Quantitative Aptitude PYQs

Shortcut Trick

Remaining food is sufficient for 360 soldiers for 40 days (50 − 10).

Total soldiers after arrival = 360 + 120 = 480 soldiers.

Using M₁D₁ = M₂D₂ logic on remaining stock: 360 × 40 = 480 × Days.

Days = 14400 ÷ 480 = 30 days.

∴ The correct answer is 30 days.

Alternate Method

Given:

Initial soldiers (M₁) = 360

Initial provision time (D₁) = 50 days

Elapsed time = 10 days

Newly joined soldiers = 120

Calculations:

⇒ Total provision units = 360 × 50 = 18,000 units

⇒ Provision consumed in 10 days = 360 × 10 = 3,600 units

⇒ Remaining provision = 18,000 − 3,600 = 14,400 units

⇒ Total soldiers now = 360 + 120 = 480

⇒ Number of days for remaining provisions = Remaining stock ÷ Total soldiers

⇒ Number of days = 14,400 ÷ 480

⇒ Number of days = 30 days

∴ The correct answer is 30 days.

Additional Information

Chain Rule (Work and Time)

The amount of work or provisions is directly proportional to the number of people and the number of days: M₁×D₁×H₁ = M₂×D₂×H₂.

Inverse Variation

For a constant amount of food, the number of days the food lasts is inversely proportional to the number of soldiers. If men increase, days decrease.

Effective Man-Days

The total quantity of provisions is often measured in man-days, which is the product of the number of men and the number of days they can be fed.

Shortcut Trick

The Mean Proportional between two numbers a and b is calculated as √(a × b).

Given a = 0.0009 and b = 0.25.

Mean proportional = √(0.0009 × 0.25) = 0.03 × 0.5 = 0.015.

∴ The correct answer is 0.015.

Alternate Method

Given: First number (a) = 0.0009 and Second number (b) = 0.25.

In a mean proportion a : x = x : b, where x is the mean proportional value.

Calculation:

⇒ x² = a × b

⇒ x² = 0.0009 × 0.25

⇒ x = √(0.0009 × 0.25)

⇒ x = √(9/10000 × 25/100)

⇒ x = √(225/1000000)

⇒ x = 15/1000

⇒ x = 0.015

∴ The correct answer is 0.015.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is such that a : b = b : c, calculated as c = b²/a.

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a : b = c : d, calculated as d = (b × c)/a.

Continued Proportion

A sequence of numbers a, b, c, d... is in continued proportion if a/b = b/c = c/d and so on.

Shortcut Trick

Average Speed = (D₁ + D₂) ÷ (D₁/S₁ + D₂/S₂)

Average Speed = (60 + 90) ÷ (60/45 + 90/60)

Average Speed = 150 ÷ (4/3 + 3/2) = 150 ÷ (17/6)

Average Speed = 900 ÷ 17 ≈ 52.94 km/hr

∴ The correct answer is 52.94 km/hr.

Alternate Method

Given:

First distance (d₁) = 60 km | Speed (s₁) = 45 km/hr

Second distance (d₂) = 90 km | Speed (s₂) = 60 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Time taken for the first part (t₁) = 60 ÷ 45 = 4/3 hours

⇒ Time taken for the second part (t₂) = 90 ÷ 60 = 3/2 hours

⇒ Total Time = t₁ + t₂ = 4/3 + 3/2

⇒ Total Time = (8 + 9) ÷ 6 = 17/6 hours

⇒ Total Distance = 60 + 90 = 150 km

⇒ Average Speed = 150 ÷ (17/6) = (150 × 6) ÷ 17

⇒ Average Speed = 900 ÷ 17 = 52.9411... km/hr

∴ The correct answer is 52.94 km/hr.

Additional Information

Average Speed for Equal Distances

If a body travels two equal distances at speeds x and y, the average speed is given by (2 × x × y) ÷ (x + y).

Average Speed for Equal Time Intervals

If a body travels at speeds x and y for equal time intervals, the average speed is simply the arithmetic mean: (x + y) ÷ 2.

