Quantitative Aptitude PYQs

Shortcut Trick

Total Monthly Cost = Daily Units × Rate × Number of Days

Total Monthly Cost = 12 × 8 × 31

Total Monthly Cost = 96 × 31 = ₹2,976

∴ The correct answer is ₹2,976.

Alternate Method

Given:

Daily consumption of electricity = 12 units

Cost per unit = ₹8

Total duration = 31-day month

Formula Used:

Total Payment = (Units per day × Total days) × Rate per unit

⇒ Total units consumed in 31 days = 12 × 31

⇒ Total units consumed = 372 units

⇒ Total payment = 372 × 8

⇒ Total payment = ₹2,976

∴ The correct answer is ₹2,976.

Additional Information

Unitary Method

A technique for solving problems by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Total Expenditure Formula

Total Expenditure = Price × Consumption. If consumption is given as a daily rate, multiply by the number of days in the specific month.

Electricity Units (kWh)

1 Unit of electricity is equivalent to 1 kiloWatt-hour (kWh). The total bill is the product of Total Units × Slab Rate.

Shortcut Trick

Percentage difference between the two values = 45% − 20% = 25%

According to the question, this 25% corresponds to 48.

To find the total number (100%), we multiply by 4:

100% = 48 × 4 = 192

∴ The correct answer is 192.

Alternate Method

Given:

First percentage of the number = 45%

Second percentage of the same number = 20%

Numerical difference between them = 48

Formula Used:

Number = (Given Difference ÷ Percentage Difference) × 100

Calculations:

⇒ Let the required number be x.

⇒ 45% of x − 20% of x = 48

⇒ (45 − 20)% of x = 48

⇒ 25% of x = 48

⇒ (25 ÷ 100) × x = 48

⇒ x ÷ 4 = 48

⇒ x = 48 × 4

⇒ x = 192

∴ The correct answer is 192.

Additional Information

Percentage to Fraction Conversion

Always remember common fractional equivalents for faster calculation: 20% = 1/5, 25% = 1/4, 40% = 2/5, 50% = 1/2, and 75% = 3/4.

Net Percentage Change

When a number is changed by two successive percentages a% and b%, the net percentage change is given by the formula: (a + b + (a × b) ÷ 100)%.

Base Value Concept

Percentage calculations are always relative to a base value (100%). If the base increases, the absolute value of the percentage also increases proportionally.

Shortcut Trick

Net percentage change for a decrease of x% followed by an increase of x% is always a decrease of (x²/100)%.

Step 1: Net change for −30% and +30% = −(30²/100) = −9%.

Step 2: Apply successive change formula for −9% and +20%: a + b + (ab/100).

⇒ −9 + 20 + (−9 × 20)/100 = 11 − 1.8 = 9.2%.

Since the result is positive, it is an increase.

∴ The correct answer is 9.2% increase.

Alternate Method

Given: First year decrease = 30%, Second year increase = 30%, Third year increase = 20%.

Formula: Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × (1 ± r₃/100)

⇒ Let the initial value of the investment be 100.

⇒ Value after the first year = 100 − 30% of 100 = 70.

⇒ Value after the second year = 70 + 30% of 70 = 70 + 21 = 91.

⇒ Value after the third year = 91 + 20% of 91 = 91 + 18.2 = 109.2.

⇒ Overall percentage change = (Final Value − Initial Value) ÷ Initial Value × 100

⇒ Net change = (109.2 − 100) ÷ 100 × 100 = 9.2%.

∴ The correct answer is 9.2% increase.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the effective net change is given by (a + b + ab/100)%.

Fraction Method for Multi-Step Changes

Convert percentages to fractions (e.g., 20% = 1/5). For an increase, multiply by (1 + 1/5) = 6/5; for a decrease, multiply by (1 − 1/5) = 4/5.

Inverse Percentage Change

If a value increases by r%, it must decrease by [r/(100 + r)] × 100% to return to the original value.

Shortcut Trick

Assume all 400 questions are correct: 400 × 4 = 1600 marks.

Actual score = 1280 marks; Difference = 1600 − 1280 = 320 marks.

Loss per wrong/unattempted question = 4 (marks not gained) + 1 (penalty) = 5 marks.

Number of wrong/unattempted questions = 320 ÷ 5 = 64.

Number of correct questions = 400 − 64 = 336.

