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Shortcut Trick

Total interest percentage for the entire duration = Rate × Time

Total Interest % = 12% × 5 = 60%

Interest earned = 60% of the Principal (₹3200)

Interest = 0.60 × 3200 = 1920

∴ The correct answer is ₹1920.

Alternate Method

Given:

Principal (P) = ₹3200

Rate of Interest (R) = 12% per annum

Time (T) = 5 years

Formula Used:

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

Calculations:

⇒ SI = (3200 × 12 × 5) ÷ 100

⇒ SI = 32 × 12 × 5

⇒ SI = 32 × 60

⇒ SI = 1920

∴ The correct answer is ₹1920.

Additional Information

Total Amount in Simple Interest

The total amount (A) payable after the time period is the sum of the Principal and the Simple Interest: A = P + SI.

Compound Interest (CI)

If the interest was compounded annually, the amount would be calculated as A = P(1 + R/100)ⁿ, where n is the number of years.

Rate and Time Consistency

Always ensure the Rate (R) and Time (T) are in the same unit (e.g., if R is per annum, T must be in years).

Shortcut Trick

The surface area of a sphere is proportional to the square of its radius (r²).

For an increase in radius by x%, the net increase in surface area is calculated using the successive percentage formula: x + x + (x × x)/100.

Given x = 20%, the calculation is 20 + 20 + (20 × 20)/100.

Net increase = 40 + 4 = 44%.

∴ The correct answer is 44%.

Alternate Method

Given:

Initial radius increase = 20%

Formula Used:

Surface Area of a spherical balloon (S) = 4πr²

Calculations:

⇒ Let the initial radius (r₁) be 5 units.

⇒ New radius (r₂) = 5 + (20% of 5) = 5 + 1 = 6 units.

⇒ Initial Surface Area (S₁) = 4π(5)² = 100π (or simply consider ratio r² = 25).

⇒ New Surface Area (S₂) = 4π(6)² = 144π (or ratio r² = 36).

⇒ Increase in Surface Area = S₂ − S₁ = 144π − 100π = 44π.

⇒ Percentage Increase = (Increase / Original) × 100

⇒ Percentage Increase = (44π / 100π) × 100 = 44%.

∴ The correct answer is 44%.

Additional Information

Volume Change of a Sphere

The volume of a sphere is 4/3πr³. If the radius increases by 20%, the volume increases by (1.2)³ − 1 = 1.728 − 1 = 72.8%.

Successive Percentage Rule

For any product based relation Z = A × B, if A changes by a% and B by b%, the net change in Z is [a + b + (ab/100)]%.

Hemisphere Properties

For a hemisphere, Curved Surface Area = 2πr² and Total Surface Area = 3πr². The percentage change logic for radius remains the same as a sphere.

Shortcut Trick

Identify the bases: a = 22, b = −6, and c = −16.

Check the sum of bases: 22 + (−6) + (−16) = 22 − 22 = 0.

Apply the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculate the result: 3 × 22 × (−6) × (−16) = 66 × 96 = 6336.

∴ The correct answer is 6336.

Alternate Method

Given: 22³ + (−6)³ + (−16)³

Formula Used: a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

⇒ Let a = 22, b = −6, and c = −16.

⇒ First, evaluate the sum (a + b + c) = 22 + (−6) + (−16) = 0.

⇒ Substituting (a + b + c) = 0 into the identity: a³ + b³ + c³ − 3abc = 0 × (a² + b² + c² − ab − bc − ca).

⇒ This simplifies to a³ + b³ + c³ = 3abc.

⇒ Required Value = 3 × 22 × (−6) × (−16).

⇒ Required Value = 66 × 96 = 6336.

∴ The correct answer is 6336.

Additional Information

Special Case of Cubic Identity

When the sum of three terms is zero (a + b + c = 0), the sum of their cubes is always equal to three times their product (3abc). This is frequently used in competitive exams to avoid long cubic calculations.

Sum of Squares Transformation

The term (a² + b² + c² − ab − bc − ca) can also be written as 1/2[(a − b)² + (b − c)² + (c − a)²], which is always non-negative for real numbers.

Common Cubic Values

Memorizing cubes up to 25 is helpful: 22³ = 10648, 16³ = 4096, and 6³ = 216.

Shortcut Trick

To find the greatest number leaving the same remainder, calculate the HCF of the differences.

Difference 1 = 321 − 186 = 135

Difference 2 = 402 − 321 = 81

The HCF of 135 and 81 is 27 since 135 = 27 × 5 and 81 = 27 × 3.

∴ The correct answer is 27.

Alternate Method

Given: The numbers are 186, 321, and 402.

