Quantitative Aptitude PYQs

Shortcut Trick

Identify the symmetrical coefficients in the equations: 2R + 3B = 78 and 3R + 2B = 72.

Simply add both equations to find the combined sum: (2 + 3)R + (3 + 2)B = 78 + 72.

5 × (Red + Blue) = 150.

Divide the total sum by 5: 150 ÷ 5 = 30.

∴ The correct answer is ₹30.

Alternate Method

Let the cost of one red pen be ₹x and the cost of one blue pen be ₹y.

Given:

⇒ 2x + 3y = 78 --- (Equation 1)

⇒ 3x + 2y = 72 --- (Equation 2)

Calculations:

Adding Equation 1 and Equation 2:

⇒ (2x + 3y) + (3x + 2y) = 78 + 72

⇒ 5x + 5y = 150

Taking 5 as a common factor:

⇒ 5(x + y) = 150

Dividing both sides by 5:

⇒ x + y = 150 ÷ 5

⇒ x + y = 30

∴ The correct answer is ₹30.

Additional Information

Simultaneous Linear Equations

When equations are in the form ax + by = m and bx + ay = n, the quickest way to find (x + y) is to add them, and (x − y) is to subtract them.

Elimination Method

To find individual costs, multiply Equation 1 by 3 and Equation 2 by 2 to eliminate x, or use substitution once (x + y) is known.

Unitary Relationship

If nx + ny = K, then the cost of a single set (x + y) is always K ÷ n.

Shortcut Trick

Fourth proportion (x) = (3 × 16) ÷ 24 = 2.

Third proportion formula: b²/a = 20²/A.

Equating both: 2 = 400 ÷ A ⇒ A = 200.

∴ The correct answer is 200.

Alternate Method

Given: Numbers for fourth proportion = 24, 3, 16; Numbers for third proportion = A, 20.

Formula: Fourth proportion (d) of a, b, c is d = (b × c)/a. Third proportion (c) of a, b is c = b²/a.

⇒ Let the fourth proportion be x.

⇒ x = (3 × 16) ÷ 24

⇒ x = 48 ÷ 24 = 2

⇒ According to the question, Third proportion of A and 20 is also 2.

⇒ 2 = 20² ÷ A

⇒ 2 = 400 ÷ A

⇒ A = 400 ÷ 2 = 200

∴ The correct answer is 200.

Additional Information

Mean Proportion

The mean proportion between two numbers a and b is given by √(a × b).

Continued Proportion

If a, b, c are in continued proportion, then a/b = b/c, which implies b² = ac.

Ratio & Proportion Basics

Product of extremes = Product of means. For a : b :: c : d, then a × d = b × c.

Shortcut Trick

Mass = Density × Volume = ρ × π(R² − r²)h

Mass = 5 × (22/7) × (7² − 5²) × 42

Mass = 5 × 22 × (49 − 25) × 6 = 110 × 24 × 6 = 110 × 144

Mass = 15840 g = 15.84 kg

∴ The correct answer is 15.84 kg.

Alternate Method

Given:

Outer radius (R) = 7 cm

Inner radius (r) = 5 cm

Length (h) = 42 cm

Density (ρ) = 5 g/cm³

Formula Used:

Volume of hollow cylinder (V) = π(R² − r²)h

Mass = Volume × Density

Calculations:

⇒ Volume of metal (V) = (22/7) × (7² − 5²) × 42

⇒ V = (22/7) × (49 − 25) × 42

⇒ V = 22 × 24 × (42 ÷ 7)

⇒ V = 22 × 24 × 6 = 3168 cm³

⇒ Mass = 3168 × 5 = 15840 g

⇒ Mass in kg = 15840 ÷ 1000 = 15.84 kg

∴ The correct answer is 15.84 kg.

Additional Information

Total Surface Area of Hollow Cylinder

The total surface area includes inner and outer curved areas plus the areas of the two rings: 2πh(R + r) + 2π(R² − r²).

