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Shortcut Trick

Solve the bracket and multiplication simultaneously for speed.

Product: 72 × 2 = 144

Bracket: 61 − 28 = 33

Combine: 144 − 19 + 33 + 35 = 125 + 68 = 193

∴ The correct answer is 193.

Alternate Method

Given: 72 × 2 − 19 + (61 − 28) + 35

Formula Used: BODMAS Rule

⇒ 72 × 2 − 19 + (33) + 35 (Solving Bracket)

⇒ 144 − 19 + 33 + 35 (Solving Multiplication)

⇒ 125 + 33 + 35 (Solving first Subtraction)

⇒ 158 + 35 (Solving first Addition)

⇒ 193

∴ The correct answer is 193.

Additional Information

BODMAS Rule Priority

The order of operations is Brackets (B), Orders (O), Division (D), Multiplication (M), Addition (A), and Subtraction (S). Note that DM and AS have equal priority and are solved from left to right.

Types of Brackets

Brackets are solved in the order: Bar/Vinculum, Parentheses ( ), Braces { }, and then Square Brackets [ ].

Shortcut Trick

Solve the expression using the VBODMAS priority order.

⇒ 76 − 33 + (28 × 2) + (36 ÷ 3) − 22 + 19

⇒ 76 − 33 + 56 + 12 − 22 + 19

⇒ (76 + 56 + 12 + 19) − (33 + 22) = 163 − 55 = 108

∴ The correct answer is 108.

Alternate Method

Given:

76 − 33 + 28 of 2 + 36 ÷ 3 − (22) + 19

Formula Used:

Follow the BODMAS rule: Brackets, Of, Division, Multiplication, Addition, Subtraction.

Calculations:

⇒ 76 − 33 + (28 × 2) + (36 ÷ 3) − 22 + 19 (Solving Of and Division)

⇒ 76 − 33 + 56 + 12 − 22 + 19

⇒ 43 + 56 + 12 − 22 + 19 (Solving Subtraction/Addition from left to right)

⇒ 99 + 12 − 22 + 19

⇒ 111 − 22 + 19

⇒ 89 + 19

⇒ 108

∴ The correct answer is 108.

Additional Information

BODMAS Rule

It specifies the order of operations: Brackets ((), {}, []), Orders (powers/roots), Division, Multiplication, Addition, and Subtraction.

The 'Of' Operator

The word 'of' represents multiplication but must be performed before regular division and multiplication in the hierarchy.

Addition and Subtraction Grouping

For faster calculation, group all positive numbers and all negative numbers separately, then find the final difference.

Shortcut Trick

Solve the expression following the BODMAS priority: Bracket, Of, Division, Multiplication, Addition, and Subtraction.

Step 1: Of → 17 × 2 = 34

Step 2: Division → 44 ÷ 4 = 11

Step 3: Multiplication → 29 × 1 = 29

Step 4: 78 − 33 + 29 + 11 + 34 − 19 = 152 − 52 = 100

∴ The correct answer is 100.

Alternate Method

Given: 78 − 33 + 29 × 1 + 44 ÷ 4 + (17 of 2) − 19

Following the BODMAS rule step-by-step:

⇒ 78 − 33 + 29 × 1 + 44 ÷ 4 + 34 − 19

⇒ 78 − 33 + 29 × 1 + 11 + 34 − 19

⇒ 78 − 33 + 29 + 11 + 34 − 19

⇒ (78 + 29 + 11 + 34) − (33 + 19)

⇒ 152 − 52

⇒ 100

∴ The correct answer is 100.

Additional Information

BODMAS Rule

It is an acronym to help remember the order of mathematical operations: Brackets, Of (orders/powers), Division, Multiplication, Addition, and Subtraction.

The 'Of' Operation

In simplification, the term of is treated as multiplication but is prioritized higher than division and standard multiplication symbols.

Calculation Tip

While performing addition and subtraction, it is often faster to group all positive terms and negative terms separately before the final subtraction.

Shortcut Trick

Identify and solve BODMAS components mentally: Bracket (18 − 14 = 4), Of (21 × 2 = 42), and Division (33 ÷ 3 = 11).

Notice the cancellation: The expression contains −42 and + (21 of 2), which is −42 + 42 = 0.

Remaining calculation: 96 + 11 + 4 = 111.

∴ The correct answer is 111.

