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Net deviation from the average must be zero.

Individual excess = 40 − 20 = +20.

To balance this, the other 5 members must have a total deficit of −20.

Average reduction for 5 members = −20 ÷ 5 = −4.

Average of remaining 5 members = 20 − 4 = 16.

∴ The correct answer is 16.

Alternate Method

Given: Average age of 6 members = 20, Age of 1 member = 40.

Formula Used: Sum of observations = Average × Number of observations.

⇒ Total sum of ages of 6 members = 6 × 20 = 120

⇒ Sum of ages of the remaining 5 members = Total sum − Age of one member

⇒ Sum of 5 members = 120 − 40 = 80

⇒ Average age of 5 members = Sum ÷ 5

⇒ Average age of 5 members = 80 ÷ 5 = 16

∴ The correct answer is 16.

Additional Information

Sum of Deviations

The sum of the deviations of all observations from their arithmetic mean is always zero. This property is used to quickly find missing values in a data set.

Weighted Average Formula

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

Effect of Constant Change

If a constant value 'k' is added to or subtracted from every term in a data set, the new average will also increase or decrease by exactly 'k'.

Shortcut Trick

Net deviation from the total average (16) must be zero.

Deviation of first 4 numbers = 4 × (14 − 16) = −8

Deviation of last 4 numbers = 4 × (18 − 16) = +8

Total deviation of 8 numbers = −8 + 8 = 0.

Since the deviation is 0, the average of the remaining 2 numbers must be equal to the total average.

∴ The correct answer is 16.

Alternate Method

Given: Average of 10 numbers = 16 | Average of first 4 = 14 | Average of last 4 = 18.

Formula Used: Sum of numbers = Average × Total numbers

⇒ Sum of 10 numbers = 10 × 16 = 160

⇒ Sum of first 4 numbers = 4 × 14 = 56

⇒ Sum of last 4 numbers = 4 × 18 = 72

⇒ Sum of middle two numbers = Total sum − (Sum of first 4 + Sum of last 4)

⇒ Sum of middle two numbers = 160 − (56 + 72) = 160 − 128 = 32

⇒ Average of middle two numbers = 32 ÷ 2 = 16

∴ The correct answer is 16.

Additional Information

Deviation Property

The sum of deviations of all observations from their arithmetic mean is always zero. This is used to find missing values in a dataset.

Weighted Average Formula

Weighted Average = (n₁x₁ + n₂x₂ + ... + n_nx_n) ÷ (n₁ + n₂ + ... + n_n), where n is the quantity and x is the value/average.

Arithmetic Mean Property

If every number in a set is increased or decreased by a constant 'k', the new average also increases or decreases by 'k'.

Shortcut Trick

Total Area of the design = Area of Quadrant + Area of Right Triangle.

Area = (1/4 × 22/7 × 14 × 14) + (1/2 × 14 × 14).

Area = 154 + 98 = 252 cm².

∴ The correct answer is 252 cm².

Alternate Method

Given: Radius (r) = 14 cm, Base (b) = 14 cm, Height (h) = 14 cm.

Formula used: Area of Quadrant = (1/4) × πr² and Area of Triangle = (1/2) × b × h.

⇒ Area of the Quadrant = (1/4) × (22/7) × 14 × 14

⇒ Area of the Quadrant = (1/4) × 22 × 2 × 14 = 11 × 14 = 154 cm².

⇒ Area of the Right Triangle = (1/2) × 14 × 14

⇒ Area of the Right Triangle = 7 × 14 = 98 cm².

⇒ Total Area of the design = Area of Quadrant + Area of Triangle

⇒ Total Area = 154 + 98 = 252 cm².

∴ The correct answer is 252 cm².

Additional Information

Area of a Sector

The area of a sector of a circle with radius r and angle θ is given by (θ/360) × πr². For a quadrant, θ = 90°.

Area of Circle

The total area of a circle with radius r is πr², where π ≈ 22/7 or 3.14.

Pythagoras Theorem

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

Shortcut Trick

Convert all values into decimal form for a quick comparison.

0.777... is equivalent to 7/9.

8/11 ≈ 0.727... and 0.78 = 0.780.

Comparing 0.727, 0.777, and 0.780, we find 0.727 is the smallest value.

∴ The correct answer is 8/11.

Alternate Method

Given values for comparison: 0.777..., 7/9, 0.78, 8/11.

Step 1: Convert each value to a decimal with three decimal places.

⇒ Value 1: 0.777... = 0.777

⇒ Value 2: 7/9 = 7 ÷ 9 = 0.777...

⇒ Value 3: 0.78 = 0.780

⇒ Value 4: 8/11 = 8 ÷ 11 = 0.7272...

Step 2: Arrange the values in ascending order.

⇒ 0.7272... < 0.777... < 0.780

⇒ 8/11 < 7/9 < 0.78

∴ The correct answer is 8/11.