Relative Speed Concept

For objects moving in the same direction, relative speed = S₁ − S₂. For opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

The Shortcut formula to find the Difference when Sum and Product are given:

Difference² = Sum² − 4 × Product

⇒ Difference² = 36² − 4 × 180

⇒ Difference² = 1296 − 720 = 576

⇒ Difference = √576 = 24

∴ The correct answer is 24.

Alternate Method

Given:

Sum of two Positive Numbers (x + y) = 36

Product of the numbers (xy) = 180

Formula Used: (x − y)² = (x + y)² − 4xy

⇒ (x − y)² = (36)² − 4 × 180

⇒ (x − y)² = 1296 − 720

⇒ (x − y)² = 576

⇒ x − y = √576

⇒ x − y = 24

∴ The correct answer is 24.

Additional Information

Algebraic Identities

(x + y)² = x² + y² + 2xy and (x − y)² = x² + y² − 2xy. Subtracting these gives (x + y)² − (x − y)² = 4xy.

Finding Individual Numbers

If Sum (S) and Difference (D) are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Sum of Squares

The sum of squares of two numbers can be found using x² + y² = (x + y)² − 2xy.

Shortcut Trick

Check the ratio of the sides: 75 : 21 : 72 ÷ 3 = 25 : 7 : 24.

The numbers 7, 24, and 25 form a Pythagorean triplet, which means the triangle is right-angled.

In a right-angled triangle, the altitude corresponding to one leg is simply the length of the other leg.

The sides forming the right angle are 21 cm and 72 cm.

∴ The correct answer is 72 cm.

Alternate Method

Given: The sides of the triangle are 75 cm, 21 cm, and 72 cm.

Formula Used: Area of a triangle = (1 ÷ 2) × base × altitude

⇒ First, identify the type of triangle using the Pythagoras Theorem: 21² + 72² = 441 + 5184 = 5625.

⇒ Since 75² = 5625, the sides satisfy a² + b² = c², making it a right-angled triangle.

⇒ In this triangle, the longest side (75 cm) is the hypotenuse, and the other two sides (21 cm and 72 cm) are the legs (base and perpendicular).

⇒ Area = (1 ÷ 2) × product of legs = (1 ÷ 2) × 21 × 72.

⇒ Also, Area = (1 ÷ 2) × base (21 cm) × altitude (h).

⇒ (1 ÷ 2) × 21 × 72 = (1 ÷ 2) × 21 × h.

⇒ h = 72 cm.

∴ The correct answer is 72 cm.

Additional Information

Pythagorean Triplets

Sets of three positive integers a, b, and c, such that a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (7, 24, 25).

Altitude of a Right Triangle

The two legs of a right-angled triangle act as altitudes to each other. The altitude to the hypotenuse can be found using: (Leg₁ × Leg₂) ÷ Hypotenuse.

Heron's Formula

For any triangle with sides a, b, and c, Area = √(s × (s − a) × (s − b) × (s − c)), where semi-perimeter s = (a + b + c) ÷ 2.

Shortcut Trick

Age of new student = New Average + (Initial count of students × Increase in average)

⇒ Increase in Average Age = 18 − 17 = 1 year

⇒ Age = 18 + (7 × 1) = 25 years

∴ The correct answer is 25 years.

Alternate Method

Given:

Initial number of students = 7

Initial average age = 17 years

New average age = 18 years

Formula Used:

Total Age = Average Age × Number of Students

Calculations:

⇒ Total age of 7 students = 7 × 17 = 119 years

⇒ Total number of students after 1 joins = 8

⇒ Total age of 8 students = 8 × 18 = 144 years

⇒ Age of new student = Total age of 8 students − Total age of 7 students

⇒ Age = 144 − 119

⇒ Age = 25 years

∴ The correct answer is 25 years.

Additional Information

Basic Average Formula

The Average of a set of numbers is the sum of the values divided by the total number of values: Average = (Sum of observations) ÷ (Total number of observations).

