Quantitative Aptitude PYQs

Shortcut Trick

Effective Simple Interest rate for 3 years at 5% = 3 × 5% = 15%.

Total Amount = Principal + Interest = 100% + 15% = 115%.

Given that 115% represents ₹6,900.

Principal (100%) = (6,900 ÷ 115) × 100 = 60 × 100 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Total Amount (A) = ₹6,900

Rate of Interest (R) = 5% per annum

Time Period (T) = 3 years

Formula Used:

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

A = P + (P × R × T) ÷ 100 ⇒ A = P × [1 + (R × T) ÷ 100]

Calculations:

⇒ 6900 = P × [1 + (5 × 3) ÷ 100]

⇒ 6900 = P × [1 + 15 ÷ 100]

⇒ 6900 = P × [1 + 0.15]

⇒ 6900 = P × 1.15

⇒ P = 6900 ÷ 1.15

⇒ P = (6900 × 100) ÷ 115

⇒ P = 60 × 100

⇒ P = ₹6,000

∴ The correct answer is ₹6,000.

Additional Information

Simple Interest (SI)

It is the interest calculated only on the original principal amount for every time period. Formula: SI = (P × R × T) ÷ 100.

Total Amount in SI

The total sum returned after the time period is A = P + SI = P(1 + RT/100). The Principal remains constant in SI calculations.

Comparison with Compound Interest

Unlike SI, Compound Interest (CI) is calculated on the principal plus the accumulated interest of previous periods. For the first year, SI and CI are equal if compounded annually.

Shortcut Trick

According to the divisibility rule of 9, the sum of all digits must be divisible by 9.

Sum of digits of 4583 = 4 + 5 + 8 + 3 = 20.

The next multiple of 9 after 20 is 27 (since 9 × 3 = 27).

Required number to be added = 27 − 20 = 7.

∴ The correct answer is 7.

Alternate Method

Given: Number = 4583, Divisor = 9

Method: Division algorithm to find the remainder.

⇒ 4583 ÷ 9

⇒ 4583 = (509 × 9) + 2

⇒ Remainder = 2

⇒ Smallest number to be added = Divisor − Remainder

⇒ 9 − 2 = 7

∴ The correct answer is 7.

Additional Information

Divisibility Rule of 9

A number is completely divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility Rule of 3

Similar to the rule of 9, a number is divisible by 3 if the sum of its digits is divisible by 3.

Digital Root

The single-digit sum obtained by repeatedly adding the digits of a number is its digital root, which is equal to the remainder when the number is divided by 9 (except when the sum is 9).

Shortcut Trick

The divisibility rule of 8 states that a number is divisible by 8 if its last three digits are divisible by 8.

For the number 76488, we only need to check the last three digits: 488.

Divide 488 by 8: 488 ÷ 8 = 61.

Since 488 is completely divisible by 8, the remainder is 0.

∴ The correct answer is 0.

Alternate Method

Given:

Dividend = 76488

Divisor = 8

Calculations:

⇒ Perform direct division of 76488 by 8:

⇒ 76 ÷ 8 ⇒ 8 × 9 = 72; Remainder = 4

⇒ Bring down 4 to make it 44: 44 ÷ 8 ⇒ 8 × 5 = 40; Remainder = 4

⇒ Bring down 8 to make it 48: 48 ÷ 8 ⇒ 8 × 6 = 48; Remainder = 0

⇒ Bring down the final 8: 8 ÷ 8 ⇒ 8 × 1 = 8; Remainder = 0

⇒ Total Quotient = 9561 and the final remainder is 0.

∴ The correct answer is 0.

Additional Information

Divisibility by 2ⁿ

A number is divisible by 2ⁿ if its last n digits are divisible by 2ⁿ. For 8 (2³), we check the last 3 digits; for 16 (2⁴), we check the last 4 digits.

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4 or if the last two digits are 00.

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.

Shortcut Trick

Difference in Profit Percentage = 48% − 16% = 32%

This 32% increase in Profit corresponds to ₹80 extra per article.

Unit Cost Price (CP) = (80 ÷ 32) × 100 = ₹250

Number of Articles = Total CP ÷ Unit CP = 8000 ÷ 250 = 32

∴ The correct answer is 32.

