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Shortcut Trick

Given Ratio = 4 : 5 and HCF = 16.

Trick: LCM = HCF × Product of the Ratio terms.

LCM = 16 × 4 × 5 = 320.

∴ The correct answer is 320.

Alternate Method

Given:

Ratio of two numbers = 4 : 5

Highest Common Factor (HCF) = 16

Formula Used:

If the ratio of two numbers is a : b and their HCF is x, then the numbers are ax and bx.

LCM of these numbers = x × a × b (where a and b are coprime).

Let the numbers be 4k and 5k.

Since the HCF is the common factor k, we have:

⇒ k = 16

⇒ First number = 4 × 16 = 64

⇒ Second number = 5 × 16 = 80

⇒ LCM(64, 80) = 16 × 4 × 5

⇒ LCM = 320

∴ The correct answer is 320.

Additional Information

Product of Two Numbers Rule

For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM: n₁ × n₂ = HCF × LCM.

HCF as a Factor of LCM

The Highest Common Factor of any set of numbers is always a factor of their Least Common Multiple.

Coprime Numbers Property

If two numbers are coprime (HCF = 1), then their LCM is simply the product of the numbers themselves.

Shortcut Trick

For any algebraic expression where xⁿ + 1/xⁿ = k, the value of the squared power term x²ⁿ + 1/x²ⁿ is given by k² − 2.

Here, the power is being doubled from 2 to 4, so n = 2 and the given value k = 16.

Applying the trick: x⁴ + 1/x⁴ = 16² − 2.

⇒ 256 − 2 = 254.

∴ The correct answer is 254.

Alternate Method

Given: x² + 1/x² = 16

Formula Used: (a + b)² = a² + b² + 2ab (Algebraic Identity for Squaring)

⇒ To find the value of x⁴ + 1/x⁴, we square the given equation on both sides:

⇒ (x² + 1/x²)² = (16)²

⇒ (x²)² + (1/x²)² + 2 × (x²) × (1/x²) = 256

⇒ x⁴ + 1/x⁴ + 2 = 256

⇒ x⁴ + 1/x⁴ = 256 − 2

⇒ x⁴ + 1/x⁴ = 254

∴ The correct answer is 254.

Additional Information

Successive Squaring

If x + 1/x = a, then x² + 1/x² = a² − 2. To find x⁴ + 1/x⁴, we apply the same rule to the new result: (a² − 2)² − 2.

Reciprocal Cube Identity

If x + 1/x = a, then the value of x³ + 1/x³ is calculated as a³ − 3a.

Negative Reciprocal Relation

If x − 1/x = a, then x² + 1/x² = a² + 2. Note the sign change in the constant term when the original expression is negative.

Shortcut Trick

Profit Ratio (X : Y) = (Capital × Time)_X : (Capital × Time)_Y

Profit Ratio = (12,000 × 12) : (18,000 × 6) = 144,000 : 108,000 = 4 : 3

Total Ratio parts = 4 + 3 = 7 units → ₹25,200

1 unit = ₹3,600

X's share = 4 × ₹3,600 = ₹14,400

∴ The correct answer is ₹14,400.

Alternate Method

Given:

Investment of X = ₹12,000 for 12 months

Investment of Y = ₹18,000 for 6 months

Total Profit = ₹25,200

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Profit Ratio (X : Y) = (12,000 × 12) : (18,000 × 6)

⇒ Profit Ratio (X : Y) = 1,44,000 : 1,08,000

⇒ Simplifying by 36,000: Ratio = 4 : 3

⇒ Total profit units = 4 + 3 = 7 units

⇒ 7 units = ₹25,200

⇒ 1 unit = ₹25,200 ÷ 7 = ₹3,600

⇒ X's share = 4 units = 4 × ₹3,600 = ₹14,400

∴ The correct answer is ₹14,400.

Additional Information

Compound Partnership

When partners invest different amounts for different periods of time, the profit is shared in the ratio of the product of their capital and time period (Capital × Time).

Ratio of Capitals

If the profit ratio and time ratio are given, the capital ratio can be found as (Profit₁/Time₁) : (Profit₂/Time₂).

Working vs Sleeping Partner

A working partner manages the business and often receives a fixed percentage of total profit as salary before the remaining profit is distributed among all partners.

Shortcut Trick

The area of a triangle is fundamentally defined by half the product of its base and corresponding altitude.

Formula: Area = 1/2 × base × altitude.

In the given options, statement (ii) specifies the fraction as 1/3, which is used for calculating the volume of pyramids or cones, not the area of a 2D triangle.

Other properties (i), (iii), and (iv) are standard geometric theorems for all triangles.

∴ The correct answer is (ii).

Alternate Method

Given: Four statements regarding the properties of a triangle.

Formula Used: Angle Sum Property, Area Formula, Perimeter Definition, and Exterior Angle Theorem.

Verification of properties:

⇒ Property (i): Sum of internal angles = ∠A + ∠B + ∠C = 180°. This is always true for any Euclidean triangle.

⇒ Property (ii): Mathematical Area of a triangle = 1/2 × base × altitude. The given statement (ii) uses 1/3, making it incorrect.