∴ The correct answer is 336.

Alternate Method

Given:

Total questions = 400

Marks for correct answer = +4

Marks for wrong/unattempted = −1

Total score obtained = 1280

Formula Used:

Total Score = (Number of Correct Answers × Marks per Correct) − (Number of Incorrect Answers × Negative Marks)

Calculations:

Let the number of correctly answered questions be C.

Let the number of wrong or unattempted questions be W.

⇒ C + W = 400 --- (i)

⇒ 4C − 1W = 1280 --- (ii)

Adding equations (i) and (ii):

⇒ (C + W) + (4C − W) = 400 + 1280

⇒ 5C = 1680

⇒ C = 1680 ÷ 5

⇒ C = 336

∴ The correct answer is 336.

Additional Information

Net Negative Marking Effect

For every wrong answer, the net loss is the sum of the positive marks and the penalty (e.g., 4 + 1 = 5). This is used in the "Assumption Method" to find the total incorrect count.

Accuracy Calculation

Accuracy % = (Number of Correct Answers ÷ Total Questions Attempted) × 100. In this case, since unattempted also carry a penalty, the base is the total 400 questions.

Alligation Method

This can be solved by comparing the average marks per question (1280 ÷ 400 = 3.2) against the individual marks (+4 and −1) to find the ratio of correct to wrong questions.

Shortcut Trick

Multiplying factor for 2 years = Amount ÷ Principal = 1.1025

Multiplying factor for 1 year = √1.1025 = 1.05

Rate of interest = (1.05 − 1) × 100 = 0.05 × 100 = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Time (n) = 2 years

Amount (A) = 1.1025 × Principal (P)

Formula Used:

A = P(1 + r/100)ⁿ

Calculations:

⇒ 1.1025 × P = P(1 + r/100)²

⇒ 1.1025 = (1 + r/100)²

⇒ √1.1025 = 1 + r/100

⇒ 1.05 = 1 + r/100

⇒ r/100 = 1.05 − 1

⇒ r/100 = 0.05

⇒ r = 5%

∴ The correct answer is 5%.

Additional Information

Successive Percentage Rule

For two years, if the annual rate is r%, the effective rate is given by r + r + (r × r)/100.

Amount for n years

When interest is compounded annually, the total amount after n years is A = P(1 + r/100)ⁿ.

Multiplying Factor

If a sum grows by r% every year, it is multiplied by (1 + r/100) each year. For 2 years, the factor is (1 + r/100)².

Shortcut Trick

Assume an Assumed Mean of 360.

Calculate deviations: (320−360), (400−360), (360−360), (340−360), (380−360)

Net Deviation = (−40) + (+40) + (0) + (−20) + (+20) = 0

Average = Assumed Mean + (Net Deviation ÷ Total Flats) = 360 + (0 ÷ 5) = 360

∴ The correct answer is 360 units.

Alternate Method

Given: Monthly consumption of five flats are 320, 400, 360, 340, and 380 units.

Formula Used: Average = Sum of all observations ÷ Total number of observations

⇒ Total Consumption = 320 + 400 + 360 + 340 + 380

⇒ Total Consumption = 1800 units

⇒ Number of Flats = 5

⇒ Average = 1800 ÷ 5

⇒ Average = 360 units

∴ The correct answer is 360 units.

Additional Information

Arithmetic Mean

It is the most common measure of central tendency, calculated as Σx ÷ n, where Σx is the sum of values and n is the count.

Properties of Mean

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

Combined Average

If two groups have sizes n₁, n₂ and averages a₁, a₂, then the combined average is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Shortcut Trick

Net profit % on 1^st half = 60 − 25 − (60 × 25) ÷ 100 = 20%

Net profit % on 2^nd half = 80 − 30 − (80 × 30) ÷ 100 = 26%

Since both halves are equal in quantity, average profit is calculated.

Overall Profit % = (20% + 26%) ÷ 2 = 23%

∴ The correct answer is 23%.

Alternate Method

Given:

Initial markup percentage = 60%

Revised markup percentage for second half = 80%

Discount on first half = 25% | Discount on second half = 30%

Formula Used:

SP = CP × (100 + Markup %)/100 × (100 − Discount %)/100

Overall Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Let the total quantity be 2 units and cost price of each unit be 100.