Additional Information

Cross-Multiplication Method

To compare two fractions a/b and c/d, calculate a × d and b × c. If a × d < b × c, then a/b < c/d.

Recurring Decimals

A repeating decimal like 0.x (where x is repeating) is expressed as the fraction x/9. Example: 0.777... = 7/9.

Proper Fractions

When comparing proper fractions, the one with the denominator significantly larger than the numerator (further from 1) is usually the smaller value.

Shortcut Trick

For two successive discounts of d₁% and d₂%, the single equivalent discount is given by:

Equivalent Discount % = d₁ + d₂ − (d₁ × d₂) ÷ 100

⇒ 12 + 18 − (12 × 18) ÷ 100

⇒ 30 − 216 ÷ 100 = 30 − 2.16 = 27.84%

∴ The correct answer is 27.84%.

Alternate Method

Given:

Marked Price (MP) = ₹ 12,000

First Discount (d₁) = 12%

Second Discount (d₂) = 18%

Formula Used: Final Selling Price = MP × (1 − d₁/100) × (1 − d₂/100)

Calculations:

⇒ Final SP = 12000 × (1 − 12/100) × (1 − 18/100)

⇒ Final SP = 12000 × (88/100) × (82/100)

⇒ Final SP = 12000 × 0.88 × 0.82

⇒ Final SP = 12000 × 0.7216 = ₹ 8659.2

⇒ Total Discount Value = MP − Final SP = 12000 − 8659.2 = ₹ 3340.8

⇒ Equivalent Discount % = (Total Discount ÷ MP) × 100

⇒ Equivalent Discount % = (3340.8 ÷ 12000) × 100 = 27.84%

∴ The correct answer is 27.84%.

Additional Information

Net Discount for Three Successive Discounts

For three successive discounts a%, b%, and c%, the net discount is calculated by first finding the equivalent of a and b, then using that result with c.

Relationship between MP and CP

The ratio of Marked Price to Cost Price is given by: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Multiplying Factor Method

Final SP = MP × MF₁ × MF₂, where MF (Multiplying Factor) for a discount d% is (100 − d) ÷ 100.

Shortcut Trick

Total ratio parts = 1 + 1 + 2 = 4 units.

Total value of the mixture = 153 × 4 = 612.

Value of first two varieties = 126 × 1 + 135 × 1 = 261.

Value of third variety = 612 − 261 = 351.

Price of third variety (for 2 units) = 351 ÷ 2 = 175.5.

∴ The correct answer is 175.5 per kg.

Alternate Method

Given:

Price of variety 1 (P₁) = 126 per kg | Weight ratio (w₁) = 1

Price of variety 2 (P₂) = 135 per kg | Weight ratio (w₂) = 1

Price of variety 3 (P₃) = x per kg | Weight ratio (w₃) = 2

Average Price (P_avg) = 153 per kg

Formula Used:

Weighted Average Price = (P₁×w₁ + P₂×w₂ + P₃×w₃) ÷ (w₁ + w₂ + w₃)

Calculations:

⇒ 153 = (126 × 1 + 135 × 1 + x × 2) ÷ (1 + 1 + 2)

⇒ 153 = (126 + 135 + 2x) ÷ 4

⇒ 153 × 4 = 261 + 2x

⇒ 612 = 261 + 2x

⇒ 2x = 612 − 261

⇒ 2x = 351

⇒ x = 351 ÷ 2

⇒ x = 175.5

∴ The correct answer is 175.5 per kg.

Additional Information

Rule of Alligation for Mixtures

It is a rule that enables us to find the ratio in which two or more ingredients at their respective prices must be mixed to produce a mixture of a specified price.

Weighted Average Concept

When elements have different weights (w_i) or frequencies, the mean is calculated as (Σw_ix_i) / Σw_i.

Mixture Replacement Formula

Final Quantity = Initial Quantity × (1 − Replacement Quantity / Total Quantity)ⁿ, where n is the number of times the process is repeated.

Shortcut Trick

Ratio of Profit Share = (Investment₁ × Time₁) : (Investment₂ × Time₂) : (Investment₃ × Time₃)

Emily : Jake : Lila = (10 × 12) : (8 × 6) : (12 × 6)

Dividing by 12: (10 × 1) : (8 ÷ 2) : (6) ⇒ 10 : 4 : 6

Simplifying the ratio: 5 : 2 : 3

∴ The correct answer is (5:2:3).

Alternate Method

Given:

Emily's Investment = ₹10,000 for 12 months

Jake's Investment = ₹8,000 for 6 months

Lila's Investment = ₹12,000 for 6 months

Total Profit = ₹24,000

Calculations:

⇒ Profit Ratio = (10,000 × 12) : (8,000 × 6) : (12,000 × 6)

⇒ Profit Ratio = 120,000 : 48,000 : 72,000

⇒ Profit Ratio = 120 : 48 : 72

⇒ Dividing each term by 24: 120÷24 : 48÷24 : 72÷24

⇒ Profit Ratio = 5 : 2 : 3

⇒ Total ratio parts = 5 + 2 + 3 = 10 units

⇒ 10 units = ₹24,000

⇒ 1 unit = ₹2,400

⇒ Emily's share = 5 × 2,400 = ₹12,000

⇒ Jake's share = 2 × 2,400 = ₹4,800

⇒ Lila's share = 3 × 2,400 = ₹7,200

∴ The correct answer is (5:2:3).