Effect of Adding a Constant

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

Weighted Average

When two groups with different averages are combined, the combined average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average of each group.

Shortcut Trick

The LCM of 12, 15, and 20 is 60.

The smallest 5-digit number is 10000.

Dividing 10000 by 60 leaves a remainder of 40.

Smallest 5-digit multiple = 10000 + (60 − 40) = 10020.

∴ The correct answer is 10020.

Alternate Method

Given: The numbers are 12, 15, and 20. We need to find the smallest 5-digit number divisible by all three.

Formula Used: Required Number = Smallest 5-digit number + (LCM − Remainder)

⇒ Prime factorization of divisors:

⇒ 12 = 2 × 2 × 3 = 2² × 3

⇒ 15 = 3 × 5

⇒ 20 = 2 × 2 × 5 = 2² × 5

⇒ LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

⇒ Smallest 5-digit number = 10000

⇒ Dividing 10000 by 60: 10000 = (166 × 60) + 40

⇒ Remainder = 40

⇒ Number to be added = 60 − 40 = 20

⇒ Required number = 10000 + 20 = 10020

∴ The correct answer is 10020.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is perfectly divisible by a given set of numbers. It is found by taking the highest power of all prime factors involved.

Divisibility by 60

A number is divisible by 60 if it satisfies the divisibility rules of 3 (sum of digits), 4 (last two digits), and 5 (ends in 0 or 5) simultaneously.

Smallest N-digit Divisible Number

To find the smallest N-digit number divisible by x, use: 10^N−1 + (x − Remainder(10^N−1 ÷ x)).

Shortcut Trick

Ratio of volumes of two cylinders: V₁/V₂ = (r₁/r₂)² × (h₁/h₂)

V₁ = 317900 × (3/5)² × (21/14)

V₁ = 317900 × (9/25) × (3/2) = 317900 × (27/50)

V₁ = 6358 × 27 = 171666 cm³

∴ The correct answer is 171666 cm³.

Alternate Method

Given:

Height of first cylinder (h₁) = 21 cm

Volume of second cylinder (V₂) = 317900 cm³

Height of second cylinder (h₂) = 14 cm

Radius of first cylinder (r₁) = 3/5 × radius of second cylinder (r₂)

Formula Used: Volume of cylinder = πr²h

Calculations:

⇒ Volume of second cylinder (V₂) = π × r₂² × h₂

⇒ 317900 = (22/7) × r₂² × 14

⇒ 317900 = 22 × r₂² × 2

⇒ 317900 = 44 × r₂²

⇒ r₂² = 317900 ÷ 44 = 7225

⇒ r₂ = √7225 = 85 cm

⇒ Radius of first cylinder (r₁) = 3/5 × 85 = 3 × 17 = 51 cm

⇒ Volume of first cylinder (V₁) = π × r₁² × h₁

⇒ V₁ = (22/7) × (51)² × 21

⇒ V₁ = 22 × 2601 × 3

⇒ V₁ = 66 × 2601 = 171666 cm³

∴ The correct answer is 171666 cm³.

Additional Information

Curved Surface Area of Cylinder

The area of the curved surface is given by the formula CSA = 2πrh.

Total Surface Area of Cylinder

The sum of the areas of the curved surface and the two circular bases: TSA = 2πr(h + r).

Volume and Ratio Property

If the radius and height of two cylinders are in the ratio r₁:r₂ and h₁:h₂, then the ratio of their volumes is (r₁/r₂)² × (h₁/h₂).

Shortcut Trick

Find the percentage difference: 45% − 20% = 25%.

The question states that 25% of the number is 36.

To get the total number (100%), multiply by 4 (since 25% × 4 = 100%).

Result = 36 × 4 = 144.

∴ The correct answer is 144.

Alternate Method

Given:

First percentage = 45%

Second percentage = 20%

Difference in values = 36

Calculations:

⇒ Let the number be x.

