Quantitative Aptitude PYQs

Shortcut Trick

Total Work (LCM of 12 and 39) = 156 units.

Efficiencies: A = 13, B = 4.

Work done by (A + B) in 4 days = (13 + 4) × 4 = 68 units.

Remaining work = 156 − 68 = 88 units.

Time taken by B = 88 ÷ 4 = 22 days.

∴ The correct answer is 22 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 39 days

Formula Used:

Efficiency = Total Work ÷ Time

Remaining Work = Total Work − Work Completed

Calculations:

⇒ Total efficiency of A and B = 13 + 4 = 17 units/day

⇒ Work done by A and B together in 4 days = 17 × 4 = 68 units

⇒ Remaining work to be done = 156 − 68 = 88 units

⇒ A leaves, so B completes the 88 units of work.

⇒ Time taken by B = 88 ÷ 4 = 22 days

∴ The correct answer is 22 days.

Additional Information

Pipes & Cisterns

Similar logic applies to pipes where inlet pipes have positive efficiency and outlet pipes have negative efficiency.

Work & Wages

Wages are distributed among workers in the ratio of their efficiency or the total work done by each.

Alternate Day Work

When people work on alternate days, calculate the work done in one complete cycle (e.g., 2 days) to find the total time.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (Efficiency ∝ 1 ÷ Time).

Shortcut Trick

Total interest percentage for 3 years = 7.5% × 3 = 22.5%

Total Amount percentage = 100% (Principal) + 22.5% (Interest) = 122.5%

Amount = 122.5% of ₹8,000

Amount = 1.225 × 8000 = ₹9,800

∴ The correct answer is ₹9800.

Alternate Method

Given:

Principal (P) = ₹8,000

Rate of Interest (R) = 7.5% per annum

Time (T) = 3 years

Formula Used:

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

Amount (A) = Principal (P) + Simple Interest (SI)

Calculations:

⇒ SI = (8000 × 7.5 × 3) ÷ 100

⇒ SI = 80 × 22.5

⇒ SI = ₹1,800

⇒ Amount = 8000 + 1800

⇒ Amount = ₹9,800

∴ The correct answer is ₹9800.

Additional Information

Simple Interest Concept

Simple Interest is calculated only on the initial principal amount throughout the entire period. SI = P × R × T / 100.

Amount in SI

The total value after adding interest to the principal is called the Amount. A = P(1 + RT/100).

SI vs CI Difference

For the first year, SI and Compound Interest (CI) are equal if interest is compounded annually. From the second year, CI > SI because interest is earned on interest.

Shortcut Trick

Convert the given target ratio 3:2 into decimal form: 3 ÷ 2 = 1.5.

Check the decimal values of the options:

60/45 ≈ 1.33 | 45/20 = 2.25 | 35/25 = 1.4 | 120/80 = 1.5.

Since 120/80 matches the decimal value 1.5, it is equivalent to 3:2.

∴ The correct answer is 120:80.

Alternate Method

Given: Target Ratio = 3 : 2

Formula Used: Simplest form of a ratio is obtained by dividing both terms by their Highest Common Factor (HCF).

⇒ Option 1: 60 : 45 (HCF = 15) ⇒ (60 ÷ 15) : (45 ÷ 15) = 4 : 3

⇒ Option 2: 45 : 20 (HCF = 5) ⇒ (45 ÷ 5) : (20 ÷ 5) = 9 : 4

⇒ Option 3: 35 : 25 (HCF = 5) ⇒ (35 ÷ 5) : (25 ÷ 5) = 7 : 5

⇒ Option 4: 120 : 80 (HCF = 40) ⇒ (120 ÷ 40) : (80 ÷ 40) = 3 : 2

∴ The correct answer is 120:80.

Additional Information

Simplest Form of Ratio

A ratio a:b is in its simplest form if the HCF of a and b is 1. To simplify, divide both terms by their common factors.

Equivalent Ratios

Ratios that represent the same relationship between numbers are called equivalent ratios. They can be found by multiplying or dividing both terms by the same non-zero number.

Proportion

When two ratios are equal, they are said to be in proportion. For a:b = c:d, the product of extremes (a×d) equals the product of means (b×c).

Shortcut Trick

In a 30°−60°−90° right-angled triangle, the sides are in the ratio 1 : √3 : 2.

The base (adjacent to 60°) corresponds to 1 unit.

Given, 1 unit = 15 m.

The ladder (hypotenuse) corresponds to 2 units.

Length of ladder = 2 × 15 = 30 m.