Curved Surface Area of Hollow Cylinder

The total curved surface area is the sum of the external and internal curved areas: 2πRh + 2πrh = 2πh(R + r).

Density-Mass Relation

Density is the mass per unit volume. For uniform materials, Mass = Volume × Density. Units must be consistent (e.g., g/cm³ with cm³ results in grams).

Shortcut Trick

Use the standard Pythagorean triplet (3, 4, 5).

Since sin θ = 3/5 = Perpendicular/Hypotenuse, the Base must be 4.

∴ cos θ = Base/Hypotenuse = 4/5.

∴ The correct answer is 4/5.

Alternate Method

Given: sin θ = 3/5

Formula Used: sin² θ + cos² θ = 1

⇒ cos² θ = 1 − sin² θ

⇒ cos² θ = 1 − (3/5)²

⇒ cos² θ = 1 − 9/25

⇒ cos² θ = (25 − 9)/25 = 16/25

⇒ cos θ = √(16/25) = 4/5 (as θ is in the 1^st quadrant, cos θ is positive)

∴ The correct answer is 4/5.

Additional Information

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Quadrant Signs (ASTC Rule)

In the 1^st quadrant, All ratios are positive. In the 2^nd, only Sine/Cosec. In the 3^rd, Tan/Cot. In the 4^th, Cos/Sec.

Shortcut Trick

Identify terms: Let a = 2x and b = 3y. Given a + b = 8 and a² + b² = 32.

Find the value of ab: ab = [(a + b)² − (a² + b²)] ÷ 2 = [64 − 32] ÷ 2 = 16.

Apply sum of cubes formula: a³ + b³ = (a + b)(a² + b² − ab).

Substitute the values: 8x³ + 27y³ = 8 × (32 − 16) = 8 × 16 = 128.

∴ The correct answer is 128.

Alternate Method

Given:

2x + 3y = 8

4x² + 9y² = 32

Formula Used:

(a + b)² = a² + b² + 2ab

a³ + b³ = (a + b)³ − 3ab(a + b)

Calculations:

⇒ Let a = 2x and b = 3y

⇒ (2x + 3y)² = (2x)² + (3y)² + 2(2x)(3y)

⇒ (8)² = 32 + 2(6xy)

⇒ 64 = 32 + 12xy

⇒ 12xy = 64 − 32 = 32

⇒ 6xy = 16

⇒ Now, 8x³ + 27y³ = (2x + 3y)³ − 3(2x)(3y)(2x + 3y)

⇒ 8x³ + 27y³ = (8)³ − 3(6xy)(8)

⇒ 8x³ + 27y³ = 512 − 3(16)(8)

⇒ 8x³ + 27y³ = 512 − 384

⇒ 8x³ + 27y³ = 128

∴ The correct answer is 128.

Additional Information

Sum of Cubes Identity

The expression a³ + b³ can be factored as (a + b)(a² − ab + b²) or written as (a + b)³ − 3ab(a + b).

Difference of Cubes Identity

The expression a³ − b³ is calculated using (a − b)(a² + ab + b²) or (a − b)³ + 3ab(a − b).

Square of a Binomial

The identity (a + b)² = a² + 2ab + b² is essential for finding the product term ab when the sum and sum of squares are known.

Shortcut Trick

Convert all decimal bases to powers of 0.5:

0.25 = 0.5² and 0.125 = 0.5³

Substitute these into the expression: (0.5²)⁶ ÷ (0.5³)² × 0.5⁴ = 0.5^x+3

⇒ 0.5¹² ÷ 0.5⁶ × 0.5⁴ = 0.5^{12 − 6 + 4} = 0.5¹⁰

Equating the powers of 0.5: 10 = x + 3 ⇒ x = 7.

∴ The correct answer is 7.