Alternate Method

Given:

Initial Profit = 16%

New Profit = 48%

Increase in Selling Price (SP) = ₹80

Total Cost Price of all articles = ₹8,000

Let the Cost Price of one article be x.

⇒ Initial Selling Price (SP₁) = 116% of x = 1.16x

⇒ New Selling Price (SP₂) = 148% of x = 1.48x

⇒ According to the question: SP₂ − SP₁ = 80

⇒ 1.48x − 1.16x = 80

⇒ 0.32x = 80

⇒ x = 80 ÷ 0.32

⇒ x = ₹250 (CP of 1 article)

⇒ Total Cost Price = Number of articles × CP of 1 article

⇒ 8000 = Number of articles × 250

⇒ Number of articles = 8000 ÷ 250 = 32

∴ The correct answer is 32.

Additional Information

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Profit and Loss are always calculated on the Cost Price unless stated otherwise.

Selling Price Formula

SP = CP × (100 + Profit %) ÷ 100 or SP = CP × (100 − Loss %) ÷ 100. This is the base for solving change-in-price problems.

Successive Percentage Change

When profit changes from P₁% to P₂%, the effective change in Selling Price is (P₂ − P₁)% of the Cost Price.

Shortcut Trick

When three numbers a, b, and c are in continued proportion, the relationship is a : b :: b : c.

This means the third proportional (c) is calculated as c = b² ÷ a.

Given: a = 4 cm and b = 6 cm. Substituting these values: c = (6 × 6) ÷ 4 = 36 ÷ 4 = 9.

∴ The correct answer is 9 cm.

Alternate Method

Given: Length of the first rod (a) = 4 cm, Length of the second rod (b) = 6 cm.

Formula Used: For three rods of lengths a, b, and c to be in continued proportion, the ratio a : b must be equal to b : c.

Let the length of the third rod be x cm.

⇒ 4 ÷ 6 = 6 ÷ x

⇒ 4 × x = 6 × 6

⇒ 4x = 36

⇒ x = 36 ÷ 4

⇒ x = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Continued Proportion

Three quantities a, b, and c 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).

Mean Proportional

In the proportion a : b :: b : c, 'b' is called the mean proportional between a and c, where b = √(a × c).

Third Proportional

In the proportion a : b :: b : c, 'c' is known as the third proportional to a and b, calculated as c = b²/a.

Fourth Proportional

For four numbers a, b, c, and d in proportion (a : b :: c : d), 'd' is the fourth proportional and d = (b × c) ÷ a.

Shortcut Trick

When two points are on opposite sides of a vertical object, the height h = D ÷ (cot α + cot β).

Given horizontal distance D = 94 m, α = 30° and β = 60°.

h = 94 ÷ (cot 30° + cot 60°) = 94 ÷ (√3 + 1/√3) = 94 ÷ (4/√3) = (94 × √3) ÷ 4.

h = 23.5 × 1.73 = 40.655 m.

∴ The correct answer is 40.655 m.

Alternate Method

Given:

Horizontal distance between two points A and B = 94 m.

Angles of elevation of the delivery drone (P) from A and B are 30° and 60° respectively.

Let the vertical height of the drone be h (PD), where D is the point on the ground directly below the drone.

Formula Used:

Tangent of an angle (θ) = Perpendicular ÷ Base

Calculations:

In right-angled triangle PAD:

⇒ tan 30° = PD ÷ AD

⇒ 1 ÷ √3 = h ÷ AD

⇒ AD = h × √3

In right-angled triangle PBD:

⇒ tan 60° = PD ÷ BD

⇒ √3 = h ÷ BD

⇒ BD = h ÷ √3

Since A and B are on opposite sides:

⇒ AB = AD + BD = 94

⇒ h√3 + h/√3 = 94

⇒ (3h + h) ÷ √3 = 94

⇒ 4h ÷ √3 = 94

⇒ h = (94 × √3) ÷ 4

⇒ h = 23.5 × 1.73

⇒ h = 40.655 m

∴ The correct answer is 40.655 m.

Additional Information

Trigonometric Ratios for 30° and 60°

Values for calculations: tan 30° = 1/√3, tan 60° = √3. Cot ratios are reciprocals: cot 30° = √3, cot 60° = 1/√3.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when the object is above the eye level.

