Quantitative Aptitude PYQs

Shortcut Trick
Ratio of Cost Price (CP) to Selling Price (SP) = 850 : 1,020.
Dividing by 170: CP : SP = 5 : 6.
Profit = 6 − 5 = 1 unit.
Profit% = (1 ÷ 5) × 100 = 20%.
∴ The correct answer is 20%.

Alternate Method
Given → CP = ₹850, SP = ₹1,020.
Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100.

⇒ Profit = 1,020 − 850 = ₹170.
⇒ Profit% = (170 ÷ 850) × 100.
⇒ Profit% = (1 ÷ 5) × 100 = 20%.
∴ The correct answer is 20%.

Additional Information
Profit and Loss Basics
Profit occurs when SP > CP. Loss occurs when CP > SP. Formula: Loss% = (Loss ÷ CP) × 100.
Selling Price Calculation
SP = CP × (100 + P%) ÷ 100 or SP = CP × (100 − L%) ÷ 100.

Shortcut Trick

Total Work (LCM of 24, 18, 36) = 72 units.

Efficiencies: A = 3, B = 4, C = 2.

A works for 4 days and leaves: 72 − (3 × 4) = 60 units remaining.

B leaves 3 days before completion, so add B's work for 3 days to the total: 60 + (4 × 3) = 72 units.

Time taken by B and C to finish this adjusted work = 72 ÷ (4 + 2) = 12 days.

∴ The correct answer is 12 days.

Alternate Method

Let the total time for the work be x days.

⇒ A worked for 4 days, B worked for (x − 3) days, and C worked for x days.

⇒ Work done by A + Work done by B + Work done by C = 72

⇒ (3 × 4) + 4(x − 3) + 2x = 72

⇒ 12 + 4x − 12 + 2x = 72

⇒ 6x = 72

⇒ x = 12

∴ The correct answer is 12 days.

Additional Information

Leaving Before Completion Shortcut

If a person leaves n days before completion, add their n days' work to the total work and divide by the combined efficiency of the remaining workers.

Total Work and Efficiency

Total Work is always assumed as the LCM of individual days. Efficiency = Total Work ÷ Individual Time.

Work & Wages

Wages are distributed in the ratio of the actual work done by each person, not just their efficiencies.

Shortcut Trick

Scheme i net discount = 48%.

Scheme ii (31% and 44%) is clearly higher than 48% as the first discount alone is 31% and the total will exceed 48% (Approx 31 + 44 − 13.6 = 61.36%).

Scheme iii (32% and 21%) net = 32 + 21 − (32 × 21 ÷ 100) = 53 − 6.72 = 46.28%.

Comparing 48%, 61.36%, and 46.28%, the least discount is 46.28%.

For a customer, the least beneficial scheme is the one offering the lowest discount percentage.

∴ The correct answer is Scheme iii.

Alternate Method

Formula: Successive Discount = a + b − (a × b ÷ 100)%

⇒ Scheme i: Discount = 48%.
⇒ Scheme ii: a = 31, b = 44. Net = 31 + 44 − (31 × 44 ÷ 100) = 75 − 13.64 = 61.36%.
⇒ Scheme iii: a = 32, b = 21. Net = 32 + 21 − (32 × 21 ÷ 100) = 53 − 6.72 = 46.28%.
⇒ Comparing all schemes: 61.36% (Best) > 48% > 46.28% (Worst).
⇒ Least beneficial to customer = lowest discount = Scheme iii.

∴ The correct answer is Scheme iii.

Additional Information

Customer vs Retailer Benefit

For a customer, the maximum discount percentage is the most beneficial. For a retailer, the minimum discount percentage is the most beneficial.

Successive Discounts Formula

Total Discount % for two successive discounts x% and y% = [x + y − (xy ÷ 100)]%.