Alternate Method

Given:

0.25⁶ ÷ 0.125² × 0.5⁴ = 0.5^x+3

Formula Used:

a^m ÷ aⁿ = a^m−n

a^m × aⁿ = a^m+n

(a^m)ⁿ = a^mn

Calculations:

0.256 ÷ 0.1252 × 0.54 = 0.5x+3

⇒ 0.25 = 1÷4 = 0.52

⇒ 0.125 = 1÷8 = 0.53

⇒ Expression = (0.52)6 ÷ (0.53)2 × (0.5)4

⇒ = 0.512 ÷ 0.56 × 0.54

⇒ = 0.512−6+4

⇒ = 0.510

⇒ 0.5x+3 = 0.510

⇒ x + 3 = 10

⇒ x = 7

∴ The correct answer is 7 units.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. In mathematical expressions with both division and multiplication, calculations are always performed from left to right sequentially.

Laws of Exponents

When multiplying terms with the identical base, add their exponents together: a^m × aⁿ = a^m+n. When dividing terms with the same base, subtract the exponent of the divisor from the dividend: a^m ÷ aⁿ = a^m−n.

Power of a Power Rule

When a number raised to a power is raised to another power, you simply multiply the inner and outer exponents: (a^m)ⁿ = a^mn.

Shortcut Trick

Assume a base mean of 50 for all observations.

Calculate deviations from 50: +2, −10, +26, +49, +2, −23, +30, −23, −26.

Net deviation sum = 2 + 26 + 49 + 2 + 30 − 10 − 23 − 23 − 26 = 27.

Average deviation = 27 ÷ 9 = 3.

Actual Arithmetic Mean = 50 + 3 = 53.

∴ The correct answer is 53.

Alternate Method

Given:

The observations are 52, 40, 76, 99, 52, 27, 80, 27, and 24.

Total count of observations (n) = 9.

Formula Used:

Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Sum = 52 + 40 + 76 + 99 + 52 + 27 + 80 + 27 + 24

⇒ Sum = 477

⇒ Arithmetic Mean = 477 ÷ 9

⇒ Arithmetic Mean = 53

∴ The correct answer is 53.

Additional Information

Sum of Deviations from Mean

The sum of the deviations of all observations from their arithmetic mean is always zero: Σ(x_i − Mean) = 0.

Effect of Change of Origin

If a constant value k is added to or subtracted from every observation, the new mean will also increase or decrease by k.

Weighted Arithmetic Mean

When different observations have different weights (importance), the mean is calculated as Σ(w_i × x_i) ÷ Σw_i.

Shortcut Trick

Identify the base value (A's marks) and the target value (B's marks).

Required Percentage = (B's marks ÷ A's marks) × 100

Calculation: (50 ÷ 75) × 100 = 2/3 × 100 = 66.666...

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by A = 75

Marks obtained by B = 50

Maximum Marks = 85 (Not required for this calculation)

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

⇒ Required Percentage = (50 ÷ 75) × 100

⇒ Dividing both numbers by 25: (2 ÷ 3) × 100

⇒ 200 ÷ 3 = 66.6666...

⇒ Rounding off to two decimal places = 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

To find how much more or less B got than A: Percentage Change = ((B − A) / A) × 100.

Fraction to Percentage Conversion

Common fractions like 1/3 = 33.33% and 2/3 = 66.67% are helpful for quick calculations in competitive exams.

Base Selection

The value following the word 'than' or 'of' in the question (A's marks) usually acts as the denominator (base).

Shortcut Trick

Speed Ratio (Normal : New) = 7 : 5

Since Distance is constant, Time Ratio (Normal : New) = 5 : 7

Difference in time units = 7 − 5 = 2 units

Given, 2 units = 22 minutes ⇒ 1 unit = 11 minutes

Usual Time = 5 units = 5 × 11 = 55 minutes

∴ The correct answer is 55 min.

Alternate Method

Given: New Speed = 5/7 of Normal Speed; Late time = 22 minutes.

Formula Used: Distance = Speed × Time; If Distance is constant, Time ∝ 1/Speed.

