Quantitative Aptitude PYQs

Shortcut Trick

In a 45°−45°−90° triangle, the ratio of sides is 1 : 1 : √2.

Base (distance) = 1 unit = 25 m.

Height of standing part = 1 unit = 25 m.

Hypotenuse (broken part) = √2 units = 25√2 m.

Total height of pole = Standing part + Broken part = 25 + 25√2 = 25(√2 + 1) m.

∴ The correct answer is 25(√2 + 1) m.

Alternate Method

Given:

Distance from foot to top (BC) = 25 m.

Angle of elevation (θ) = 45°.

Formula Used:

tan θ = Perpendicular / Base

cos θ = Base / Hypotenuse

Calculations:

⇒ In ΔABC, tan 45° = AB / BC

⇒ 1 = AB / 25

⇒ Standing part (AB) = 25 m

⇒ cos 45° = BC / AC

⇒ 1 / √2 = 25 / AC

⇒ Broken part (AC) = 25√2 m

⇒ Total length of pole = AB + AC

⇒ Total length = 25 + 25√2 = 25(√2 + 1) m

∴ The correct answer is 25(√2 + 1) m.

Additional Information

Trigonometric Ratios for 45°

For a 45° angle, sin 45° = 1/√2, cos 45° = 1/√2, and tan 45° = 1.

Pythagoras Theorem

In a right-angled triangle, Hypotenuse² = Base² + Perpendicular².

Height and Distance Logic

The total height of a broken object is always the sum of the vertical part still standing and the hypotenuse formed by the broken part.

Shortcut Trick

Net percentage change formula = a + b + (a × b)/100

Here, a = −25 (decrease) and b = +12 (increase)

Net change = −25 + 12 + (−25 × 12)/100

Net change = −13 − 3 = −16%

∴ The correct answer is Decreases by 16%.

Alternate Method

Given:

First decrease = 25%

Second increase = 12%

Calculations:

⇒ Let the initial salary of Sonali be 100 units.

⇒ Salary after 25% decrease = 100 − (25% of 100) = 75 units.

⇒ Salary after 12% increase = 75 + (12% of 75)

⇒ 75 + (12 × 75)/100 = 75 + 9 = 84 units.

⇒ Net change = Final salary − Initial salary = 84 − 100 = −16 units.

⇒ Net percentage effect = (−16 / 100) × 100 = −16%.

∴ The correct answer is Decreases by 16%.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the net effective change is given by [x + y + (xy/100)]%.

Base Value Concept

Percentage change is always calculated on the preceding value in a series of successive changes, not the original base unless specified.

Ratio Method

For 25% decrease, ratio is 4:3. For 12% increase, ratio is 25:28. Multiplying ratios (4×25):(3×28) = 100:84, which shows a 16% decrease.

Shortcut Trick

If each number in a data set is decreased by a constant value k, the average of the set is also decreased by k.

Given: Old Average = 25 and Decrease (k) = 3.

New Average = Old Average − k

New Average = 25 − 3 = 22.

∴ The correct answer is 22.

Alternate Method

Given:

Total numbers (n) = 7

Initial Average = 25

Decrease in each number = 3

Formula Used:

Average = Sum of observations ÷ Total number of observations

Calculations:

⇒ Total Sum of 7 numbers = 7 × 25 = 175

⇒ Total decrease across all numbers = 7 × 3 = 21

⇒ New Total Sum = 175 − 21 = 154

⇒ New Average = 154 ÷ 7

⇒ New Average = 22

∴ The correct answer is 22.

Additional Information

Properties of Averages

If every term in a set is increased by k, the average increases by k. If every term is multiplied or divided by k, the average is also multiplied or divided by k respectively.

Sum of Observations

Total Sum = Average × Total number of observations. This is the fundamental relation used for most average-based problems.

Average of Consecutive Numbers

For a set of consecutive numbers (AP), the average is always the middle term or (First term + Last term) ÷ 2.

