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Shortcut Trick

If each number in a set is decreased by a constant value k, the average of the set also decreases by the same value k.

Given Old Average = 45 and Decrease per number = 3.

New Average = 45 − 3 = 42.

∴ The correct answer is 42.

Alternate Method

Given:

Total numbers = 8

Initial Average = 45

Formula Used:

Average = Sum of observations ÷ Number of observations

Calculations:

⇒ Initial Sum = Average × Count = 45 × 8 = 360

⇒ Total Reduction = 8 × 3 = 24

⇒ New Sum = 360 − 24 = 336

⇒ New Average = 336 ÷ 8

⇒ New Average = 42

∴ The correct answer is 42.

Additional Information

Addition Rule

If k is added to every observation, the new average is Old Average + k.

Multiplication Rule

If every observation is multiplied by a constant n, the new average becomes Old Average × n.

Division Rule

If every observation is divided by a non-zero constant m, the new average is Old Average ÷ m.

Shortcut Trick

Percentage difference = 45% − 20% = 25%

The question states that this 25% difference is equal to 30.

Using the Unitary method:

25% → 30

100% → 30 × (100 ÷ 25) = 30 × 4 = 120

∴ The correct answer is 120.

Alternate Method

Given: 45% of a number exceeds 20% of the same number by 30.

Formula Used: Percentage of a number = (Percent ÷ 100) × Number

⇒ Let the number be x. ⇒ (45% of x) − (20% of x) = 30 ⇒ (45/100) × x − (20/100) × x = 30 ⇒ (45 − 20)/100 × x = 30 ⇒ (25/100) × x = 30 ⇒ (1/4) × x = 30 ⇒ x = 30 × 4 ⇒ x = 120

∴ The correct answer is 120.

Additional Information

Percentage to Fraction Conversion

To solve quickly, remember common fractions: 25% = 1/4, 20% = 1/5, 50% = 1/2, and 75% = 3/4.

Base Value Property

When comparing two percentages of the same number, you can directly subtract or add the percentage values before applying them to the base.

Percentage Change Formula

Percentage change = (Difference ÷ Original Value) × 100. This is used to find how much one value is greater or smaller than another in percentage terms.

Shortcut Trick

Let the smaller number be x. Then the larger number is 54 − x.

Given condition: 3 × x − 2 × (54 − x) = 6

⇒ 3x − 108 + 2x = 6 ⇒ 5x = 114 ⇒ x = 22.8.

Larger number = 54 − 22.8 = 31.2.

Difference = 31.2 − 22.8 = 8.4.

∴ The correct answer is 8.4.

Alternate Method

Given: Sum of two numbers = 54. Let smaller number = x and larger number = y.

Formulas: (i) x + y = 54 | (ii) 3x − 2y = 6

⇒ From (i), y = 54 − x.

⇒ Substitute y in (ii): 3x − 2(54 − x) = 6

⇒ 3x − 108 + 2x = 6

⇒ 5x = 114 ⇒ x = 22.8

⇒ y = 54 − 22.8 = 31.2

⇒ Difference = y − x = 31.2 − 22.8 = 8.4

∴ The correct answer is 8.4.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c can be solved using the Elimination Method or Substitution Method to find unique values for the variables.

Sum and Difference Relation

If the Sum (S) and Difference (D) of two numbers are known, the Larger number = (S + D) ÷ 2 and the Smaller number = (S − D) ÷ 2.

Algebraic Modeling

Key phrases like exceeds or more than indicate subtraction (Larger − Smaller = Difference) when setting up equations.

Shortcut Trick

In a right-angled triangle with angles 30°−60°−90°, the ratio of sides is 1 : √3 : 2.

Here, the side opposite to 60° (Height of wall) corresponds to √3 parts.

The hypotenuse (Ladder length) corresponds to 2 parts = 8 m.

⇒ 1 part = 8 ÷ 2 = 4 m.

⇒ Height (opposite to 60°) = 4 × √3 = 4√3 m.

∴ The correct answer is 4√3 m.

Alternate Method

Given:

Length of the ladder (Hypotenuse, AC) = 8 m

Angle of elevation with the ground (θ) = 60°

Let the height of the wall reached by the ladder be AB.