Complementary Angles Rule

If angles of elevation from two points are complementary (α + β = 90°), then height h = √(AD × BD).

Heights and Distances (Same Side)

If points are on the same side of the drone, the horizontal distance D = h × (cot α − cot β).

Shortcut Trick

Assume Total Capacity is the LCM of 24 and 32 = 96 units.

Efficiency of M = 4 units/min and N = 3 units/min.

Combined efficiency (M + N + O) = 96 ÷ 32 = 3 units/min.

Efficiency of Outlet O = 3 − (4 + 3) = −4 units/min (negative sign denotes drainage).

Time for O to drain = 96 ÷ 4 = 24 min.

∴ The correct answer is 24 min.

Alternate Method

Given:

Time taken by Inlet M = 24 min

Time taken by Inlet N = 32 min

Time taken by (M + N + Outlet O) = 32 min

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total capacity = LCM(24, 32) = 96 units

⇒ Efficiency of Inlet M = 96 ÷ 24 = 4 units/min

⇒ Efficiency of Inlet N = 96 ÷ 32 = 3 units/min

⇒ Net efficiency when all three operate = 96 ÷ 32 = 3 units/min

⇒ Efficiency of M + N + O = Efficiency of M + Efficiency of N + Efficiency of O

⇒ 3 = 4 + 3 + Efficiency of O

⇒ Efficiency of O = 3 − 7 = −4 units/min

⇒ Time taken by Outlet O alone = Total Capacity ÷ |Efficiency of O|

⇒ Time = 96 ÷ 4 = 24 min

∴ The correct answer is 24 min.

Additional Information

Pipes and Cisterns Basics

Inlets fill the tank (positive work) and Outlets empty the tank (negative work). If a pipe fills a tank in 'x' hours, its efficiency is 1/x per hour.

Combined Rate Formula

For two inlets A and B and one outlet C, the net part filled in 1 hour = (1/A + 1/B − 1/C).

Emptying vs. Filling

If the net efficiency is negative, the tank will be emptied; if positive, the tank will be filled.

Given:

Expression = (((10)/(18)))/(((10)/(9))) ÷ ((4)/(9) × (18)/(12) + (5)/(6)) + (9)/(9) ÷ (27)/(15) of (15)/(9)

Formula used:

According to the BODMAS rule, operations are performed in the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

Step 1: Simplify the first term

⇒ (((10)/(18)))/(((10)/(9))) = 10/18 × 9/10 = 1/2

Step 2: Simplify the expression inside the bracket

⇒ ((4/9) × (18/12) + 5/6) = (2/3 + 5/6)

⇒ (4/6 + 5/6) = 9/6 = 3/2

Step 3: Simplify the part containing of and division

⇒ (9/9) ÷ (27/15 of 15/9) = 1 ÷ (27/15 × 15/9)

⇒ 1 ÷ (27/9) = 1 ÷ 3 = 1/3

Step 4: Combine all simplified parts into the main expression

⇒ (1/2) ÷ (3/2) + 1/3

⇒ (1/2 × 2/3) + 1/3

⇒ 1/3 + 1/3 = 2/3

∴ The correct answer is option (4).

Shortcut Trick

Ratio of Expenditure to Savings = 70 : 30 = 7 : 3.

Using Weighted Average formula: Overall Income % = (E_ratio × E% + S_ratio × S%) ÷ Total_ratio

20 = (7 × 25 + 3 × x) ÷ 10

200 = 175 + 3x ⇒ 3x = 25 ⇒ x = 8¹/₃%.

∴ The correct answer is 8¹/₃% increase.

Alternate Method

Given: Initial Expenditure = 70% of Income. Income increase = 20%. Expenditure increase = 25%.

Formula Used: Income = Expenditure + Savings

⇒ Let Initial Income = 100

⇒ Initial Expenditure = 70% of 100 = 70

⇒ Initial Savings = 100 − 70 = 30

⇒ New Income = 100 + 20% of 100 = 120

⇒ New Expenditure = 70 + 25% of 70 = 70 + 17.5 = 87.5

⇒ New Savings = 120 − 87.5 = 32.5

⇒ Change in Savings = 32.5 − 30 = 2.5

⇒ Percentage Change = (2.5 ÷ 30) × 100 = 25/3 = 8¹/₃%

∴ The correct answer is 8¹/₃% increase.