⇒ Property (iii): Perimeter = Length(AB) + Length(BC) + Length(CA). This is the standard definition of perimeter for any polygon.

⇒ Property (iv): According to the Exterior Angle Theorem, the exterior angle at any vertex equals the sum of the two opposite interior angles.

⇒ Therefore, statement (ii) is not a property of a triangle.

∴ The correct answer is (ii).

Additional Information

Heron's Formula

For a triangle with sides a, b, and c, the area is √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must always be strictly greater than the length of the third side.

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

Shortcut Trick

वृद्धि राशि ज्ञात करने के लिए सीधे प्रतिशत निकालें:

300 का 5% = 300 × 0.05

⇒ 3 × 5 = 15

∴ The correct answer is ₹15.

Alternate Method

दिया गया है:

पुस्तक की मूल कीमत = ₹300

कीमत में वृद्धि प्रतिशत = 5%

सूत्र: वृद्धि राशि = मूल कीमत × (वृद्धि प्रतिशत ÷ 100)

⇒ वृद्धि राशि = 300 × (5 ÷ 100)

⇒ वृद्धि राशि = 300 × 0.05

⇒ वृद्धि राशि = 15

अतः, नई कीमत मूल कीमत से ₹15 अधिक होगी।

∴ The correct answer is ₹15.

Additional Information

प्रतिशत वृद्धि (Percentage Increase)

जब किसी वस्तु के मूल्य में प्रतिशत वृद्धि होती है, तो बढ़ी हुई राशि (मूल मूल्य × दर) ÷ 100 होती है।

नया मूल्य (New Price)

वृद्धि के बाद नया मूल्य = मूल मूल्य + वृद्धि राशि। इस प्रश्न में नया मूल्य 300 + 15 = ₹315 होगा।

प्रतिशत कमी (Percentage Decrease)

यदि कीमत में 5% की कमी होती, तो कमी की राशि ₹15 होती और नया मूल्य ₹285 होता।

Shortcut Trick

If x = (√a + √b)/(√a − √b) and y is its reciprocal, then x + y = 2(a + b)/(a − b) and xy = 1.

Here, a = 5 and b = 4 (since 2 = √4), so x + y = 2(5 + 4) / (5 − 4) = 18.

Using the algebraic identity, x² + y² = (x + y)² − 2xy.

x² + y² = 18² − 2(1) = 324 − 2 = 322.

∴ The correct answer is 322.

Alternate Method

Given: x = (√5 + 2)/(√5 − 2) and y = (√5 − 2)/(√5 + 2)

Formula Used: x² + y² = (x + y)² − 2xy

⇒ Rationalizing x: x = [(√5 + 2) × (√5 + 2)] / [(√5 − 2) × (√5 + 2)]

⇒ x = (√5 + 2)² / (5 − 4) = 5 + 4 + 4√5 = 9 + 4√5

⇒ Rationalizing y: y = [(√5 − 2) × (√5 − 2)] / [(√5 + 2) × (√5 − 2)]

⇒ y = (√5 − 2)² / (5 − 4) = 5 + 4 − 4√5 = 9 − 4√5

⇒ x + y = (9 + 4√5) + (9 − 4√5) = 18

⇒ xy = (9 + 4√5)(9 − 4√5) = 81 − 16(5) = 81 − 80 = 1

⇒ x² + y² = (18)² − 2(1)

⇒ x² + y² = 324 − 2 = 322

∴ The correct answer is 322.

Additional Information

Sum of Squares Identity

The expression a² + b² can be written as (a + b)² − 2ab or (a − b)² + 2ab depending on the given values.

Conjugate Surds

The conjugate of (√a + b) is (√a − b). Their product (√a + b)(√a − b) = a − b² results in a rational number.

Reciprocal Sum Property

If x = √a + √b and y = √a − √b where a − b = 1, then x and y are reciprocals of each other, meaning xy = 1.

Shortcut Trick

Relative Speed = (54 + 72) × 5/18 = 35 m/s.

Total Length (P + Q) = 35 × 30 = 1050 m.

Length of Train P = (3/5) × 1050 = 630 m.

Total Distance covered by P in 90 sec = 15 × 90 = 1350 m.

Length of Bridge = 1350 − 630 = 720 m.

∴ The correct answer is 720 m.

Alternate Method

Given:

Ratio of lengths of Train P and Train Q = 3 : 2

Speed of Train P (S_P) = 54 km/h | Speed of Train Q (S_Q) = 72 km/h

Time to cross each other = 30 sec | Time for P to cross bridge = 90 sec

Formula Used:

Distance = Speed × Time | Relative Speed (Opposite) = S₁ + S₂

Calculations:

⇒ Let Length of Train P = 3x and Train Q = 2x

⇒ S_P in m/s = 54 × (5/18) = 15 m/s

⇒ S_Q in m/s = 72 × (5/18) = 20 m/s

⇒ Relative Speed = 15 + 20 = 35 m/s

⇒ (3x + 2x) = 35 × 30

⇒ 5x = 1050 ⇒ x = 210

⇒ Length of Train P = 3 × 210 = 630 m

⇒ When P crosses bridge of length L:

⇒ 630 + L = 15 × 90

⇒ 630 + L = 1350

⇒ L = 1350 − 630 = 720 m

∴ The correct answer is 720 m.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is S₁ + S₂. In the same direction, it is |S₁ − S₂|.