⇒ Total Cost Price = 2 × 100 = 200

⇒ Selling Price of 1^st unit = 100 × (160/100) × (75/100) = 120

⇒ Selling Price of 2^nd unit = 100 × (180/100) × (70/100) = 126

⇒ Total Selling Price = 120 + 126 = 246

⇒ Total Profit = 246 − 200 = 46

⇒ Overall Profit Percentage = (46 ÷ 200) × 100 = 23%

∴ The correct answer is 23%.

Additional Information

Net Percentage Change

When a value changes by x% and then by y%, the net percentage change is given by the formula x + y + (xy ÷ 100). In profit-loss, markup is positive and discount is negative.

CP, MP, and Profit Relationship

The ratio of Marked Price to Cost Price can be directly calculated as MP ÷ CP = (100 + Profit %) ÷ (100 − Discount %).

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − d₁d₂ ÷ 100)%.

Shortcut Trick

The difference between a 12% Loss and an 8% Profit is 12% + 8% = 20%.

Given that this 20% difference is equal to ₹84.

20% of Cost Price (CP) = ₹84.

100% (Total CP) = 84 × 5 = ₹420.

∴ The correct answer is ₹420.

Alternate Method

Given:

Initial Loss = 12%

New Profit = 8%

Increase in Selling Price (SP) = ₹84

Formula Used:

SP = CP × (100 ± Profit/Loss)%

Calculations:

⇒ Let the Cost Price (CP) be 100 units.

⇒ Initial Selling Price (SP₁) = 100 − 12 = 88 units.

⇒ New Selling Price (SP₂) = 100 + 8 = 108 units.

⇒ Difference in Selling Price = 108 − 88 = 20 units.

⇒ 20 units = ₹84.

⇒ 1 unit = 84 ÷ 20 = ₹4.2.

⇒ 100 units (CP) = 4.2 × 100 = ₹420.

⇒ Therefore, the Cost Price of the article is ₹420, which corresponds to Option 2.

∴ The correct answer is ₹420.

Additional Information

Profit and Loss on Cost Price

Unless stated otherwise, profit or loss percentage is always calculated on the Cost Price (CP). Formula: Profit % = (Profit ÷ CP) × 100.

Relation between CP and SP

If there is a profit of P%, then SP = CP × (100 + P)/100. If there is a loss of L%, then SP = CP × (100 − L)/100.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount. MP = CP + Markup.

Shortcut Trick

The Volume of a cylinder is calculated using the formula πr²h.

Since π = 22/7, the final result must be a multiple of 11.

Checking the options for divisibility by 11: In 4158, the sum of digits at odd positions (4+5=9) minus the sum of digits at even positions (1+8=9) is 0.

Thus, 4158 is the only option that satisfies the divisibility rule of 11.

∴ The correct answer is 4158 cm³.

Alternate Method

Given:

Radius (r) = 7 cm

Height (h) = 27 cm

π = 22/7

Formula Used:

Volume of a Cylinder = π × r² × h

Calculations:

⇒ Volume = (22/7) × 7 × 7 × 27

⇒ Volume = 22 × 7 × 27

⇒ Volume = 154 × 27

⇒ Volume = 4158 cm³

∴ The correct answer is 4158 cm³.

Additional Information

Curved Surface Area (CSA) of a Cylinder

The area of the curved surface only, excluding the top and bottom circles, is given by 2πrh.

Total Surface Area (TSA) of a Cylinder

The total area including the two circular bases and the curved surface: 2πr(h + r).

Volume of a Cone

If a cone has the same base radius and height as the cylinder, its volume is exactly one-third: (1/3)πr²h.

Given:

((3⁵ × 6⁴ × 4³) ÷ (3² × 6² × 4¹))

Formula used:

The Law of Exponents for division states: a^m ÷ aⁿ = a^m−n

Calculation:

Subtracting the powers of the terms with the same base:

⇒ 3⁵⁻² × 6⁴⁻² × 4³⁻¹

⇒ 3³ × 6² × 4²

Evaluating the exponential values:

⇒ 27 × 36 × 16

⇒ 972 × 16

⇒ 15552

∴ The correct answer is option (1).