Additional Information

Income-Expenditure Identity

Total Income is always the sum of Expenditure and Savings (I = E + S).

Percentage Change Formula

% Change = [(New Value − Old Value) ÷ Old Value] × 100.

Weighted Average in Percentages

If a total is divided into two parts, the overall percentage change is the weighted average of the percentage changes of the individual parts.

Shortcut Trick

Identify the bases: a = 20, b = −16, and c = −4.

Check the sum of the bases: 20 + (−16) + (−4) = 0.

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculation: 3 × 20 × (−16) × (−4) = 60 × 64 = 3840.

∴ The correct answer is 3840.

Alternate Method

Given:

The expression to evaluate is 20³ + (−16)³ + (−4)³.

Formula Used:

Direct calculation of cubes of the given numbers.

Calculations:

⇒ 20³ = 20 × 20 × 20 = 8000

⇒ (−16)³ = (−16) × (−16) × (−16) = −4096

⇒ (−4)³ = (−4) × (−4) × (−4) = −64

⇒ Sum = 8000 + (−4096) + (−64)

⇒ Sum = 8000 − 4160

⇒ Result = 3840

∴ The correct answer is 3840.

Additional Information

Algebraic Cube Identity

The expression a³ + b³ + c³ − 3abc is equal to (a + b + c)(a² + b² + c² − ab − bc − ca).

Special Case

When a + b + c = 0, the identity simplifies to a³ + b³ + c³ = 3abc, which is useful for simplifying complex sums of cubes.

Negative Number Powers

Any negative number raised to an odd power results in a negative value, while an even power results in a positive value.

Shortcut Trick

Total Work = LCM(8, 22, 33) = 264 units.

Efficiencies: Mike = 33, Connor = 12, Floyd = 8.

Adjusted Work = Total Work − (Mike's work) + (Connor's potential work for the 2 days he left early).

Adjusted Work = 264 − (33 × 4) + (12 × 2) = 264 − 132 + 24 = 156 units.

Total Days = Adjusted Work ÷ (Connor's efficiency + Floyd's efficiency) = 156 ÷ (12 + 8) = 7.8 days.

∴ The correct answer is 7.8 days.

Alternate Method

Given: Mike finishes in 8 days, Connor in 22 days, and Floyd in 33 days. Mike leaves after 4 days and Connor leaves 2 days before the end.

Let the total number of days taken to complete the work be T.

⇒ Mike's work + Connor's work + Floyd's work = Total Work

⇒ (33 × 4) + 12 × (T − 2) + (8 × T) = 264

⇒ 132 + 12T − 24 + 8T = 264

⇒ 108 + 20T = 264

⇒ 20T = 156

⇒ T = 156 ÷ 20 = 7.8 days

∴ The correct answer is 7.8 days.

Additional Information

Leaving Work Logic

If a person leaves after x days, subtract their x days of work from the total. If a person leaves before the completion, add their work for those missing days to the total work and treat them as present until the end.

Pipes and Cisterns

Similar to time and work, but outlet pipes have negative efficiency. Total capacity is equivalent to total work.

Work and Wages

Wages are always distributed in the ratio of the work done by each person. If they work for the same amount of time, the ratio of wages is the same as the ratio of their efficiencies.

Shortcut Trick

Since the Original Price is the same for both Article M and Article N (₹10,000), the difference in reduction percentages is proportional to the difference in actual reduction amounts.

Difference in reduction amount = ₹3,000 − ₹2,500 = ₹500.

Difference (R1 − R2) = (Difference in Amount ÷ Common Original Price) × 100.

Difference = (500 ÷ 10,000) × 100 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Original Price (OP) = ₹10,000 (for both articles)

Reduction for Article M = ₹3,000

Reduction for Article N = ₹2,500

Calculations:

⇒ Reduction percentage for Article M (R1) = (Reduction ÷ Original Price) × 100

⇒ R1 = (3,000 ÷ 10,000) × 100 = 30%

⇒ Reduction percentage for Article N (R2) = (Reduction ÷ Original Price) × 100

⇒ R2 = (2,500 ÷ 10,000) × 100 = 25%

⇒ Difference between R1 and R2 = 30% − 25% = 5%

∴ The correct answer is 5.

