Quantitative Aptitude PYQs

Shortcut Trick

Memorize common percentage to fraction conversions for faster calculation.

12 ¹/₂% = 1/8 and 66 ²/₃% = 2/3

⇒ Expression = (1/8) × 560 + (2/3) × 240

⇒ Expression = 70 + 160 = 230

∴ The correct answer is 230 km.

Alternate Method

Given:

12 ¹/₂% of 560 km + 66 ²/₃% of 240 km

Calculation:

⇒ 12 ¹/₂% = 25/2 % = (25/2) × (1/100) = 25/200 = 1/8

⇒ 66 ²/₃% = 200/3 % = (200/3) × (1/100) = 200/300 = 2/3

Substituting the fractional values into the original expression:

⇒ (1/8 × 560) + (2/3 × 240)

⇒ 70 + 160

⇒ 230

∴ The correct answer is 230 km.

Additional Information

Common Percentage-Fraction Equivalents

Memorizing standard equivalents saves calculation time: 16 ²/₃% = 1/6, 14 ²/₇% = 1/7, 33 ¹/₃% = 1/3, and 37 ¹/₂% = 3/8.

BODMAS Rule

Always follow the order of operations: Brackets, Orders, Division/Multiplication (left to right), Addition/Subtraction (left to right). The word 'of' indicates multiplication and is solved before division.

Percentage Reversibility

X% of Y is always equal to Y% of X. This property can be used interchangeably to make calculations easier when numbers are reversed.

Shortcut Trick

Daily commission from shirts = 3 × (2% of 500) = 3 × 10 = ₹30

Daily commission from ties = 4 × (12% of 150) = 4 × 18 = ₹72

Total daily commission = 30 + 72 = ₹102

Total commission earned in 30 days = 102 × 30 = ₹3060

∴ The correct answer is ₹3060.

Alternate Method

Given:

Price of a shirt = ₹500, Commission on a shirt = 2%

Price of a tie = ₹150, Commission on a tie = 12%

Items sold per day = 3 shirts and 4 ties

Total days = 30

Shirts Price: ₹500 | Comm: 2% Comm/item: ₹10 Daily (x3): ₹30

Ties Price: ₹150 | Comm: 12% Comm/item: ₹18 Daily (x4): ₹72

Total in 30 Days = ₹3060

Calculations:

⇒ Commission earned on one shirt = 2% of 500 = (2 ÷ 100) × 500 = ₹10

⇒ Commission earned on 3 shirts per day = 3 × 10 = ₹30

⇒ Commission earned on one tie = 12% of 150 = (12 ÷ 100) × 150 = ₹18

⇒ Commission earned on 4 ties per day = 4 × 18 = ₹72

⇒ Total commission earned per day = 30 + 72 = ₹102

⇒ Total commission earned in 30 days = 102 × 30 = ₹3060

∴ The correct answer is ₹3060.

Additional Information

Percentage Application

Percentage implies 'out of 100'. Thus, x% of y is calculated as (x ÷ 100) × y. In practical applications like commissions, discounts, or taxes, the percentage is applied directly to the base value.

Commission and Base Amount

Commission is generally paid as a percentage of the total sales value. If multiple items are sold with varying commission rates, they must be calculated individually before summation.

Unitary Method

Finding the value for a single unit and then multiplying it by the total number of units required is called the unitary method. It is highly effective for scaling daily quantities to monthly or annual values.

Shortcut Trick

Let the Cost Price (CP) of 100 units of apples be ₹100.

Selling Price (SP) at a claimed 5% loss = ₹95.

Actual weight used is 20% less, so the seller gives only 80 units (True CP = ₹80).

Profit earned by the seller = SP − CP = 95 − 80 = ₹15.

Total Profit % = (15 ÷ 80) × 100 = 18.75%.

∴ The correct answer is 18.75%.

Alternate Method

Given:

Loss percentage on Cost Price = 5%

False weight used = 20% less than actual weight

Formula Used:

Profit % = (Profit ÷ Cost Price) × 100

Actual CP800 gm = ₹800Selling Price1000 gm claimed = ₹950Profit = ₹150Profit % = (150 ÷ 800) × 100 = 18.75%

Calculations:

⇒ Let the cost price of 1000 gm of apples be ₹1000.

⇒ Selling price at a 5% loss = 1000 − (5% of 1000) = 1000 − 50 = ₹950.

