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Shortcut Trick

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding altitudes.

Ratio of areas = (Ratio of altitudes)²

⇒ Area Ratio = (2 ÷ 5)² = 4 : 25

∴ The correct answer is 4:25.

Alternate Method

Given:

Ratio of corresponding altitudes of two similar triangles (h₁ : h₂) = 2 : 5

Formula Used:

Area of Δ₁ ÷ Area of Δ₂ = (h₁ ÷ h₂)²

h1 = 2 Area = A1

h2 = 5 Area = A2

A1 : A2 = h1² : h2²

Calculations:

⇒ Area of Δ₁ ÷ Area of Δ₂ = (2 ÷ 5)²

⇒ Area of Δ₁ ÷ Area of Δ₂ = 4 ÷ 25

⇒ Ratio of their areas = 4 : 25

∴ The correct answer is 4:25.

Additional Information

Area of Similar Triangles

The ratio of the areas of two similar triangles is precisely equal to the square of the ratio of their corresponding sides, altitudes, or medians.

Perimeter of Similar Triangles

The ratio of the perimeters of two similar triangles is directly equal to the ratio of their corresponding sides, altitudes, or medians.

Inradius & Circumradius Ratio

For two similar triangles, the ratio of their corresponding inradii or circumradii follows the exact same ratio as their corresponding sides.

Shortcut Trick

Directly apply the algebraic identity: (x + y)² + (x − y)² = 2(x² + y²)

Here, replacing x with a and y with 1:

⇒ (a + 1)² + (a − 1)² = 2(a² + 1²)

Substitute a = √5 ⇒ a² = 5.

⇒ 2(5 + 1) = 2 × 6 = 12

∴ The correct answer is 12.

Alternate Method

Given:

a = √5

Formula Used:

(x + y)² = x² + 2xy + y²

(x − y)² = x² − 2xy + y²

Calculation:

⇒ (a + 1)² = a² + 2a + 1

⇒ (a − 1)² = a² − 2a + 1

Adding both expanded expressions together:

⇒ (a² + 2a + 1) + (a² − 2a + 1)

⇒ 2a² + 2

Substitute the given value a = √5:

⇒ 2(√5)² + 2

⇒ 2 × 5 + 2

⇒ 10 + 2 = 12

∴ The correct answer is 12.

Additional Information

Important Algebraic Identities

Another useful identity for subtraction is: (a + b)² − (a − b)² = 4ab.

Difference of Squares

The product of sum and difference is: a² − b² = (a − b)(a + b).

Square Root Properties

The square of a square root returns the original non-negative value: (√x)² = x. Also, roots can be multiplied directly: √x × √y = √(xy).

Shortcut Trick

Directly extract perfect squares from the square roots:

⇒ √50 + √18 − √8

⇒ √(25 × 2) + √(9 × 2) − √(4 × 2)

⇒ 5√2 + 3√2 − 2√2 = 6√2

∴ The correct answer is 6√2.

Alternate Method

Given:

Expression: √50 + √18 − √8

Method Used:

Prime factorization of the numbers inside the roots to extract terms outside the square root.

Calculations:

⇒ √50 = √(2 × 5 × 5) = 5√2

⇒ √18 = √(2 × 3 × 3) = 3√2

⇒ √8 = √(2 × 2 × 2) = 2√2

Substitute these values back into the expression:

⇒ 5√2 + 3√2 − 2√2

Take √2 as common:

⇒ (5 + 3 − 2) × √2

⇒ 6√2

∴ The correct answer is 6√2.

Additional Information

Similar Surds

Surds which have the same irrational factor are called similar surds. Only similar surds can be added or subtracted. Example: 3√2 and 5√2 are similar surds.

Multiplication and Division of Surds

Unlike addition and subtraction, surds with the same order can be multiplied or divided regardless of their base. √a × √b = √(ab) and √a ÷ √b = √(a/b).

Simplifying Surds

To simplify a square root, find the largest perfect square factor of the number inside the root, and bring its square root outside. Example: √(x²y) = x√y.

Shortcut Trick

The longest chord of a circle is its diameter.

Given the diameter of the circle is 25 cm.