Additional Information

Percentage Reduction Formula

The rate of reduction or discount is calculated as: (Actual Reduction ÷ Original Marked Price) × 100.

Percentage Difference with Same Base

When comparing two percentages of the same base value, the difference between the percentages is equal to the percentage of the difference between the absolute values.

Successive Discounts

If an article is reduced by a% and then further by b%, the effective reduction is given by the formula: [a + b − (a × b) ÷ 100]%.

Shortcut Trick

Effective rate of compound interest for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%

Effective rate of simple interest for 2 years at 10% = 10 + 10 = 20%

Given that 21% of principal corresponds to ₹220.5.

⇒ 20% of principal = (220.5 ÷ 21) × 20 = 10.5 × 20 = 210

∴ The correct answer is ₹210.

Alternate Method

Given: Rate (R) = 10% per annum, Time (T) = 2 years, Compound Interest (CI) = ₹220.5.

Formula Used: CI = P[(1 + R/100)^T − 1] and SI = (P × R × T) ÷ 100

⇒ 220.5 = P[(1 + 10/100)² − 1]

⇒ 220.5 = P[(1.1)² − 1]

⇒ 220.5 = P[1.21 − 1]

⇒ 220.5 = 0.21P

⇒ P = 220.5 ÷ 0.21 = 1050

⇒ Now, Simple Interest (SI) for 2 years = (1050 × 10 × 2) ÷ 100

⇒ SI = 1050 × 0.2 = 210

∴ The correct answer is ₹210.

Additional Information

Difference between CI and SI for 2 Years

The difference between compound and simple interest for two years is given by D = P × (R/100)².

Effective Interest Rate

For successive percentage increases of x% and y%, the effective change is x + y + (x × y)/100.

Compound Interest Formula

The amount A after n years is A = P × (1 + R/100)ⁿ, where CI = A − P.

Shortcut Trick

Let the Larger Number be L and the Smaller Number be S.

According to the conditions: 2S + 3L = 55 and L + 5 = 2S.

Substitute 2S directly into the first equation: (L + 5) + 3L = 55.

4L + 5 = 55 ⇒ 4L = 50 ⇒ L = 12.5.

∴ The correct answer is 12.5.

Alternate Method

Given:

Let the Smaller Number be x and the Larger Number be y.

Equation 1: 2x + 3y = 55

Equation 2: y + 5 = 2x

Calculations:

⇒ 2x + 3y = 55 --- (1)

⇒ 2x − y = 5 --- (2)

Substitute the value of 2x from (2) into (1):

⇒ (y + 5) + 3y = 55

⇒ 4y + 5 = 55

⇒ 4y = 55 − 5

⇒ 4y = 50

⇒ y = 50 ÷ 4

⇒ y = 12.5

∴ The correct answer is 12.5.

Additional Information

Linear Equation Substitution

This method involves expressing one variable in terms of another from one equation and substituting it into the second equation to reduce it to a single-variable equation.

Elimination Method

This approach involves multiplying equations by suitable constants so that adding or subtracting them eliminates one variable, allowing for the solving of the remaining one.

Consistency of Equations

A system of Linear Equations has a unique solution if the ratio of the coefficients of the variables is not equal (a₁/a₂ ≠ b₁/b₂).

Shortcut Trick

Calculate semi-perimeter (s) = (51 + 35 + 26) ÷ 2 = 56 cm.

Calculate Area using Heron's formula: √[56 × (56−51) × (56−35) × (56−26)] = √(56 × 5 × 21 × 30) = 420 cm².

Use the area formula: Altitude = (2 × Area) ÷ Base.

Altitude = (2 × 420) ÷ 35 = 840 ÷ 35 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given: Sides of the triangle are a = 51 cm, b = 35 cm, and c = 26 cm.

Formula Used: Heron's Formula for Area = √[s(s − a)(s − b)(s − c)] and Area of Triangle = (1 ÷ 2) × Base × Altitude.

