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Shortcut Trick

Solve using the BODMAS rule by prioritising brackets and multiplication.

Calculate Bracket: (41 + 23) = 64.

Calculate Multiplication: 64 × 2 = 128.

Final expression: 56 + 74 − 52 + 128 − 96 = 110.

∴ The correct answer is 110.

Alternate Method

Given: 56 + 74 − 52 + (41 + 23) × 2 − 96

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

⇒ 56 + 74 − 52 + (64) × 2 − 96

⇒ 56 + 74 − 52 + 128 − 96

⇒ (56 + 74 + 128) − (52 + 96)

⇒ 258 − 148

⇒ 110

∴ The correct answer is 110.

Additional Information

The BODMAS Rule

It defines the order of operations: Brackets (), Of (orders like powers/roots), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Left-to-Right Rule

When operators of the same priority level appear (like + and −), solve the expression from left to right to maintain accuracy.

Shortcut Trick

Solve the expression using the BODMAS rule by identifying operations in order.

72 + 38 − (96 ÷ 12) + 29 of 2 − 14

⇒ 72 + 38 − 8 + 58 − 14

⇒ 110 + 50 − 14

⇒ 160 − 14 = 146

∴ The correct answer is 146.

Alternate Method

Given:

72 + 38 − (96 ÷ 12) + 29 of 2 − 14

Formula Used:

Apply the VBODMAS rule (Vinculum, Bracket, Of, Division, Multiplication, Addition, Subtraction).

Calculations:

⇒ 72 + 38 − 8 + 29 of 2 − 14 (Solving Bracket: 96 ÷ 12 = 8)

⇒ 72 + 38 − 8 + 58 − 14 (Solving Of: 29 × 2 = 58)

⇒ 110 − 8 + 58 − 14 (Solving Addition: 72 + 38 = 110)

⇒ 102 + 58 − 14 (Solving Subtraction: 110 − 8 = 102)

⇒ 160 − 14 (Solving Addition: 102 + 58 = 160)

⇒ 146 (Solving Subtraction: 160 − 14 = 146)

∴ The correct answer is 146.

Additional Information

VBODMAS Rule

The order of priority is Vinculum (bar), Brackets (), Of (multiplication), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

'Of' Operator

The 'of' operator is treated as multiplication but must be performed before Division and Multiplication according to standard mathematical hierarchy.

Left-to-Right Rule

When operations of the same priority (like Addition and Subtraction) appear, it is best to solve them from left to right to avoid confusion.

Shortcut Trick

Separate Whole Numbers and Fractions: (7 + 9 − 5) + (2/7 + 1/3 − 3/4)

Integer part: 11

Fraction part: LCM(7, 3, 4) = 84

Fractional sum: (24 + 28 − 63) / 84 = −11/84

Combine: 11 − 11/84 = 10 + (1 − 11/84) = 10(73/84)

∴ The correct answer is 10(73/84).

Alternate Method

Given: 7(2/7) + 9(1/3) − 5(3/4)

Method: Mixed fraction decomposition into Integer and Proper Fraction.

⇒ (7 + 2/7) + (9 + 1/3) − (5 + 3/4) ⇒ (7 + 9 − 5) + (2/7 + 1/3 − 3/4) ⇒ 11 + [(2 × 12 + 1 × 28 − 3 × 21) / 84] ⇒ 11 + [(24 + 28 − 63) / 84] ⇒ 11 + [−11 / 84] ⇒ 11 − 11/84 ⇒ 10 + 1 − 11/84 ⇒ 10 + (84 − 11) / 84 ⇒ 10 + 73/84 ⇒ 10(73/84)

∴ The correct answer is 10(73/84).

Additional Information

Mixed Fractions

A mixed fraction is the sum of a non-zero integer and a proper fraction. For any mixed fraction A(b/c), it is calculated as (A × c + b) / c.

LCM of Denominators

When performing addition or subtraction of fractions, a common denominator must be established using the Least Common Multiple (LCM) of the given denominators.

Negative Fractional Parts

If the sum of fractional parts is negative, borrow 1 from the integer part (1 = c/c) to make the resulting fractional part positive.

Shortcut Trick

Solve the square roots first: √0.09 = 0.3 and √0.36 = 0.6.

Substitute back: (0.3/0.6) × (0.4/0.6) × (1.8/0.6).

Simplify each fraction: (1/2) × (2/3) × 3.

Multiply the simplified values: 0.5 × 2 = 1.