Formula Used:

In a right-angled triangle, sin θ = Perpendicular (Height) ÷ Hypotenuse

Calculations:

⇒ sin 60° = AB ÷ AC

⇒ √3 ÷ 2 = AB ÷ 8

⇒ AB = 8 × (√3 ÷ 2)

⇒ AB = 4 × √3

⇒ AB = 4√3 m

∴ The correct answer is 4√3 m.

Additional Information

Trigonometric Ratios

For an angle θ, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.

Standard Trigonometric Values

Key values include sin 30° = 1/2, sin 45° = 1/√2, and sin 60° = √3/2.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides: a² + b² = c².

Shortcut Trick

Total units in the ratio = 3 + 3 + 4 = 10 units.

Sum of angles in a triangle = 180°.

10 units → 180°

1 unit → 18°

Measure of the largest angle (4 units) = 4 × 18° = 72°.

∴ The correct answer is 72degree.

Alternate Method

Given:

Ratio of three angles = 3 : 3 : 4

Formula Used:

The Angle-Sum Property of a triangle states that the sum of all interior angles is exactly 180°.

Calculations:

⇒ Let the three angles of the triangle be 3x, 3x, and 4x.

⇒ 3x + 3x + 4x = 180°

⇒ 10x = 180°

⇒ x = 180° / 10 = 18°

⇒ Measure of the first angle = 3 × 18° = 54°

⇒ Measure of the second angle = 3 × 18° = 54°

⇒ Measure of the third (largest) angle = 4 × 18° = 72°

∴ The correct answer is 72degree.

Additional Information

Angle-Sum Property

For any polygon with n sides, the sum of interior angles is (n − 2) × 180°.

Isosceles Triangle

When a triangle has two equal angles, the sides opposite those angles are also equal.

Classification by Angles

A triangle where all angles are less than 90° is an acute triangle; if one is exactly 90°, it is a right triangle.

Shortcut Trick

Area of Rectangle = Area of Circle ⇒ 98 × 77 = (22/7) × r²

r² = (98 × 77 × 7) ÷ 22 ⇒ r² = 49 × 11 × 7 × 7 ÷ 11 × 2 ÷ 2 (Simplifying: 98 = 49 × 2 and 77 = 11 × 7)

r² = 49 × 49 ⇒ r = 49 m

Perimeter of Circle = 2 × (22/7) × 49 = 44 × 7 = 308 m

∴ The correct answer is 308 m.

Alternate Method

Given: Length of rectangular field = 98 m, Breadth = 77 m. Area of circle = Area of rectangle.

⇒ Area of Rectangular Field = Length × Breadth

⇒ Area = 98 × 77 = 7546 m²

⇒ Area of Circular Field = πr²

⇒ Since Area of Circle = Area of Rectangle:

⇒ (22/7) × r² = 7546

⇒ r² = (7546 × 7) ÷ 22

⇒ r² = 343 × 7 = 2401

⇒ Radius (r) = √2401 = 49 m

⇒ Perimeter (Circumference) of Circular Field = 2 × π × r

⇒ Perimeter = 2 × (22/7) × 49 = 2 × 22 × 7 = 308 m

∴ The correct answer is 308 m.

Additional Information

Area and Perimeter of a Rectangle

The Area is length × breadth (l × b) and the Perimeter is 2(l + b).

Circle Mensuration Formulas

For a circle of radius r, the Area = πr² and the Circumference (Perimeter) = 2πr.

Relation Between Radius and Area

If the radius is multiplied by a factor k, the Area increases by k² and the Perimeter increases by k.

Shortcut Trick

Total Work = LCM(14, 21, 42) = 42 units.

Efficiencies: P = 3, Q = 2, R = 1. Combined (P+Q+R) = 6.

Work done in 4 days = 6 × 4 = 24 units.

Remaining work = 42 − 24 = 18 units.

Time for Q and R to finish = 18 ÷ (2 + 1) = 6 days.

Total time = 4 + 6 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given: Time taken by P = 14 days, Q = 21 days, and R = 42 days.