⇒ Semi-perimeter (s) = (a + b + c) ÷ 2

⇒ s = (51 + 35 + 26) ÷ 2 = 112 ÷ 2 = 56 cm

⇒ Area = √[56 × (56 − 51) × (56 − 35) × (56 − 26)]

⇒ Area = √[56 × 5 × 21 × 30] = √[176400] = 420 cm²

⇒ To find the altitude (h) corresponding to the side of 35 cm:

⇒ Area = (1 ÷ 2) × 35 × h

⇒ 420 = (35 × h) ÷ 2

⇒ h = (420 × 2) ÷ 35

⇒ h = 840 ÷ 35 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Heron's Formula

It is used to find the area of a triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.

Altitude of a Triangle

The altitude is the perpendicular distance from a vertex to the opposite side. The area can be expressed as (1/2) × base × corresponding altitude.

Relationship between Sides and Altitudes

In any triangle, the side lengths are inversely proportional to their corresponding altitudes. The longest side has the shortest altitude.

Shortcut Trick

Ratio of the diagonals = 5 : 12

Product of ratio terms = 5 × 12 = 60

We know, (1/2) × Product = Area ⇒ (1/2) × 60x² = 240 ⇒ x² = 8 ⇒ x = 2√2

Required diagonals = 5 × 2√2 = 10√2 cm and 12 × 2√2 = 24√2 cm.

∴ The correct answer is 10√2 cm and 24√2 cm.

Alternate Method

Given:

Ratio of the diagonals of a rhombus = 5 : 12

Area of the rhombus = 240 cm²

Formula Used:

Area of Rhombus = (1/2) × d₁ × d₂

Where d₁ and d₂ are the lengths of the diagonals.

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Shortcut Trick

When two trains of equal lengths cross a pole in t₁ and t₂ seconds, the time taken to cross each other in opposite directions is given by:

Time = (2 × t₁ × t₂) ÷ (t₁ + t₂)

Time = (2 × 12 × 16) ÷ (12 + 16) = 384 ÷ 28

Time = 96 ÷ 7 ≈ 13.71 sec

∴ The correct answer is 13.71 sec.

Alternate Method

Given:

Time taken by Train 1 (t₁) = 12 sec

Time taken by Train 2 (t₂) = 16 sec

Let the length of each train be the LCM of 12 and 16 = 48 units.

Formula Used:

Speed = Distance ÷ Time

Time to cross each other = Total Distance ÷ Relative Speed

Calculations:

⇒ Speed of Train 1 (v₁) = 48 ÷ 12 = 4 units/sec

⇒ Speed of Train 2 (v₂) = 48 ÷ 16 = 3 units/sec

⇒ Total distance to cover = Length of Train 1 + Length of Train 2 = 48 + 48 = 96 units

⇒ Relative Speed (opposite direction) = v₁ + v₂ = 4 + 3 = 7 units/sec

⇒ Required Time = 96 ÷ 7

⇒ Required Time = 13.71428... sec

⇒ Required Time ≈ 13.71 sec

∴ The correct answer is 13.71 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (v₁ + v₂). If they move in the same direction, it is the difference (|v₁ − v₂|).

Distance while Crossing

To cross a stationary point (pole, man, tree), the distance covered is equal to the length of the train. To cross another train or platform, the distance is the sum of their lengths (L₁ + L₂).

Time for Same Direction

If the trains were running in the same direction, the time taken to cross each other would be (2 × t₁ × t₂) ÷ (t₂ − t₁), where t₂ > t₁.

Shortcut Trick

Ratio of Men to Women = 58 : 42 = 29 : 21.

Let the assumed average age be 42 years.

Net deviation from Men = 29 × (57 − 42) = +435.

Total distribution of deviation = 435 ÷ (29 + 21) = 8.7.

Total Average Age = 42 + 8.7 = 50.7 years.

∴ The correct answer is 50.7 years.

Alternate Method

Given:

Percentage of Men = 58% | Percentage of Women = 42%

Average age of Men = 57 years | Average age of Women = 42 years

Formula Used:

Average of groups = (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂)

Calculations:

⇒ Let Total Members = 100

⇒ Total age of Men = 58 × 57 = 3306 years

⇒ Total age of Women = 42 × 42 = 1764 years

⇒ Sum of ages of all members = 3306 + 1764 = 5070 years

⇒ Average age of all members = 5070 ÷ 100

⇒ Average age = 50.7 years

∴ The correct answer is 50.7 years.

