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Shortcut Trick

Solve the expression by following the BODMAS rule from the innermost bracket outwards.

1. Innermost Bracket: (10 ÷ 2 × 3 − 4 + 7) = 5 × 3 − 4 + 7 = 18
2. Curly Bracket: {14 ÷ 2 − 18} = 7 − 18 = −11
3. Square Bracket: [10 − 3 × (−11)] = 10 + 33 = 43
4. Final Calculation: 36 − 43 = −7

∴ The correct answer is -7.

Alternate Method

Given Expression: 36 − [10 − 3 {14 ÷ 2 − (10 ÷ 2 × 3 − 4 + 7)}]

Formula Used: BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction)

⇒ 36 − [10 − 3 {14 ÷ 2 − (5 × 3 − 4 + 7)}]

⇒ 36 − [10 − 3 {7 − (15 − 4 + 7)}]

⇒ 36 − [10 − 3 {7 − 18}]

⇒ 36 − [10 − 3 {−11}]

⇒ 36 − [10 + 33]

⇒ 36 − 43

⇒ −7

∴ The correct answer is -7.

Additional Information

BODMAS Rule

The acronym stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction, which defines the hierarchy of mathematical operations.

Bracket Priority

Always prioritize parentheses () first, followed by curly braces {}, and finally square brackets [].

Multiplication of Signs

Remember that multiplying two negative numbers results in a positive product (− × − = +).

Shortcut Trick

Relative Speed in opposite directions = 60 + 75 = 135 km/h.

Convert to m/s: 135 × 5/18 = 37.5 m/s.

Total Distance = Sum of lengths = 120 + 180 = 300 m.

Time = 300 ÷ 37.5 = 8 sec.

∴ The correct answer is 8 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 120 m

Length of Train 2 (L₂) = 180 m

Speed of Train 1 (S₁) = 60 km/h

Speed of Train 2 (S₂) = 75 km/h

Formula Used:

Time = (Total Distance) ÷ (Relative Speed)

Calculations:

⇒ Total Distance = 120 + 180 = 300 m

⇒ Relative Speed (Opposite Directions) = S₁ + S₂

⇒ Relative Speed = 60 + 75 = 135 km/h

⇒ Convert speed to m/s = 135 × (5 ÷ 18)

⇒ Relative Speed = 7.5 × 5 = 37.5 m/s

⇒ Time taken = 300 ÷ 37.5

⇒ Time taken = 3000 ÷ 375 = 8 sec

∴ The correct answer is 8 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, relative speed is S₁ + S₂. When they move in the same direction, relative speed is |S₁ − S₂|.

Unit Conversion

To convert speed from km/h to m/s, multiply by 5/18. To convert from m/s to km/h, multiply by 18/5.

Crossing Stationary Objects

Time to cross a pole or man = Train Length ÷ Speed. Time to cross a platform or bridge = (Train Length + Platform Length) ÷ Speed.

Shortcut Trick

Using the Net Deviation method from the Average (20):

Deviation of 11 = 11 − 20 = −9

Deviation of 22 = 22 − 20 = +2

Third number deviation = 0 − (−9 + 2) = +7

Third number = 20 + 7 = 27

∴ The correct answer is 27.

Alternate Method

Given:

Average of three numbers = 20

First number = 11 | Second number = 22

Formula Used:

Sum of numbers = Average × Total count

⇒ Sum of 3 numbers = 20 × 3 = 60

⇒ 11 + 22 + Third Number = 60

⇒ 33 + Third Number = 60

⇒ Third Number = 60 − 33

⇒ Third Number = 27

∴ The correct answer is 27.

Additional Information

Arithmetic Mean (Average)

The average is the central value of a set of numbers, calculated by dividing the total sum by the count of numbers.

Property of Deviations

The algebraic sum of the deviations of a set of values from their arithmetic mean is always zero.

Weighted Average Formula

When groups have different sizes: Weighted Avg = (n₁×x₁ + n₂×x₂) ÷ (n₁ + n₂).

Shortcut Trick

For numbers a, b, c to be in continued proportion by adding x, use the shortcut: x = (b² − ac) / (a + c − 2b).

Here, a = 16, b = 20, c = 30 ⇒ x = (20² − 16 × 30) / (16 + 30 − 2 × 20) = (400 − 480) / (46 − 40) = −80 / 6 = −40/3.

Smallest number = 16 − 40/3 = 8/3 | Largest number = 30 − 40/3 = 50/3.

Mean proportional = √(Smallest × Largest) = √(8/3 × 50/3) = √(400/9) = 20/3.

∴ The correct answer is 20/3.

Alternate Method

Given: Initial numbers are 16, 20, 30. A number x is added to each.

Formula Used: If a, b, c are in continued proportion, then b² = ac. The mean proportional of m and n is √(m × n).