Shortcut Trick

In right-angled triangle BDC, ∠C = 90° − ∠DBC = 90° − 33° = 57°.

In triangle AEC, ∠AEB is the exterior angle at vertex E.

By the Exterior Angle Theorem, ∠AEB = ∠CAE + ∠C.

⇒ ∠AEB = 45° + 57° = 102°.

∴ The correct answer is 102°.

Alternate Method

Given:

In triangle ABC, BD ⊥ AC at D, ∠DBC = 33°, and ∠CAE = 45° where E is on BC.

Formula Used:

1. Angle Sum Property of a triangle: Sum of angles = 180°.
2. Linear Pair: Sum of angles on a straight line = 180°.

Calculations:

⇒ In right-angled triangle BDC, ∠BDC = 90°.

⇒ ∠BCD + ∠DBC + ∠BDC = 180°

⇒ ∠BCD + 33° + 90° = 180°

⇒ ∠BCD = 180° − 123° = 57°

⇒ Now, consider triangle AEC. The sum of interior angles is 180°.

⇒ ∠CAE + ∠ACE + ∠AEC = 180°

⇒ 45° + 57° + ∠AEC = 180°

⇒ 102° + ∠AEC = 180°

⇒ ∠AEC = 180° − 102° = 78°

⇒ Since B, E, and C are collinear, ∠AEB and ∠AEC form a linear pair.

⇒ ∠AEB + ∠AEC = 180°

⇒ ∠AEB + 78° = 180°

⇒ ∠AEB = 180° − 78° = 102°

∴ The correct answer is 102°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of the interior angles of any triangle is always 180°.

Linear Pair Postulate

If two angles form a linear pair, then the measures of the angles add up to 180°.

Shortcut Trick

Use the direct algebraic identity: 2ab = (a + b)² − (a² + b²)

Substitute the given values: 2ab = 50² − 1300

2ab = 2500 − 1300 = 1200

ab = 1200 ÷ 2 = 600

∴ The correct answer is 600.

Alternate Method

Given:

Total number of widgets produced (a + b) = 50

Sum of squares of widgets produced (a² + b²) = 1300

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ (50)² = 1300 + 2ab ⇒ 2500 = 1300 + 2ab ⇒ 2ab = 2500 − 1300 ⇒ 2ab = 1200 ⇒ ab = 1200 ÷ 2 ⇒ ab = 600

∴ The correct answer is 600.

Additional Information

Square of Difference Identity

(a − b)² = a² + b² − 2ab. This can be used to find the difference between two quantities when their sum of squares and product are known.

Sum and Product Relation

The relationship (a + b)² − (a − b)² = 4ab is frequently used in quantitative aptitude to solve problems involving two variables without finding individual values.

Alternate Expression for Sum of Squares

a² + b² = (a + b)² − 2ab or a² + b² = (a − b)² + 2ab. These are fundamental algebraic identities for competitive exams.

Shortcut Trick

Let Total Capacity of the tank be LCM of 4, 6, and 8 = 24 units.

Net efficiency of (A + B − C) = 6 + 4 − 3 = 7 units/hr.

Work done in first 3 hours = 7 × 3 = 21 units.

Remaining work = 24 − 21 = 3 units, to be filled by (A + B) with efficiency 10.

Additional time = 3/10 hours = 18 minutes.

Total time = 3 hours + 18 minutes = 3 hr 18 min.

∴ The correct answer is 3 hr 18 min.

Alternate Method

Given: Filling time for Pipe A = 4 hr, Pipe B = 6 hr, and Emptying time for Pipe C = 8 hr.

