square of a sum of three terms is given by (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

To find the ratio between the first term (a) and the last term (c), directly multiply the given ratios.

a/c = (a/b) × (b/c)

a/c = (12/13) × (4/5) = 48/65

∴ The correct answer is 48:65.

Alternate Method

Given: The ratio a:b is 12:13 and b:c is 4:5.

To find a:c, we can express the ratios as fractions:

⇒ a/b = 12/13

⇒ b/c = 4/5

⇒ Multiply the two equations: (a/b) × (b/c) = (12/13) × (4/5)

⇒ The common term 'b' cancels out: a/c = (12 × 4) / (13 × 5)

⇒ a/c = 48/65

This result matches Option 3 provided in the question.

∴ The correct answer is 48:65.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is called a compound ratio. For ratios a:b and c:d, the compound ratio is (a × c) : (b × d).

Continued Proportion

To find the combined ratio a:b:c, make the common term 'b' equal by multiplying the ratios by suitable constants. Here, a:b:c = (12×4) : (13×4) : (13×5) = 48:52:65.

Duplicate Ratio

The duplicate ratio of a:b is simply the ratio of their squares, i.e., a² : b².

Shortcut Trick

Net percentage change formula = x + y + (xy / 100)

Net % change = 20 − 25 − (20 × 25 / 100) = −5 − 5 = −10%

Final Price = 90% of 600 = 0.9 × 600 = 540

∴ The correct answer is ₹540.

Alternate Method

Given: Original price of the jacket = ₹600, First increase = 20%, Second decrease = 25%.

⇒ First price increase = 20% of 600 = (20/100) × 600 = ₹120

⇒ Price after increase = 600 + 120 = ₹720

⇒ Price decrease = 25% of 720 = (25/100) × 720 = ₹180

⇒ Final price of the jacket = 720 − 180 = ₹540

∴ The correct answer is ₹540.

Additional Information

Successive Percentage Change

When a value is changed by x% and then y%, the net change is given by (x + y + xy/100)%. Use positive values for increase and negative for decrease.

Fractional Multiplier Method

For a 20% increase, multiply by 1.2 or 6/5. For a 25% decrease, multiply by 0.75 or 3/4. Final Value = Initial Value × Multiplier₁ × Multiplier₂.

Base Change Property

A change of +20% followed by −20% does not return the value to its original state; the net change would be a 4% decrease (loss).

Shortcut Trick

Net deviation from the overall mean (42) must be zero.

Sum of deviations = (30 − 42) + (38 − 42) + (50 − 42) + 6 × (45 − 42)

⇒ −12 − 4 + 8 + (6 × 3) = −16 + 8 + 18 = +10.

Since the existing 9 numbers are +10 above the mean, the 10^th number must be 10 below the mean.

⇒ Missing observation = 42 − 10 = 32.

∴ The correct answer is 32.

Alternate Method

Given:

Total observations = 10 | Total Arithmetic Mean = 42

First 3 observations = 30, 38, 50 | Average of next 6 = 45

Calculations:

⇒ Total Sum = Mean × Number of observations

⇒ Total Sum = 42 × 10 = 420

⇒ Sum of 3 known observations = 30 + 38 + 50 = 118

⇒ Sum of next 6 observations = 45 × 6 = 270

⇒ Sum of 9 observations = 118 + 270 = 388

⇒ Missing observation = Total Sum − Sum of 9 observations

⇒ Missing observation = 420 − 388

⇒ Missing observation = 32.

∴ The correct answer is 32.

Additional Information

Arithmetic Mean Formula

Mean = (Sum of all observations) ÷ (Number of observations). To find a missing value, subtract the sum of known values from the total sum.

Sum of Deviations

The algebraic sum of the deviations of a set of values from their arithmetic mean is always zero. ∑(x_i − mean) = 0.

Weighted Mean

When different groups have different averages, the combined mean is (n₁ × x₁ + n₂ × x₂ + ...) ÷ (n₁ + n₂ + ...).