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Shortcut Trick

Identify operations using BODMAS: Bracket → Multiplication → Addition/Subtraction.

91 − 56 + 78 − (54) + 90

Combine positive terms: 91 + 78 + 90 = 259

Combine negative terms: 56 + 54 = 110

Net result: 259 − 110 = 149

∴ The correct answer is 149.

Alternate Method

Given expression: 91 − 56 + 78 − (31 + 23) + 45 × 2

Following the BODMAS order of operations:

⇒ 91 − 56 + 78 − 54 + 45 × 2 (Solved Bracket: 31 + 23 = 54)

⇒ 91 − 56 + 78 − 54 + 90 (Solved Multiplication: 45 × 2 = 90)

⇒ 35 + 78 − 54 + 90 (Subtraction: 91 − 56 = 35)

⇒ 113 − 54 + 90 (Addition: 35 + 78 = 113)

⇒ 59 + 90 (Subtraction: 113 − 54 = 59)

⇒ 149

∴ The correct answer is 149.

Additional Information

BODMAS Rule

It defines the sequence to solve mathematical expressions: Brackets (), Of (orders/powers), Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Handling Signs

While performing addition and subtraction, grouping like-signed numbers (all positives together and all negatives together) reduces the risk of calculation errors.

Unit Digit Check

In competitive exams, verify the unit digit: 1 − 6 + 8 − 4 + 0 = −1. Since the result is positive, 10 − 1 = 9. The answer must end in 9.

Shortcut Trick

Total interest to be earned = 16,800 − 12,000 = 4,800

Interest for 1 year = 8% of 12,000 = 960

Time required = Total Interest ÷ Yearly Interest

Time = 4,800 ÷ 960 = 5 years

∴ The correct answer is 5 years.

Alternate Method

Given:

Principal (P) = ₹12,000

Total Amount (A) = ₹16,800

Rate of Interest (R) = 8% per annum

Formula Used:

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

Amount (A) = Principal (P) + Simple Interest (SI)

⇒ SI = Amount − Principal

⇒ SI = 16,800 − 12,000 = 4,800

⇒ 4,800 = (12,000 × 8 × T) ÷ 100

⇒ 4,800 = 120 × 8 × T

⇒ 4,800 = 960 × T

⇒ T = 4,800 ÷ 960

⇒ T = 5 years

∴ The correct answer is 5 years.

Additional Information

Simple Interest Concept

Simple interest is calculated only on the initial principal amount throughout the entire period of the loan or investment.

Total Amount Formula

The total amount can be expressed as A = P(1 + RT/100), where R is the annual rate and T is the time in years.

Rate and Time Relation

For a fixed principal and interest, the rate of interest is inversely proportional to the time period (R ∝ 1/T).

Shortcut Trick

Total Surface Area (TSA) of hemisphere = 3πr² = 4158 m².

⇒ πr² = 4158 ÷ 3 = 1386 m².

New radius (R) = r/3. New Curved Surface Area (CSA) = 2πR².

⇒ New CSA = 2π(r/3)² = (2/9) × πr².

⇒ New CSA = (2/9) × 1386 = 2 × 154 = 308 m².

∴ The correct answer is 308 m².

Alternate Method

Given:

Total Surface Area (TSA) of hemisphere = 4158 m²

New radius (R) = Original radius (r) ÷ 3

Formula Used:

TSA of hemisphere = 3πr²

CSA of hemisphere = 2πr²

Calculations:

⇒ 3 × (22/7) × r² = 4158

⇒ (66/7) × r² = 4158

⇒ r² = (4158 × 7) ÷ 66

⇒ r² = 63 × 7 = 441

⇒ r = √441 = 21 m

Now, new radius R = 21 ÷ 3 = 7 m

⇒ New CSA = 2 × π × R²

⇒ New CSA = 2 × (22/7) × 7 × 7

⇒ New CSA = 2 × 22 × 7

⇒ New CSA = 44 × 7 = 308 m²

∴ The correct answer is 308 m².

Additional Information

Volume of a Hemisphere

The amount of space occupied by a hemisphere is given by the formula V = (2/3)πr³.

Surface Area of a Sphere

A sphere has only one surface area (TSA = CSA), which is calculated as 4πr².

Volume of a Sphere

The volume of a complete sphere is exactly double that of a hemisphere: V = (4/3)πr³.

Shortcut Trick

Given ratio: 42 : 5 = 7x : 10

Compare denominators: 10 is 5 × 2.

Therefore, compare numerators: 7x must be 42 × 2.

7x = 84 ⇒ x = 84 ÷ 7 = 12.

∴ The correct answer is 12.