Step-by-step Calculations:

⇒ Total efficiency when all pipes are open = (6 + 4 − 3) = 7 units/hr

⇒ Tank filled in first 3 hours = 7 × 3 = 21 units

⇒ Remaining tank capacity to be filled = 24 − 21 = 3 units

⇒ Efficiency of (A + B) only = 6 + 4 = 10 units/hr

⇒ Time taken by A and B to fill remaining part = 3 ÷ 10 = 0.3 hr

⇒ Converting to minutes: 0.3 × 60 = 18 min

⇒ Total time required = 3 hr + 18 min = 3 hr 18 min

∴ The correct answer is 3 hr 18 min.

Additional Information

Net Efficiency Concept

In pipes and cisterns, efficiency of an inlet pipe is positive (+) and efficiency of an outlet/leak pipe is negative (−).

Time and Work Relation

Total Work = Time taken × Efficiency. To find total time, add the durations of different operational stages separately.

Unit Conversion

To convert hours into minutes, multiply the fractional or decimal part of the hour by 60.

Shortcut Trick

Average Speed = (2 × s₁ × s₂) ÷ (s₁ + s₂) = (2 × 5 × 4) ÷ (5 + 4) = 40 ÷ 9 km/hr.

Total Time = 4 hours 30 minutes = 4.5 hours = 9/2 hours.

Total Distance (both ways) = Average Speed × Total Time = (40 ÷ 9) × (9 ÷ 2) = 20 km.

One-way Distance = 20 ÷ 2 = 10 km.

∴ The correct answer is 10 km.

Alternate Method

Given: Speed while going (s₁) = 5 km/hr, Speed while returning (s₂) = 4 km/hr, Total time (T) = 4.5 hours.

Formula Used: Distance (D) = [ (s₁ × s₂) ÷ (s₁ + s₂) ] × Total Time

⇒ Total Time (T) = 4 + 30/60 = 4.5 = 9/2 hours

⇒ D = [ (5 × 4) ÷ (5 + 4) ] × (9 ÷ 2)

⇒ D = (20 ÷ 9) × (9 ÷ 2)

⇒ D = 10 km

∴ The correct answer is 10 km.

Additional Information

Average Speed Formula

When a person travels equal distances at speeds x and y, the average speed for the whole journey is (2xy) ÷ (x + y).

Time, Speed, and Distance Relationship

Distance = Speed × Time. When distance is constant, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Conversion of Time

To convert minutes to hours, divide the value by 60 (e.g., 30 mins = 30/60 = 0.5 hours).

Shortcut Trick

Convert Volume: 21.560 litres × 1000 = 21560 cm³.

Let Radius (r) = 2x and Height (h) = 5x ⇒ Volume = (22/7) × (2x)² × 5x = 21560.

Solving: (22/7) × 4x² × 5x = 21560 ⇒ 110/7 × 4x³ = 21560 ⇒ x³ = 343 ⇒ x = 7.

Dimensions: r = 14 cm, h = 35 cm ⇒ Total Surface Area = 2πr(r + h) = 88 × 49 = 4312 cm².

∴ The correct answer is 4312 cm².

Alternate Method

Given:

Volume (V) = 21.560 litres

Radius (r) = 2/5 × Height (h)

Formula Used:

1 litre = 1000 cm³

Volume of Cylinder = πr²h

Total Surface Area (TSA) = 2πr(r + h)

Calculations:

⇒ Volume = 21.560 × 1000 = 21560 cm³

⇒ r = (2/5)h → h = (5/2)r

⇒ 21560 = (22/7) × r² × (5/2)r

⇒ 21560 = (110/14) × r³

⇒ r³ = (21560 × 14) ÷ 110

⇒ r³ = 196 × 14 = 2744

⇒ r = √³2744 = 14 cm

⇒ h = (5 × 14) ÷ 2 = 35 cm

⇒ TSA = 2 × (22/7) × 14 × (14 + 35)

⇒ TSA = 2 × 22 × 2 × 49

⇒ TSA = 88 × 49 = 4312 cm²

∴ The correct answer is 4312 cm².

Additional Information

Cylinder Surface Areas

The Curved Surface Area (CSA) of a cylinder is 2πrh. The Total Surface Area (TSA) includes the area of two circular bases: TSA = 2πrh + 2πr² = 2πr(r + h).