⇒ The seller uses a false weight that is 20% less than the actual weight.

⇒ Actual weight given by the seller = 1000 − (20% of 1000) = 1000 − 200 = 800 gm.

⇒ Actual cost price incurred by the seller (for 800 gm) = ₹800.

⇒ Profit earned = Selling Price − Actual Cost Price

⇒ Profit = 950 − 800 = ₹150.

⇒ Total Profit % = (Profit ÷ Cost Price) × 100

⇒ Total Profit % = (150 ÷ 800) × 100 = 18.75%.

∴ The correct answer is 18.75%.

Additional Information

Dishonest Dealer (Selling at Cost Price)

If a trader professes to sell his goods at cost price but uses false weights, then his Profit % = [Error ÷ (True Value − Error)] × 100.

Successive Percentage Method

The overall profit can be evaluated using successive percentages. Giving 20% less weight means a gain of (20 ÷ 80) × 100 = 25%. Combined with a 5% loss on price, the Net % = 25 − 5 − (25 × 5)÷100 = 20 − 1.25 = 18.75%.

Ratio Method for Dishonest Dealer

Write the price ratio and quantity ratio separately. Price ratio (SP:CP) = 95:100. Quantity given ratio (SP:CP) = 100:80. Net Ratio = (95 × 100) : (100 × 80) = 95 : 80, which results in a profit of 15 on a base of 80.

Shortcut Trick

If the profit at one selling price equals the loss at another, the Cost Price is exactly halfway between them.

CP = (SP₁ + SP₂) ÷ 2

CP = (2200 + 1800) ÷ 2 = ₹2000

New SP = ₹2050 ⇒ Profit = 2050 − 2000 = ₹50

Profit % = (50 ÷ 2000) × 100 = 2.5%

∴ The correct answer is Profit of 2.5%.

Alternate Method

Given:

Selling Price 1 (SP₁) = ₹2200

Selling Price 2 (SP₂) = ₹1800

Profit at SP₁ = Loss at SP₂

Formula Used:

Profit = SP − CP

Loss = CP − SP

Profit % = (Profit ÷ CP) × 100

₹1800 CP = ₹2000 ₹2200

New SP = ₹2050

Loss (L)

Profit (P = L)

Profit = ₹50

Calculations:

Let the cost price of the item be CP.

⇒ Profit at SP₁ = Loss at SP₂

⇒ 2200 − CP = CP − 1800

⇒ 2CP = 2200 + 1800

⇒ 2CP = 4000

⇒ CP = ₹2000

Now, the item is sold for a new SP = ₹2050.

⇒ Profit = New SP − CP = 2050 − 2000 = ₹50

⇒ Profit % = (50 ÷ 2000) × 100

⇒ Profit % = 2.5%

∴ The correct answer is Profit of 2.5%.

Additional Information

Marked Price and Successive Discount

Successive discounts of a% and b% are equivalent to a single net discount of (a + b − (a × b) ÷ 100)%.

Relation between CP and MP

The exact ratio of Cost Price to Marked Price is directly proportional to the discount and profit given: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Dishonest Dealer

If a dealer claims to sell goods at cost price but uses a false weight, the actual profit percentage is (Error ÷ False Weight) × 100.

Shortcut Trick

Cost Price (CP) to Selling Price (SP1) ratio at 20% profit = 100 : 120 = 5 : 6.

Assuming CP = 5 units, then initial SP1 = P = 6 units.

Given the festive Marked Price (MP) = 1.5P ⇒ MP = 1.5 × 6 = 9 units.

A discount of 10% is given, so New SP = 9 × 0.9 = 8.1 units.

Profit % = [(New SP − CP) ÷ CP] × 100 = [(8.1 − 5) ÷ 5] × 100 = 62%.

∴ The correct answer is 62% (or 0.62).

Alternate Method

Given:

Initial Selling Price = ₹P

Initial Profit = 20%

New Marked Price (MP) = 1.5P

Festive Discount = 10%

Formula Used:

CP = SP × [100 ÷ (100 + Profit%)]

New SP = MP × [100 − Discount%] ÷ 100

Profit % = [(New SP − CP) ÷ CP] × 100

CP 5P/6 SP1 P SP2 1.35P MP 1.5P +20% -10% New Profit = 62%

Calculations:

⇒ CP = P × (100 ÷ 120) = 5P/6

⇒ New SP = 1.5P × (90 ÷ 100) = 1.35P = 27P/20

⇒ Profit = New SP − CP = (27P/20) − (5P/6)

⇒ LCM of 20 and 6 is 60.