Alternate Method

Given:

Ratio 1 = 42 : 5

Ratio 2 = 7x : 10

Ratios are equal: 42 : 5 = 7x : 10

Formula Used:

If a : b = c : d, then a/b = c/d

Product of extremes = Product of means (a × d = b × c)

Calculations:

⇒ 42 ÷ 5 = 7x ÷ 10

⇒ 42 × 10 = 5 × 7x

⇒ 420 = 35x

⇒ x = 420 ÷ 35

⇒ x = 12

∴ The correct answer is 12.

Additional Information

Proportion Rule

Four numbers a, b, c, d are in proportion if a : b = c : d, which implies a × d = b × c.

Fourth Proportional

If a : b = c : x, then x is the fourth proportional to a, b, and c, calculated as x = (b × c) ÷ a.

Simplifying Ratios

A ratio remains unchanged if both its terms are multiplied or divided by the same non-zero number.

Shortcut Trick

Perimeter of square = 12 × 4 = 48 m.

Rectangle perimeter ratio (L : W = 2 : 1) → 2 × (2 + 1) = 6 units.

6 units = 48 m → 1 unit = 8 m.

Dimensions of rectangle = 16 m and 8 m.

Difference in area = 12² − (16 × 8) = 144 − 128 = 16 m².

∴ The correct answer is 16 m².

Alternate Method

Given:

Side of square (s) = 12 m

Length of rectangle (L) = 2 × Width (W)

Perimeter of square = Perimeter of rectangle

Formula Used:

Perimeter of square = 4 × side

Perimeter of rectangle = 2 × (length + width)

Area of square = side²

Area of rectangle = length × width

⇒ Perimeter of square = 4 × 12 = 48 m

⇒ Perimeter of rectangle = 2(L + W) = 48 m

⇒ 2(2W + W) = 48

⇒ 6W = 48 → W = 8 m

⇒ L = 2 × 8 = 16 m

⇒ Area of square = 12² = 144 m²

⇒ Area of rectangle = 16 × 8 = 128 m²

⇒ Difference in area = 144 − 128 = 16 m²

∴ The correct answer is 16 m².

Additional Information

Maximum Area Property

For a fixed perimeter, a square always has a larger area than any other rectangle.

Perimeter and Side Relationship

If a square and rectangle have the same perimeter, the side of the square is the arithmetic mean of the rectangle dimensions: s = (L + W) ÷ 2.

Area Formulas

Area of Square = side² and Area of Rectangle = length × width. Units are always in square measure (m², cm², etc.).

Shortcut Trick

When two articles are sold at the same price, one at x% profit and the other at x% loss, there is always an overall loss.

Shortcut formula: Overall Loss % = (x / 10)²

Given: x = 10%

Overall Loss % = (10 / 10)² = 1² = 1%

∴ The correct answer is 1% loss.

Alternate Method

Given: SP₁ = SP₂, Profit % = 10%, Loss % = 10%

Formula: CP = SP × [100 / (100 ± Profit/Loss %)]

Let the Selling Price (SP) of each article be ₹99.

⇒ CP₁ = 99 × (100 / 110) = ₹90

⇒ CP₂ = 99 × (100 / 90) = ₹110

⇒ Total CP = 90 + 110 = ₹200

⇒ Total SP = 99 + 99 = ₹198

⇒ Overall Loss = 200 − 198 = ₹2

⇒ Overall Loss % = (2 / 200) × 100 = 1%

∴ The correct answer is 1% loss.

Additional Information

Equal Cost Price Case

If two articles are sold at the same Cost Price, one at x% profit and another at x% loss, the net result is always 0% (No Profit, No Loss).

Successive Percentage Formula

Net change for two successive changes of a% and b% is given by [a + b + (ab / 100)]%.

Same SP with Different %

If two items are sold at same SP with x% profit and y% loss, net % = [100(x−y) − 2xy] / [200+x−y].

Shortcut Trick

Profit Ratio M : N = (28 × 4 + 20 × 8) : (32 × 4 + 40 × 8)

⇒ 272 : 448 = 17 : 28

Total Ratio Parts = 17 + 28 = 45 units

45 units = ₹54,000 → 1 unit = ₹1,200

N's Share = 28 units × 1,200 = ₹33,600

∴ The correct answer is ₹33,600.