Unit Conversions

Capacity and Volume relationship: 1 Litre = 1000 cm³ = 1 dm³; 1 m³ = 1000 Litres.

Volume of Cylinder

The amount of space occupied by a 3D cylinder is given by the base area times the height: V = πr²h.

Shortcut Trick

List the first 17 prime numbers and sum them manually as there is no arithmetic progression formula for primes.

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.

Sum = (2+3+5+7) + (11+13+17+19) + (23+29+31+37) + (41+43+47) + (53+59)

Sum = 17 + 60 + 120 + 131 + 112 = 440.

∴ The correct answer is 440.

Alternate Method

Given: The first 17 prime numbers.

Formula: Sum = p₁ + p₂ + ... + p₁₇

⇒ Prime numbers are natural numbers greater than 1 with exactly two factors: 1 and itself.

⇒ Sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59

⇒ Sum = (2+3+5+7+11+13+17+19+23+29) + (31+37+41+43+47+53+59)

⇒ Sum = 129 + 311

⇒ Sum = 440

∴ The correct answer is 440.

Additional Information

Definition of Prime Numbers

A prime number is a whole number greater than 1 whose only factors are 1 and itself. 2 is the only even prime number.

Sum of Primes Property

There is no general algebraic formula to find the sum of the first n prime numbers; it requires direct addition of the specific terms.

Composite Numbers

Numbers that have more than two factors are composite numbers. Note that 1 is neither prime nor composite.

Shortcut Trick

Total Effective Cost Price (CP) = 20,000 + 2,000 + 3,000 = ₹25,000.

Selling Price (SP) = ₹30,000.

Ratio of CP : SP = 25,000 : 30,000 = 5 : 6.

Profit = 6 − 5 = 1 unit.

Profit % = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: Purchase Price = ₹20,000, Repair = ₹2,000, Transportation = ₹3,000, Selling Price (SP) = ₹30,000.
Formulas: Total CP = Purchase Price + Overheads | Profit = SP − CP | Profit % = (Profit ÷ CP) × 100

⇒ Total Cost Price (CP) = 20,000 + 2,000 + 3,000 = ₹25,000.
⇒ Profit = SP − CP = 30,000 − 25,000 = ₹5,000.
⇒ Profit % = (5,000 ÷ 25,000) × 100
⇒ Profit % = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Effective Cost Price

Any overhead expenses like repairs, labor, or freight are always added to the original purchase price to calculate the actual CP.

Profit and Loss Calculation

Both Profit Percentage and Loss Percentage are always calculated on the Cost Price (CP) unless otherwise stated.

Shortcut Trick

Apply BODMAS: First solve the innermost product, then the brackets, and finally the subtraction.

Inner curly bracket: 3/4 − (1/2 × 8/3) = 3/4 − 4/3 = −7/12.

Outer square bracket: 7/3 ÷ (−7/12) = 7/3 × (−12/7) = −4.

Final calculation: 11/2 − (−4) = 11/2 + 4 = 19/2.

∴ The correct answer is 19/2.

Alternate Method

Given: 5 1/2 − [ 2 1/3 ÷ { 3/4 − 1/2 × 8/3 } ]

Method: BODMAS Rule (Brackets, Of, Division, Multiplication, Addition, Subtraction)

⇒ Convert mixed fractions: 5 1/2 = 11/2 and 2 1/3 = 7/3

⇒ 11/2 − [ 7/3 ÷ { 3/4 − (1/2 × 8/3) } ]

⇒ 11/2 − [ 7/3 ÷ { 3/4 − 4/3 } ]

⇒ 11/2 − [ 7/3 ÷ { (9 − 16) / 12 } ]

⇒ 11/2 − [ 7/3 ÷ { −7/12 } ]

⇒ 11/2 − [ 7/3 × (−12/7) ]

⇒ 11/2 − [ −4 ]

⇒ 11/2 + 4

⇒ (11 + 8) / 2 = 19/2

∴ The correct answer is 19/2.

