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Shortcut Trick

Ratio of half-diagonals = (224/2) : (420/2) = 112 : 210 = 8 : 15 (divided by 14).

Using Pythagorean triplet (8, 15, 17), the side ratio is 17.

Actual side of rhombus = 17 × 14 = 238 m.

Perimeter = 4 × Side = 4 × 238 = 952 m.

∴ The correct answer is 952 m.

Alternate Method

Given:

Diagonal 1 (d₁) = 224 m

Diagonal 2 (d₂) = 420 m

Formula Used:

Side of rhombus (a) = √[(d₁/2)² + (d₂/2)²]

Perimeter of rhombus = 4 × a

Calculations:

⇒ Half of Diagonal 1 = 224 ÷ 2 = 112 m

⇒ Half of Diagonal 2 = 420 ÷ 2 = 210 m

⇒ Side (a) = √(112² + 210²)

⇒ Side (a) = √(12544 + 44100)

⇒ Side (a) = √56644

⇒ Side (a) = 238 m

⇒ Perimeter = 4 × 238

⇒ Perimeter = 952 m

∴ The correct answer is 952 m.

Additional Information

Properties of a Rhombus

A rhombus is a quadrilateral with all sides equal. Its diagonals bisect each other at right angles (90°).

Area of Rhombus

The area of a rhombus can be calculated using diagonals as: Area = 1/2 × d₁ × d₂.

Relation between Side and Diagonals

For a rhombus with side 'a' and diagonals d₁ and d₂, the relation is: 4a² = d₁² + d₂².

Shortcut Trick

When time is constant, the distance covered is directly proportional to the speed.

Ratio of Speeds = 5 : 7 ⇒ Ratio of Distances = 5 : 7.

Given Faster car (7 units) covers 140 km.

1 unit = 140 ÷ 7 = 20 km.

Slower car (5 units) distance = 5 × 20 = 100 km.

∴ The correct answer is 100 km.

Alternate Method

Given:

Ratio of Speeds = 5 : 7

Faster car distance = 140 km | Time = 2 hr

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ Speed of the faster car = 140 ÷ 2 = 70 kmph

⇒ According to the ratio, if the speeds are 5x and 7x, then:

⇒ 7x = 70 kmph

⇒ x = 10 kmph

⇒ Speed of the slower car = 5 × 10 = 50 kmph

⇒ Distance covered by the slower car in 2 hours = Speed × Time

⇒ Distance = 50 × 2 = 100 km

∴ The correct answer is 100 km.

Additional Information

Direct Proportionality of Distance

If time is constant, the ratio of distances covered is equal to the ratio of speeds: D₁/D₂ = S₁/S₂.

Inverse Proportionality of Time

If distance is constant, the ratio of speeds is inversely proportional to the ratio of time taken: S₁/S₂ = T₂/T₁.

Average Speed Formula

When a person travels equal distances at speeds x and y, the average speed is (2 × x × y) ÷ (x + y).

Shortcut Trick

For complementary angles, sin α = cos β implies α + β = 90°.

Equating the angles: 6A + 12A = 90° ⇒ 18A = 90°.

Dividing by 2 gives the target angle: 9A = 45°.

Substitute the value: tan 45° + cot 45° = 1 + 1 = 2.

∴ The correct answer is 2.

Alternate Method

Given: sin 6A = cos 12A

Formula Used: sin θ = cos(90° − θ) or cos θ = sin(90° − θ)

⇒ sin 6A = cos 12A

⇒ sin 6A = sin(90° − 12A)

⇒ 6A = 90° − 12A

⇒ 18A = 90°

⇒ A = 5°

⇒ 9A = 9 × 5° = 45°

⇒ tan 9A + cot 9A = tan 45° + cot 45°

⇒ Since tan 45° = 1 and cot 45° = 1

⇒ 1 + 1 = 2

∴ The correct answer is 2.

Additional Information

Co-function Identities

Trigonometric functions of complementary angles are related as sin θ = cos(90° − θ), tan θ = cot(90° − θ), and sec θ = cosec(90° − θ).

Reciprocal Relations

Tangent and cotangent are reciprocals of each other, where tan θ × cot θ = 1.

Sum of tan and cot

The expression tan θ + cot θ can be simplified to 2 / sin 2θ, which equals 2 when θ = 45°.

Shortcut Trick

In a trapezoid (where AB || DC), the length of the line segment joining the midpoints of the diagonals is always equal to half the absolute difference between the lengths of the parallel sides.