Given:

((23/8))¹² × ((8/23))¹⁵ × ((23/8))⁹ = ((8/23))^{8m + 8}

Formula used:

(a/b)ⁿ = (b/a)⁻ⁿ

a^x × a^y × a^z = a^{(x + y + z)}

If a^p = a^q, then p = q

Calculations:

The given equation is:

((23/8))¹² × ((8/23))¹⁵ × ((23/8))⁹ = ((8/23))^{8m + 8}

Convert all terms to a common base (8/23):

⇒ ((8/23))⁻¹² × ((8/23))¹⁵ × ((8/23))⁻⁹ = ((8/23))^{8m + 8}

Combine the exponents on the left side:

⇒ ((8/23))^{(−12 + 15 − 9)} = ((8/23))^{8m + 8}

⇒ ((8/23))⁻⁶ = ((8/23))^{8m + 8}

Compare the exponents of the same base:

⇒ −6 = 8m + 8

⇒ 8m = −6 − 8

⇒ 8m = −14

⇒ m = −14/8

⇒ m = −7/4

∴ The correct answer is option (4).

Shortcut Trick

Total Work (LCM of 12 and 33) = 132 units.

Efficiency of A = 11, B = 4. Combined = 15 units/day.

Work done by A and B in 4 days = 15 × 4 = 60 units.

Remaining Work = 132 − 60 = 72 units.

Time taken by B to complete remaining work = 72 ÷ 4 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 33 days

Time for which both worked = 4 days

Formula Used:

Total Work = LCM of time taken | Efficiency = Work ÷ Time

Calculations:

⇒ Total Work = LCM(12, 33) = 132 units.

⇒ Efficiency of A = 132 ÷ 12 = 11 units/day.

⇒ Efficiency of B = 132 ÷ 33 = 4 units/day.

⇒ Total work done by (A + B) in 4 days = (11 + 4) × 4 = 60 units.

⇒ Remaining work = 132 − 60 = 72 units.

⇒ Time taken by B for remaining work = 72 ÷ 4 = 18 days.

∴ The correct answer is 18 days.

Additional Information

Work and Efficiency Relationship

The Efficiency of a person is inversely proportional to the Time taken to complete the work. Efficiency = Work ÷ Time.

Total Work by LCM Method

For multiple workers, taking the LCM of their individual time periods as the Total Work simplifies fractional calculations into whole numbers.

Combined Work Equation

If multiple people work together, their Total Efficiency is the sum of their individual efficiencies: E_total = E₁ + E₂ + ... + E_n.

Shortcut Trick

Use the effective price factor method: Profit Factor = (1 + Markup%) × (1 − Discount%) × (True Weight ÷ Faulty Weight).

Net Selling Price Factor = 1.25 × 0.80 × (1000 ÷ 800) = 1 × 1.25 = 1.25.

Effective Selling Price per kg = 1.25 × ₹100 = ₹125.

Profit per kg = ₹125 − ₹100 = ₹25.

∴ The correct answer is ₹25.

Alternate Method

Given:

Cost Price (CP) of 1 kg = ₹100

Markup = 25% | Discount = 20%

Actual Weight sold as 1 kg = 800 g

Formula Used:

Marked Price (MP) = CP × (100 + Markup%) ÷ 100

Selling Price (SP) = MP × (100 − Discount%) ÷ 100

Profit = Final Revenue − Initial Investment

Calculations:

⇒ Marked Price (MP) per kg = 100 + (25% of 100) = ₹125

⇒ Selling Price (SP) for a claimed kg = 125 − (20% of 125) = 125 − 25 = ₹100

⇒ In reality, the Merchant receives ₹100 for selling only 800 g of tea.

⇒ Total Revenue from selling 1000 g = (100 ÷ 800) × 1000 = ₹125

⇒ Profit per kg = Total Revenue − Total Cost Price = 125 − 100 = ₹25

∴ The correct answer is ₹25.

Additional Information

Dishonest Dealer Profit

When a dealer uses a faulty weight, Profit% = [Error ÷ (True Value − Error)] × 100.

Successive Percentage Change

The net change for markup and discount is given by [M − D − (MD ÷ 100)]%.

Relation between CP and MP

MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%). This is used to find markup directly when profit and discount are known.

Shortcut Trick

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

Here, a = 4 and b = 64.

