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Shortcut Trick

The third proportional (c) of two numbers a and b is given by the relation: c = b² ÷ a.

Given: Second number (b) = 16 and Third proportional (c) = 32.

Substituting values: 32 = 16² ÷ a ⇒ 32 = 256 ÷ a.

a = 256 ÷ 32 = 8.

∴ The correct answer is 8.

Alternate Method

Given:

Second number (b) = 16

Third proportional (c) = 32

Let the first number be a.

Formula Used:

If three numbers a, b, and c are in continued proportion, then a : b = b : c.

Calculations:

⇒ a ÷ 16 = 16 ÷ 32

⇒ a × 32 = 16 × 16 (Product of extremes = Product of means)

⇒ 32a = 256

⇒ a = 256 ÷ 32

⇒ a = 8

∴ The correct answer is 8.

Additional Information

Mean Proportional

The mean proportional between two numbers x and y is defined as √(x × y).

Fourth Proportional

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

Continued Proportion Property

For three numbers to be in continued proportion, the middle term squared must equal the product of the first and third terms: b² = a × c.

Shortcut Trick

Equate the number of pens by multiplying the first set by 2 and the second set by 3.

First Set: 6 Pens + 4 Notebooks = ₹240

Second Set: 6 Pens + 12 Notebooks = ₹420

Difference: 8 Notebooks = ₹180

Price of 1 Notebook = 180 ÷ 8 = ₹22.50

∴ The correct answer is ₹22.50.

Alternate Method

Given:

Cost of 3 pens and 2 notebooks = ₹120

Cost of 2 pens and 4 notebooks = ₹140

Calculations:

Let the price of one pen be P and one notebook be N.

⇒ 3P + 2N = 120 --- (Eq. 1)

⇒ 2P + 4N = 140 --- (Eq. 2)

⇒ From (Eq. 2), simplify by dividing by 2: P + 2N = 70

⇒ Express P in terms of N: P = 70 − 2N

⇒ Substitute P in (Eq. 1): 3 × (70 − 2N) + 2N = 120

⇒ 210 − 6N + 2N = 120

⇒ 210 − 4N = 120

⇒ 4N = 210 − 120

⇒ 4N = 90

⇒ N = 90 ÷ 4 = 22.50

∴ The correct answer is ₹22.50.

Additional Information

Linear Equations in Two Variables

Equations in the form ax + by = c that can be solved using substitution, elimination, or cross-multiplication.

Elimination Method

Making the coefficients of one variable equal in both equations and then adding or subtracting them to solve for the other variable.

Substitution Method

Expressing one variable in terms of the other from one equation and substituting this value into the second equation.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) ⇒ 100 : 111 1/9 = 9 : 10

Ratio of Marked Price (MP) to Selling Price (SP) ⇒ 1 : [(1 − 1/6) × (1 − 1/5)] = 1 : (5/6 × 4/5) = 1 : 2/3 = 3 : 2

Combined ratio CP : SP : MP ⇒ 9 : 10 : 15 (as SP is 10 and 2, common SP = 10)

Given CP = 9 units = ₹1800 ⇒ 1 unit = ₹200

Marked Price (MP) = 15 units = 15 × 200 = ₹3000

∴ The correct answer is ₹3000.

Alternate Method

Given:

Cost Price (CP) of the article = ₹1800

Profit percentage = 11 1/9% = 1/9

Successive discounts = 16 2/3% (1/6) and 20% (1/5)

Formula Used:

1. Selling Price (SP) = CP × (1 + Profit%)
2. Selling Price (SP) = MP × (1 − Discount₁%) × (1 − Discount₂%)

Calculations:

⇒ Profit amount = 1/9 × 1800 = ₹200

⇒ Selling Price (SP) = 1800 + 200 = ₹2000

⇒ Total multiplier for successive discounts = (1 − 1/6) × (1 − 1/5) = 5/6 × 4/5 = 4/6 = 2/3

⇒ Selling Price (SP) = Marked Price (MP) × 2/3

⇒ 2000 = MP × 2/3

⇒ MP = (2000 × 3) ÷ 2 = ₹3000

∴ The correct answer is ₹3000.

Additional Information

Effective Successive Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

CP and MP Relation

The direct ratio of Marked Price to Cost Price is given by MP/CP = (100 + P%) / (100 − D%), where P is profit and D is discount.

Fraction Conversions

Standard percentages to fractions: 11 1/9% = 1/9, 16 2/3% = 1/6, 20% = 1/5, 25% = 1/4, and 33 1/3% = 1/3.

Shortcut Trick

Arrange the data in ascending order: 12, 17, 22, 27, 31, 36.

Total number of terms (n) = 6 (which is even).

Median = Average of the two middle terms (3^rd and 4^th terms).

Median = (22 + 27) ÷ 2 = 49 ÷ 2 = 24.5.

∴ The correct answer is 24.5.