Alternate Method

Given: 96 − 42 + 33 ÷ 3 + 21 of 2 + (18 − 14)

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

⇒ 96 − 42 + 33 ÷ 3 + 21 of 2 + 4 (Solving Bracket: 18 − 14)

⇒ 96 − 42 + 33 ÷ 3 + 42 + 4 (Solving Of: 21 × 2)

⇒ 96 − 42 + 11 + 42 + 4 (Solving Division: 33 ÷ 3)

⇒ 54 + 11 + 42 + 4 (Solving Subtraction: 96 − 42)

⇒ 65 + 42 + 4 (Solving Addition: 54 + 11)

⇒ 107 + 4 = 111

∴ The correct answer is 111.

Additional Information

BODMAS Hierarchy

The rule ensures a unique solution to mathematical expressions by following the sequence: Brackets, Orders/Of, Division, Multiplication, Addition, and Subtraction.

Sign Convention

When multiple addition and subtraction operations are present, they are performed from left to right, although results remain the same if all additions are grouped first.

Of Operator Significance

The 'of' operator acts like multiplication but holds a higher precedence than division, meaning it must be calculated before any division in the expression.

Shortcut Trick

In a parallelogram, the sum of any two adjacent angles is always 180° (they are supplementary).

Adjacent Angle = 180° − Given Angle

Adjacent Angle = 180° − 60° = 120°

∴ The correct answer is 120°.

Alternate Method

Given: One angle of the parallelogram = 60°

Formula Used: Consecutive interior angles formed by a transversal crossing parallel lines are supplementary. For a parallelogram ABCD, ∠A + ∠B = 180°.

Calculations:

⇒ Let the angles of the parallelogram be ∠A, ∠B, ∠C, and ∠D.

⇒ Given ∠A = 60°.

⇒ Since AD || BC and AB || DC, the adjacent angles are supplementary.

⇒ ∠A + ∠B = 180°

⇒ 60° + ∠B = 180°

⇒ ∠B = 180° − 60°

⇒ ∠B = 120°

∴ The correct answer is 120°.

Additional Information

Opposite Angles Property

In any parallelogram, the opposite angles are always equal in measure. If one angle is 60°, its opposite angle is also 60°.

Angle Sum Property

The sum of all four interior angles of a parallelogram (or any quadrilateral) is always 360°.

Consecutive Angles

Any two consecutive angles of a parallelogram are supplementary, meaning their sum is 180°.

Shortcut Trick

The Least Common Multiple (LCM) of a set of numbers can be found by taking the LCM of the LCMs of its subsets.

LCM(A, B, C, D) = LCM[LCM(A, B), LCM(C, D)]

Given: LCM(A, B) = 15 and LCM(C, D) = 25

Required LCM = LCM(15, 25)

LCM(15, 25) = 75

∴ The correct answer is 75.

Alternate Method

Given:

LCM of A and B = 15

LCM of C and D = 25

Formula Used:

The Associative Property of LCM states: LCM(a, b, c, d) = LCM(LCM(a, b), LCM(c, d)).

Calculations:

⇒ To find the LCM of 15 and 25, we use Prime Factorization.

⇒ 15 = 3¹ × 5¹

⇒ 25 = 5 × 5 = 5²

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

⇒ LCM(15, 25) = 3¹ × 5²

⇒ LCM = 3 × 25

⇒ LCM = 75

∴ The correct answer is 75.

Additional Information

Fundamental LCM Property

The Least Common Multiple (LCM) is the smallest positive integer that is perfectly divisible by every number in the given set.

LCM and HCF Relationship

For any two numbers x and y, the product of their HCF and LCM is equal to the product of the numbers: HCF(x, y) × LCM(x, y) = x × y.

LCM of Co-prime Numbers

If two numbers are co-prime (having no common factors other than 1), their LCM is simply the product of the two numbers.

Shortcut Trick

To find the LCM of three numbers A, B, and C, we use the property: LCM(A, B, C) = LCM[LCM(A, B), C].

Given LCM(A, B) = 30 and C = 18.

We need to find the LCM of 30 and 18.

Check the multiples of the larger number (30): 30 × 1 = 30 (not divisible by 18), 30 × 2 = 60 (not divisible by 18), 30 × 3 = 90 (divisible by 18).

∴ The correct answer is 90.

Alternate Method

Given:

LCM of A and B = 30

Value of C = 18

Formula Used:

The Least Common Multiple (LCM) of three numbers is the LCM of the LCM of any two of those numbers and the third number.