Value of x = √(4 × 64) = √256 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Length of the rectangular garden = 16 m

Width of the rectangular garden = x m

Numbers for mean proportional = 4 m and 64 m

Formula Used:

If x is the mean proportional between a and b, then a : x :: x : b

⇒ a/x = x/b

⇒ x² = a × b

Calculations:

⇒ x² = 4 × 64

⇒ x² = 256

⇒ x = √256

⇒ x = 16

∴ The correct answer is 16.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is the square root of their product: x = √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a : b :: b : c), then c is the third proportional: c = b² ÷ a.

Fourth Proportional

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

Shortcut Trick

Ratio of products = (p₁ × p₂ × p₃) ÷ (p₂ × p₃ × p₄) = 105 ÷ 385.

Common factors p₂ and p₃ cancel out, leaving p₁ ÷ p₄ = 105 ÷ 385.

Simplifying the fraction: 105 ÷ 385 = 21 ÷ 77 = 3 ÷ 11.

Since the numbers are prime numbers listed in order, we have p₁ = 3 and p₄ = 11.

The largest prime number is p₄ = 11.

∴ The correct answer is 11.

Alternate Method

Given:

Product of the first three prime numbers = 105

Product of the last three prime numbers = 385

Calculations:

⇒ Let the four prime numbers be p₁, p₂, p₃, and p₄.

⇒ Step 1: Factorise 105 into prime numbers ⇒ 105 = 3 × 5 × 7.

⇒ Step 2: Factorise 385 into prime numbers ⇒ 385 = 5 × 7 × 11.

⇒ Step 3: By comparing the two sets of factors, the common primes are 5 and 7.

⇒ The unique prime from the first set is 3, and the unique prime from the last set is 11.

⇒ The sequence of four prime numbers is 3, 5, 7, 11.

⇒ The largest prime number is 11.

∴ The correct answer is 11.

Additional Information

Prime Numbers

A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7, 11...).

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a unique product of prime numbers, regardless of the order of factors.

Co-prime Numbers

Two numbers are called co-prime if their Highest Common Factor (HCF) is 1. All distinct prime numbers are co-prime to each other.

Shortcut Trick

The ratio of sugar to flour is 7 : 11.

Given sugar = 154 g ⇒ 7 units = 154 g.

1 unit = 154 ÷ 7 = 22 g.

Flour required = 11 units = 11 × 22 = 242 g.

∴ The correct answer is 242 g.

Alternate Method

Given:

Sugar : Flour = 7 : 11

Amount of sugar = 154 g

Let the amount of sugar be 7x and the amount of flour be 11x.

⇒ 7x = 154

⇒ x = 154 ÷ 7

⇒ x = 22

⇒ Quantity of flour = 11x

⇒ 11 × 22 = 242 g

∴ The correct answer is 242 g.

Additional Information

Direct Proportion

When two quantities increase or decrease in the same ratio, they are in direct proportion. Formula: x₁/y₁ = x₂/y₂.

Property of Ratios

Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio (a : b = ka : kb).

Unitary Method

Finding the value of a single unit first and then multiplying it by the required number of units to find the total value.

Shortcut Trick

Divide the whole equation sin θ + cos θ = √2 cos θ by cos θ.

(sin θ ÷ cos θ) + (cos θ ÷ cos θ) = √2 (cos θ ÷ cos θ)

tan θ + 1 = √2

tan θ = √2 − 1

∴ The correct answer is √2 − 1.

Alternate Method

Given: sin θ + cos θ = √2 cos θ

Formula Used: tan θ = sin θ ÷ cos θ

Calculations:

⇒ sin θ + cos θ = √2 cos θ

⇒ sin θ = √2 cos θ − cos θ

⇒ sin θ = cos θ × (√2 − 1)

⇒ sin θ ÷ cos θ = √2 − 1

⇒ tan θ = √2 − 1

∴ The correct answer is √2 − 1.

Additional Information

Trigonometric Ratio Definition

The tan θ ratio represents the slope of the hypotenuse and is calculated as Opposite ÷ Adjacent.

Relationship between Ratios

For any angle θ, tan θ = sin θ ÷ cos θ and cot θ = cos θ ÷ sin θ.

Complementary Angles

In a right triangle, the ratios of complementary angles are related such that tan(90° − θ) = cot θ.

Shortcut Trick

Ratio of Boys : Girls : Teachers = 25 : 55 : (100 − 25 − 55) = 25 : 55 : 20 = 5 : 11 : 4.