Additional Information

BODMAS / VBODMAS Rule

Operations must be performed in the order: Vinculum, Brackets ( (), {}, [] ), Of, Division, Multiplication, Addition, and Subtraction.

Mixed Fraction Conversion

A mixed fraction A b/c is converted to an improper fraction using the formula: (A × c + b) / c.

Division of Fractions

To divide one fraction by another, multiply the first fraction by the reciprocal of the second: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

Convert all numbers to base 2.

16 = 2⁴, 8 = 2³, and (1/2)³ = 2⁻³.

For multiplication add powers, and for division subtract powers.

Resulting power = 4 + 5 + 3 − (−3) = 15.

∴ The correct answer is 2¹⁵.

Alternate Method

Given:

Expression: 16 × 2⁵ × 8 ÷ (1/2)³

Formula Used:

a^m × aⁿ = a^m+n

a^m ÷ aⁿ = a^m−n

(1/a)ⁿ = a⁻ⁿ

Calculation:

⇒ First, express each term as a power of 2.

⇒ 16 = 2⁴

⇒ 8 = 2³

⇒ (1/2)³ = 2⁻³

⇒ Substitute these values into the given expression:

⇒ 2⁴ × 2⁵ × 2³ ÷ 2⁻³

⇒ Using the multiplication rule (add exponents):

⇒ 2^{(4 + 5 + 3)} ÷ 2⁻³

⇒ 2¹² ÷ 2⁻³

⇒ Using the division rule (subtract exponents):

⇒ 2^{12 − (−3)}

⇒ 2^{12 + 3} = 2¹⁵

∴ The correct answer is 2¹⁵.

Additional Information

BODMAS Rule

Simplification follows the BODMAS order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Laws of Exponents (Multiplication & Division)

When multiplying terms with the same base, add their exponents: a^m × aⁿ = a^m+n. When dividing, subtract their exponents: a^m ÷ aⁿ = a^m−n.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent: a⁻ⁿ = 1 ÷ aⁿ.

Shortcut Trick

Expenditure Ratio = (Price Ratio) × (Quantity Ratio)

Expenditure on Jowar : Gram = (1 × 5) : (3 × 11)

Expenditure ratio = 5 : 33

∴ The correct answer is 5 : 33.

Alternate Method

Given:

Ratio of Market Price (Jowar : Gram) = 1 : 3

Ratio of Quantity Consumed (Jowar : Gram) = 5 : 11

Formula Used:

Expenditure = Price per unit × Quantity

Calculations:

⇒ Let the prices be 1x and 3x respectively.

⇒ Let the quantities consumed be 5y and 11y respectively.

⇒ Expenditure on Jowar = 1x × 5y = 5xy

⇒ Expenditure on Gram = 3x × 11y = 33xy

⇒ Ratio of Expenditure = 5xy : 33xy

⇒ Ratio = 5 : 33

∴ The correct answer is 5 : 33.

Additional Information

Product of Ratios

When a value depends on the product of two independent variables, its ratio is the product of the individual ratios of those variables.

Compound Ratio

The compound ratio of (a : b) and (c : d) is (ac : bd). This is used in calculating total Expenditure, Revenue, or Work.

Inverse Relation

If Expenditure is constant, the Price is inversely proportional to the Quantity (P₁/P₂ = Q₂/Q₁).

Shortcut Trick

Assume Savings = 100 units.

Expenditure = 100 + 150% of 100 = 250 units.

Net change in Income = (Change in Expenditure + Change in Savings) ÷ Initial Income.

Net change = (−12% × 250 + 37% × 100) ÷ 350 = (−30 + 37) ÷ 350 = 7 ÷ 350 = 1/50 = 2%.

∴ The correct answer is 2%.