Shortcut Trick

Use identity: 1/(a+b) − 1/(a−b) = −2b/(a² − b²)

⇒ Here a = 2, b = √2

⇒ 1/(2+√2) − 1/(2−√2) = −2√2/(4 − 2) = −√2

Now expression = √2 + 1/√2 − √2

⇒ = 1/√2

∴ Answer = \( \frac{1}{\sqrt{2}} \)

Alternate Method

Given:

√2 + 1/√2 + 1/(2+√2) − 1/(2−√2)

Concept Used:

Rationalization and BODMAS rule

Calculation:

⇒ Rationalize 1/(2+√2): (2−√2)/(4−2) = (2−√2)/2

⇒ Rationalize 1/(2−√2): (2+√2)/(4−2) = (2+√2)/2

⇒ Expression = √2 + 1/√2 + (2−√2)/2 − (2+√2)/2

⇒ = √2 + 1/√2 + [2−√2 − 2 − √2]/2

⇒ = √2 + 1/√2 + (−2√2)/2

⇒ = √2 + 1/√2 − √2

⇒ = 1/√2

∴ The correct answer is \( \frac{1}{\sqrt{2}} \).

Additional Information

Rationalization

Multiply numerator and denominator by conjugate to remove root from denominator.

Conjugate Identity

(a + b)(a − b) = a² − b²

Simplification Tip

Combine similar radical terms to reduce expression quickly.

Shortcut Trick

Combined efficiency of P and Q = 1/9

Efficiency of P alone = 1/12

Efficiency of Q alone = 1/9 − 1/12 = (4 − 3)/36 = 1/36

Time for Q to fill 1/4^th cistern = (1/4) ÷ (1/36) = 36 ÷ 4 = 9 hours

∴ The correct answer is 9 hours.

Alternate Method

Given: Time taken by (P+Q) = 9 hours, Time taken by P = 12 hours.

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

⇒ Total capacity of cistern = LCM(9, 12) = 36 units.

⇒ Efficiency of (P + Q) = 36 ÷ 9 = 4 units/hour.

⇒ Efficiency of P = 36 ÷ 12 = 3 units/hour.

⇒ Efficiency of Q = Efficiency of (P + Q) − Efficiency of P = 4 − 3 = 1 unit/hour.

⇒ Work to be done = 1/4 of Total Work = 1/4 × 36 = 9 units.

⇒ Time taken by Q = 9 units ÷ 1 unit/hour = 9 hours.

∴ The correct answer is 9 hours.

Additional Information

Pipes & Cisterns Efficiency

The rate of work done by a pipe is its efficiency. If a pipe fills a tank in 'x' hours, its 1-hour work is 1/x of the tank capacity.

Inlet and Outlet Pipes

Inlet pipes (filling) are considered to have positive efficiency, while outlet pipes (emptying/leakage) are considered to have negative efficiency.

Combined Work Formula

If pipe A fills in x hours and pipe B in y hours, they fill together in (x × y)/(x + y) hours.

Shortcut Trick

Convert ratio 1/2 : 1/6 : 1/4 to whole numbers by multiplying with LCM(2, 6, 4) = 12.

New Ratio = 6 : 2 : 3. Total units = 6 + 2 + 3 = 11 units.

11 units = 649 ⇒ 1 unit = 59.

First monkey = 6 units = 6 × 59 = 354.

∴ The correct answer is 354 bananas.

Alternate Method

Given: Total bananas = 649, Ratio = 1/2 : 1/6 : 1/4

⇒ Ratio of bananas = 1/2 : 1/6 : 1/4

⇒ Multiply each by 12 → (1/2 × 12) : (1/6 × 12) : (1/4 × 12) = 6 : 2 : 3

⇒ Sum of ratio parts = 6 + 2 + 3 = 11

⇒ Share of first monkey = (6/11) × 649

⇒ Share = 6 × 59 = 354

∴ The correct answer is 354 bananas.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d to a whole number ratio, multiply both terms by the LCM of the denominators (b and d).

Proportional Division

If a quantity Z is divided in ratio a:b:c, the share of the first part is calculated as [a / (a + b + c)] × Z.

Inverse Ratio

The inverse ratio of a : b is 1/a : 1/b, which simplifies to b : a for two terms.

Shortcut Trick

Equate increase in volumes: π × (r + 14)² × h − πr²h = πr² × (h + 30) − πr²h

Given Height (h) = 24 cm ⇒ 24 × (28r + 196) = 30r²

Solving the quadratic 5r² − 112r − 784 = 0 gives Radius (r) = 28 cm.

Calculate Q = 30 × π × r² = 30 × π × 28² = 23,520π cm³.

∴ The correct answer is 23,520π cm³.