Use Weighted Average method: Average = Σ(Ratio × Value) ÷ ΣRatio

Average weight = (5 × 50 + 11 × 55 + 4 × 65) ÷ (5 + 11 + 4) = (250 + 605 + 260) ÷ 20

Average weight = 1115 ÷ 20 = 55.75 kg.

∴ The correct answer is 55.75 kg.

Alternate Method

Given:

Percentage of Boys = 25% | Average weight = 50 kg

Percentage of Girls = 55% | Average weight = 55 kg

Percentage of Teachers = 100% − (25% + 55%) = 20% | Average weight = 65 kg

Calculations:

⇒ Let the total number of people in the group = 100.

⇒ Total weight of Boys = 25 × 50 = 1250 kg.

⇒ Total weight of Girls = 55 × 55 = 3025 kg.

⇒ Total weight of Teachers = 20 × 65 = 1300 kg.

⇒ Sum of all weights = 1250 + 3025 + 1300 = 5575 kg.

⇒ Average weight of the group = Total weight ÷ Total number

⇒ Average weight = 5575 ÷ 100 = 55.75 kg.

∴ The correct answer is 55.75 kg.

Additional Information

Weighted Average Formula

When a group is divided into subgroups, the overall average is given by (n₁w₁ + n₂w₂ + ... + n_kw_k) ÷ (n₁ + n₂ + ... + n_k).

Combined Average Shortcut

Instead of using actual numbers, you can use the simplest ratio of the quantities to calculate the combined average weight without changing the final result.

Deviation Method

Assume a base average (e.g., 55) and calculate net deviation as Σ(Ratio × Difference from base) ÷ Total Ratio to find the adjustment to the assumed average.

Shortcut Trick

Combined efficiency of Pipe A and Pipe B = 1/20 + 1/30 = 5/60 = 1/12 part per min.

Time taken to fill the full tank = 12 min.

Time taken to fill 2/3 of the tank = 12 × 2/3 = 8 min.

∴ The correct answer is 8 min.

Alternate Method

Given:

Time taken by Pipe A = 20 min

Time taken by Pipe B = 30 min

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total capacity (LCM of 20, 30) = 60 units

⇒ Target volume to fill = 2/3 of total capacity = 2/3 × 60 = 40 units

⇒ Combined efficiency of (A + B) = 3 + 2 = 5 units/min

⇒ Time required to fill 40 units = 40 ÷ 5 = 8 min

∴ The correct answer is 8 min.

Additional Information

Pipes and Cisterns Logic

Filling pipes are considered to have positive efficiency, while emptying pipes (leaks) are considered to have negative efficiency.

Net Efficiency

The net work done per unit time is the algebraic sum of the efficiencies of all pipes operating simultaneously.

Time and Work Relation

Total Work = Efficiency × Time. If work is constant, Time is inversely proportional to Efficiency.

Shortcut Trick

The Area of any Quadrilateral whose diagonals intersect at right angles (90°) is given by the formula:

Area = 1/2 × d₁ × d₂

Given diagonals are 12 m and 16 m.

⇒ Area = 1/2 × 12 × 16 = 6 × 16 = 96 m².

∴ The correct answer is 96 m².

Alternate Method

Given:

Length of first diagonal (d₁) = 12 m

Length of second diagonal (d₂) = 16 m

Angle of intersection = 90°

Formula Used:

Area of an orthodiagonal quadrilateral = (1/2) × Product of diagonals

Calculations:

⇒ Area = 1/2 × d₁ × d₂

⇒ Area = 1/2 × 12 × 16

⇒ Area = 6 × 16

⇒ Area = 96 m²

∴ The correct answer is 96 m².

Additional Information

Orthodiagonal Quadrilateral

Any quadrilateral where diagonals cross at right angles is called orthodiagonal. Common examples include the rhombus, square, and kite.

Area of a Rhombus

Since a rhombus is a special type of orthodiagonal quadrilateral, its area is also 1/2 × d₁ × d₂, where d₁ and d₂ are the perpendicular diagonals.

Sides and Diagonals Relationship

In an orthodiagonal quadrilateral, the sum of the squares of opposite sides is equal: a² + c² = b² + d².

Area of a General Quadrilateral

If the angle θ between diagonals is not 90°, Area = 1/2 × d₁ × d₂ × sin(θ).