Alternate Method

Given:

Expenditure = 150% more than Savings.

Decrease in Expenditure = 12%.

Increase in Savings = 37%.

Formula Used:

Income = Expenditure + Savings

Let Savings = 100

⇒ Expenditure = 100 + (150/100) × 100 = 250

⇒ Initial Income = 250 + 100 = 350

⇒ New Expenditure = 250 − 12% of 250 = 250 − 30 = 220

⇒ New Savings = 100 + 37% of 100 = 100 + 37 = 137

⇒ New Income = 220 + 137 = 357

⇒ Increase in Income = 357 − 350 = 7

⇒ Percentage Increase = (7 ÷ 350) × 100 = 2%

∴ The correct answer is 2%.

Additional Information

Income and Expenditure Relationship

The fundamental equation is Income = Expenditure + Savings. Any percentage change in components affects the total income proportionally.

Percentage Change

Percentage Increase/Decrease = (Final Value − Initial Value) ÷ Initial Value × 100.

Weighted Average in Income

Income change % = (E % × E_ratio + S % × S_ratio) ÷ (Total ratio). Here (250 × −12 + 100 × 37) ÷ 350 = 2%.

Shortcut Trick

Effective Compound Interest rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount percentage = 100% (Principal) + 21% (Interest) = 121%.

Amount = 121% of ₹6,500 = 1.21 × 6,500.

Amount = ₹7,865.

∴ The correct answer is ₹7,865.

Alternate Method

Given:

Principal (P) = ₹6,500

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P × (1 + R/100)ⁿ

Calculations:

⇒ A = 6,500 × (1 + 10/100)²

⇒ A = 6,500 × (1 + 0.1)²

⇒ A = 6,500 × (1.1)²

⇒ A = 6,500 × 1.21

⇒ A = 65 × 121

⇒ A = ₹7,865

∴ The correct answer is ₹7,865.

Additional Information

CI vs SI Difference for 2 Years

The difference between Compound Interest and Simple Interest for 2 years is given by the formula: Difference = P × (R/100)².

Half-Yearly Compounding

If interest is compounded half-yearly, the rate is halved (R/2) and the time period is doubled (2n).

Compound Interest Formula

Compound Interest (CI) alone can be calculated as CI = A − P = P[(1 + R/100)ⁿ − 1].

Shortcut Trick

Age ratio of Father : Son = 5 : 1

Difference in ratio parts = 5 − 1 = 4 units

Given age difference = 30 years

4 units = 30 years → 1 unit = 30 ÷ 4 = 7.5 years

∴ The correct answer is 7.5 years.

Alternate Method

Let the current age of the son be x years.

The father's age is 5× that of the son ⇒ 5x years.

Given that the father is 30 years older than the son.

⇒ 5x − x = 30

⇒ 4x = 30

⇒ x = 30 ÷ 4

⇒ x = 7.5

∴ The correct answer is 7.5 years.

Additional Information

Age Difference Constant

The difference between the ages of two persons remains constant at any point in time.

Ratio Method

If the ratio of ages is a:b and the difference is D, then 1 unit = D ÷ (a−b).

Linear Equations in Ages

Problems involving 'n' years ago or 'n' years hence require adding or subtracting 'n' from the current age variables.

Shortcut Trick

20% of number + 60 = number

⇒ 80% of number = 60

⇒ Number = 60 ÷ 0.8 = 75

75% of number = 0.75 × 75 = 56.25

∴ The correct answer = 56.25

Alternate Method

Given:

20% of a number + 60 = number

Formula Used:

Percentage = (Value ÷ 100) × Number

Calculations:

⇒ 0.2x + 60 = x

⇒ x − 0.2x = 60

⇒ 0.8x = 60

⇒ x = 75

⇒ 75% of x = 0.75 × 75 = 56.25

∴ The correct answer is 56.25.