⇒ Profit = (81P − 50P) ÷ 60 = 31P/60

⇒ Profit % = [(31P/60) ÷ (5P/6)] × 100

⇒ Profit % = (31/60) × (6/5) × 100

⇒ Profit % = (31/50) × 100 = 62%

⇒ Expressed as a decimal fraction, it is 0.62.

∴ The correct answer is 62% (or 0.62).

Additional Information

Successive Percentage Change

When a value undergoes two successive changes of a% and b%, the net percentage change is given by: Net Change = a + b + (a × b) ÷ 100.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). The Selling Price is derived as: SP = MP × [100 − Discount%] ÷ 100.

Relationship between CP, MP, Profit and Discount

A direct relationship ratio is given by: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Using the effective successive percentage change formula:

Net Profit % = M − D − (M × D) / 100

Where Markup (M) = 25% and Discount (D) = 10%

⇒ Profit % = 25 − 10 − (25 × 10) / 100

⇒ Profit % = 15 − 2.5 = 12.5%

∴ The correct answer is 12.50%.

Alternate Method

Given:

Markup = 25% above Cost Price (CP)

Discount = 10% on Marked Price (MP)

CP = 100

SP = 112.5

MP = 125

+25% Markup

−10%

+12.5% Profit

Calculations:

⇒ Let the Cost Price (CP) = 100

⇒ Marked Price (MP) = CP + 25% of CP = 100 + 25 = 125

⇒ Discount = 10% of MP = (10 / 100) × 125 = 12.5

⇒ Selling Price (SP) = MP − Discount = 125 − 12.5 = 112.5

⇒ Profit = SP − CP = 112.5 − 100 = 12.5

⇒ Profit Percentage = (Profit / CP) × 100 = (12.5 / 100) × 100 = 12.5%

∴ The correct answer is 12.50%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). The formula is: Selling Price (SP) = MP − Discount.

Successive Percentage Rule

When a value is increased by A% and then decreased by B%, the effective change percentage is given by the formula: A − B − (A × B) / 100.

Profit & Loss Basics

Profit or Loss percentage is always calculated on the original Cost Price (CP) unless specifically stated otherwise.

Shortcut Trick

Let CP = 100%. Then Marked Price (MP) = 160% (as it is 60% above CP).

Selling Price (SP) after 25% discount = 160% − (25% of 160%) = 120%.

Actual Final SP = 100% + 14% profit = 114%.

Difference in SPs = 120% − 114% = 6%, which corresponds to the ₹150 cash discount.

⇒ 6% of CP = 150 ⇒ CP = 150 × 100 ÷ 6 = ₹2500.

∴ The correct answer is ₹2500.

Alternate Method

Given:

Markup = 60% above Cost Price

Discount = 25%

Additional Cash Discount = ₹150

Overall Profit = 14%

Formula Used:

Selling Price (SP) = Marked Price × (100 − Discount%) ÷ 100

Final SP = Cost Price × (100 + Profit%) ÷ 100

CP = 100%

+60%

MP = 160%

-25%

SP = 120%

+14% Profit

Final SP = 114%

• ₹150

120% - 114% = 6% → ₹150   ∴ 100% = ₹2500

Calculations:

⇒ Let the Cost Price (CP) be 100x.

⇒ Marked Price (MP) = 100x + (60% of 100x) = 160x.

⇒ Selling Price before cash discount = 160x − (25% of 160x) = 160x − 40x = 120x.

⇒ Final Selling Price = 120x − 150.

⇒ Based on the profit percentage, Final SP = CP + (14% of CP) = 114x.

⇒ Equating both values of Final SP:

⇒ 120x − 150 = 114x

⇒ 120x − 114x = 150

⇒ 6x = 150

⇒ x = 25

⇒ Cost Price = 100x = 100 × 25 = ₹2500.

∴ The correct answer is ₹2500.

Additional Information

Marked Price & Discount

Discount is exclusively given on the Marked Price (MP), never on Cost Price. It is simply a reduction in price directly impacting the Selling Price. Formula: SP = MP × (100 − Discount%) ÷ 100.