⇒ All three started together, so combined efficiency = 3 + 2 + 1 = 6 units/day.

⇒ Work completed by P, Q, and R in 4 days = 6 × 4 = 24 units.

⇒ Remaining work to be done = 42 − 24 = 18 units.

⇒ Since P left, Q and R complete the remaining work.

⇒ Combined efficiency of Q and R = 2 + 1 = 3 units/day.

⇒ Time taken by Q and R = 18 ÷ 3 = 6 days.

⇒ Total days taken for the wall to be painted = 4 + 6 = 10 days.

∴ The correct answer is 10 days.

Additional Information

Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a task. If the ratio of time is a:b, then the ratio of efficiency is b:a.

Work and Wages

Wages are generally distributed in the ratio of the total work done by each individual. If they work for the same time, it is the ratio of their efficiencies.

Pipes and Cisterns

This concept follows the same logic as Time & Work, where an inlet pipe does positive work and an outlet pipe (leakage) does negative work.

Shortcut Trick

Assume the mean 13 as the balance point for all deviations.

Net deviation from 13: (6.5 − 13) + (10.8 − 13) + (20 − 13) + (a − 13) = 0

⇒ −6.5 − 2.2 + 7.0 + (a − 13) = 0

⇒ −1.7 + (a − 13) = 0

⇒ a − 13 = 1.7 ⇒ a = 13 + 1.7 = 14.7

∴ The correct answer is 14.7.

Alternate Method

Given: Numbers = 6.5, 10.8, 20, a | Total Count (n) = 4 | Average = 13

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum of observations = Average × Count

⇒ 6.5 + 10.8 + 20 + a = 13 × 4

⇒ 37.3 + a = 52

⇒ a = 52 − 37.3

⇒ a = 14.7

∴ The correct answer is 14.7.

Additional Information

Arithmetic Mean Properties

If each observation is increased or decreased by a constant k, the new average also increases or decreases by k.

Sum of Deviations

The sum of deviations of all observations from their mean is always zero (Σ(x_i − x̄) = 0).

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Length of 4 identical cubes placed in a row = 256 cm.

Side of a single cube (a) = 256 ÷ 4 = 64 cm.

Volume of the cube = a³ = 64 × 64 × 64.

Volume = 262144 cm³.

∴ The correct answer is 262144 cm³.

Alternate Method

Given:

Total length of the cuboid formed by 4 cubes = 256 cm.

Formula Used:

Volume of a cube = (side)³

Calculations:

⇒ Let the side of each cube be a.

⇒ When 4 cubes are placed side-by-side, length (L) = 4a.

⇒ 4a = 256

⇒ a = 256 ÷ 4

⇒ a = 64 cm

⇒ Volume of a single cube = a × a × a

⇒ Volume = 64 × 64 × 64

⇒ Volume = 262144 cm³

∴ The correct answer is 262144 cm³.

Additional Information

Total Surface Area of Cube

The total surface area of a cube with side a is given by 6a².

Diagonal of a Cube

The length of the longest diagonal in a cube is a√3.

Volume of a Cuboid

For a cuboid with dimensions L, B, and H, the volume is L × B × H.

Shortcut Trick

When two trains move in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 8 + 10 = 18 m/s.

Combined Length = Relative Speed × Time = 18 × 8 = 144 m.

∴ The correct answer is 144 m.

Alternate Method

Given:

Speed of the first train (v₁) = 8 m/s

Speed of the second train (v₂) = 10 m/s

Time taken to pass each other (t) = 8 sec

Formula Used:

Total Distance (Sum of lengths) = Relative Speed × Time

Calculations:

⇒ When two objects move in opposite directions, Relative Speed (S_R) = v₁ + v₂

⇒ S_R = 8 + 10 = 18 m/s

⇒ Combined Length (L₁ + L₂) = S_R × t

⇒ Combined Length = 18 × 8 = 144 m

∴ The correct answer is 144 m.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction with speeds u and v (where v), the relative speed is u − v.

Crossing a Stationary Object

When a train crosses a stationary object of negligible length (like a pole), the distance covered is the length of the train.

Unit Conversion Shortcut

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