∴ The correct answer is 30 m.

Alternate Method

Given:

Angle of elevation of the ladder with the ground (θ) = 60°

Distance of the foot of the ladder from the wall (Base) = 15 m

Formula Used:

In a right-angled triangle, cos θ = Base / Hypotenuse

Calculations:

Let the length of the ladder be L.

⇒ cos 60° = 15 / L

⇒ 1 / 2 = 15 / L

⇒ L = 15 × 2

⇒ L = 30 m

∴ The correct answer is 30 m.

Additional Information

Trigonometric Ratios

Fundamental ratios in a right triangle: sin θ = P/H, cos θ = B/H, and tan θ = P/B (where P = Perpendicular, B = Base, H = Hypotenuse).

Standard Values of Cosine

Common values include cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, and cos 90° = 0.

Pythagoras Theorem

In any right-angled triangle, the relationship between the sides is given by H² = P² + B².

30-60-90 Triangle Rule

For a triangle with angles 30°, 60°, and 90°, the side lengths are always in the ratio x : x√3 : 2x.

Shortcut Trick

When Distance is constant, Speed is inversely proportional to Time.

Ratio of Time = 5 : 6 → Ratio of Speed = 6 : 5.

Original Speed = 300 ÷ 5 = 60 km/h.

Here, 6 units = 60 km/h ⇒ 1 unit (change) = 10 km/h.

Since the ratio decreases from 6 to 5, the speed decreases.

∴ The correct answer is 10 km/h decrease.

Alternate Method

Given:

Distance (D) = 300 km

Initial Time (T₁) = 5 hr

New Time (T₂) = 6 hr

Formula Used:

Speed (S) = Distance (D) ÷ Time (T)

Calculations:

⇒ Initial Speed (S₁) = 300 ÷ 5 = 60 km/h

⇒ New Speed (S₂) = 300 ÷ 6 = 50 km/h

⇒ Change in Speed = Initial Speed − New Speed

⇒ Change = 60 − 50 = 10 km/h

⇒ Since the new speed is lower, there is a decrease of 10 km/h.

∴ The correct answer is 10 km/h decrease.

Additional Information

Speed-Time Relation

For a fixed distance, Speed and Time are inversely proportional. If time increases by a certain factor, speed must decrease by the same factor to cover the same distance.

Average Speed

Average Speed is calculated as the Total Distance divided by the Total Time taken for the entire journey. Formula: Average Speed = (Total Distance) ÷ (Total Time).

Unit Conversion

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

Shortcut Trick

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Check 3,762: (2 + 7) − (6 + 3) = 9 − 9 = 0.

Since the result is 0, the number 3,762 is divisible by 11.

∴ The correct answer is 3,762.

Alternate Method

Given: Numbers are 3,762, 7,393, 4,915, and 6,184.

Divisibility Rule of 11: Let the digits of a number be checked from right to left. The difference (Sum of digits at positions 1, 3, 5...) − (Sum of digits at positions 2, 4, 6...) must be 0, 11, 22, etc.

⇒ For 3,762: (2 + 7) − (6 + 3) = 9 − 9 = 0 (Divisible)

⇒ For 7,393: (3 + 3) − (9 + 7) = 6 − 16 = −10 (Not divisible)

⇒ For 4,915: (5 + 9) − (1 + 4) = 14 − 5 = 9 (Not divisible)

⇒ For 6,184: (4 + 1) − (8 + 6) = 5 − 14 = −9 (Not divisible)

∴ The correct answer is 3,762.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively.

Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4.

Divisibility by 8

A number is divisible by 8 if the last three digits of the number are divisible by 8.

Shortcut Trick

Check if the sum of terms is zero: 23 + (−14) + (−9) = 23 − 23 = 0.

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

Calculation: 3 × 23 × (−14) × (−9) = 3 × 23 × 126.

Result: 69 × 126 = 8694.

∴ The correct answer is 8694.

Alternate Method

Given:

a = 23

b = −14

c = −9

Formula Used:

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

Calculations:

⇒ First, evaluate the sum (a + b + c):

⇒ 23 + (−14) + (−9) = 23 − 14 − 9 = 0

⇒ Since the sum is zero, the right side of the identity becomes zero.

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

⇒ a³ + b³ + c³ = 3abc

⇒ Substitute the values: 3 × 23 × (−14) × (−9)

⇒ 3 × 23 × 126

⇒ 69 × 126 = 8694

∴ The correct answer is 8694.