Alternate Method

Given:

Initial Capital: M = ₹28,000, N = ₹32,000

After 4 months: M's Capital = 28,000 − 8,000 = ₹20,000; N's Capital = 32,000 + 8,000 = ₹40,000

Total Profit = ₹54,000, Total Time = 12 months

Formula Used:

Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Calculations:

⇒ Total Investment of M = (28,000 × 4) + (20,000 × 8)

⇒ M's Share = 1,12,000 + 1,60,000 = ₹2,72,000

⇒ Total Investment of N = (32,000 × 4) + (40,000 × 8)

⇒ N's Share = 1,28,000 + 3,20,000 = ₹4,48,000

⇒ Ratio of M's profit to N's profit = 2,72,000 : 4,48,000 = 17 : 28

⇒ N's share of profit = (28 ÷ (17 + 28)) × 54,000

⇒ N's share = (28 ÷ 45) × 54,000

⇒ N's share = 28 × 1,200 = ₹33,600

∴ The correct answer is ₹33,600.

Additional Information

Simple Partnership

When investments of all partners are for the same time, profit is distributed in the ratio of their capitals: P₁ : P₂ = C₁ : C₂.

Compound Partnership

When investments are for different time periods, profit ratio is the product of capital and time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Working vs. Sleeping Partner

A working partner manages the business and may receive a fixed salary or percentage of profit before the remaining profit is divided.

Shortcut Trick

Total work (LCM of 10 and 15) = 30 units.

Efficiency: A = 3, B = 2. Combined efficiency = 5 units/day.

Work done in 3 days = 5 × 3 = 15 units.

Remaining work = 30 − 15 = 15 units.

More days for B = 15 ÷ 2 = 7.5 days.

∴ The correct answer is 7.5 days.

Alternate Method

Given:

Time taken by A = 10 days

Time taken by B = 15 days

They work together for 3 days.

Calculations:

⇒ Efficiency of A = 30 ÷ 10 = 3 units/day

⇒ Efficiency of B = 30 ÷ 15 = 2 units/day

⇒ Total work done by A and B in 1 day = 3 + 2 = 5 units

⇒ Work done in 3 days = 5 × 3 = 15 units

⇒ Remaining work = Total work − Work done together

⇒ Remaining work = 30 − 15 = 15 units

⇒ Extra time taken by B = Remaining work ÷ B's efficiency

⇒ Time = 15 ÷ 2 = 7.5 days

∴ The correct answer is 7.5 days.

Additional Information

LCM Method for Time & Work

Total work is assumed as the LCM of individual days, making efficiency (Work ÷ Time) a simple integer for easier calculation.

Work and Wages

Wages are always distributed in the ratio of the actual work done by each person, or the ratio of their efficiencies if they work for the same duration.

Alternate Day Work

In alternate day problems, calculate the work done in one complete cycle (e.g., 2 days) and then find the total cycles needed to approach the total work.

Shortcut Trick

Net interest rate for 3 years = Rate × Time = 5% × 3 = 15%.

Given that 15% of the Principal is equivalent to ₹180.

Value of 1% = 180 ÷ 15 = 12.

Value of 100% (Principal) = 12 × 100 = ₹1200.

∴ The correct answer is ₹1200.

Alternate Method

Given:

Simple Interest (SI) = ₹180

Time (T) = 3 years

Rate of Interest (R) = 5% per annum

Formula Used:

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

Where P is the Principal.

Calculations:

⇒ 180 = (P × 5 × 3) ÷ 100

⇒ 180 = (P × 15) ÷ 100

⇒ P = (180 × 100) ÷ 15

⇒ P = 12 × 100

⇒ P = 1200

∴ The correct answer is ₹1200.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, the interest is calculated only on the initial principal. In Compound Interest, interest is calculated on the principal plus any accumulated interest from previous periods.

Total Amount Formula

Amount (A) = Principal (P) + Simple Interest (SI). It represents the total money returned to the depositor after the time period T.

Rate and Time Units

Ensure that the Rate (R) and Time (T) are in the same time units (usually annual). If time is given in months, divide by 12 to convert to years.

Shortcut Trick

Apply BODMAS: First solve the bracket (21 + 18) = 39.

Multiply the result by 2: 39 × 2 = 78.

Combine all terms: 77 − 41 + 34 + 78 − 55 = 189 − 96 = 93.

∴ The correct answer is 93.

Alternate Method

Given expression: 77 − 41 + 34 + (21 + 18) × 2 − 55

Method: BODMAS Rule

⇒ 77 − 41 + 34 + (39) × 2 − 55 ⇒ 77 − 41 + 34 + 78 − 55 ⇒ (77 + 34 + 78) − (41 + 55) ⇒ 189 − 96 ⇒ 93

∴ The correct answer is 93.

Additional Information

BODMAS Rule

It stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction. It dictates the order of operations in any mathematical expression.

Bracket Types

Brackets should be solved in the order: small brackets (), then curly brackets {}, and finally big brackets [].

Addition and Subtraction Priority

Addition and Subtraction have the same priority level and can be performed from left to right or by grouping like-signed terms.