∴ The correct answer is 1.

Alternate Method

Given: (√0.09/0.6) × (0.4/√0.36) × (1.8/√0.36)

Method: BODMAS and Square Root simplification

⇒ Step 1: Find the square roots of the decimals.

⇒ √0.09 = √(9/100) = 3/10 = 0.3

⇒ √0.36 = √(36/100) = 6/10 = 0.6

⇒ Step 2: Substitute values into the expression.

⇒ Result = (0.3/0.6) × (0.4/0.6) × (1.8/0.6)

⇒ Step 3: Simplify the ratios.

⇒ Result = (1/2) × (2/3) × 3

⇒ Result = (1/2) × 2 = 1

∴ The correct answer is 1.

Additional Information

Square Root of Decimals

To find the square root of a decimal, express it as a fraction (e.g., 0.09 = 9/100) and then take the square root of both numerator and denominator.

BODMAS Rule

Always follow the BODMAS order: Brackets, Orders (powers/roots), Division and Multiplication (left to right), and Addition and Subtraction.

Fraction Simplification

When dealing with decimal fractions, you can remove decimals by multiplying the numerator and denominator by 10ⁿ, where n is the number of decimal places.

Shortcut Trick

Solve the expression under the square root by calculating the square and products first.

18² = 324, 8 × 8 = 64, and 5 × 22 = 110.

Combine the results: 324 + 64 − 110 + 11 = 289.

The square root of 289 is 17.

∴ The correct answer is 17.

Alternate Method

Given: √(18² + 8 × 8 − 5 × 22 + 11)

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

⇒ √(324 + 64 − 110 + 11)

⇒ √(388 − 110 + 11)

⇒ √(278 + 11)

⇒ √289

⇒ 17

∴ The correct answer is 17.

Additional Information

BODMAS Rule

The BODMAS acronym defines the correct sequence for solving mathematical operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

Perfect Squares

Memorizing perfect squares from 1 to 30 (e.g., 17² = 289, 18² = 324) is crucial for improving speed in simplification and quantitative aptitude sections.

Square Root Calculation

The square root (√) is the inverse operation of squaring a number; if x² = y, then √y = x.

Shortcut Trick

Second diagonal (d₂) = (2 × 216) ÷ 18 = 24 cm.

Half-diagonals are 9 cm and 12 cm.

Using the Pythagorean triplet (3, 4, 5) scaled by 3 → (9, 12, 15).

Therefore, the side of the rhombus = 15 cm.

Perimeter = 4 × 15 = 60 cm.

∴ The correct answer is 60 cm.

Alternate Method

Given: Area of rhombus = 216 cm² and one diagonal (d₁) = 18 cm.

Formula: Area = (1/2) × d₁ × d₂

⇒ 216 = (1/2) × 18 × d₂

⇒ 216 = 9 × d₂

⇒ d₂ = 216 ÷ 9 = 24 cm.

The diagonals of a rhombus bisect each other at right angles (90°).

⇒ Side (s) = √((d₁/2)² + (d₂/2)²)

⇒ s = √(9² + 12²) = √(81 + 144) = √225 = 15 cm.

⇒ Perimeter = 4 × s = 4 × 15 = 60 cm.

∴ The correct answer is 60 cm.

Additional Information

Diagonal Properties

The diagonals of a rhombus bisect each other at 90° and bisect the vertex angles.

Side-Diagonal Relation

The relationship between the side (s) and diagonals is given by: 4s² = d₁² + d₂².

Rhombus Area

Area of a rhombus can also be calculated as Base × Height if those values are known.

Shortcut Trick

Area of rhombus = (1/2) × d₁ × d₂

384 = (1/2) × 24 × d₂ ⇒ d₂ = 32 cm.

Half-diagonals are 12 and 16. Using Pythagorean Triplet (3, 4, 5) scaled by 4 ⇒ Side = 20 cm.

Perimeter = 4 × Side = 4 × 20 = 80 cm.

∴ The correct answer is 80 cm.

Alternate Method

Given: Area = 384 cm², Diagonal 1 (d₁) = 24 cm.

Formula Used:

1. Area = (1/2) × d₁ × d₂
2. Side (a) = √((d₁/2)² + (d₂/2)²)
3. Perimeter = 4a

Calculations:

⇒ Area = (1/2) × 24 × d₂

⇒ 384 = 12 × d₂

⇒ d₂ = 384 ÷ 12 = 32 cm

⇒ Side a = √((24/2)² + (32/2)²)

⇒ Side a = √(12² + 16²)

⇒ Side a = √(144 + 256) = √400

⇒ Side a = 20 cm

⇒ Perimeter = 4 × 20 = 80 cm

∴ The correct answer is 80 cm.