Additional Information

Conditional Algebraic Identity

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

General Expansion

The expression a² + b² + c² − ab − bc − ca can be rewritten as 1/2[(a−b)² + (b−c)² + (c−a)²], which is always non-negative for real numbers.

Trinomial Square

Another useful identity is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

Surface area of a sphere = π × d²

Area = 3.14 × 10 × 10 = 314 m²

Total cost = Area × rate = 314 × 5

∴ The correct answer is ₹1570.

Alternate Method

Given: Internal Diameter (d) = 10 m, Rate of coating = ₹5 per m², π = 3.14

Formula Used: Surface Area of a Sphere = 4πr²; Total Cost = Area × Rate

Calculations:

⇒ Radius (r) = Diameter ÷ 2 = 10 ÷ 2 = 5 m

⇒ Internal Surface Area = 4 × 3.14 × (5)²

⇒ Area = 4 × 3.14 × 25

⇒ Area = 100 × 3.14 = 314 m²

⇒ Total Cost = 314 × 5

⇒ Total Cost = ₹1570

∴ The correct answer is ₹1570.

Additional Information

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius.

Hemisphere Surface Area

For a hemisphere, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Surface Area in terms of Diameter

The total surface area of a sphere can also be written as πd², which often simplifies calculations when the diameter is known.

Shortcut Trick

Use the Ratio Method for successive changes:

Year 2015 (10% increase): 10 → 11

Year 2016 (15% decrease): 20 → 17 (since 15% = 3/20)

Overall Ratio: (10 × 20) : (11 × 17) = 200 : 187

Given final value: 187 units = 1,87,000 ⇒ 1 unit = 1,000

Initial Population = 200 × 1,000 = 2,00,000

∴ The correct answer is 2,00,000.

Alternate Method

Let the Initial Population in the beginning of 2015 be P.

⇒ Population after 10% increase in 2015 = P × (1 + 10/100) = 1.1P

⇒ Population after 15% decrease in 2016 = 1.1P × (1 − 15/100)

⇒ Population at the end of 2016 = 1.1P × 0.85 = 0.935P

⇒ According to the question: 0.935P = 1,87,000

⇒ P = 1,87,000 ÷ 0.935

⇒ P = (1,87,000 × 1,000) ÷ 935

⇒ P = 200 × 1,000 = 2,00,000

∴ The correct answer is 2,00,000.

Additional Information

Successive Percentage Change Formula

For two successive changes of a% and b%, the effective net change is [a + b + (a × b)/100]%. Use positive for increase and negative for decrease.

Multiplier Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). This is widely used in population growth and depreciation problems.

Population Growth

If the population increases at a constant rate r% per annum, the population after n years is P(1 + r/100)ⁿ.

Shortcut Trick

In an isosceles triangle, angles opposite to equal sides are equal.

Given XY = XZ, therefore the base angles ∠Y = ∠Z.

Sum of angles in a triangle = 180°.

∠X + 2 × ∠Z = 180° ⇒ 40° + 2 × ∠Z = 180°.

2 × ∠Z = 140° ⇒ ∠Z = 70°.

∴ The correct answer is 70°.

Alternate Method

Given:

In ΔXYZ, XY = XZ and vertex angle ∠X = 40°.

Formula Used:

1. Isosceles Triangle Property: Angles opposite to equal sides are equal.
2. Angle Sum Property: The sum of all interior angles of a triangle is 180°.

Calculations:

⇒ In ΔXYZ, since XY = XZ, it is an isosceles triangle.

⇒ ∠Y = ∠Z (Property of isosceles triangles)

⇒ ∠X + ∠Y + ∠Z = 180° (Angle sum property)

⇒ 40° + ∠Z + ∠Z = 180°

⇒ 40° + 2 × ∠Z = 180°

⇒ 2 × ∠Z = 180° − 40°

⇒ 2 × ∠Z = 140°

⇒ ∠Z = 140° ÷ 2

⇒ ∠Z = 70°

∴ The correct answer is 70°.

Additional Information

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Equal sides ↔ Equal angles).

Triangle Angle Sum Theorem

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

Exterior Angle Property

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

Shortcut Trick

Marked Price per pair = ₹2800.

Discounted Price (DP) per pair = 82% of 2800 = ₹2296.

Special offer allows 5 pairs at full DP and 2 pairs at 50% DP (which is 1 pair at full DP).

Total effective cost for 7 pairs = (5 + 1) × 2296 = 6 × 2296.

Total Cost = ₹13,776.

∴ The correct answer is ₹13,776.