⇒ Resulting numbers: (16 + x), (20 + x), (30 + x)

⇒ According to the condition: (16 + x) / (20 + x) = (20 + x) / (30 + x)

⇒ (16 + x)(30 + x) = (20 + x)²

⇒ 480 + 16x + 30x + x² = 400 + 40x + x²

⇒ 480 + 46x = 400 + 40x

⇒ 46x − 40x = 400 − 480

⇒ 6x = −80 ⇒ x = −40/3

⇒ Smallest number = 16 − 40/3 = (48 − 40) / 3 = 8/3

⇒ Largest number = 30 − 40/3 = (90 − 40) / 3 = 50/3

⇒ Mean proportional = √(8/3 × 50/3) = √(400/9) = 20/3

∴ The correct answer is 20/3.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is defined as x = √(ab), such that a : x = x : b.

Third Proportional

If three numbers a, b, and c are in continued proportion (a/b = b/c), then c is called the third proportional and c = b²/a.

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b = c : d), then d is the fourth proportional and d = (b × c) / a.

Shortcut Trick

Direct formula connecting Volume (V) and Lateral Surface Area (LSA): V = (LSA²) ÷ (4 × π × h)

V = (140.1 × 140.1) ÷ (4 × 3.14 × 3)

V = 19628.01 ÷ 37.68

V ≈ 520.913...

∴ The correct answer is 520.91 cm³.

Alternate Method

Given:

Lateral Surface Area (LSA) = 140.1 cm²

Height (h) = 3 cm

π = 3.14

Formula Used:

LSA of Cylinder = 2 × π × r × h

Volume of Cylinder = π × r² × h

Step 1: Find the radius (r)

⇒ 2 × 3.14 × r × 3 = 140.1

⇒ 18.84 × r = 140.1

⇒ r = 140.1 ÷ 18.84

⇒ r ≈ 7.4363 cm

Step 2: Find the volume

⇒ Volume = π × r² × h

⇒ Volume = 3.14 × (7.4363)² × 3

⇒ Volume = 9.42 × 55.2985...

⇒ Volume ≈ 520.9123...

∴ The correct answer is 520.91 cm³.

Additional Information

Total Surface Area (TSA) of a Cylinder

It includes the lateral area and the area of the two circular bases: TSA = 2πrh + 2πr² = 2πr(r + h).

Volume of a Hollow Cylinder

The volume of the material in a hollow cylinder with outer radius R and inner radius r is: V = πh(R² − r²).

Curved Surface Area vs Lateral Surface Area

For a cylinder, the curved surface area (CSA) is the same as the lateral surface area (LSA), excluding the base areas.

Shortcut Trick

Total sum after 2 years = 36 + (2 × 2) = 40 years (both ages increase by 2).

Ratio after 2 years (Sharad : Adarsh) = 3 : 1.

Total parts = 3 + 1 = 4 units → 40 years ∴ 1 unit = 10 years.

Sharad's age after 2 years = 3 units = 3 × 10 = 30 years.

Sharad's present age = 30 − 2 = 28 years.

∴ The correct answer is 28 years.

Alternate Method

Given:

Sum of present ages of Sharad and Adarsh = 36 years.

Relationship after 2 years: Sharad = 3 × Adarsh.

Calculations:

⇒ Let the present age of Adarsh be x years.

⇒ Present age of Sharad = (36 − x) years.

⇒ After 2 years, Adarsh's age = (x + 2).

⇒ After 2 years, Sharad's age = (36 − x) + 2 = (38 − x).

⇒ According to the condition: (38 − x) = 3 × (x + 2)

⇒ 38 − x = 3x + 6

⇒ 4x = 32

⇒ x = 8 (Adarsh's present age)

⇒ Sharad's present age = 36 − 8 = 28 years.

∴ The correct answer is 28 years.

Additional Information

Constant Age Difference

The difference between the ages of two people remains the same at any point in time, whether in the past, present, or future.

Sum of Ages Rule

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

Ratio Unitary Method

When a ratio A:B and the total value V are known, the value of 1 unit is calculated as V ÷ (A + B).

Shortcut Trick

The Gain per year is the difference between the lending rate and the borrowing rate applied to the Principal.

Rate difference = 8(1/4)% − 5% = 8.25% − 5% = 3.25%

Gain per year = 3.25% of ₹10,000

Gain per year = (3.25 ÷ 100) × 10,000 = 3.25 × 100 = ₹325

∴ The correct answer is ₹325.

Alternate Method 

Given:

Principal (P) = ₹10,000

Borrowing Rate (R₁) = 5% per annum

Lending Rate (R₂) = 8(1/4)% = 8.25% per annum

Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ Total Interest paid in 3 years = (10,000 × 5 × 3) ÷ 100

⇒ Total Interest paid = 100 × 15 = ₹1,500

⇒ Total Interest earned in 3 years = (10,000 × 8.25 × 3) ÷ 100

⇒ Total Interest earned = 100 × 24.75 = ₹2,475

⇒ Total Gain in 3 years = Interest earned − Interest paid

⇒ Total Gain = 2,475 − 1,500 = ₹975

⇒ Gain in the transaction per year = 975 ÷ 3

⇒ Gain per year = ₹325

∴ The correct answer is ₹325.