Additional Information

Properties of Rhombus

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

Relation between Side and Diagonals

The side 'a' and diagonals 'd₁' and 'd₂' are related by the equation 4a² = d₁² + d₂².

Area of Rhombus

Area can also be calculated as Base × Height if the altitude is known.

Shortcut Trick

The Least Common Multiple (LCM) follows the associative property, which states that LCM(A, B, C, D) = LCM[LCM(A, B), LCM(C, D)].

⇒ Given LCM(A, B) = 20 and LCM(C, D) = 30.

⇒ Required LCM = LCM(20, 30).

⇒ The smallest number divisible by both 20 and 30 is 60.

∴ The correct answer is 60.

Alternate Method

Given:

LCM of A and B = 20

LCM of C and D = 30

Formula Used:

LCM(a, b, c, d) = LCM(LCM(a, b), LCM(c, d))

Calculations:

⇒ We need to find the LCM of 20 and 30.

⇒ Prime factorization of 20 = 2 × 2 × 5 = 2² × 5

⇒ Prime factorization of 30 = 2 × 3 × 5

⇒ LCM is the product of the highest powers of all prime factors involved.

⇒ LCM = 2² × 3 × 5

⇒ LCM = 4 × 3 × 5 = 60

∴ The correct answer is 60.

Additional Information

Associative Property of LCM

The LCM of a set of numbers can be found by grouping them: LCM(a, b, c) = LCM(a, LCM(b, c)).

Relationship between HCF and LCM

For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.

LCM of Fractions

LCM of (a/b, c/d) = LCM of numerators (a, c) ÷ HCF of denominators (b, d).

Shortcut Trick

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

Given LCM = 90, the potential HCF must divide 90 exactly without a remainder.

Checking options: 90 ÷ 28 (No) | 90 ÷ 24 (No) | 90 ÷ 36 (No) | 90 ÷ 15 = 6 (Yes).

Since only 15 is a factor of 90, it is the only possible HCF.

∴ The correct answer is 15.

Alternate Method

Given:

LCM of two numbers = 90

Formula/Property:

For any two integers, LCM = HCF × k (where k is a positive integer representing the product of the non-common prime factors).

Calculations:

⇒ Let the HCF be 'h'. Then 90/h must be an integer.

⇒ Option 1: 90 ÷ 28 ≈ 3.21 (Not an integer)

⇒ Option 2: 90 ÷ 24 = 3.75 (Not an integer)

⇒ Option 3: 90 ÷ 36 = 2.5 (Not an integer)

⇒ Option 4: 90 ÷ 15 = 6 (Integer)

⇒ Since 15 divides 90 perfectly, it can be the HCF.

∴ The correct answer is 15.

Additional Information

Fundamental Relation

For any two positive integers a and b: Product of numbers = HCF(a, b) × LCM(a, b).

HCF vs LCM Magnitude

The HCF is always less than or equal to the LCM. Specifically, HCF ≤ smaller number ≤ larger number ≤ LCM.

Co-prime Numbers

If two numbers are co-prime, their HCF is 1 and their LCM is simply the product of the two numbers.

Divisibility Rule

The LCM of a set of numbers is the smallest number that is divisible by all the numbers in that set.

Shortcut Trick

To find 10% of any value, simply divide the number by 10 or shift the decimal point one place to the left.

Profit = 480 ÷ 10

Profit = Rs. 48

∴ The correct answer is Rs. 48.

Alternate Method

Given:

Cost Price (CP) of the article = Rs. 480

Profit Percentage = 10%

Formula Used:

Profit = (Profit% ÷ 100) × Cost Price

Calculations:

⇒ Profit = (10 ÷ 100) × 480

⇒ Profit = 0.1 × 480

⇒ Profit = Rs. 48

∴ The correct answer is Rs. 48.

Additional Information

Selling Price (SP) Calculation

The selling price is the sum of cost price and profit: SP = CP + Profit. In this case, SP = 480 + 48 = Rs. 528.

Loss Calculation

If the article is sold for less than the CP, a loss occurs. Loss = CP − SP. Loss% = (Loss ÷ CP) × 100.

Direct SP Formula

To find SP directly when Profit% is known: SP = CP × (100 + Profit%) ÷ 100.