Formula Used: The greatest number that divides x, y, and z leaving the same remainder in each case is HCF(|x − y|, |y − z|, |z − x|).

⇒ Difference (d₁) = 321 − 186 = 135

⇒ Difference (d₂) = 402 − 321 = 81

⇒ Difference (d₃) = 402 − 186 = 216

⇒ Prime factorization of 135 = 3 × 3 × 3 × 5 = 3³ × 5

⇒ Prime factorization of 81 = 3 × 3 × 3 × 3 = 3⁴

⇒ Prime factorization of 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 3³

⇒ Highest Common Factor (HCF) = 3³ = 27

∴ The correct answer is 27.

Additional Information

HCF with Given Remainders

To find the greatest number that divides x, y, and z leaving remainders a, b, and c respectively, find the HCF of (x−a), (y−b), and (z−c).

LCM with Same Remainder

The smallest number which when divided by x, y, and z leaves the same remainder r in each case is given by [LCM(x, y, z) + r].

Property of HCF

The HCF of any two or more numbers must also divide the difference between those numbers.

Shortcut Trick

When a single discount (D) equals the sum of two successive discounts (x + y), the difference in discount amount is always (x × y / 100)% of the Marked Price.

Here, D = 30%, x = 25%, and y = 5%. Note that 30 = 25 + 5.

Difference % = (25 × 5) / 100 = 1.25%.

Difference amount = 1.25% of ₹800 = 1.25 × 8 = ₹10.

∴ The correct answer is ₹10.

Alternate Method

Given:

Marked Price (MP) = ₹800

Offer 1: Single Discount = 30%

Offer 2: Successive Discounts = 25% and 5%

Formula Used:

Total Discount = MP × Discount%

Effective Discount for x% and y% successive = x + y − (xy / 100)

Calculations:

⇒ Offer 1 Discount amount = 30% of 800 = (30/100) × 800 = ₹240

⇒ Offer 2 Effective Discount% = 25 + 5 − (25 × 5 / 100)

⇒ Offer 2 Effective Discount% = 30 − 1.25 = 28.75%

⇒ Offer 2 Discount amount = 28.75% of 800 = (28.75 / 100) × 800 = 28.75 × 8 = ₹230

⇒ Difference in Discount amount = 240 − 230 = ₹10

∴ The correct answer is ₹10.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the net discount is (a + b − ab/100)%. For three discounts a, b, and c, apply the formula twice.

Marked Price and Selling Price

Selling Price = Marked Price × (100 − Discount%)/100. The discount is always calculated on the Marked Price (MP) unless specified otherwise.

Single vs Successive Discount

A single discount of (x + y)% is always greater than two successive discounts of x% and y% because in the successive case, the second discount is applied to a reduced amount.

Shortcut Trick

Simplify the ratio of the numbers: 6 : 4 → 3 : 2 (co-prime terms).

For co-prime ratio parts 3 and 2, and HCF = 49:

LCM = Product of ratio terms × HCF

LCM = 3 × 2 × 49 = 6 × 49 = 294.

∴ The correct answer is 294.

Alternate Method

Given:

Ratio of the two numbers = 6 : 4

HCF of the numbers = 49

Formula used:

If two numbers are in the ratio a : b, then the numbers can be written as ax and bx, where x is the HCF of the numbers.

LCM of ax and bx = a × b × x

Shortcut Trick

Total cost of Notebooks and Pencils = (20 × 50) + (50 × 10) = 1000 + 500 = ₹1500.

Remaining amount for Pens = 3500 − 1500 = ₹2000.

Average cost of one Pen = Total Pen cost ÷ Number of Pens = 2000 ÷ 100 = ₹20.

∴ The correct answer is ₹20.

Alternate Method

Given:

Total quantity: 20 Notebooks, 100 Pens, and 50 Pencils.

Total Expenditure = ₹3,500.

Average cost of Notebook = ₹50 | Average cost of Pencil = ₹10.

Calculations:

⇒ Total cost of 20 Notebooks = 20 × 50 = ₹1,000

⇒ Total cost of 50 Pencils = 50 × 10 = ₹500

⇒ Combined cost of Notebooks and Pencils = 1000 + 500 = ₹1,500

⇒ Total cost for 100 Pens = Total Expenditure − Combined cost

⇒ Total cost for 100 Pens = 3500 − 1500 = ₹2,000

⇒ Average cost of a Pen = Total cost of Pens ÷ Total number of Pens

⇒ Average cost of a Pen = 2000 ÷ 100 = ₹20

∴ The correct answer is ₹20.

Additional Information

Fundamental Average Formula

The Average of a set of values is the sum of the values divided by the number of values: Average = Total Sum ÷ Number of Items.