Alternate Method

Given: Height (h) = 24 cm, increase in radius = 14 cm, increase in height = 30 cm, Increase in volume = Q cm³.

Formula Used: Volume of Cylinder = πr²h

⇒ Case 1: Increase in volume for vessel 1 (radius increases by 14)

⇒ Q = π(r + 14)²h − πr²h

⇒ Q = πh[(r + 14)² − r²] = πh[r² + 28r + 196 − r²] = πh(28r + 196)

⇒ Case 2: Increase in volume for vessel 2 (height increases by 30)

⇒ Q = πr²(h + 30) − πr²h = 30πr²

⇒ Equating Case 1 and Case 2: 24π(28r + 196) = 30πr²

⇒ 4(28r + 196) = 5r²

⇒ 112r + 784 = 5r² ⇒ 5r² − 112r − 784 = 0

⇒ Roots: r = [112 ± √(12544 + 15680)] / 10 = [112 ± 168] / 10

⇒ r = 280 / 10 = 28 cm (ignoring negative value)

⇒ Q = 30 × π × (28)² = 30 × 784 × π = 23,520π

∴ The correct answer is 23,520π cm³.

Additional Information

Volume of Cylinder

The volume of a cylinder is calculated as V = πr²h, where r is the radius and h is the height.

Total Surface Area of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πrh + 2πr² = 2πr(r + h).

Curved Surface Area (CSA)

The area of the lateral surface of the cylinder: CSA = 2πrh.

Shortcut Trick

Number of terms divisible by n in the range [1, L] is given by floor(L ÷ n).

Required count = floor(650 ÷ 7) − floor(351 ÷ 7)

Required count = 92 − 50 = 42

∴ The correct answer is 42.

Alternate Method

Given: Range is between 351 and 650. We need to find numbers divisible by 7.

Formula Used: n^th term of Arithmetic Progression (AP): a_n = a + (n − 1)d

⇒ First number after 351 divisible by 7: 351 ÷ 7 = 50.14 → 7 × 51 = 357 (a)

⇒ Last number before 650 divisible by 7: 650 ÷ 7 = 92.85 → 7 × 92 = 644 (a_n)

⇒ 644 = 357 + (n − 1) × 7

⇒ 644 − 357 = (n − 1) × 7

⇒ 287 = (n − 1) × 7

⇒ 41 = n − 1

⇒ n = 42

∴ The correct answer is 42.

Additional Information

Arithmetic Progression (AP)

A sequence where the difference between consecutive terms is constant. Sum S_n = n/2[2a + (n−1)d].

Divisibility Rule of 7

Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the number is divisible by 7.

Inclusive vs Exclusive Ranges

Between A and B usually excludes A and B. From A to B usually includes both endpoints.

Shortcut Trick

Difference in percentage between Candidate A and Candidate B = 60% − 30% = 30%.

Difference in votes = 30% of 1,20,000.

Difference = 0.30 × 1,20,000 = 36,000.

∴ The correct answer is 36,000 votes.

Alternate Method

Given:

Total votes = 1,20,000

Candidate A share = 60%

Candidate B share = 30%

Formula Used: Number of votes = (Percentage ÷ 100) × Total votes

Calculations:

⇒ Votes for Candidate A = (60 ÷ 100) × 1,20,000 = 72,000

⇒ Votes for Candidate B = (30 ÷ 100) × 1,20,000 = 36,000

⇒ Required difference = 72,000 − 36,000

⇒ Difference = 36,000

∴ The correct answer is 36,000 votes.

Additional Information

Percentage Change

To find the percentage difference between two values, use: |Value₁ − Value₂| ÷ Reference Value × 100.

Base Change in Election Problems

Always ensure all individual percentages (60% + 30% + 10%) sum up to 100% of the total valid votes.

Successive Percentage

If values change successively, the net percentage change is given by a + b + (ab ÷ 100).