Maximum length of the chord = Diameter = 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Diameter of the circle = 25 cm

Concept Used:

A chord is a line segment whose endpoints lie on the circumference of the circle.

As a chord moves closer to the center of the circle, its length continuously increases.

The chord that passes exactly through the center is the longest possible chord, which is known as the diameter.

Shorter Chord

Longest Chord (Diameter) = 25 cm Center O

Calculation:

⇒ Maximum length of chord = Length of diameter

⇒ Maximum length of chord = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Perpendicular from Center to Chord

A perpendicular drawn from the center of a circle to a chord perfectly bisects the chord into two equal parts.

Equal Chords and Subtended Angles

Equal chords of a circle subtend equal angles at the center of the circle.

Distance of Chords from Center

Equal chords of a circle are always equidistant from the center. Conversely, chords that are equidistant from the center are exactly equal in length.

Shortcut Trick

⇒ sin A + cos A = √2 sin A

⇒ cos A = sin A(√2 − 1) ⇒ cot A = √2 − 1

⇒ tan A = 1 ÷ cot A = 1 ÷ (√2 − 1)

⇒ Rationalizing the denominator gives tan A = √2 + 1

∴ The correct answer is √2 + 1.

Alternate Method

Given:

sin A + cos A = √2 sin A

Formula Used:

tan A = sin A ÷ cos A

(a − b)(a + b) = a² − b²

Calculations:

⇒ Divide the entire given equation by cos A:

⇒ (sin A ÷ cos A) + (cos A ÷ cos A) = √2(sin A ÷ cos A)

⇒ tan A + 1 = √2 tan A

⇒ 1 = √2 tan A − tan A

⇒ 1 = tan A(√2 − 1)

⇒ tan A = 1 ÷ (√2 − 1)

⇒ Multiply numerator and denominator by (√2 + 1):

⇒ tan A = (√2 + 1) ÷ [(√2 − 1)(√2 + 1)]

⇒ tan A = (√2 + 1) ÷ [(√2)² − (1)²]

⇒ tan A = (√2 + 1) ÷ (2 − 1)

⇒ tan A = √2 + 1

∴ The correct answer is √2 + 1.

Additional Information

Basic Trigonometric Ratios

tan A = sin A ÷ cos A and cot A = cos A ÷ sin A.

Pythagorean Identities

sin² A + cos² A = 1 | 1 + tan² A = sec² A | 1 + cot² A = cosec² A.

Algebraic Rationalization

To rationalize a denominator of the form (a − b), multiply numerator and denominator by its conjugate (a + b) using the identity (a − b)(a + b) = a² − b².

Shortcut Trick

To find the ratio of the first and last terms (A : D), simply multiply all the given ratios together.

A/D = (A/B) × (B/C) × (C/D)

A/D = (7/5) × (2/3) × (4/5)

A/D = (7 × 2 × 4) / (5 × 3 × 5)

A/D = 56 / 75

∴ The correct answer is 56 : 75.

Alternate Method

Given:

A : B = 7 : 5

B : C = 2 : 3

C : D = 4 : 5

Method Used:

We can use the shift method to find the combined ratio A : B : C : D by copying the adjacent numbers into the empty spaces of each row.

ABCD75552233444556406075

Calculations:

⇒ Multiply down the columns to get the final combined ratio components:

⇒ A = 7 × 2 × 4 = 56

⇒ B = 5 × 2 × 4 = 40

⇒ C = 5 × 3 × 4 = 60

⇒ D = 5 × 3 × 5 = 75

⇒ The combined ratio is A : B : C : D = 56 : 40 : 60 : 75

⇒ Therefore, the required ratio is A : D = 56 : 75

∴ The correct answer is 56 : 75.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is called a compound ratio. For a:b and c:d, the compound ratio is (a × c) : (b × d).

Inverse Ratio

For any ratio a : b, its inverse ratio is b : a. The inverse of a compound ratio a : b : c is evaluated as 1/a : 1/b : 1/c.

Duplicate & Triplicate Ratio

The duplicate ratio of a : b is given by a² : b². The triplicate ratio of a : b is given by a³ : b³.