⇒ Let Normal Speed be S and Normal Time be T.

⇒ New Speed = (5/7)S and New Time = (T + 22)

⇒ S × T = (5/7)S × (T + 22)

⇒ T = (5/7)(T + 22)

⇒ 7T = 5T + 110

⇒ 2T = 110

⇒ T = 55 minutes

∴ The correct answer is 55 min.

Additional Information

Inversely Proportional Relation

When the distance is constant, the ratio of speeds is the inverse of the ratio of time taken: S₁/S₂ = T₂/T₁.

Usual Time Shortcut

If a person walks at a/b of his usual speed and is 't' minutes late, the usual time taken is [a / (b − a)] × t.

Average Speed

Average speed for the same distance at two different speeds x and y is given by 2xy / (x + y).

Shortcut Trick

Speed = Distance ÷ Time

Speed = 270 ÷ 4.5 = 2700 ÷ 45

Speed = 60 km/h

∴ The correct answer is 60 km/h.

Alternate Method

Given:

Distance = 270 km

Time = 4.5 hours

Formula Used:

Speed = Distance / Time

Calculations:

⇒ Speed = 270 / 4.5

⇒ Speed = 270 / (9/2)

⇒ Speed = (270 × 2) / 9

⇒ Speed = 30 × 2

⇒ Speed = 60 km/h

∴ The correct answer is 60 km/h.

Additional Information

Unit Conversion (km/h to m/s)

To convert speed from km/h to m/s, multiply the value by 5/18.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances at speeds x and y, Average Speed = 2xy/(x + y).

Relative Speed

When two objects move in opposite directions, relative speed is the sum (s₁ + s₂). In the same direction, it is the difference (s₁ − s₂).

Shortcut Trick

Given tan θ = (1 − p²) / 2p, which represents Perpendicular / Base.

Identify the standard Pythagorean triplet: (1 − p²), 2p, and (1 + p²).

Here, Perpendicular = 1 − p², Base = 2p, and Hypotenuse = 1 + p².

Since sin θ = Perpendicular / Hypotenuse, the value is (1 − p²) / (1 + p²).

∴ The correct answer is (1 − p²) / (1 + p²).

Alternate Method

Given: tan θ = (1 − p²) / 2p

Formula Used: sec²θ = 1 + tan²θ and sin θ = tan θ / sec θ

⇒ Hypotenuse² = Perpendicular² + Base²

⇒ H² = (1 − p²)² + (2p)²

⇒ H² = 1 + p⁴ − 2p² + 4p²

⇒ H² = 1 + p⁴ + 2p² = (1 + p²)²

⇒ H = 1 + p²

⇒ sin θ = Perpendicular / Hypotenuse = (1 − p²) / (1 + p²)

∴ The correct answer is (1 − p²) / (1 + p²).

Additional Information

Trigonometric Triplets

For any variable p, the expressions (1 − p²), 2p, and (1 + p²) always satisfy the Pythagoras Theorem.

Reciprocal Identities

Remember the basic relations: sin θ = 1/cosec θ, cos θ = 1/sec θ, and tan θ = sin θ / cos θ.

Square Identities

Fundamental identities include sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

Shortcut Trick

Using Successive Multipliers: Initial Price = 100%.

Final Price = 100 × (1/2) × (3/5) × (4/5) × (3/2) = 36%.

Net change = 100% − 36% = 64% decrease.

∴ The correct answer is Decreases by 64%.

Alternate Method

Given: Initial fuel price decreases by 50%, 40%, 20% and then increases by 50%.

Assume original price of fuel = ₹100

⇒ Price after 1^st month (50% decrease) = 100 − 50 = ₹50

⇒ Price after 2^nd month (40% decrease) = 50 − (40% of 50) = 50 − 20 = ₹30

⇒ Price after 3^rd month (20% decrease) = 30 − (20% of 30) = 30 − 6 = ₹24

⇒ Price after 4^th month (50% increase) = 24 + (50% of 24) = 24 + 12 = ₹36

⇒ Total Percentage Change = [(Original − Final) ÷ Original] × 100

⇒ Change = [(100 − 36) ÷ 100] × 100 = 64%

∴ The correct answer is Decreases by 64%.