Total Value Derivation

To find the Total Value when the average is known, use the formula: Total Value = Average × Number of Items.

Weighted Average Concept

When multiple groups are combined, the Overall Average is calculated as: (n₁×a₁ + n₂×a₂ + ...) ÷ (n₁ + n₂ + ...).

Shortcut Trick

Identify the triangle type: Since 5² + 12² = 13² (25 + 144 = 169), it is a right-angled triangle.

In a right-angled triangle, the perpendicular from the right-angle vertex to the hypotenuse (h) is given by:

h = (Product of legs) ÷ Hypotenuse

h = (5 × 12) ÷ 13 = 60 ÷ 13 ≈ 4.615...

∴ The correct answer is 4.62 cm.

Alternate Method

Given: Sides of the triangle are 5 cm, 12 cm, and 13 cm.

Formula: Area of triangle = 1/2 × base × height

⇒ First, calculate Area using the two legs: Area = 1/2 × 5 × 12

⇒ Area = 30 cm²

⇒ Now, consider the longest side (13 cm) as the base and 'h' as the perpendicular height.

⇒ Area = 1/2 × 13 × h

⇒ 30 = 1/2 × 13 × h

⇒ 60 = 13 × h

⇒ h = 60 ÷ 13 ≈ 4.6153... cm

⇒ Rounding to two decimal places, h = 4.62 cm.

∴ The correct answer is 4.62 cm.

Additional Information

Pythagorean Triplet

A set of three positive integers a, b, and c such that a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Altitude of a Right Triangle

The altitude (h) to the hypotenuse (c) of a right triangle with legs a and b is given by h = (a × b) ÷ c.

Right Triangle Area

Area can be calculated either as (1/2 × leg₁ × leg₂) or (1/2 × hypotenuse × altitude to hypotenuse).

Shortcut Trick

Change in students = 40 − 25 = 15; Change in bill = 25,000 − 17,500 = ₹7,500.

Variable charge per student = 7,500 ÷ 15 = ₹500.

To find the bill for 60 students, add the cost of 20 more students to the 40-student bill.

Total bill for 60 students = 25,000 + (20 × 500) = 25,000 + 10,000 = ₹35,000.

∴ The correct answer is ₹35,000.

Alternate Method

Given: Total bill for 25 students = ₹17,500 and for 40 students = ₹25,000.

Formula Used: Total Charge = Fixed Part (F) + (Number of Students (n) × Variable Part per student (v))

⇒ Let Fixed Charge be F and Variable Charge per student be v.

⇒ F + 25v = 17,500 ... (i)

⇒ F + 40v = 25,000 ... (ii)

⇒ Subtracting (i) from (ii): (40v − 25v) = 25,000 − 17,500

⇒ 15v = 7,500 ⇒ v = 500

⇒ Substituting v in (i): F + 25(500) = 17,500 ⇒ F + 12,500 = 17,500

⇒ Fixed Charge (F) = 17,500 − 12,500 = ₹5,000

⇒ Total bill for 60 students = F + 60v = 5,000 + (60 × 500)

⇒ Total bill = 5,000 + 30,000 = ₹35,000

∴ The correct answer is ₹35,000.

Additional Information

Linear Variation

This problem follows a linear equation pattern y = mx + c, where 'c' is the fixed cost and 'm' is the variable rate per unit.

Fixed and Variable Costs

Fixed costs remain constant regardless of student count, while variable costs increase proportionally with the number of students.

System of Linear Equations

Problems involving two unknowns (Fixed and Variable) are typically solved using the Elimination method or Substitution method.

Shortcut Trick

Profit = 575 − 500 = 75

Profit % = (Profit ÷ CP) × 100

Profit % = (75 ÷ 500) × 100 = 15%

∴ The correct answer is 15%.

Alternate Method

Given: Cost Price (CP) of the article = ₹500 and Selling Price (SP) = ₹575.

Formula used: Profit Percentage = [(Selling Price − Cost Price) ÷ Cost Price] × 100

⇒ Profit = Selling Price − Cost Price

⇒ Profit = 575 − 500 = ₹75

⇒ Profit Percentage = (75 ÷ 500) × 100

⇒ Profit Percentage = 15%

∴ The correct answer is 15%.

Additional Information

Loss Percentage

If Selling Price is less than Cost Price, then Loss % = [(CP − SP) ÷ CP] × 100.

Selling Price Calculation

To find Selling Price when Profit % is given: SP = CP × (100 + Profit %) ÷ 100.

Cost Price Calculation

To find Cost Price when Profit % is given: CP = [SP × 100] ÷ (100 + Profit %).