Additional Information

Compound Partnership

When partners invest different amounts for different time periods, the profit is shared in the ratio of the product of their capital and time.

Profit Sharing Formula

Profit Ratio (P₁:P₂) = (C₁ × T₁) : (C₂ × T₂), where C is Capital and T is Time.

Calculating Capital or Time

If the profit ratio and one variable are known, Capital = Profit ÷ Time and Time = Profit ÷ Capital.

Shortcut Trick

The difference between the actual and measured value is 200 − 198 = 2 ml.

Error Percentage = (Error ÷ Actual Value) × 100.

Error % = (2 ÷ 200) × 100 = 1%.

∴ The correct answer is 1%.

Alternate Method

Given:

Actual Volume of solution = 200 ml

Measured Volume of solution = 198 ml

Formula Used:

Percentage Error = (|Actual Value − Measured Value| ÷ Actual Value) × 100

Calculations:

⇒ Absolute Error = 200 − 198 = 2 ml

⇒ Relative Error = 2 ÷ 200

⇒ Percentage Error = (2 ÷ 200) × 100

⇒ Percentage Error = 1 ÷ 100 × 100

⇒ Percentage Error = 1%

∴ The correct answer is 1%.

Additional Information

Absolute Error

It is the numerical difference between the measured value and the true value, expressed as |v − v_approx|.

Relative Error

It is the ratio of the absolute error to the actual value, often used to show the scale of the error relative to the size of the object.

Percentage Change

Percentage change is calculated as (Change ÷ Original Value) × 100, which is the same principle used for finding measurement error.

Shortcut Trick

Ratio of investments: (1/4)P₁ = (1/5)P₂ ⇒ P₁ : P₂ = 4 : 5.

Total interest = [4 × 6% × 2] + [5 × 7% × 2] = 48% + 70% = 118% units.

118% units = 354 ⇒ 1% unit = 3 ⇒ 100% unit = 300.

Total investment = (4 + 5) × 300 = 2700.

∴ The correct answer is Rs. 2700.

Alternate Method

Given:

Rate 1 (R₁) = 6%, Rate 2 (R₂) = 7%, Time (T) = 2 years, Total Interest = Rs. 354.

Relationship: (1/4) × Amount_Govt = (1/5) × Amount_Company.

Calculations:

⇒ Let investment in Govt bond be 4x and Company bond be 5x.

⇒ Total Investment = 4x + 5x = 9x.

⇒ Interest from Govt bond = (4x × 6 × 2) / 100 = 48x / 100.

⇒ Interest from Company bond = (5x × 7 × 2) / 100 = 70x / 100.

⇒ Total Interest = (48x + 70x) / 100 = 118x / 100.

⇒ 118x / 100 = 354.

⇒ 118x = 35400.

⇒ x = 35400 ÷ 118 = 300.

⇒ Total amount invested = 9 × 300 = 2700.

∴ The correct answer is Rs. 2700.

Additional Information

Simple Interest Formula

SI = (P × R × T) ÷ 100, where P is Principal, R is Rate per annum, and T is Time in years.

Ratio of Investments

When interests are equal for two different rates and times, the ratio of principals is P₁ : P₂ = (1 / R₁T₁) : (1 / R₂T₂).

Effective Rate of Interest

The effective interest rate for a period is simply R × T for Simple Interest.

Shortcut Trick

Recognize the Standard Trigonometric Identity for the Double Angle of cosine.

The expression cos² A − sin² A is the fundamental definition of cos 2A.

By inspection, we identify this as the Cosine Addition Formula where both angles are equal.

∴ The correct answer is cos 2A.

Alternate Method

Given: cos² A − sin² A

Formula Used: cos(A + B) = cos A × cos B − sin A × sin B

⇒ We start with the Compound Angle Formula for cosine:

⇒ cos(A + B) = cos A × cos B − sin A × sin B

⇒ To find the double angle, we let B = A:

⇒ cos(A + A) = (cos A × cos A) − (sin A × sin A)

⇒ cos 2A = cos² A − sin² A

∴ The correct answer is cos 2A.

Additional Information

Other Forms of cos 2A

cos 2A can also be written as 2 cos² A − 1 or 1 − 2 sin² A using the identity sin² A + cos² A = 1.

Double Angle of Sine

The formula for sin 2A is 2 × sin A × cos A.

Double Angle of Tangent

The formula for tan 2A is (2 × tan A) ÷ (1 − tan² A).

Cosine Double Angle in terms of Tan

cos 2A = (1 − tan² A) ÷ (1 + tan² A).