Shortcut Trick

Convert division to multiplication by taking the reciprocal of the second fraction in each bracket.

⇒ (7/5 × 15/14) × (10/7 × 3/5)

⇒ (3/2) × (6/7)

⇒ 18/14 = 9/7

∴ The correct answer is 9/7.

Alternate Method

Given:

Expression: (7/5 ÷ 14/15) × (10/7 ÷ 5/3)

Formula Used:

BODMAS Rule: Solve Brackets first, then Division (by converting to multiplication with reciprocal), then Multiplication.

Fraction Division: a/b ÷ c/d = a/b × d/c

Calculation:

Solving the first bracket:

⇒ 7/5 ÷ 14/15 = 7/5 × 15/14

⇒ (7 × 15) / (5 × 14) = 105 / 70 = 3/2

Solving the second bracket:

⇒ 10/7 ÷ 5/3 = 10/7 × 3/5

⇒ (10 × 3) / (7 × 5) = 30 / 35 = 6/7

Multiplying the results of both brackets:

⇒ 3/2 × 6/7

⇒ (3 × 6) / (2 × 7)

⇒ 18 / 14 = 9/7

∴ The correct answer is 9/7.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. Expressions must be solved in this exact sequential order.

Fraction Division

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

Simplifying Fractions

Always cancel out common factors from the numerator and denominator before multiplying to make calculations faster and avoid large numbers.

Shortcut Trick

Convert terms to simple fractions for faster and error-free calculation:

(3/2 + 3/8)³ − (9/5 × 5/4)

⇒ (15/8)³ − 9/4

⇒ 3375/512 − 1152/512 = 2223/512

⇒ 2223 ÷ 512 ≈ 4.3418

∴ The correct answer is 4.3418.

Alternate Method

Given:

(1.5 + 3/8)³ − (9/5 × 1.25)

Calculations:

⇒ Convert decimals to fractions to easily simplify the expression:

⇒ 1.5 = 15/10 = 3/2

⇒ 1.25 = 125/100 = 5/4

⇒ Substitute the fraction values back into the expression:

⇒ (3/2 + 3/8)³ − (9/5 × 5/4)

⇒ Make denominators equal inside the bracket:

⇒ (12/8 + 3/8)³ − 9/4

⇒ Add the terms together inside the bracket:

⇒ (15/8)³ − 9/4

⇒ Calculate the cube of the fraction (15³ = 3375, 8³ = 512):

⇒ 3375/512 − 9/4

⇒ Make denominators equal to perform subtraction (multiply 9/4 by 128/128):

⇒ 3375/512 − (9 × 128)/(4 × 128)

⇒ 3375/512 − 1152/512

⇒ Subtract the numerators:

⇒ (3375 − 1152) / 512

⇒ 2223 / 512

⇒ Convert the final fraction back to a decimal:

⇒ 2223 ÷ 512 = 4.341796875

⇒ Rounding to four decimal places, we get 4.3418.

∴ The correct answer is 4.3418.

Additional Information

BODMAS Rule

Always simplify complex mathematical expressions in the exact order of Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Fraction to Decimal Conversion

Memorizing common fraction values makes calculations significantly faster. For example, 1/8 = 0.125, 3/8 = 0.375, and 1/4 = 0.25.

Rounding Decimals

When rounding to 4 decimal places, check the 5^th decimal digit. If it is 5 or more, round up the 4^th digit by 1. Otherwise, leave it unchanged.

Shortcut Trick

Convert the given fractions directly to decimals for faster calculation:

⇒ 3/4 = 0.75 and 1/2 = 0.5

Substitute these values into the expression:

⇒ ((1.5 + 0.75) ÷ (2.25 − 0.5)) + 0.6

⇒ (2.25 ÷ 1.75) + 0.6 = (9 ÷ 7) + 0.6 ≈ 1.2857 + 0.6 = 1.8857

∴ The correct answer is 1.8857.