Additional Information

Percentage Basics

20% = 1/5 and 75% = 3/4

Equation Formation

Translate statements into algebraic equations

Shortcut Logic

If x − 20% of x = constant, then remaining % = constant

Shortcut Trick

The Surface Area of a sphere is 4 × π × (radius)².

Given Radius = 35 cm.

Surface Area = 4 × π × (35)² = 4 × π × 1225 = 4900π cm².

∴ The correct answer is 4900π cm².

Alternate Method

Given:

Radius (r) of the Sphere = 35 cm.

Formula Used:

Surface Area (A) of a sphere = 4πr²

Calculations:

⇒ Surface Area = 4 × π × (35)²

⇒ Surface Area = 4 × π × (35 × 35)

⇒ Surface Area = 4 × π × 1225

⇒ Surface Area = 4900π cm²

∴ The correct answer is 4900π cm².

Additional Information

Volume of a Sphere

The amount of space inside a sphere is given by the formula V = (4/3)πr³.

Curved Surface Area of Hemisphere

For a hemisphere (half-sphere), the curved surface area is exactly half of a full sphere: CSA = 2πr².

Total Surface Area of Hemisphere

A solid hemisphere includes the base circle (πr²), making the total area: TSA = 2πr² + πr² = 3πr².

Shortcut Trick

Simplified ratio: 1/5 : 1/2 : 11/16

Multiply by LCM (80): 16 : 40 : 55

Difference (Greatest − Smallest): 55 − 16 = 39 units

39 units = 78 ⇒ 1 unit = 2

Largest number (55 units) = 55 × 2 = 110

∴ The correct answer is 110.

Alternate Method

Given: Ratio of three numbers = 1/5 : 9/18 : 11/16

Formula: To convert fractional ratios to integers, multiply each term by the LCM of denominators.

⇒ First, simplify 9/18 = 1/2

⇒ Ratio = 1/5 : 1/2 : 11/16

⇒ LCM of 5, 2, 16 = 80

⇒ Simplified Ratio = (1/5 × 80) : (1/2 × 80) : (11/16 × 80) = 16 : 40 : 55

⇒ Let numbers be 16x, 40x, and 55x

⇒ Difference = 55x − 16x = 39x

⇒ Given 39x = 78 ⇒ x = 2

⇒ Largest number = 55x = 55 × 2 = 110

∴ The correct answer is 110.

Additional Information

Ratio Simplification

To simplify a ratio of fractions a/b : c/d, multiply both terms by the LCM of b and d to obtain a ratio of whole numbers.

Proportional Constant

The 'x' in ratios like ax : bx represents the highest common factor (HCF) of the original numbers being compared.

Compound Ratio

The compound ratio of a:b and c:d is (a × c) : (b × d). This is useful when combining multiple relationships.

Shortcut Trick

Net Selling Price (SP) = Marked Price (MP) × (100 − d₁)% × (100 − d₂)%

SP = 7500 × (100 − 14)% × (100 − 28)%

SP = 7500 × 86% × 72%

SP = 7500 × (86 ÷ 100) × (72 ÷ 100) = 75 × 0.86 × 72

SP = 5400 × 0.86 = 4644

∴ The correct answer is ₹4,644.

Alternate Method

Given: Marked Price (MP) = ₹7,500, First Discount (d₁) = 14%, Second Discount (d₂) = 28%.

Formulas: Successive Discount = a + b − (a × b ÷ 100) | SP = MP × (100 − Total Discount%) ÷ 100

⇒ Price after first discount = 7500 × (100 − 14) ÷ 100 = 7500 × 0.86 = ₹6,450

⇒ Final Selling Price (SP) = 6450 × (100 − 28) ÷ 100 = 6450 × 0.72

⇒ SP = ₹4,644

∴ The correct answer is ₹4,644.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Selling Price = MP − Discount.

Successive Discount Calculation

Two successive discounts of x% and y% are equivalent to a single discount of [x + y − (xy ÷ 100)]%.