Additional Information

Weighted Average Concept

It is used when groups have different sizes. Formula: A_avg = (Sum of all values) ÷ (Total number of items).

Alligation Method

The ratio of quantities (n₁/n₂) is equal to (Average₂ − Average_mix) / (Average_mix − Average₁).

Deviation Method for Averages

Assume a temporary average and calculate the net deviation from each group to find the final result quickly.

Shortcut Trick

Equate the volumes: Radius² × Height of Solid Cylinder = (Outer Radius² − Inner Radius²) × Height of Hollow Cylinder.

6² × 14 = [(2x)² − x²] × 7

36 × 2 = 3x² ⇒ x² = 24 ⇒ x = 2√6 cm

Outer radius = 2x = 2 × 2√6 = 4√6 cm.

∴ The correct answer is 4√6 cm.

Alternate Method

Given:

Solid cylinder: Radius (r₁) = 6 cm, Height (h₁) = 14 cm.

Hollow cylinder: Height (h₂) = 14/2 = 7 cm, Outer radius (R) = 2 × Inner radius (r).

Formula Used:

Volume of Solid Cylinder = πr²h

Volume of Hollow Cylinder = π(R² − r²)h

Calculations:

⇒ Volume of solid cylinder = π × 6² × 14 = 504π cm³

⇒ Let inner radius = x, then outer radius = 2x

⇒ Volume of hollow cylinder = π × [(2x)² − x²] × 7

⇒ 21πx² = 504π

⇒ x² = 504 / 21

⇒ x² = 24

⇒ x = √24 = 2√6 cm

⇒ Outer radius = 2x = 2 × 2√6 = 4√6 cm

∴ The correct answer is 4√6 cm.

Additional Information

Volume Conservation

When a solid is melted and recast into a different shape, the total Volume of the material remains the same.

Hollow Cylinder Properties

The volume of a hollow cylinder with outer radius R, inner radius r, and height h is V = π(R² − r²)h.

Total Surface Area of Hollow Cylinder

The total surface area is given by 2π(R + r)(h + R − r).

Shortcut Trick

Compare effective discount percentages for 10 toys:

Scheme A: 38 + 39 − (38 × 39)/100 = 77 − 14.82 = 62.18%

Scheme B: 42 + 38 − 15.96 = 64.04% → 64.04 + 20 − 12.8 = 71.23%

Scheme C: [ (2 × 11) + (8 × 42) ] ÷ 10 = (22 + 336) ÷ 10 = 35.8%

Scheme D: [ 9 ÷ (10 + 9) ] × 100 ≈ 47.37%

Comparing all: 35.8% (C) is the lowest discount, hence least beneficial.

∴ The correct answer is Scheme C.

Alternate Method

Assume Marked Price (MP) of each toy = ₹100. Total MP for 10 toys = ₹1000.

Detailed Calculations:

⇒ Scheme A: Successive 38% and 39%.

⇒ Net Discount % = 38 + 39 − (38 × 39)/100 = 77 − 14.82 = 62.18%

⇒ Scheme B: Successive 42%, 38% and 20%.

⇒ Net of 42% and 38% = 42 + 38 − (42 × 38)/100 = 80 − 15.96 = 64.04%

⇒ Net with 20% = 64.04 + 20 − (64.04 × 20)/100 = 84.04 − 12.808 = 71.232%

⇒ Scheme C: 11% on first 2, 42% on next 8 toys.

⇒ Total Discount = (2 × 11% of 100) + (8 × 42% of 100) = 22 + 336 = ₹358

⇒ Effective Discount % = (358 ÷ 1000) × 100 = 35.8%

⇒ Scheme D: 9 toys free on buying 10.

⇒ Discount % = [Free Items ÷ Total Items] × 100 = [9 ÷ (10 + 9)] × 100 ≈ 47.37%

⇒ Comparing all discounts: 35.8% is the minimum value.

∴ The correct answer is Scheme C.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the equivalent single discount is given by (a + b − ab/100)%.

Buy X Get Y Free Scheme

The effective discount percentage in a 'Buy X Get Y Free' offer is calculated as [Y ÷ (X + Y)] × 100.

Benefit to Customer

A scheme is most beneficial when the discount percentage is highest and least beneficial when the discount percentage is lowest.