Formula: EF = (AB − DC) ÷ 2

⇒ EF = (74 − 26) ÷ 2

⇒ EF = 48 ÷ 2 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given: Parallel sides AB = 74 cm and DC = 26 cm. E and F are midpoints of diagonals AC and BD.

Formula Used: In ▵ADC and ▵ADB, the line segment joining the midpoints of two sides is parallel to the third side and half its length.

Let M be the midpoint of the non-parallel side AD.

⇒ In ▵ADC, M and E are midpoints of AD and AC respectively.

⇒ ME = DC ÷ 2 = 26 ÷ 2 = 13 cm.

⇒ In ▵ADB, M and F are midpoints of AD and BD respectively.

⇒ MF = AB ÷ 2 = 74 ÷ 2 = 37 cm.

⇒ Since AB || DC, the points M, E, and F are collinear.

⇒ EF = MF − ME

⇒ EF = 37 − 13 = 24 cm.

∴ The correct answer is 24 cm.

Additional Information

Median of a Trapezoid

The segment connecting the midpoints of the non-parallel sides is called the median, and its length is (a + b) ÷ 2.

Midsegment Theorem

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.

Area of a Trapezoid

The area is calculated as ½ × (sum of parallel sides) × height, or Area = ½ × (a + b) × h.

Shortcut Trick

Find the LCM of divisors 7, 6, and 10, which is 210.

Identify the largest 4-digit multiple of 210 by dividing 9999 ÷ 210.

9999 ÷ 210 = 47 with a remainder of 129.

Subtract the remainder from 9999 to get 9870, then add the common remainder 5.

Required Number = 9870 + 5 = 9875.

∴ The correct answer is 9875.

Alternate Method

Given: Divisors are 7, 6, and 10. The common remainder in each case is 5.

Formula Used: Required Number = (LCM of divisors × k) + Remainder, where k is a constant to find the largest 4-digit number.

Step-by-step Calculation:

⇒ LCM of 7, 6, and 10:

7 = 7

6 = 2 × 3

10 = 2 × 5

⇒ LCM = 2 × 3 × 5 × 7 = 210.

⇒ Largest 4-digit number = 9999.

⇒ Divide 9999 by 210: 9999 ÷ 210 = 47.614...

⇒ Largest multiple of 210 below 9999 = 210 × 47 = 9870.

⇒ Adding the remainder: 9870 + 5 = 9875.

∴ The correct answer is 9875.

Additional Information

General Form for LCM Remainders

If a number leaves the same remainder r when divided by x, y, and z, the general form of the number is LCM(x, y, z) × k + r.

Finding N-digit Largest Number

To find the largest n-digit number divisible by L, calculate (Largest n-digit number − Remainder of [Largest n-digit number ÷ L]).

Dividend Rule

Dividend = (Divisor × Quotient) + Remainder. This rule helps verify divisibility and remainder constraints.

Shortcut Trick

First ratio (a + 2) : (b − 2) = (6 + 2) : (8 − 2) = 8 : 6 = 4 : 3.

Second ratio (c + 4) : (d − 4) = (12 + 4) : (d − 4) = 16 : (d − 4).

Since the ratios are equal: 4 / 3 = 16 / (d − 4).

By comparison, 16 is 4 × 4, so (d − 4) must be 3 × 4 = 12.

Solving for d: d = 12 + 4 = 16.

∴ The correct answer is 16.

Alternate Method

Given: a = 6, b = 8, c = 12 and the proportion (a + 2) / (b − 2) = (c + 4) / (d − 4).

Formula: If x : y = p : q, then x/y = p/q.

Calculations:

⇒ (6 + 2) / (8 − 2) = (12 + 4) / (d − 4)

⇒ 8 / 6 = 16 / (d − 4)

⇒ 4 / 3 = 16 / (d − 4)

⇒ 4 × (d − 4) = 16 × 3

⇒ 4d − 16 = 48

⇒ 4d = 48 + 16

⇒ 4d = 64

⇒ d = 64 / 4 = 16

∴ The correct answer is 16.

Additional Information

Definition of Proportion

A proportion is an equation that states that two ratios are equal. If a : b = c : d, then a × d = b × c.

Cross-Multiplication Rule

For any fractional equation a/b = c/d, the product of the extremes (a and d) equals the product of the means (b and c).

Fourth Proportional

In the proportion a : b = c : d, the term d is called the fourth proportional to a, b, and c, calculated as d = (b × c) / a.