Shortcut Trick

Since bases are same ⇒ equate powers

(x⁴ + x² + 1)/(x² + x + 1) = 2x + 5

Factor: x⁴ + x² + 1 = (x² − x + 1)(x² + x + 1)

⇒ LHS = x² − x + 1

So x² − x + 1 = 2x + 5

⇒ x² − 3x − 4 = 0

⇒ (x − 4)(x + 1) = 0

Positive value ⇒ x = 4

∴ Answer = 4

Alternate Method

Given:

3^((x⁴ + x² + 1)/(x² + x + 1)) = 3^(2x + 5)

Concept Used:

If a^m = a^n ⇒ m = n

Calculation:

⇒ (x⁴ + x² + 1)/(x² + x + 1) = 2x + 5

⇒ x⁴ + x² + 1 = (x² + x + 1)(2x + 5)

⇒ Simplify ⇒ x² − 3x − 4 = 0

⇒ Solve ⇒ x = 4, −1

⇒ Positive value ⇒ x = 4

∴ The correct answer is 4.

Additional Information

Exponent Rule

Same base ⇒ powers are equal

Factorization

x⁴ + x² + 1 = (x² − x + 1)(x² + x + 1)

Root Selection

Choose positive solution as required

Shortcut Trick

Relative Speed when moving towards each other = Sum of speeds = 13 + 17 = 30 km/hr.

Time to meet = Distance ÷ Relative Speed = 75 ÷ 30 = 2.5 hours.

Convert to minutes: 2.5 × 60 = 150 min.

∴ The correct answer is 150 min.

Alternate Method

Given:

Distance (D) = 75 km

Speed of Engine 1 (S₁) = 13 km/hr

Speed of Engine 2 (S₂) = 17 km/hr

Formula Used:

Time = Distance ÷ Relative Speed

Relative Speed (Opposite Direction) = S₁ + S₂

Calculations:

⇒ Relative Speed = 13 + 17 = 30 km/hr

⇒ Time (in hours) = 75 ÷ 30

⇒ Time (in hours) = 2.5 hours

⇒ Time (in minutes) = 2.5 × 60

⇒ Time = 150 min

∴ The correct answer is 150 min.

Additional Information

Relative Speed Concept

When two objects move toward each other, their relative speed is the sum of their individual speeds (S₁ + S₂). If they move in the same direction, it is the difference |S₁ − S₂|.

Unit Conversion

To convert hours to minutes, multiply by 60. To convert km/hr to m/s, multiply by 5/18.

Meeting Point Distance

Distance from starting point of Engine 1 = S₁ × Time = 13 × 2.5 = 32.5 km.

Shortcut Trick

For a Mean Proportional between two numbers x and y, the value of A is √(x × y).

Given values are 9 and 4.

A = √(9 × 4) = √36 = 6.

∴ The correct answer is 6.

Alternate Method

Given: 9 : A :: A : 4

Formula Used: In a proportion a : b :: c : d, the product of extremes is equal to the product of means (a × d = b × c).

⇒ 9 ÷ A = A ÷ 4

⇒ A × A = 9 × 4

⇒ A² = 36

⇒ A = √36

⇒ A = 6

∴ The correct answer is 6.

Additional Information

Mean Proportional

If three numbers a, b, and c are in continued proportion (a : b :: b : c), then b is called the mean proportional and b² = ac.

Third Proportional

In the continued proportion a : b :: b : c, the term c is known as the third proportional to a and b, calculated as c = b² ÷ a.

Fourth Proportional

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

Shortcut Trick

Apply the algebraic identity (a + b)² = a² + 2ab + b².

First term square: (3x)² = 9x²

Middle term (2 × Product): 2 × 3x × 2 = 12x

Last term square: 2² = 4

Combine all terms: 9x² + 12x + 4

∴ The correct answer is 9x² + 12x + 4.

Alternate Method

Given:

Expression to simplify: (3x + 2)²

Formula Used:

The standard expansion for a binomial square is (a + b)² = a² + 2ab + b².

Calculations:

⇒ Identify the values: a = 3x and b = 2.

⇒ Substitute into the expansion formula: (3x + 2)² = (3x)² + 2 × (3x) × (2) + (2)²

⇒ Square the first term: (3x)² = 9x²

⇒ Multiply the middle terms: 2 × 3x × 2 = 12x

⇒ Square the second term: (2)² = 4

⇒ Simplify the full expression: 9x² + 12x + 4

∴ The correct answer is 9x² + 12x + 4.

Additional Information

Binomial Square (Subtraction)

The expansion for the difference of a binomial is (a − b)² = a² − 2ab + b².

Difference of Two Squares

Multiplying the sum and difference of the same terms results in (a + b)(a − b) = a² − b².

