Search Previous Year Questions

Shortcut Trick

The percentage area of a ring relative to the full circle is independent of π.

Required Percentage = [(R² − r²) ÷ R²] × 100

⇒ Required Percentage = [(10² − 6²) ÷ 10²] × 100

⇒ Required Percentage = [(100 − 36) ÷ 100] × 100 = 64%

∴ The correct answer is 64%.

Alternate Method

r = 6 cm

R = 10 cm Metallic Ring Area

Given:

Outer radius of the metallic sheet (R) = 10 cm

Inner radius of the metallic sheet (r) = 6 cm

Formula Used:

Area of a circle = π × R²

Area of the ring = Area of outer circle − Area of inner circle

Calculation:

⇒ Area of the full outer circle = π × 10² = 100π cm²

⇒ Area of the inner empty circle = π × 6² = 36π cm²

⇒ Area of the ring = 100π − 36π = 64π cm²

⇒ Percentage of the full circle's area = (Area of ring ÷ Area of full circle) × 100

⇒ Percentage = (64π ÷ 100π) × 100

⇒ Percentage = 64%

∴ The correct answer is 64%.

Additional Information

Concentric Circles

Two or more circles that share the exact same center point but have different radii are called concentric circles.

Area of an Annulus (Ring)

The flat shape bounded by two concentric circles is known as an annulus. Its area is calculated using the formula π(R² − r²).

Width of the Ring

The uniform width or thickness of any circular ring is strictly the difference between the outer radius and the inner radius: Width = R − r.

Shortcut Trick

The area of the pond is directly given as 452.16 m².

Area of a circular pond = π × r²

⇒ 3.14 × r² = 452.16

⇒ r² = 452.16 ÷ 3.14 = 144

⇒ r = √144 = 12 m

∴ The correct answer is 12 m.

Alternate Method

Given:

Area of the pond = 452.16 m²

Total area of pond and path = 616 m²

Width of the path = 2 m

Formula Used:

Area of a circle = π × radius²

     r   2 m   R  Pond Area = 452.16 m² Total Area = 616 m²

Calculation:

Method 1: Using Pond Area

⇒ Area of Pond = π × r²

⇒ 3.14 × r² = 452.16

⇒ r² = 452.16 ÷ 3.14

⇒ r² = 144

⇒ r = √144 = 12 m

Method 2: Using Total Area

Let the outer radius be R. (where R = r + 2)

⇒ Total Area = π × R² = 616

⇒ (22 ÷ 7) × R² = 616

⇒ R² = (616 × 7) ÷ 22

⇒ R² = 28 × 7 = 196

⇒ R = √196 = 14 m

Since R = r + 2,

⇒ r = R − 2 = 14 − 2 = 12 m

Both calculations confirm the radius of the pond.

∴ The correct answer is 12 m.

Additional Information

Area of a Circular Path

The area of a circular ring or path is calculated as π × (R² − r²), where R is the outer radius and r is the inner radius.

Circumference of a Circle

The total length of the outer boundary of a given circle is computed using the formula C = 2 × π × r.

Area of a Sector

The area of a circle's sector with a central angle θ (in degrees) is derived by the formula (θ ÷ 360) × π × r².

Shortcut Trick

The ratio of the circumferences of two circles is directly equal to the ratio of their radii.

Ratio of radii = 3 : 5

3 parts = 9 m (radius of the smaller flower bed)

1 part = 9 ÷ 3 = 3 m

5 parts = 5 × 3 = 15 m (radius of the larger flower bed)

∴ The correct answer is 15 m.

Alternate Method

Given:

Ratio of circumferences of two circular flower beds = 3 : 5

Radius of the smaller flower bed (r₁) = 9 m

Formula Used:

Circumference of a circle (C) = 2πr

Ratio of Circumferences = 3 : 5

r1 = 9 m

r2 = ?

Calculations:

⇒ Let the circumferences be C₁ and C₂, and their respective radii be r₁ and r₂.