⇒ According to the problem: 45% of x − 20% of x = 36

⇒ (45 − 20)% of x = 36

⇒ 25% of x = 36

⇒ (25 ÷ 100) × x = 36

⇒ (1 ÷ 4) × x = 36

⇒ x = 36 × 4

⇒ x = 144

∴ The correct answer is 144.

Additional Information

Percentage Formula

The value of P% of a number N is calculated as (P × N) ÷ 100.

Finding the Base Number

If x% of a number is V, the original number is (V ÷ x) × 100.

Common Fraction Equivalents

Knowing common fractions like 20% = 1/5, 25% = 1/4, and 50% = 1/2 helps in faster calculation during competitive exams.

Shortcut Trick

Tap B (fills in 9 hr) and Tap C (empties in 9 hr) have equal rates and opposite effects.

Their net Efficiency is zero, meaning only Tap A effectively fills the cistern.

Tap A takes 4 hr to fill the complete cistern.

Time to fill one-fourth of the cistern = 4 hr × 1/4 = 1 hr.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap A = 4 hr

Time taken by Tap B = 9 hr

Time taken by Tap C (Emptying) = 9 hr

Formula Used:

Efficiency = Total Work ÷ Time

Calculations:

⇒ Total Efficiency of all taps = 9 + 4 − 4 = 9 units/hr.

⇒ Target to fill one-fourth of the cistern = 36 ÷ 4 = 9 units.

⇒ Time required = 9 units ÷ 9 units/hr = 1 hr.

∴ The correct answer is 1 hr.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills the tank (positive efficiency), while an outlet pipe empties the tank (negative efficiency).

Net Efficiency

When multiple pipes work together, the net filling or emptying rate is the algebraic sum of their individual efficiencies.

LCM Method

The capacity of the tank is usually assumed to be the Least Common Multiple (LCM) of the individual time periods given in the problem.

Shortcut Trick

Price after 40% discount = 60% of 1200 = ₹720.

Final Sale Price = ₹520.

Additional Discount value = 720 − 520 = ₹200.

Additional Discount % = (200 ÷ 720) × 100 = (5 ÷ 18) × 100 = 250/9% = 27(7)/(9)%.

∴ The correct answer is 27(7)/(9)%.

Alternate Method

Given:

Marked Price (MP) = ₹1,200

First Discount = 40%

Final Sale Price (SP) = ₹520

Formula:

Price after discount = MP × (100 − Discount%) ÷ 100

Calculations:

⇒ Price after 40% discount = 1200 × (1 − 0.40)

⇒ Price after 40% discount = 1200 × 0.60 = ₹720

⇒ Total Additional Discount needed = 720 − 520 = ₹200

⇒ Additional Discount % = (Additional Discount ÷ Mid Price) × 100

⇒ Additional Discount % = (200 ÷ 720) × 100

⇒ Additional Discount % = (5 ÷ 18) × 100 = 500/18%

⇒ Additional Discount % = 250/9% = 27 7/9%

∴ The correct answer is 27(7)/(9)%.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the equivalent single discount is calculated as: [a + b − (ab ÷ 100)]%.

Marked Price (MP)

The price printed on the product. Discounts are always calculated on the Marked Price unless stated otherwise.

Selling Price (SP)

The actual price at which an item is sold after all discounts. SP = MP − Discount.

Given:

$\sqrt{\sqrt{\sqrt{256} + \sqrt{144} + (2\sqrt{2})^2} + \sqrt{841} \times (\sqrt{2})^2}$

Formula used:

√x² = x

(a√b)² = a² × b

Calculations:

⇒ $\sqrt{\sqrt{16 + 12 + 8} + 29 \times 2}$

⇒ $\sqrt{\sqrt{36} + 58}$

⇒ $\sqrt{6 + 58}$

⇒ $\sqrt{64}$

⇒ 8

Now, checking the options:

A. (√2)² = 2

B. (2√2)² = 4 × 2 = 8

C. (4√2)² = 16 × 2 = 32

D. (3√2)² = 9 × 2 = 18

∴ The correct answer is option (2).