Square of a Trinomial

For three terms, the formula is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Shortcut Trick

The reasoning involves applying the successive discount formula iteratively to combine multiple percentage reductions into a single equivalent value.

Successive discount for 20% and 4% = 20 + 4 − (20 × 4) ÷ 100 = 23.2%.

Successive discount for 23.2% and 14% = 23.2 + 14 − (23.2 × 14) ÷ 100.

Result = 37.2 − 3.248 = 33.952%.

∴ The correct answer is 33.95%.

Alternate Method

Given → Discounts = 20%, 4%, 14%.
Formulas: Net SP = MP × (100 − d₁)% × (100 − d₂)% × (100 − d₃)% | Single Discount = 100 − Net SP.

⇒ Let Marked Price (MP) = 100.
⇒ After 20% discount: 100 × (80/100) = 80.
⇒ After 4% discount: 80 × (96/100) = 76.8.
⇒ After 14% discount: 76.8 × (86/100) = 66.048.
⇒ Single Discount = 100 − 66.048 = 33.952%.
∴ The correct answer is 33.95%.

Additional Information

Successive Discounts

When a series of discounts are applied, the subsequent discount is calculated on the reduced price, not the original Marked Price.

Formula for Two Discounts

Equivalent single discount for a% and b% = (a + b − ab/100)%.

Shortcut Trick

In any quadrilateral PQRS, the angle formed by the bisectors of two adjacent angles (∠R and ∠S) is equal to half the sum of the other two angles (∠P and ∠Q).

Logic: ∠STR = (∠P + ∠Q) ÷ 2

Calculation: 14° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 28°

Given ratio ∠P : ∠Q = 2 : 5. Total parts = 7.

Difference = (5 − 2) ÷ 7 × 28° = 3 × 4° = 12°.

∴ The correct answer is 12°.

Alternate Method

Given:

∠STR = 14° (Angle between bisectors of ∠R and ∠S)

Ratio ∠P : ∠Q = 2 : 5

Formula Used:

Sum of interior angles of a quadrilateral = 360°

In ΔSTR, ∠TSR + ∠TRS + ∠STR = 180°

Calculations:

⇒ In ΔSTR: (∠S ÷ 2) + (∠R ÷ 2) + 14° = 180°

⇒ (∠S + ∠R) ÷ 2 = 18

Shortcut Trick

Calculate the excess runs scored above the old average: 119 − 71 = 48 runs.

Distribute these excess runs equally across all 12 innings to find the average increase.

Increase in average = 48 ÷ 12 = 4 runs.

∴ The correct answer is 4 runs.

Alternate Method

Given:

Average of 11 innings = 71 runs

Score in 12^th inning = 119 runs

Formula Used:

Total Sum = Average × Number of items

Increase in Average = New Average − Old Average

Calculations:

⇒ Total runs in 11 innings = 11 × 71 = 781 runs

⇒ Total runs after 12 innings = 781 + 119 = 900 runs

⇒ New average score = 900 ÷ 12 = 75 runs

⇒ Increase in average = 75 − 71 = 4 runs

∴ The correct answer is 4 runs.

Additional Information

Net Change in Average

When a new value is added, the change in average = (New Value − Old Average) ÷ Total New Count.

Weighted Average Formula

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the individual average.

Effect of Constant Addition

If a constant value 'k' is added to every observation in a data set, the new average also increases by exactly 'k'.

Shortcut Trick

Saving % = 100% − 30% = 70%

Daily Saving = 70% of ₹500

⇒ 0.70 × 500 = 350

∴ The correct answer is ₹350.

Alternate Method

Given:

Daily Earning = ₹500

Expenditure Percentage = 30%

Formula Used:

Saving = Earning − Expenditure

Calculations:

⇒ Daily Expenditure = 30% of ₹500

⇒ Daily Expenditure = (30/100) × 500 = ₹150

⇒ Daily Saving = Total Earning − Daily Expenditure

⇒ Daily Saving = ₹500 − ₹150 = ₹350

∴ The correct answer is ₹350.

Additional Information

Percentage Change

If a value is decreased by x%, the remaining value is (100 − x)% of the original.

Saving-Income Relation

Income = Expenditure + Savings. This fundamental equation is used for budget and tax problems.