Additional Information

Successive Percentage Formula

For two successive changes of x% and y%, the effective change is (x + y + xy/100)%.

Fraction Equivalents

50% = 1/2, 40% = 2/5, 20% = 1/5. Use multiplying factors (1±r/100) for faster calculation in series.

Base Value Selection

When calculating percentage increase or decrease, always use 100 as the base for the initial value to simplify the final calculation.

Shortcut Trick

Direct application of the Empirical Formula: 3 × Median = Mode + 2 × Mean

3 × Median = 51.3 + 2 × 86.7

3 × Median = 51.3 + 173.4 = 224.7

Median = 224.7 ÷ 3 = 74.9

∴ The correct answer is 74.9.

Alternate Method

Given:

Mean = 86.7

Mode = 51.3

Formula Used:

Mode = 3(Median) − 2(Mean)

Calculations:

⇒ Mode = 3 × Median − 2 × Mean

⇒ 51.3 = 3 × Median − 2 × 86.7

⇒ 51.3 = 3 × Median − 173.4

⇒ 3 × Median = 51.3 + 173.4

⇒ 3 × Median = 224.7

⇒ Median = 224.7 ÷ 3

⇒ Median = 74.9

∴ The correct answer is 74.9.

Additional Information

The Empirical Formula

For a moderately skewed distribution, the relationship is Mode ≈ 3 × Median − 2 × Mean. This is often used when one of the three measures of central tendency is missing.

Symmetric Distribution

In a perfectly Symmetric Distribution (like the Normal Distribution), the values of Mean, Median, and Mode are all equal (Mean = Median = Mode).

Skewed Distributions

In a Positively Skewed distribution, Mean > Median > Mode. In a Negatively Skewed distribution, Mode > Median > Mean.

Karl Pearson's Coefficient of Skewness

It measures the degree of asymmetry: Skewness = (Mean − Mode) ÷ Standard Deviation or 3 × (Mean − Median) ÷ Standard Deviation.

Shortcut Trick

If the sum of three terms is zero (a + b + c = 0), then the sum of their cubes equals thrice their product: a³ + b³ + c³ = 3abc.

Required value = (a³ + b³ + c³)²

Substitute 3abc: (3abc)² = 9 a² b² c².

∴ The correct answer is 9 a² b² c².

Alternate Method

Given: a + b + c = 0

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

⇒ a³ + b³ + c³ − 3abc = (0) × (a² + b² + c² − ab − bc − ca)

⇒ a³ + b³ + c³ − 3abc = 0

⇒ a³ + b³ + c³ = 3abc

Now, we need to find the value of (a³ + b³ + c³)² by squaring both sides:

⇒ (a³ + b³ + c³)² = (3abc)²

⇒ (a³ + b³ + c³)² = 3² × a² × b² × c²

⇒ (a³ + b³ + c³)² = 9 a² b² c²

∴ The correct answer is 9 a² b² c².

Additional Information

Special Case of Cubic Identity

The identity a³ + b³ + c³ = 3abc holds true if either a + b + c = 0 or a = b = c.

Square of a Trinomial

The expansion of (a + b + c)² is a² + b² + c² + 2(ab + bc + ca).

BODMAS Rule

Always follow Brackets, Orders, Division, Multiplication, Addition, and Subtraction for simplification steps.

Shortcut Trick

Total amount to be divided in ratio = 10511 − (87 + 42 + 12) = 10511 − 141 = 10370.

Ratio of remaining shares X : Y : Z = 6 : 16 : 12.

Sum of ratio parts = 6 + 16 + 12 = 34 units.

Difference between Y and Z in ratio = 16 − 12 = 4 units.