⇒ The ratio of their circumferences is given as:

⇒ C₁ / C₂ = 2πr₁ / 2πr₂ = r₁ / r₂

⇒ Therefore, r₁ / r₂ = 3 / 5

⇒ Substituting r₁ = 9 m in the equation:

⇒ 9 / r₂ = 3 / 5

⇒ 3 × r₂ = 9 × 5

⇒ 3 × r₂ = 45

⇒ r₂ = 45 / 3

⇒ r₂ = 15 m

∴ The correct answer is 15 m.

Additional Information

Relation between Circumference and Radius

Since the circumference formula is C = 2πr, the value of C is directly proportional to the radius. Therefore, the ratio of circumferences of any two circles is always equal to the ratio of their radii (C₁ : C₂ = r₁ : r₂).

Relation between Areas

The area of a circle is calculated using A = πr². The ratio of the areas of two circles is the square of the ratio of their radii (A₁ : A₂ = r₁² : r₂²).

Relation with Diameter

Since the diameter is twice the radius (D = 2r), the ratio of the diameters of two circles is also identical to the ratio of their radii and circumferences (D₁ : D₂ = r₁ : r₂).

Shortcut Trick

The line passes through the y-axis at (0, 3), which means its y-intercept (c) is 3.

The given slope (m) is −1.

Using the slope-intercept form equation: y = mx + c

Substituting the given values: y = (−1)x + 3 ⇒ y = −x + 3

∴ The correct answer is y = −x + 3.

Alternate Method

Given:

Point on the line (x₁, y₁) = (0, 3)

Slope of the line (m) = −1

Formula Used:

The point-slope form of a linear equation is: y − y₁ = m(x − x₁)

X-axisY-axisO(0, 3)(3, 0)y = −x + 3

Calculations:

⇒ y − y₁ = m(x − x₁)

⇒ y − 3 = −1 × (x − 0)

⇒ y − 3 = −x

⇒ y = −x + 3

∴ The correct answer is y = −x + 3.

Additional Information

Slope-Intercept Form

The equation of a straight line is represented as y = mx + c, where m is the slope of the line and c is the y-intercept.

Intercept Form

When a line makes intercepts a and b on the x and y axes respectively, the equation is given by x/a + y/b = 1.

Two-Point Form

When two distinct points (x₁, y₁) and (x₂, y₂) are known, the equation is given by y − y₁ = [(y₂ − y₁) ÷ (x₂ − x₁)] × (x − x₁).

Shortcut Trick

If x + 1/x = k, then x² + 1/x² = k² − 2.

Here, k = 5.

x² + 1/x² = 5² − 2 = 25 − 2 = 23

∴ The correct answer is 23.

Alternate Method

Given:

x + 1/x = 5

Formula Used:

(a + b)² = a² + b² + 2ab

Calculations:

⇒ x + 1/x = 5

Squaring both sides, we get:

⇒ (x + 1/x)² = 5²

⇒ x² + 1/x² + 2 × x × (1/x) = 25

⇒ x² + 1/x² + 2 = 25

⇒ x² + 1/x² = 25 − 2

⇒ x² + 1/x² = 23

∴ The correct answer is 23.

Additional Information

Minus Identity Shortcut

If x − 1/x = k, then x² + 1/x² = k² + 2.

Cubic Identity (Plus)

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Cubic Identity (Minus)

If x − 1/x = k, then x³ − 1/x³ = k³ + 3k.

Power of 4 Identity

If x + 1/x = k, then x⁴ + 1/x⁴ = (k² − 2)² − 2.

Shortcut Trick

Let the value of A be 100.

Since B is 25% more than A, B = 100 + 25 = 125.

Since C is 20% more than B, C = 125 + (20 / 100) × 125 = 125 + 25 = 150.

The ratio of A : C = 100 : 150 = 2 : 3.

∴ The correct answer is 2 : 3.

Alternate Method

Given:

B is 25% more than A.

C is 20% more than B.

Method Used:

Fractional representation of percentage increase.

A 100

+25%

B 125

+20%

C 150

A : C = 100 : 150 = 2 : 3

Calculations:

⇒ 25% expressed as a fraction = 1 / 4

⇒ B = A + (1 / 4) × A = 5 / 4 × A

⇒ Ratio of A / B = 4 / 5

⇒ 20% expressed as a fraction = 1 / 5

⇒ C = B + (1 / 5) × B = 6 / 5 × B

⇒ Ratio of B / C = 5 / 6

⇒ To find the overall ratio A : C, multiply the individual ratios:

⇒ A / C = (A / B) × (B / C)

⇒ A / C = (4 / 5) × (5 / 6) = 20 / 30 = 2 / 3

∴ The correct answer is 2 : 3.