Alternate Method

Given Data: 12, 36, 22, 27, 31, 17

Formula: If n is even, Median = [ (n/2)^th term + (n/2 + 1)^th term ] ÷ 2

Calculations:

⇒ Arranging in Ascending order: 12, 17, 22, 27, 31, 36

⇒ Number of observations (n) = 6

⇒ Median = [ (6/2)^th term + (6/2 + 1)^th term ] ÷ 2

⇒ Median = [ 3^rd term + 4^th term ] ÷ 2

⇒ Median = [ 22 + 27 ] ÷ 2

⇒ Median = 49 ÷ 2

⇒ Median = 24.5

∴ The correct answer is 24.5.

Additional Information

Median for Odd Number of Terms

When n is odd, the median is the exactly middle term: ((n + 1) ÷ 2)^th term.

Arithmetic Mean

The average of all values: Mean = (Sum of all observations) ÷ (Total number of observations).

Mode

The value that appears most frequently in a data set. A set can have one mode, more than one, or none.

Range

The difference between the maximum and minimum values in the data set: Range = Max − Min.

Shortcut Trick

The product of the two positive numbers is the square of their mean proportional: 30² = 900.

Find factors of 900 whose difference is 45: 60 × 15 = 900 and 60 − 15 = 45.

The numbers are 60 and 15, where 15 is the smaller number.

∴ The correct answer is 15.

Alternate Method

Given:

Mean proportional = 30

Difference between numbers = 45

Formula Used:

Mean Proportional = √(a × b)

(a + b)² = (a − b)² + 4ab

Calculations:

⇒ Let the smaller number be b and the larger number be a.

⇒ √(ab) = 30 ⇒ ab = 900

⇒ a − b = 45

⇒ (a + b)² = (a − b)² + 4ab

⇒ (a + b)² = (45)² + 4 × 900

⇒ (a + b)² = 2025 + 3600 = 5625

⇒ a + b = √5625 = 75

⇒ Subtracting (a − b) from (a + b): (a + b) − (a − b) = 75 − 45

⇒ 2b = 30

⇒ b = 15

∴ The correct answer is 15.

Additional Information

Mean Proportional

The mean proportional between two numbers x and y is given by the geometric mean, which is √(xy).

Third Proportional

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

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a:b = c:d, which gives d = (b × c) ÷ a.

Algebraic Identity

The relation between sum, difference, and product of two numbers is (a + b)² − (a − b)² = 4ab.

Shortcut Trick

Net Discount % for each scheme:

(A) 39 + 3 − (39 × 3 / 100) = 42 − 1.17 = 40.83%

(B) Successive 36%, 13%, 8% → First two = 44.32%, Total ≈ 44.3 + 8 − 3.5 = 48.8%

(C) Weighted Average = (18 × 2 + 41 × 5) ÷ 7 = 241 ÷ 7 ≈ 34.4%

(D) Buy 7 Get 5 Free → (5 ÷ 12) × 100 ≈ 41.7%

Comparing the values: 34.4% (C) < 40.83% (A) < 41.7% (D) < 48.8% (B). The least beneficial scheme is the one with the minimum discount.

∴ The correct answer is C.

Alternate Method

Given: Total toys to buy = 7. Marked Price (MP) per toy = ₹100. Total MP = ₹700.

Calculation of Selling Price (SP) and Discount for each scheme:

Scheme A: Successive 39% and 3%

⇒ SP = 700 × (61/100) × (97/100) = 414.19

⇒ Total Discount = 700 − 414.19 = 285.81

⇒ Discount % = (285.81 ÷ 700) × 100 = 40.83%

Scheme B: Successive 36%, 13%, and 8%

⇒ SP = 700 × 0.64 × 0.87 × 0.92 = 358.58

⇒ Total Discount = 700 − 358.58 = 341.42

⇒ Discount % = (341.42 ÷ 700) × 100 = 48.77%

Scheme C: 18% on first 2, 41% on remaining 5

⇒ Discount on 2 toys = 2 × (18% of 100) = 36

⇒ Discount on 5 toys = 5 × (41% of 100) = 205

⇒ Total Discount = 36 + 205 = 241

⇒ Discount % = (241 ÷ 700) × 100 = 34.43%

Scheme D: 5 toys free on buying 7 toys (Standard Buy 7 Get 5 Free)

⇒ Total items = 7 + 5 = 12

⇒ Discount % = (Free items ÷ Total items) × 100 = (5 ÷ 12) × 100 = 41.67%

Scheme C provides the lowest discount percentage (34.43%), making it the least beneficial for the customer.

∴ The correct answer is C.

Additional Information

Successive Discount Formula

For two successive discounts x% and y%, the single equivalent discount is (x + y − xy/100)%.

Buy X Get Y Free

The effective discount percentage for this offer is calculated as: [Y ÷ (X + Y)] × 100.

