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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.