Base Value

In profit, loss, or saving calculations, the Income or Cost Price is usually treated as the 100% base.

Shortcut Trick

Net efficiency = Filling rate − Emptying rate

Total capacity = LCM(5, 10) = 10 units

Filling pipe efficiency = 10 ÷ 5 = 2 units/min; Emptying pipe efficiency = 10 ÷ 10 = 1 unit/min

Net efficiency = 2 − 1 = 1 unit/min

Time to fill half tank (5 units) = 5 ÷ 1 = 5 minutes

∴ The correct answer is 5 min.

Alternate Method

Given: Filling time = 5 min, Emptying time = 10 min.

Formula: Total Work = Time × Efficiency

⇒ Total Capacity of tank = LCM of 5 and 10 = 10 units

⇒ Capacity to be filled (one-half) = 10 ÷ 2 = 5 units

⇒ Combined Efficiency = 2 + (−1) = 1 unit/min

⇒ Required Time = Half Capacity ÷ Combined Efficiency

⇒ Required Time = 5 ÷ 1 = 5 minutes

∴ The correct answer is 5 min.

Additional Information

Pipes and Cisterns Logic

Filling pipes have positive efficiency while outlet or emptying pipes have negative efficiency.

Net Work Formula

If pipe A fills in 'x' hours and pipe B empties in 'y' hours, net work in 1 hour = (1/x − 1/y).

Time and Work Relation

Total Work = Efficiency × Time. For partial filling, multiply the total capacity by the required fraction.

Shortcut Trick

Total days from 10 Feb to 24 Apr 2023 = 18 (Feb) + 31 (Mar) + 24 (Apr) = 73 days.

Time in years = 73 ÷ 365 = 1/5 year.

Simple Interest = (2000 × 7 × 1) ÷ (100 × 5) = 20 × 7 ÷ 5 = 4 × 7 = ₹28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate (R) = 7% per annum

Time period: 10 February 2023 to 24 April 2023

Step-by-step Calculation:

⇒ Number of days in Feb = 28 − 10 = 18 days (2023 is not a leap year)

⇒ Number of days in March = 31 days

⇒ Number of days in April = 24 days

⇒ Total days = 18 + 31 + 24 = 73 days

⇒ Time (T) in years = 73 ÷ 365 = 1/5 year

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

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (400 × 7) ÷ 100

⇒ SI = 2800 ÷ 100 = ₹28

∴ The correct answer is ₹28.

Additional Information

Standard Day Conversion

In simple interest problems involving days, 73 days is frequently used as it equals exactly 1/5 of a year (73 × 5 = 365).

Leap Year Check

Always check if February falls in a leap year (divisible by 4). If yes, February has 29 days; otherwise, it has 28 days.

Date Calculation Rule

While counting days, generally the day on which money is deposited is excluded and the day on which money is withdrawn is included.

Shortcut Trick

For a solid cuboid, if the areas of three adjacent faces are A₁, A₂, and A₃, then its volume (V) is given by:

V = √(A₁ × A₂ × A₃)

V = √(92 × 69 × 75) = √((4 × 23) × (3 × 23) × (3 × 25))

V = √(2² × 23² × 3² × 5²) = 2 × 23 × 3 × 5

V = 690 cm³.

∴ The correct answer is 690 cm³.

Alternate Method

Given:

Area of face 1 (l × w) = 92 cm²

Area of face 2 (w × h) = 69 cm²

Area of face 3 (l × h) = 75 cm²

Formula Used:

Volume (V) = length (l) × width (w) × height (h)

Calculations:

⇒ (l × w) × (w × h) × (h × l) = 92 × 69 × 75

⇒ l² × w² × h² = 92 × 69 × 75

⇒ (l × w × h)² = 92 × 69 × 75

⇒ V² = 476100

⇒ V = √476100

⇒ V = 690 cm³

∴ The correct answer is 690 cm³.

Additional Information

Total Surface Area of a Cuboid

The total surface area is the sum of the areas of all 6 rectangular faces: TSA = 2(lw + wh + lh).

Diagonal of a Cuboid

The longest line segment connecting two opposite vertices: d = √(l² + w² + h²).

Volume of a Cube

For a cube where all sides are equal (a), the volume is V = a³.