Additional Information

Percentage to Fraction Conversion

Converting percentages directly to simplified fractions helps in speeding up ratio calculations. Always memorize common fractions like 25% = 1/4 and 20% = 1/5.

Compound Ratio

The compound ratio of two sequential ratios a:b and b:c is calculated by multiplying them: (a/b) × (b/c) = a/c. This provides the direct relationship between the first and last terms.

Successive Percentage Change

When a value undergoes multiple changes sequentially, you can multiply the base with successive scaling factors. For example: Final = Base × (1 + %1) × (1 + %2).

Shortcut Trick

Convert the decimal rate to a fraction for faster calculation:

0.375 L = 3/8 L

Total volume filled in 8 seconds = (3/8) × 8

Total amount = 3 L

∴ The correct answer is 3.0 L.

Alternate Method

Given:

Rate at which the machine fills water = 0.375 L/s

Total time = 8 seconds

Formula Used:

Total Volume = Rate × Time

Calculations:

⇒ Total Volume = 0.375 × 8

⇒ Total Volume = (375/1000) × 8

⇒ Total Volume = 3000/1000

⇒ Total Volume = 3.0 L

∴ The correct answer is 3.0 L.

Additional Information

Fraction to Decimal Equivalents

Memorising standard fractional values helps speed up calculations significantly. Common equivalents include: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, and 7/8 = 0.875.

Unitary Method

The unitary method is a technique for solving arithmetic problems by first finding the value of a single unit (1 second in this case), and then finding the required value by multiplying the single unit value.

Work and Time Relationship

The total amount of work done or quantity processed is consistently given by the product of the rate of work and the time taken (Quantity = Rate × Time).

Shortcut Trick

Given ratio of lengths = 2/5 : 3/10 : 1/2

Multiply by LCM of denominators (10) to get integer ratio → 4 : 3 : 5

Total parts = 4 + 3 + 5 = 12 parts

Smallest piece represents 3 parts out of total 12 parts.

Length of smallest piece = (3 ÷ 12) × 6 = 1.5 m

∴ The correct answer is 1.5 m.

Alternate Method

Given:

Total length of rod = 6 m

Ratio of division = 2/5 : 3/10 : 1/2

Total Length = 6 m (12 parts)

4 parts 3 parts 5 parts

Smallest Piece = ?

Calculations:

⇒ To convert fractions into simplest integer ratio, find the LCM of denominators 5, 10, and 2.

⇒ LCM of 5, 10, 2 = 10

⇒ Multiply each fraction by 10:

⇒ First piece = 2/5 × 10 = 4 parts

⇒ Second piece = 3/10 × 10 = 3 parts

⇒ Third piece = 1/2 × 10 = 5 parts

⇒ Simplified length ratio = 4 : 3 : 5

⇒ Total ratio parts = 4 + 3 + 5 = 12 parts

⇒ Total length equivalent to 12 parts = 6 m

⇒ Value of 1 part = 6 ÷ 12 = 0.5 m

⇒ The smallest piece corresponds to the smallest ratio value, which is 3 parts.

⇒ Length of smallest piece = 3 × 0.5 m = 1.5 m

∴ The correct answer is 1.5 m.

Additional Information

Simplifying Fractional Ratios

Always multiply each term of a fractional ratio by the Least Common Multiple (LCM) of their denominators to easily convert them into standard integers.

Dividing Quantity in a Given Ratio

If a total quantity T is divided in the ratio a : b : c, the individual value of the first segment is calculated as (a ÷ (a + b + c)) × T.

Inverse Ratio Property

The inverse ratio of a : b is given as 1/a : 1/b, which simplifies mathematically to b : a. For three continuous terms a : b : c, the respective inverse ratio is expressed as 1/a : 1/b : 1/c.