Value of 4 units = (10370 ÷ 34) × 4 = 305 × 4 = 1220.

Difference between original shares = (Y − 42) − (Z − 12) = (Y − Z) − 30. Since Y − Z (after deduction) is 1220, original difference = 1220 + (42 − 12) = 1250.

∴ The correct answer is ₹1,250.

Alternate Method

Given: Total = ₹10,511. Reductions = ₹87, ₹42, ₹12. Ratio = 6 : 16 : 12.

⇒ Total reduction = 87 + 42 + 12 = ₹141

⇒ Remaining amount = 10511 − 141 = ₹10,370

⇒ 34k = 10370

⇒ k = 10370 ÷ 34 = 305

⇒ Y's original share = 16k + 42 = 16 × 305 + 42 = 4880 + 42 = ₹4,922

⇒ Z's original share = 12k + 12 = 12 × 305 + 12 = 3660 + 12 = ₹3,672

⇒ Difference = 4922 − 3672 = ₹1,250

∴ The correct answer is ₹1,250.

Additional Information

Ratio and Proportion

If a total quantity A is divided in the ratio a:b:c, then the parts are [a/(a+b+c)]×A, [b/(a+b+c)]×A, and [c/(a+b+c)]×A.

Handling Deductions

When fixed amounts are deducted from shares to obtain a ratio, always subtract the total deduction from the original sum before applying the ratio units.

Shortcut Trick

Combine all percentages and fractions into a single multiplier: 0.4 × 0.5 × 0.5 = 0.1.

Let the number be N.

0.1 × N = 900

N = 9000

∴ The correct answer is 9000.

Alternate Method

Given:

Percentage 1 = 40% = 40/100 = 2/5

Percentage 2 = 50% = 50/100 = 1/2

Fraction = 1/2

Resulting Value = 900

Calculations:

Let the unknown number be 'x'.

⇒ (40/100) × (50/100) × (1/2) × x = 900

⇒ (2/5) × (1/2) × (1/2) × x = 900

⇒ (2/20) × x = 900

⇒ (1/10) × x = 900

⇒ x = 900 × 10

⇒ x = 9000

∴ The correct answer is 9000.

Additional Information

Percentage to Fraction Conversion

To convert percentage to fraction, divide by 100. Example: x% = x/100.

Successive Percentages

When multiple percentages are applied 'of' each other, they are multiplied together as fractions or decimals.

Base Change Rule

If the result is known and the initial value is required, divide the result by the product of all percentage multipliers.

Shortcut Trick

In problems where Sum and Difference of two quantities are given:

Larger Value (Raju) = (Sum + Difference) ÷ 2

Raju's amount = (350 + 50) ÷ 2 = 400 ÷ 2 = 200

∴ The correct answer is ₹200.

Alternate Method

Given:

Total amount (Raju + Riya) = ₹350

Difference (Raju − Riya) = ₹50

Calculations:

⇒ Let Riya have ₹x.

⇒ Then Raju has ₹(x + 50).

⇒ x + (x + 50) = 350

⇒ 2x + 50 = 350

⇒ 2x = 300

⇒ x = 150

⇒ Raju's share = 150 + 50 = 200

∴ The correct answer is ₹200.

Additional Information

Sum and Difference Formula

If A + B = S and A − B = D, then the larger value A = (S + D) ÷ 2 and the smaller value B = (S − D) ÷ 2.

Linear Equations

In problems involving two variables, substitution or elimination methods are standard techniques to find individual values.

Verification

Always verify: Raju (200) + Riya (150) = 350; Raju (200) − Riya (150) = 50. Conditions satisfied.

Shortcut Trick

For fourth proportional to P, 15 and 20 to be 1:

P / 15 = 20 / 1

P = 15 × 20 = 300

∴ The correct answer is 300.

Alternate Method

Given: First term = P, Second term = 15, Third term = 20, Fourth term = 1.

Formula Used: If a, b, c, d are in proportion, then a : b = c : d or a/b = c/d.