Unit Conversion

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

Train Crossing Objects

Total distance = Train length + Object length (if object has length like a bridge, platform, or tunnel). For a pole or man, distance = Train length only.

Shortcut Trick

Total years added to the sum of ages in 14 years = 14 × 2 = 28 years.

Given: Future Sum = 2 × Present Sum ⇒ Present Sum + 28 = 2 × Present Sum.

⇒ Present Sum (A + B) = 28 years.

Age difference (A − B) = 8 years.

A's present age = (Sum + Difference) ÷ 2 = (28 + 8) ÷ 2 = 18 years.

∴ The correct answer is 18.

Alternate Method

Given:

Difference between A's age and B's age = 8 years.

Condition: Age sum 14 years later = 2 × (Age sum now).

Let B's present age be x years.

Then A's present age = (x + 8) years.

According to the question:

⇒ [(x + 8) + 14] + [x + 14] = 2 × [(x + 8) + x]

⇒ (x + 22) + (x + 14) = 2 × (2x + 8)

⇒ 2x + 36 = 4x + 16

⇒ 4x − 2x = 36 − 16

⇒ 2x = 20

⇒ x = 10

A's present age = x + 8 = 10 + 8 = 18 years.

∴ The correct answer is 18.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals remains the same throughout their lives, regardless of how many years pass.

Sum and Difference Rule

If the Sum (S) and Difference (D) of two ages are known, then Elder age = (S + D) ÷ 2 and Younger age = (S − D) ÷ 2.

Ratio Method in Ages

When ages are given in ratios at different time periods, ensure the ratio difference between the two individuals is made equal to solve for the multiplier.

Shortcut Trick

Solve the inner brackets first: 3 + 2 × 3 = 9 and 40 − 12 = 28.

Simplify the first major term: (36 + 4 × 9) ÷ 9 ⇒ 72 ÷ 9 = 8.

Simplify the second major term: 28 + (16 − 6) ⇒ 28 + 10 = 38.

Combine the results: 8 + 38 = 46.

∴ The correct answer is 46.

Alternate Method

Given: [ ( 36 + 4 × ( 3 + 2 × ( 5 − 2 ) ) ) ÷ ( 6 + 3 ) ] + [ ( 40 − ( 6 + 3 × 2 ) ) + ( 8 × 2 − 6 ) ]

According to the BODMAS rule, we solve in the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

⇒ [ ( 36 + 4 × ( 3 + 2 × 3 ) ) ÷ 9 ] + [ ( 40 − ( 6 + 6 ) ) + ( 16 − 6 ) ]

⇒ [ ( 36 + 4 × 9 ) ÷ 9 ] + [ ( 40 − 12 ) + 10 ]

⇒ [ ( 36 + 36 ) ÷ 9 ] + [ 28 + 10 ]

⇒ [ 72 ÷ 9 ] + 38

⇒ 8 + 38 = 46

∴ The correct answer is 46.

Additional Information

BODMAS / VBODMAS Rule

It defines the sequence of operations: Vinculum (Bar), Brackets, Of (multiplication), Division, Multiplication, Addition, and Subtraction.

Order of Brackets

Always simplify parentheses ( ) first, then curly braces { }, and finally square brackets [ ].

Arithmetic Precedence

In expressions with only addition and subtraction or only multiplication and division, solve from left to right.

Shortcut Trick

Convert all terms to simple fractions: (5/6 + 2/6) ÷ (4/10 + 3/10) × 1/2 + 5/6

Solve brackets: (7/6) ÷ (7/10) × 1/2 + 5/6

Apply division and multiplication: (7/6 × 10/7) × 1/2 + 5/6 = 10/6 × 1/2 + 5/6

Final addition: 5/6 + 5/6 = 10/6 = 5/3

∴ The correct answer is 5/3.

Alternate Method

Given: ((5/6) + (1/3)) ÷ ((2/5) + 0.3) × 0.5 + (5/6)

Method Used: BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ Step 1 (Solve first bracket): (5/6 + 2/6) = 7/6

⇒ Step 2 (Solve second bracket): (2/5 + 3/10) = (4/10 + 3/10) = 7/10

⇒ Step 3 (Substitute values): (7/6) ÷ (7/10) × 0.5 + 5/6

⇒ Step 4 (Perform division): 7/6 × 10/7 = 10/6

⇒ Step 5 (Perform multiplication): 10/6 × 1/2 = 5/6

⇒ Step 6 (Final addition): 5/6 + 5/6 = 10/6

⇒ Simplify: 5/3

∴ The correct answer is 5/3.

Additional Information

BODMAS Rule

This rule defines the order of operations: Brackets (), Of (multiplication), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Fraction Division

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction: a/b ÷ c/d = a/b × d/c.

Decimal to Fraction Conversion

Convert decimals to fractions by placing the number over a power of 10. For example, 0.3 = 3/10 and 0.5 = 5/10 = 1/2.