Alternate Method

Given:

Marked Price (MP) of a pair of shoes = ₹2800

Trade Discount = 18%

Quantity 1 = 5 pairs at discounted price

Quantity 2 = 2 pairs at 50% of discounted price

Formula Used:

Discounted Price = MP × (100 − Discount%) / 100

Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂)

Calculations:

⇒ Trade Discount amount = 18% of 2800 = 0.18 × 2800 = ₹504

⇒ Discounted Price (DP) per pair = 2800 − 504 = ₹2296

⇒ Cost of first 5 pairs = 5 × 2296 = ₹11,480

⇒ Cost of next 2 pairs = 2 × (50% of 2296) = 2 × 1148 = ₹2,296

⇒ Total Cost for 7 pairs = 11480 + 2296 = ₹13,776

∴ The correct answer is ₹13,776.

Additional Information

Marked Price and Selling Price

Selling Price (SP) is calculated as MP − Discount. If multiple items have different discounts, the total SP is the sum of individual SPs.

Successive Discounts

When two discounts d₁% and d₂% are applied successively, the effective discount is (d₁ + d₂ − (d₁×d₂)/100)%.

Trade Discount vs Cash Discount

Trade discount is given on the list price at the time of sale, while cash discount is offered for immediate or early payment.

Shortcut Trick

Direct calculation: Age of A = 2 × (Age of B) + 5

Given Person B is 15 years old.

Age of Person A = (2 × 15) + 5 = 30 + 5

Age of Person A = 35 years

∴ The correct answer is 35 years.

Alternate Method

Given:

Present age of Person B = 15 years

Present age of Person A = 5 years more than twice the age of Person B

Formula Used:

Age_A = (2 × Age_B) + 5

Calculations:

⇒ Age of Person B = 15 years

⇒ Twice the age of Person B = 2 × 15 = 30 years

⇒ Age of Person A = 30 + 5

⇒ Age of Person A = 35 years

∴ The correct answer is 35 years.

Additional Information

Linear Equations in Age Problems

Age relations such as 'twice as old' or 'n years older' are formulated as linear equations (y = mx + c) where m is the multiplier and c is the constant difference.

Constant Difference Rule

The difference between the ages of two individuals always remains constant, regardless of the number of years that pass.

Ratio Method

For complex age problems, ratios are used by finding a common multiplier (k) to satisfy the given conditions for different time periods.

Shortcut Trick

Break the expression into three parts using the VBODMAS rule.

Part 1: {(16/36) ÷ (16/4)} = (4/9) ÷ 4 = 1/9.

Part 2: ((2/7) × (14/6) + 5/12) = (2/3 + 5/12) = 13/12.

Part 3: (4/7) ÷ (39/12 of 12/7) = (4/7) ÷ (39/7) = 4/39.

Final: (1/9 ÷ 13/12) + 4/39 = (1/9 × 12/13) + 4/39 = 4/39 + 4/39 = 8/39.

∴ The correct answer is 8/39.

Alternate Method

Given: {(16/36) ÷ (16/4)} ÷ ((2/7) × (14/6) + (5/12)) + (4/7) ÷ (39/12) of (12/7)

Formula Used: Order of operations follows VBODMAS (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction).

⇒ Simplify first bracket: (16/36) ÷ (16/4) = 4/9 ÷ 4 = 4/9 × 1/4 = 1/9

⇒ Simplify inner part of second group: (2/7 × 14/6) = 2/7 × 7/3 = 2/3

⇒ Complete second bracket: 2/3 + 5/12 = 8/12 + 5/12 = 13/12

⇒ Simplify Of operation: (39/12) of (12/7) = 39/12 × 12/7 = 39/7

⇒ Remaining expression: (1/9 ÷ 13/12) + (4/7 ÷ 39/7)

⇒ Perform divisions: (1/9 × 12/13) + (4/7 × 7/39) = 4/39 + 4/39

⇒ Final Addition: 8/39

∴ The correct answer is 8/39.

Additional Information

The VBODMAS Rule

It ensures mathematical consistency by prioritizing Brackets then Of (which is a form of multiplication) over Division and Multiplication.

Fraction Division

Dividing by a fraction is equivalent to multiplying by its reciprocal (inverted form): a/b ÷ c/d = a/b × d/c.

Common Denominators

When adding fractions like 2/3 and 5/12, find the LCM of denominators (12) to create a common base before adding numerators.

Shortcut Trick

Identify common base 5 and use the Net Power rule.

Net Power = (Power of term 1 + Power of term 2) − Power of denominator.