Weighted Average Discount

When different quantities have different discounts, use (n₁d₁ + n₂d₂) ÷ (n₁ + n₂) to find the total discount percentage.

Shortcut Trick

Effective Interest Rate for 4 years = 6% × 4 = 24%.

Interest earned = 24% of the Principal (₹25,200).

Interest = 0.24 × 25200 = 24 × 252 = 6048.

∴ The correct answer is ₹6,048.

Alternate Method

Given:

Principal (P) = ₹25,200

Rate (R) = 6% per annum

Time (T) = 4 years

Formula Used:

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

⇒ SI = (25200 × 6 × 4) ÷ 100

⇒ SI = 25200 × 24 ÷ 100

⇒ SI = 252 × 24

⇒ SI = ₹6,048

∴ The correct answer is ₹6,048.

Additional Information

Total Amount Calculation

The total sum payable at the end of the tenure is Amount = Principal + Simple Interest. Here, A = 25200 + 6048 = ₹31,248.

Calculation of Rate or Time

The rate (R) can be found using R = (SI × 100) ÷ (P × T) and time (T) using T = (SI × 100) ÷ (P × R).

Simple Interest vs Compound Interest

For the first year, Simple Interest and Compound Interest are identical if the compounding is annual.

Shortcut Trick

Simplify the equation: 2cos²x − 1 = 1/2 ⇒ cos 2x = 1/2.

Identify the angle: cos 60° = 1/2, so 2x = 60° and x = 30°.

Substitute x = 30° into the expression sec 2x + cosec x + cot²x.

Calculation: sec 60° + cosec 30° + cot² 30° = 2 + 2 + 3 = 7.

∴ The correct answer is 7.

Alternate Method

Given: 2cos²x − 1 = 1/2 and 0° < x < 90°

Formula Used: cos 2x = 2cos²x − 1

⇒ 2cos²x − 1 = 1/2

⇒ cos 2x = 1/2

⇒ 2x = 60° (Since cos 60° = 1/2)

⇒ x = 30°

⇒ sec 2x = sec 60° = 2

⇒ cosec x = cosec 30° = 2

⇒ cot²x = (cot 30°)² = (√3)² = 3

⇒ Value of expression = 2 + 2 + 3 = 7

∴ The correct answer is 7.

Additional Information

Double Angle Formulas

Fundamental identities include cos 2x = 2cos²x − 1 = 1 − 2sin²x = cos²x − sin²x.

Reciprocal Identities

Trigonometric ratios are related as sec x = 1/cos x, cosec x = 1/sin x, and cot x = 1/tan x.

Standard Trigonometric Values

For 30°: sin = 1/2, cos = √3/2, tan = 1/√3. For 60°: sin = √3/2, cos = 1/2, tan = √3.

Shortcut Trick

Identify factors of 176 that add up to 30.

Pairs of factors for 176: (1, 176), (2, 88), (4, 44), (8, 22), (11, 16).

Check the Sum: 8 + 22 = 30.

Required Positive Difference = 22 − 8 = 14.

∴ The correct answer is 14.

Alternate Method

Given:

Sum of numbers (x + y) = 30

Product of numbers (xy) = 176

Formula Used:

(x − y)² = (x + y)² − 4xy

Calculations:

⇒ (x − y)² = (30)² − 4 × 176

⇒ (x − y)² = 900 − 704

⇒ (x − y)² = 196

⇒ (x − y) = √196

⇒ x − y = 14

∴ The correct answer is 14.

Additional Information

Quadratic Relation

If the sum (S) and product (P) of two numbers are known, the numbers are the roots of the quadratic equation z² − Sz + P = 0.

Algebraic Identity

The relationship between the square of the sum and square of the difference is given by: (a + b)² − (a − b)² = 4ab.

Difference of Squares

a² − b² = (a − b)(a + b). This is useful when the difference and sum are both known to find the difference of their squares.

Shortcut Trick

Given side (a) = 86 cm.

Total Surface Area (TSA) = 6 × a² = 6 × 86².

Apply unit digit method: 6 × (...6 × ...6) = 6 × 36 = 216 → last digit is 6.

The final result must end with the digit 6.

Only the value 44376 satisfies this condition.

∴ The correct answer is 44376 cm².

Alternate Method

Given:

Side (a) of the cube = 86 cm

Formula Used:

Total Surface Area (TSA) of a cube = 6 × a²

Calculations:

⇒ TSA = 6 × (86)²

⇒ TSA = 6 × (86 × 86)

⇒ TSA = 6 × 7396

⇒ TSA = 44376 cm²

∴ The correct answer is 44376 cm².

Additional Information

Lateral Surface Area (LSA)

It is the sum of the areas of all the side faces of the cube, excluding the top and bottom: LSA = 4 × a².

Volume of a Cube

The total space occupied by the cube in three dimensions is given by the formula: Volume = a³.

Diagonal of a Cube

The longest distance between any two corners of the cube is calculated as: Diagonal = a√3.