Alternate Method

Given:

Expression = ((1.5 + 3/4) ÷ (2.25 − 1/2)) + 0.6

Calculations:

To avoid decimal division, convert all values to fractions:

⇒ 1.5 = 3/2

⇒ 2.25 = 9/4

⇒ 0.6 = 3/5

Substitute the fractions back into the expression:

⇒ ((3/2 + 3/4) ÷ (9/4 − 1/2)) + 3/5

Make denominators equal to add and subtract the inner terms:

⇒ ((6/4 + 3/4) ÷ (9/4 − 2/4)) + 3/5

⇒ (9/4 ÷ 7/4) + 3/5

⇒ (9/4 × 4/7) + 3/5

⇒ 9/7 + 3/5

Take the LCM of 7 and 5, which is 35:

⇒ (45 + 21) / 35

⇒ 66 / 35 ≈ 1.885714...

⇒ Rounding to 4 decimal places, we get 1.8857.

∴ The correct answer is 1.8857.

Additional Information

BODMAS Rule

Always follow the sequence of Brackets, Orders, Division/Multiplication, and Addition/Subtraction while simplifying expressions.

Fraction to Decimal Conversion

Memorising common fraction-to-decimal values like 1/2 = 0.5, 3/4 = 0.75, and 1/5 = 0.2 helps in quick calculation without the need to take an LCM.

Shortcut Trick

Equating the initial and new ratios for liquids A and B: (2x + 6) ÷ (3x + 12) = 3 ÷ 5 ⇒ 10x + 30 = 9x + 36 ⇒ x = 6.

New quantity of A = 2(6) + 6 = 18 liters (which represents 3 parts of the new ratio).

Since 3 parts = 18 liters, 1 part = 6 liters, and new C (8 parts) = 8 × 6 = 48 liters.

Initial quantity of C = 5x = 5(6) = 30 liters.

Quantity of C added = Final C − Initial C = 48 − 30 = 18 liters.

∴ The correct answer is 18 liters.

Alternate Method

Given:

Initial ratio of A, B, and C = 2 : 3 : 5

Quantity added to A = 6 liters

Quantity added to B = 12 liters

New ratio of A, B, and C = 3 : 5 : 8

Formula Used:

New Quantity = Initial Quantity + Quantity Added

Liquid A2x → 12 L+ 6 L18 L (3 parts)Liquid B3x → 18 L+ 12 L30 L (5 parts)Liquid C5x → 30 L+ y L48 L (8 parts)Initial Ratio = 2 : 3 : 5y = 48 − 30 = 18 Liters

Calculations:

⇒ Let the initial quantities of A, B, and C be 2x, 3x, and 5x respectively.

⇒ After adding 6 liters to A and 12 liters to B, their respective new quantities are:

⇒ New A = 2x + 6

⇒ New B = 3x + 12

⇒ According to the new given ratio of A to B (3 : 5), we can set up the equation:

⇒ (2x + 6) ÷ (3x + 12) = 3 ÷ 5

⇒ 5 × (2x + 6) = 3 × (3x + 12)

⇒ 10x + 30 = 9x + 36

⇒ 10x − 9x = 36 − 30

⇒ x = 6

⇒ Initial quantity of C = 5x = 5 × 6 = 30 liters

⇒ New quantity of A = 2(6) + 6 = 12 + 6 = 18 liters

⇒ According to the new ratio (3 : 5 : 8), 3 parts correspond to 18 liters of liquid A.

⇒ 1 part = 18 ÷ 3 = 6 liters

⇒ The new quantity of C corresponds to 8 parts of this ratio.

⇒ New C = 8 × 6 = 48 liters

⇒ Quantity of C added = New quantity of C − Initial quantity of C

⇒ Quantity of C added = 48 − 30 = 18 liters

∴ The correct answer is 18 liters.

Additional Information

Ratio Multiplier Concept

When components of a mixture are given in a ratio a : b : c, their absolute quantities can be modeled as ax, bx, and cx where x acts as a common variable multiplier.

Handling Independent Alterations

When distinct amounts are added to specific mixture components, forming an equation using only the known ratio components will yield the multiplier first, before trying to evaluate the unknown variables.

Cross-Multiplication Method

For simple rational equations structured as a ÷ b = c ÷ d, you can quickly simplify and solve by using direct cross-multiplication: a × d = b × c.