Calculation: [3x + 2(x − 1)] − (4x − 1).

⇒ (3x + 2x − 2) − 4x + 1 = 5x − 2 − 4x + 1 = x − 1.

The resulting expression is 5^x-1.

∴ The correct answer is 5^x-1.

Alternate Method

Given: [5^3x × (5²)^(x-1)] ÷ 5^4x-1

Formula Used: (a^m)ⁿ = a^m×n | a^m × aⁿ = a^m+n | a^m ÷ aⁿ = a^m−n

Step-by-step Simplification:

⇒ [5^3x × 5^2(x-1)] ÷ 5^4x-1

⇒ [5^3x × 5^2x-2] ÷ 5^4x-1

⇒ 5^{(3x + 2x − 2)} ÷ 5^4x-1

⇒ 5^{(5x − 2)} ÷ 5^4x-1

⇒ 5^{(5x − 2) − (4x − 1)}

⇒ 5^{5x − 2 − 4x + 1}

⇒ 5^{x − 1}

∴ The correct answer is 5^x-1.

Additional Information

Product and Quotient Laws

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

Power of a Power Rule

Raising a power to another power requires multiplying the exponents: (a^m)ⁿ = a^m×n.

Negative Exponents

Any non-zero base raised to a negative power is the reciprocal: a⁻ⁿ = 1/aⁿ.

Zero Exponent Rule

Any non-zero base raised to the power of zero is always 1: a⁰ = 1.

Shortcut Trick

Identify the increase in average: 25,500 − 24,000 = ₹1,500.

Income of new employee = New average + (Initial number of employees × Increase in average).

Income = 25,500 + (6 × 1,500) = 25,500 + 9,000 = ₹34,500.

∴ The correct answer is ₹34,500.

Alternate Method

Given:

Initial number of employees = 6 | Initial average income = ₹24,000

Final number of employees = 7 | Final average income = ₹25,500

Formula Used:

Total Income = Average Income × Number of Employees

Calculations:

⇒ Total monthly income of 6 employees = 6 × 24,000 = ₹1,44,000

⇒ Total monthly income of 7 employees = 7 × 25,500 = ₹1,78,500

⇒ Monthly income of the new employee = Total (7 employees) − Total (6 employees)

⇒ Monthly income = 1,78,500 − 1,44,000 = ₹34,500

∴ The correct answer is ₹34,500.

Additional Information

Net Deviation Concept

The sum of deviations of all observations from their arithmetic mean is always zero.

Weighted Average

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

Inclusion and Exclusion

If adding a new member increases the average, the new member's value is higher than the original average.

Shortcut Trick

Divisor = 5 × Remainder = 5 × 56 = 280

Quotient = Divisor ÷ 14 = 280 ÷ 14 = 20

Dividend = (Divisor × Quotient) + Remainder

Dividend = (280 × 20) + 56 = 5600 + 56 = 5656

∴ The correct answer is 5656.

Alternate Method

Given:

Remainder (r) = 56

Divisor (d) = 5 × Remainder

Divisor (d) = 14 × Quotient (q)

Formula Used:

Dividend = (Divisor × Quotient) + Remainder

Calculations:

⇒ Divisor = 5 × 56 = 280

⇒ 14 × Quotient = 280

⇒ Quotient = 280 ÷ 14 = 20

⇒ Dividend = (280 × 20) + 56

⇒ Dividend = 5600 + 56

⇒ Dividend = 5656

∴ The correct answer is 5656.

Additional Information

Division Algorithm

For any two whole numbers, the dividend is always equal to the product of the divisor and quotient plus the remainder.

Remainder Property

The remainder must always be greater than or equal to 0 and strictly less than the divisor (0 ≤ r < d).

Euclid's Division Lemma

It states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r.

Shortcut Trick

The ratio of surface areas of two spheres is equal to the square of the ratio of their radii.

Radius of Sphere A : Radius of Sphere B = 6 : 3 = 2 : 1

Surface Area Ratio = (2)² : (1)² = 4 : 1

∴ The correct answer is 4:1.

Alternate Method

Given:

Radius of Sphere A (r₁) = 6 cm

Radius of Sphere B (r₂) = 3 cm

Formula Used:

Surface Area (S) of a sphere = 4πr²

Calculations:

⇒ Surface Area of Sphere A (S₁) = 4 × π × (6)²

⇒ S₁ = 4 × π × 36 = 144π cm²

⇒ Surface Area of Sphere B (S₂) = 4 × π × (3)²

⇒ S₂ = 4 × π × 9 = 36π cm²

⇒ Ratio = S₁ : S₂ = 144π : 36π

⇒ Ratio = 144 / 36 = 4 / 1

∴ The correct answer is 4:1.