Shortcut Trick

Effective rate of Compound Interest for 2 years at 2% = 2 + 2 + (2 × 2)/100 = 4.04%.

Given Compound Interest (4.04%) = ₹404.

⇒ Principal (y) = (404 ÷ 4.04) × 100 = ₹10,000.

Total Amount = Principal + Interest = 10,000 + 404 = ₹10,404.

∴ The correct answer is ₹10,404.

Alternate Method

Given: Interest (CI) = ₹404, Rate (r) = 2% per annum, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + r/100)ⁿ and CI = A − P

⇒ CI = y[(1 + 2/100)² − 1]

⇒ 404 = y[(51/50)² − 1]

⇒ 404 = y[2601/2500 − 1]

⇒ 404 = y[101/2500]

⇒ y = (404 × 2500) ÷ 101

⇒ y = 4 × 2500 = 10,000

⇒ Total Amount (A) = Principal (y) + Interest

⇒ Total Amount (A) = 10,000 + 404 = 10,404

∴ The correct answer is ₹10,404.

Additional Information

Effective Percentage Rate

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

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is P(r/100)².

Amount Property

In Compound Interest, the Total Amount of the previous year becomes the Principal for the next year.

Shortcut Trick

A natural number is a positive integer starting from 1 (1, 2, 3, ...).

Evaluate the expression 2x + 1 sequentially:

For x = 1, 2, 3 → 2x + 1 = 3, 5, 7 (all are prime numbers)

For x = 4 → 2(4) + 1 = 9 (composite number, not prime)

∴ The correct answer is 4.

Alternate Method

Given:

An algebraic expression: 2x + 1

We need to find the smallest natural number x such that 2x + 1 is not a prime number.

Definition and Method:

A natural number belongs to the set {1, 2, 3, 4, ...}.

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

We substitute sequential natural numbers for x into the expression and check for primality.

Calculations:

⇒ For x = 1:

⇒ 2(1) + 1 = 3 (Prime number)

⇒ For x = 2:

⇒ 2(2) + 1 = 5 (Prime number)

⇒ For x = 3:

⇒ 2(3) + 1 = 7 (Prime number)

⇒ For x = 4:

⇒ 2(4) + 1 = 9 (Composite number, since 9 = 3 × 3)

Since 9 is divisible by 3, it is not a prime number. Hence, the smallest natural number satisfying the given condition is x = 4.

∴ The correct answer is 4.

Additional Information

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, etc.

Composite Numbers

A composite number is a positive integer greater than 1 that has more than two distinct positive divisors. For example, 4, 6, 8, 9, 10 are composite numbers.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, i.e., {1, 2, 3, 4, ...}. Note that 0 is not considered a natural number in this context.

Primality of 2x + 1

For any integer x, the expression 2x + 1 always represents an odd number. Thus, we are looking for the smallest odd composite number greater than 1, which is 9.

Shortcut Trick

When the sum (S) and difference (D) of two ages are known:

Son's age (Smaller value) = (Sum − Difference) ÷ 2

Son's age = (56 − 24) ÷ 2

Son's age = 32 ÷ 2 = 16 years

∴ The correct answer is 16 years.

Alternate Method

Given:

⇒ Sum of Father and Son's ages = 56 years

⇒ Difference between Father and Son's ages = 24 years

⇒ Let the age of the son be x years.

⇒ According to the question, the father is 24 years older than the son.

⇒ Father's age = (x + 24) years

⇒ Sum of their ages: x + (x + 24) = 56

⇒ 2x + 24 = 56

⇒ 2x = 56 − 24

⇒ 2x = 32

⇒ x = 32 ÷ 2

⇒ x = 16

∴ The correct answer is 16 years.

Additional Information

Linear Equations in Ages

Age problems are typically solved by forming a linear equation where the unknown age is represented by a variable like 'x'.

Constant Age Difference

The difference between the ages of two individuals never changes over time. If the father is 24 years older now, he will always be 24 years older.