Additional Information

Simple Interest Principle

Simple interest is calculated only on the initial principal amount of a loan or deposit. Formula: SI = P × R × T ÷ 100.

Effective Gain Rate

When a person borrows at R₁ and lends at R₂, the net profit rate per year is R₂ − R₁, provided the principal and time basis are the same.

Compound Interest vs Simple Interest

Unlike SI, Compound Interest is calculated on the principal plus any accumulated interest from previous periods. Amount = P(1 + R/100)ⁿ.

Shortcut Trick

Use the Successive Multiplying Factor method.

Net Multiplying Factor = 0.80 (20% decrease) × 1.25 (25% increase) × 0.90 (10% decrease)

Net Factor = (0.80 × 1.25) × 0.90 = 1.00 × 0.90 = 0.90

Overall Change % = (Final Factor − 1) × 100 = (0.90 − 1) × 100 = −10%

A negative sign indicates a percentage decrease.

∴ The correct answer is 10% decrease.

Alternate Method

Given:

First Change = 20% Depreciation (−)

Second Change = 25% Appreciation (+)

Third Change = 10% Depreciation (−)

Let the Initial Value of the machine be 100.

⇒ Value after 20% depreciation = 100 − (20% of 100) = 100 − 20 = 80

⇒ Value after 25% appreciation = 80 + (25% of 80) = 80 + 20 = 100

⇒ Value after 10% depreciation = 100 − (10% of 100) = 100 − 10 = 90

⇒ Total change in value = 100 − 90 = 10

⇒ Overall Percentage Change = (Total Change ÷ Initial Value) × 100

⇒ Overall Percentage Change = (10 ÷ 100) × 100 = 10%

∴ The correct answer is 10% decrease.

Additional Information

Successive Percentage Change Formula

For two successive percentage changes of a% and b%, the effective change is given by the formula: [a + b + (ab/100)]%.

Fractional Equivalents

Calculations become faster using fractions: 20% = 1/5, 25% = 1/4, and 10% = 1/10. For depreciation, multiply by (1 − fraction), and for appreciation, multiply by (1 + fraction).

Net Change Principle

The order of percentage changes does not affect the final result. A 20% decrease followed by a 25% increase results in the same final value as a 25% increase followed by a 20% decrease.

Shortcut Trick

Use the empirical relationship formula: Mode = 3 × Median − 2 × Mean.

Substituting the given values: Mode = 3 × 44 − 2 × 48.5

Mode = 132 − 97 = 35.

∴ The correct answer is 35.

Alternate Method

Given:

• Mean = 48.5

• Median = 44

Formula Used:

The relationship between the three measures of central tendency is given by: Mode = 3 × Median − 2 × Mean.

Calculations:

⇒ Mode = 3 × 44 − 2 × 48.5

⇒ Mode = 132 − 97

⇒ Mode = 35

∴ The correct answer is 35.

Additional Information

Empirical Relationship Constraints

This formula is an approximation and works best for moderately skewed frequency distributions. It is not applicable for highly skewed or J-shaped distributions.

Symmetrical Distribution

In a perfectly symmetrical distribution (like the Normal distribution), the Mean, Median, and Mode are all equal: Mean = Median = Mode.

Positively vs Negatively Skewed

If Mean > Median > Mode, the data is positively skewed. If Mean < Median < Mode, the data is negatively skewed.

Shortcut Trick

Area of trapezium = ½ × (Sum of parallel sides) × Height

Given: Side 1 = 8 cm, Side 2 = h, Height = h, Area = 120 cm²

⇒ 120 = ½ × (8 + h) × h ⇒ h(h + 8) = 240

By checking options: If h = 12, then 12 × (12 + 8) = 12 × 20 = 240 (Matches!)

∴ The correct answer is 12 cm.

Alternate Method

Given:

⇒ Parallel side a = 8 cm

⇒ Parallel side b = Distance h

⇒ Area = 120 cm²

Formula Used:

⇒ Area of Trapezium = ½ × (a + b) × h

Calculations:

⇒ 120 = ½ × (8 + h) × h

⇒ 240 = 8h + h²

⇒ h² + 8h − 240 = 0

⇒ h² + 20h − 12h − 240 = 0

⇒ h(h + 20) − 12(h + 20) = 0

⇒ (h − 12)(h + 20) = 0

⇒ Since h cannot be negative, h = 12 cm.

∴ The correct answer is 12 cm.

Additional Information

Area of Trapezium

It is defined as the product of the average of the parallel sides and the vertical distance between them: Area = ½ × (sum of parallel sides) × h.

Median of a Trapezium

The median is a line segment connecting the midpoints of the non-parallel sides. Its length L = ½ × (a + b). Area can also be written as Area = L × h.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal in length and the base angles are equal. The diagonals of an isosceles trapezium are always equal.