Additional Information

Volume of a Sphere

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

Ratio of Volumes

The ratio of the volumes of two spheres is equal to the cube of the ratio of their radii: V₁/V₂ = (r₁/r₂)³.

Hemisphere Properties

For a hemisphere, the Curved Surface Area = 2πr² and the Total Surface Area = 3πr².

Shortcut Trick

Summing both conditions: (8 + 5) Mangoes + (5 + 8) Oranges = 164 + 148

⇒ 13 (Mangoes + Oranges) = ₹312

⇒ 1 (Mango + Orange) = 312 ÷ 13 = ₹24

⇒ 3 (Mangoes + Oranges) = 3 × 24 = ₹72

∴ The correct answer is ₹72.

Alternate Method

Given:

Cost of 8 mangoes and 5 oranges = ₹164

Cost of 5 mangoes and 8 oranges = ₹148

Formula Used:

Symmetrical Linear Equations: Add equations when coefficients are interchanged.

Calculations:

⇒ Let cost of 1 mango be M and 1 orange be O.

⇒ 8M + 5O = 164 --- (Equation 1)

⇒ 5M + 8O = 148 --- (Equation 2)

⇒ Adding Eq(1) and Eq(2): (8M + 5M) + (5O + 8O) = 164 + 148

⇒ 13M + 13O = 312

⇒ 13(M + O) = 312

⇒ M + O = 312 ÷ 13 = 24

⇒ Cost of 3 mangoes and 3 oranges = 3 × (M + O)

⇒ 3 × 24 = 72

∴ The correct answer is ₹72.

Additional Information

Linear Equations in Two Variables

Equations expressed in the form ax + by = c, where a and b are coefficients and c is a constant.

Symmetrical Coefficients Property

When coefficients are interchanged (ax + by = m and bx + ay = n), adding them directly gives a simplified value for (x + y).

Unitary Method

A problem-solving technique where you first find the value of a single unit and then multiply it by the required quantity.

Shortcut Trick

In a distribution with two values, the mean acts as the balance point.

Difference between first value (10) and mean (12.5) = 12.5 − 10 = 2.5

Difference between second value (15) and mean (12.5) = 15 − 12.5 = 2.5

Since the deviations are equal in magnitude (2.5), the frequencies must also be equal.

Given first frequency = 5, therefore p = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Mean (χ) = 12.5

Values (x_i): 10, 15

Frequencies (f_i): 5, p

Formula Used:

Arithmetic Mean (χ) = ∑f_ix_i ÷ ∑f_i

Calculations:

⇒ 12.5 = (10 × 5 + 15 × p) ÷ (5 + p)

⇒ 12.5 × (5 + p) = 50 + 15p

⇒ 62.5 + 12.5p = 50 + 15p

⇒ 62.5 − 50 = 15p − 12.5p

⇒ 12.5 = 2.5p

⇒ p = 12.5 ÷ 2.5

⇒ p = 5

∴ The correct answer is 5.

Additional Information

Properties of Arithmetic Mean

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

Weighted Average Formula

When different values have different importance or frequency, the weighted mean is calculated as χ_w = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n).

Change of Origin and Scale

If every observation is increased by 'a', the mean increases by 'a'. If every observation is multiplied by 'k', the mean is multiplied by 'k'.

Shortcut Trick

The Third Proportional of a and b is calculated as b² ÷ a.

First, find m: m = (20 × 20) ÷ 4 = 100.

Then, find n: n = (50 × 50) ÷ 100 = 25.

∴ The correct answer is 25.

Alternate Method

Given:

First pair: 4, 20 and m are in continuous proportion.

Second pair: m, 50 and n are in continuous proportion.

Formula Used:

If a, b, c are in continuous proportion, then a : b = b : c ⇒ c = b² ÷ a.

Calculations:

⇒ For the first set: 4 : 20 = 20 : m

⇒ m = (20 × 20) ÷ 4

⇒ m = 400 ÷ 4 = 100

⇒ For the second set: 100 : 50 = 50 : n

⇒ n = (50 × 50) ÷ 100

⇒ n = 2500 ÷ 100 = 25

∴ The correct answer is 25.

Additional Information

Mean Proportional

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

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b :: c : d), then the fourth proportional d = (b × c) ÷ a.

Continued Proportion

Three numbers are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third (a/b = b/c).