Sum and Difference Rule

If (A + B) = S and (A − B) = D, then the larger value A = (S + D) ÷ 2 and the smaller value B = (S − D) ÷ 2.

Shortcut Trick

Error in total sum = 37 − 22 = 15 marks.

Given average marks increase = 5/4 = 1.25.

Number of students = Total Increase in Sum ÷ Increase in Average.

⇒ 15 ÷ 1.25 = (15 × 4) ÷ 5 = 12.

∴ The correct answer is 12.

Alternate Method

Given:

Incorrect entry = 37

Correct entry = 22

Average increase = 5/4

Formula used:

Average = Total Sum of Marks ÷ Number of Students

Total Increase in Sum = Number of Students × Increase in Average

Calculations:

⇒ Net effect on total sum = 37 − 22 = 15.

⇒ Let the total number of students be n.

⇒ (New Sum − Old Sum) ÷ n = 5/4

⇒ 15 ÷ n = 5/4

⇒ 5 × n = 15 × 4

⇒ 5 × n = 60

⇒ n = 60 ÷ 5 = 12.

∴ The correct answer is 12.

Additional Information

Average Property (Constant Change)

If every value in a data set is increased or decreased by a constant k, the average also increases or decreases by k.

Relationship between Sum and Average

The change in total sum is always equal to the number of terms × the change in average (i.e., ΔSum = n × ΔAvg).

Correction of Errors

Correct Average = [Incorrect Sum + (Correct Value − Incorrect Value)] ÷ Total Number of Items.

Shortcut Trick

Ratio of numbers = 19 : 7 : 13

Since 19, 7, and 13 are all prime numbers, they are co-prime to each other.

For co-prime ratios, LCM = (Product of Ratios) × HCF

8645 = (19 × 7 × 13) × HCF ⇒ 8645 = 1729 × HCF

HCF = 8645 ÷ 1729 = 5

∴ The correct answer is 5.

Alternate Method

Given:

Ratio of three numbers = 19 : 7 : 13

Least Common Multiple (LCM) = 8645

Formula Used:

Let the HCF be x. Then the numbers are ax, bx, and cx.

LCM of (ax, bx, cx) = x × LCM(a, b, c)

Calculations:

⇒ Let the three numbers be 19x, 7x, and 13x, where x is the HCF.

⇒ Since 19, 7, and 13 are prime numbers, their LCM is the product of these numbers and their common factor x.

⇒ LCM = 19 × 7 × 13 × x

⇒ 8645 = 1729 × x

⇒ x = 8645 ÷ 1729

⇒ x = 5

Therefore, the Highest Common Factor (HCF) of the numbers is 5, which corresponds to Option 4.

∴ The correct answer is 5.

Additional Information

Relationship between HCF and LCM

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

HCF of Co-prime Numbers

The HCF of two or more co-prime numbers (numbers that have no common factor other than 1) is always 1.

LCM and HCF of Fractions

LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators). HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators).

Shortcut Trick

Use Option Verification to identify the number satisfying both conditions.

If the larger number is 9, then the smaller number is 9 − 8 = 1.

Check the second condition: (2 × 9) − (5 × 1) = 18 − 5 = 13.

The result matches the condition given in the problem.

∴ The correct answer is 9.

Alternate Method

Given:

Difference between numbers = 8

Result of (2 × Larger − 5 × Smaller) = 13

Formula Used:

Method of Linear Equations using Substitution.

Calculations:

⇒ Let the larger number be x and the smaller number be y.

⇒ From the first condition: x − y = 8 → y = x − 8

⇒ From the second condition: 2x − 5y = 13

⇒ Substitute the value of y in the second equation:

⇒ 2x − 5(x − 8) = 13

⇒ 2x − 5x + 40 = 13

⇒ −3x = 13 − 40

⇒ −3x = −27

⇒ x = −27 ÷ (−3) = 9

∴ The correct answer is 9.

Additional Information

Linear Equations in One Variable

Equations where the highest power of the variable is 1, generally expressed as ax + b = 0.