Shortcut TrickCost Price per kg of mixture = (120 × 1 + 180 × 2) ÷ (1 + 2)⇒ CP = (120 + 360) ÷ 3 = 480 ÷ 3 = ₹160Given Selling Price (SP) = ₹152Loss = CP − SP = 160 − 152 = ₹8Loss % = (8 ÷ 160) × 100 = 5%∴ The correct answer is 5% Loss.Alternate MethodGiven:Cost of 1^st type of coffee = ₹120 per kgCost of 2^nd type of coffee = ₹180 per kgRatio of mixing = 1 : 2Selling Price (SP) of mixture = ₹152 per kgFormula Used:Total Cost Price (CP) = (Q1 × P1) + (Q2 × P2)Loss % = (Loss ÷ CP) × 100₹ 120Type 1 Cost₹ 180Type 2 Cost1Ratio Part2Ratio PartCP = ₹ 160SP = ₹ 152 → Loss = ₹ 8 (5%)Calculations:Let the quantities mixed be 1 kg and 2 kg respectively.⇒ Total CP of 3 kg mixture = (1 × 120) + (2 × 180)⇒ Total CP = 120 + 360 = ₹480⇒ CP per kg = 480 ÷ 3 = ₹160⇒ SP per kg is given as ₹152.⇒ Since SP < CP, there is a loss in the transaction.⇒ Loss per kg = 160 − 152 = ₹8⇒ Loss % = (8 ÷ 160) × 100⇒ Loss % = (1 ÷ 20) × 100 = 5%∴ The correct answer is 5% Loss.Additional InformationAlligation RuleWhen two ingredients of different prices are mixed, the ratio of their quantities is determined using: (Cheaper Qty) ÷ (Dearer Qty) = (Cost of Dearer − Mean Price) ÷ (Mean Price − Cost of Cheaper).Profit and LossProfit occurs when Selling Price > Cost Price (Profit % = Profit ÷ CP × 100). Loss occurs when Cost Price > Selling Price (Loss % = Loss ÷ CP × 100).

Shortcut Trick

Profit ratio = Capital ratio (since time is identical for all partners).

Ratio of A : B : C = 40,000 : 60,000 : 1,00,000 = 2 : 3 : 5.

Total profit ratio parts = 2 + 3 + 5 = 10 parts.

10 parts = ₹40,000 ⇒ 1 part = ₹4,000.

A's share = 2 parts = 2 × ₹4,000 = ₹8,000.

∴ The correct answer is ₹8,000.

Alternate Method

Investment & Profit Ratio

₹40k Ratio: 2 A

₹60k Ratio: 3 B

₹100k Ratio: 5 C

Total Profit = ₹40,000 (10 Ratio Units)

Given:

Investment of A = ₹40,000

Investment of B = ₹60,000

Investment of C = ₹1,00,000

Total Profit at the end of the year = ₹40,000

Formula Used:

Profit Ratio = (Capital of A × Time) : (Capital of B × Time) : (Capital of C × Time)

Calculations:

⇒ Since the time period is 1 year for all, the profit ratio relies entirely on their initial investments.

⇒ Profit Ratio of A, B, and C = 40,000 : 60,000 : 1,00,000

⇒ Dividing each term by 20,000, we get the simplified ratio = 2 : 3 : 5

⇒ Sum of profit ratios = 2 + 3 + 5 = 10 units

⇒ Total profit representing 10 units = ₹40,000

⇒ 1 unit of profit = 40,000 ÷ 10 = ₹4,000

⇒ A's share of profit is 2 units.

⇒ A's share = 2 × 4,000 = ₹8,000

∴ The correct answer is ₹8,000.

Additional Information

Simple Partnership

When all partners invest their capital for the exact same time period, their profits or losses are distributed directly in the ratio of their invested amounts.

Compound Partnership

When partners invest their capital for different time periods, the profit ratio is determined by the product of their respective capital and the total time it was invested (Profit = Capital × Time).

Sleeping Partner

A sleeping partner provides capital but does not actively participate in running the business. Their share of the profit is generated purely based on the financial investment ratio.

Working Partner

An active partner who manages the daily operations of the business. They often receive a fixed salary or a specific percentage of the total profit for their management before the rest of the profits are divided.