Calculations:

⇒ Prime factorization of 30 = 2 × 3 × 5

⇒ Prime factorization of 18 = 2 × 3 × 3 = 2 × 3²

⇒ To find the LCM, take the highest power of each prime factor present in both numbers:

⇒ Highest power of 2 = 2¹

⇒ Highest power of 3 = 3²

⇒ Highest power of 5 = 5¹

⇒ LCM(30, 18) = 2¹ × 3² × 5¹

⇒ LCM = 2 × 9 × 5 = 90

∴ The correct answer is 90.

Additional Information

LCM and HCF Relation

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

LCM of Fractions

To calculate the LCM of fractions, use the formula: LCM = LCM(Numerators) ÷ HCF(Denominators).

Divisibility Rule

The HCF of a set of numbers always completely divides their LCM.

Shortcut Trick

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

Given LCM = 150.

Check the options to see which number divides 150 exactly: 150 ÷ 15 = 10 (Integer).

Other options (20, 35, 28) do not divide 150 without leaving a remainder.

∴ The correct answer is 15.

Alternate Method

Given:

LCM = 150

Property: In any pair of numbers, the LCM is always a multiple of the HCF. Conversely, the HCF must divide the LCM completely.

Let us test each option:

⇒ Option (1): 150 ÷ 15 = 10 (Divisible)

⇒ Option (2): 150 ÷ 20 = 7.5 (Not divisible)

⇒ Option (3): 150 ÷ 35 = 4.28 (Not divisible)

⇒ Option (4): 150 ÷ 28 = 5.35 (Not divisible)

⇒ Since only 15 is a factor of 150, it can be the HCF.

∴ The correct answer is 15.

Additional Information

Product Rule of LCM and HCF

For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF(A, B) × LCM(A, B).

HCF of Coprime Numbers

The Highest Common Factor of two coprime numbers is always 1, and their LCM is simply the product of the two numbers.

Range of HCF and LCM

The HCF is always less than or equal to the smallest number in the set, while the LCM is always greater than or equal to the largest number in the set.

Shortcut Trick

Loss of 18% = ₹81

1% = 81 ÷ 18 = 4.5

Selling Price (SP) = 100% − 18% = 82%

Selling Price (SP) = 82 × 4.5 = ₹369

∴ The correct answer is ₹369.

Alternate Method

Given:

Loss = ₹81

Loss Percentage = 18%

Formula Used:

Loss % = (Loss ÷ Cost Price) × 100

Selling Price (SP) = Cost Price (CP) − Loss

Calculations:

⇒ 18 = (81 ÷ CP) × 100

⇒ CP = (81 × 100) ÷ 18

⇒ CP = 4.5 × 100 = ₹450

⇒ SP = 450 − 81

⇒ SP = ₹369

∴ The correct answer is ₹369.

Additional Information

Cost Price (CP)

The price at which an article is purchased. It is the base for calculating Profit % or Loss %.

Profit Percentage Formula

Profit % = [(Selling Price − Cost Price) ÷ Cost Price] × 100.

Selling Price for Profit

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

Shortcut Trick

Ratio of Smaller number (S) to Larger number (L) is inversely proportional to their percentage values.

S : L = 20% : 40% = 1 : 2

Sum of ratios = 1 + 2 = 3 units

3 units = 750 ⇒ 1 unit = 250

Larger number = 2 units = 2 × 250 = 500

∴ The correct answer is 500.

Alternate Method

Given: Sum of numbers = 750, 40% of Smaller = 20% of Larger.

Formula Used: Percentage = (Value / Total) × 100

Let the smaller number be S and the larger number be L.

⇒ 40% × S = 20% × L

⇒ (40/100) × S = (20/100) × L

⇒ 2 × S = 1 × L ⇒ L = 2S

⇒ S + L = 750

⇒ S + 2S = 750

⇒ 3S = 750 ⇒ S = 250

⇒ Larger number (L) = 2 × 250 = 500

∴ The correct answer is 500.

Additional Information

Percentage Equivalence

If x% of A = y% of B, then the ratio of the numbers is A : B = y : x. This is a common shortcut for direct ratio conversion.

Dividing a Sum into Ratios

To divide a total sum T into ratio a : b, the parts are calculated as [a / (a + b)] × T and [b / (a + b)] × T.

Linear Equations in Two Variables

Problems involving the sum of two numbers can always be modeled using x + y = Total and solved by substituting one variable in terms of the other.