Substitution Method

A technique to solve simultaneous equations by expressing one variable in terms of another and substituting it into the other equation.

Verification Technique

Fastest method for objective exams where you plug option values into the given constraints to see which one satisfies all conditions.

Shortcut Trick

Equate the total work done in both cases: (15M + 19W) × 8 days = (9M + 4W) × 14 days.

Divide both sides by 2: (15M + 19W) × 4 = (9M + 4W) × 7.

Expand and group terms: 60M + 76W = 63M + 28W ⇒ 3M = 48W.

Thus, 1 man is equivalent to 16 women.

∴ The correct answer is 16.

Alternate Method

Given:

Case 1: (15 men + 19 women) can do the work in 8 days.

Case 2: (9 men + 4 women) can do the same work in 14 days.

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ 15M + 19W = 7 ... (i)

⇒ 9M + 4W = 4 ... (ii)

⇒ To find the ratio 1M/1W, use cross-multiplication or elimination.

⇒ From (ii), 4W = 4 − 9M.

⇒ Substitute into (i) by multiplying (i) by 4: 60M + 19(4W) = 28.

⇒ 60M + 19(4 − 9M) = 28.

⇒ 60M + 76 − 171M = 28.

⇒ 111M = 48 ⇒ M = 48/111 = 16/37.

⇒ Substituting M in 4W = 4 − 9M ⇒ 4W = 4 − 9(16/37) = (148 − 144)/37 = 4/37.

⇒ W = 1/37.

⇒ Ratio M/W = (16/37) ÷ (1/37) = 16.

⇒ 1 man = 16 women.

∴ The correct answer is 16.

Additional Information

Work-Efficiency Relation

Efficiency (E) is inversely proportional to Time (T). For the same work, E₁ × T₁ = E₂ × T₂.

Man-Day Concept

Total work is often measured in man-days, calculated as (Number of persons × Number of days taken).

Composite Efficiency

When different types of workers (men, women, children) work together, their individual efficiencies are summed to find the group's total efficiency.

Shortcut Trick

Net Profit Ratio = (True Weight ÷ Faulty Weight) × (Markup Factor) × (Discount Factor)

Net Ratio = (1000 ÷ 960) × (120 ÷ 100) × (90 ÷ 100)

Net Ratio = (25/24) × (6/5) × (9/10) = 45/40 = 9/8

Profit % = [(9 − 8) ÷ 8] × 100 = 1/8 × 100 = 12.5%

∴ The correct answer is 12.5% profit.

Alternate Method

Given:

Cost Price (CP) per kg = ₹48

Markup percentage = 20%

Discount percentage = 10%

Faulty weight used = 960 g instead of 1000 g

Calculations:

⇒ Marked Price (MP) for 1000 g = 48 + (20 ÷ 100) × 48 = 48 + 9.6 = ₹57.6

⇒ Selling Price (SP) for 1000 g = 57.6 − (10 ÷ 100) × 57.6 = 57.6 − 5.76 = ₹51.84

⇒ The shopkeeper delivers only 960 g, so his actual cost is for 960 g.

⇒ Actual CP for 960 g = (48 ÷ 1000) × 960 = 0.048 × 960 = ₹46.08

⇒ Total Profit = SP for 1000 g (received from customer) − Actual CP for 960 g

⇒ Total Profit = 51.84 − 46.08 = ₹5.76

⇒ Profit % = (Total Profit ÷ Actual CP) × 100

⇒ Profit % = (5.76 ÷ 46.08) × 100 = 1/8 × 100 = 12.5%

∴ The correct answer is 12.5% profit.

Additional Information

Dishonest Dealer Formula

Profit % = [ (True Weight − False Weight) ÷ False Weight ] × 100 when selling at cost price.

Successive Discounts/Markups

Effective change of two percentages x and y is given by the formula: [ x + y + (xy ÷ 100) ].

General Profit Relation

Selling Price (SP) = CP × (100 + Profit%) ÷ 100 = MP × (100 − Discount%) ÷ 100.