⇒ P ÷ 15 = 20 ÷ 1

⇒ P × 1 = 15 × 20

⇒ P = 300

∴ The correct answer is 300.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

The third proportional to a and b is calculated as (b²) ÷ a.

Basic Proportion Property

In any proportion, the product of the extremes is always equal to the product of the means (ad = bc).

Shortcut Trick

The radius (r) of a sphere just immersed in a cone of radius (R), height (H), and slant height (L) is given by: r = (R × H) ÷ (R + L).

Given R = 12 cm, H = 16 cm ⇒ L = √(12² + 16²) = 20 cm.

Radius of sphere (r) = (12 × 16) ÷ (12 + 20) = 192 ÷ 32 = 6 cm.

Volume ratio = V_sphere : (V_cone − V_sphere) = (4/3 π r³) : (1/3 π R²H − 4/3 π r³).

Volume ratio = (4 × 6³) : (12² × 16 − 4 × 6³) = 864 : (2304 − 864) = 864 : 1440 = 3 : 5.

∴ The correct answer is 3 : 5.

Alternate Method

Given: Radius of conical vessel (R) = 12 cm, Height (H) = 16 cm.

Formula Used:

1. Slant height (L) of cone = √(R² + H²)
2. Volume of cone (V_c) = (1/3) × π × R² × H
3. Volume of sphere (V_s) = (4/3) × π × r³

Calculations:

⇒ Slant height (L) = √(12² + 16²) = √(144 + 256) = √400 = 20 cm.

⇒ Let radius of sphere be r. Since it touches the sides and top, it is the incircle of the triangular cross-section.

⇒ Using similarity of triangles: r ÷ (H − r) = R ÷ L

⇒ r ÷ (16 − r) = 12 ÷ 20 = 3 ÷ 5

⇒ 5r = 48 − 3r ⇒ 8r = 48 ⇒ r = 6 cm.

⇒ Volume of water overflown = Volume of sphere (V_s) = (4/3) × π × 6³ = 288π cm³.

⇒ Total volume of cone (V_c) = (1/3) × π × 12² × 16 = 768π cm³.

⇒ Volume of water left = V_c − V_s = 768π − 288π = 480π cm³.

⇒ Required Ratio = 288π : 480π = 288 : 480 = 3 : 5.

∴ The correct answer is 3 : 5.

Additional Information

Volume of Archimedes' Displacement

When an object is fully submerged in a liquid, the volume of liquid overflown or displaced is exactly equal to the volume of the object.

Inscribed Sphere in a Cone

For a sphere of radius r inscribed in a cone of radius R and height H, touching the base, the radius is r = [R × H] ÷ [R + √(R² + H²)].

Surface Area Formulas

Surface area of sphere = 4πr² | Curved surface area of cone = πRL.

Shortcut Trick

Net efficiency = Filling pipe − Emptying pipe

Net efficiency = 1/6 − 1/12 = 1/12 per minute.

Time to fill full tank = 12 minutes.

Time to fill one-half (1/2) tank = 12 ÷ 2 = 6 minutes.

∴ The correct answer is 6 minutes.

Alternate Method

Given: Filling pipe time = 6 min, Emptying pipe time = 12 min.

Formula Used: Total Work = LCM of time; Efficiency = Work ÷ Time.

⇒ Net Efficiency when both opened = +2 − 1 = 1 unit/min

⇒ Required Volume (half tank) = 12 ÷ 2 = 6 units

⇒ Time taken = Required Volume ÷ Net Efficiency

⇒ Time taken = 6 ÷ 1 = 6 minutes

∴ The correct answer is 6 minutes.

Additional Information

Net Work Principle

When pipes work together, filling pipes have positive efficiency and emptying pipes (outlets) have negative efficiency. Total time = Capacity ÷ (Sum of efficiencies).

Fractional Work

If a tank is filled in T minutes, the time to fill 'n' fraction of the tank is simply n × T.