Shortcut Trick

Difference in percentage = 40% − 25% = 15%

According to the question, 15% of the number corresponds to 36.

⇒ 1% of the number = 36 ÷ 15 = 2.4

⇒ 100% of the number = 2.4 × 100 = 240

∴ The correct answer is 240.

Alternate Method

Given:

Percentage 1 = 40%

Percentage 2 = 25%

Difference in values = 36

Formula Used:

Number = (Difference in values × 100) ÷ (Difference in percentages)

Calculations:

Let the number be x.

⇒ 40% of x − 25% of x = 36

⇒ 0.40x − 0.25x = 36

⇒ 0.15x = 36

⇒ x = 36 ÷ 0.15

⇒ x = 3600 ÷ 15

⇒ x = 240

∴ The correct answer is 240.

Additional Information

Percentage Change Formula

Percentage change = (Difference ÷ Original Value) × 100. This is useful for finding percentage increases or decreases.

Fractional Conversion

Converting percentages to fractions can simplify calculations: 25% = 1/4 and 40% = 2/5. The difference (2/5 − 1/4) = 3/20.

Base Value Concept

In problems where a number is unknown, always assume the base number to be 100% or 100 units for easier calculation.

Shortcut Trick

Net Interest Percentage = Rate × Time

Net Interest Percentage = 5% × 2 = 10%

Simple Interest = 10% of 1000

Simple Interest = ₹100

∴ The correct answer is ₹100.

Alternate Method

Given:

Principal (P) = ₹1000

Rate (R) = 5% per annum

Time (T) = 2 years

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ SI = (1000 × 5 × 2) ÷ 100

⇒ SI = (1000 × 10) ÷ 100

⇒ SI = 10000 ÷ 100

⇒ SI = ₹100

∴ The correct answer is ₹100.

Additional Information

Amount Formula

The total sum payable after a specific time is the Amount, calculated as A = Principal + Simple Interest.

SI vs CI for First Year

For the first year, if interest is compounded annually, both Simple Interest and Compound Interest remain exactly the same.

Relationship of SI to Variables

Simple Interest is directly proportional to the Principal, Rate, and Time. If any of these double, the interest also doubles.

Shortcut Trick

Let the age of the younger sibling be x.

Equation based on condition: x + (2x − 4) = 44 years.

⇒ 3x − 4 = 44 ⇒ 3x = 48 ⇒ x = 16 years.

The two ages are 16 and (44 − 16) = 28 years.

Difference = 28 − 16 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given:

Total sum of ages = 44 years

Relation: One sibling = 2 × (Other sibling) − 4 years

Calculations:

⇒ Let the age of the younger sibling = x

⇒ Age of the older sibling = 2x − 4

⇒ x + 2x − 4 = 44

⇒ 3x = 44 + 4

⇒ 3x = 48

⇒ x = 16

⇒ Younger sibling age = 16 years

⇒ Older sibling age = 2(16) − 4 = 32 − 4 = 28 years

⇒ Difference = 28 − 16 = 12 years

∴ The correct answer is 12 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals always remains constant, regardless of how many years pass.

Age Ratio Problems

If the current ages are in the ratio a:b, then after n years, their ages will be (a+n) and (b+n). The ratio changes but the difference (b−a) stays same.

Sum of Ages in 'n' Years

If the current sum of ages of k people is S, then after n years, the new sum will be S + (k × n).

Shortcut Trick

Solve the expression from innermost to outermost using VBODMAS logic.

Vinculum and Small Bracket: (3.2 − 0.4) = 2.8

Curly Braces: {3.3 − 2.8} = 0.5

Square Brackets: [6.5 − 0.5] = 6.0

Final Step: 10.5 − 6.0 = 4.5

∴ The correct answer is 4.5.

Alternate Method

Given: 10.5 − [6.5 − {3.3 − (3.2 − (2.2 − 1.8))}]

Method: Using the VBODMAS rule (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction).

⇒ 10.5 − [6.5 − {3.3 − (3.2 − 0.4)}]

⇒ 10.5 − [6.5 − {3.3 − 2.8}]

⇒ 10.5 − [6.5 − 0.5]

⇒ 10.5 − 6.0

⇒ 4.5

∴ The correct answer is 4.5.

Additional Information

VBODMAS Rule

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

Brackets Order

Solve brackets in this specific order: Round Brackets ( ) → Curly Braces { } → Square Brackets [ ].

Vinculum (Bar)

A horizontal line over an expression indicates that it must be simplified first, even before the round brackets.