Direct Formula for CP and MP

When Profit% and Discount% are known, the direct ratio between Cost Price and Marked Price is efficiently calculated as CP / MP = (100 − Discount%) / (100 + Profit%).

Successive Discounts

When multiple flat discounts x% and y% are given sequentially, the single equivalent discount percentage is evaluated using: (x + y − x × y ÷ 100)%.

Shortcut Trick

First, find the Cost Price (CP) of the mixture from its Selling Price (SP) and Gain %.

CP of mixture = SP ÷ (1 + Gain%) = 43.70 ÷ 1.15 = ₹38/kg

Using Alligation, Ratio of quantities = (38 − 32) : (45 − 38) = 6 : 7

Since 7 parts correspond to 25 kg, 6 parts will correspond to: (25 ÷ 7) × 6 = 150 ÷ 7 ≈ 21.43 kg

∴ The correct answer is 21 kg.

Alternate Method

Given:

CP of first type of sugar = ₹45/kg

CP of second type of sugar = ₹32/kg

Quantity of second type of sugar = 25 kg

SP of mixture = ₹43.70/kg

Profit on mixture = 15%

Formula Used:

CP = SP × [100 ÷ (100 + Gain%)]

Sugar 1 (₹45)Sugar 2 (₹32)₹3838 − 32 = 645 − 38 = 7Quantity Ratio = 6 : 7

Calculations:

⇒ Mean CP of mixture = 43.70 × [100 ÷ (100 + 15)]

⇒ Mean CP = (43.70 × 100) ÷ 115

⇒ Mean CP = 4370 ÷ 115 = ₹38/kg

⇒ Using the rule of alligation (as shown in the diagram), the ratio of the quantities is:

⇒ (Mean CP − CP₂) : (CP₁ − Mean CP) = (38 − 32) : (45 − 38) = 6 : 7

⇒ Let the required quantity of the first type of sugar be x kg.

⇒ x ÷ 25 = 6 ÷ 7

⇒ x = (25 × 6) ÷ 7

⇒ x = 150 ÷ 7 ≈ 21.43 kg

⇒ Therefore, approximately 21 kg of the first type of sugar must be mixed.

∴ The correct answer is 21 kg.

Additional Information

Rule of Alligation

It is a fast cross-calculation method used to find the ratio in which two ingredients at given unit prices must be mixed to produce a mixture of a desired target price. The target price must always lie exactly between the two given prices.

Mean Price Requirement

The central value placed in the alligation cross is called the Mean Price. It must always be the Cost Price (CP) of the mixture. Never place Selling Price (SP) in the alligation diagram.

Cost Price from Selling Price

When Selling Price (SP) and Gain Percentage (P%) are given, Cost Price (CP) is computed backwards using the relation: CP = [100 ÷ (100 + P%)] × SP.

Shortcut Trick

In a right circular cone, the height, base radius, and slant height always form a right-angled triangle.

Using the common Pythagorean triplet (7, 24, 25), where the two legs are 7 and 24, the hypotenuse is directly 25.

Since the given radius is 7 cm and the height is 24 cm, the slant height is directly 25 cm.

∴ The correct answer is 25 cm.

Alternate Method

Given:

Radius of the base (r) = 7 cm

Height of the cone (h) = 24 cm

Formula Used:

Slant height (l) = √(r² + h²)

h = 24 cm r = 7 cm l = ?

Calculations:

⇒ l = √(7² + 24²)

⇒ l = √(49 + 576)

⇒ l = √(625)

⇒ l = 25 cm

∴ The correct answer is 25 cm.

Additional Information

Volume of a Cone

The volume is one-third of the volume of a cylinder with the same base and height: V = (1 ÷ 3) × π × r² × h.

Curved Surface Area (CSA)

The area of the curved surface alone is calculated using the slant height: CSA = π × r × l.

Total Surface Area (TSA)

The total surface area includes both the base area and the curved surface area: TSA = π × r × (r + l).

Shortcut Trick

Equivalent principal for 1 year = (5000 × 2) + (3000 × 4) = 10000 + 12000 = 22000

Total Simple Interest received = 2200

Rate = (Total Interest ÷ Equivalent Principal) × 100

Rate = (2200 ÷ 22000) × 100

Rate = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Principal lent to B = ₹5000, Time = 2 years

Principal lent to C = ₹3000, Time = 4 years

Total simple interest = ₹2200

Formula Used:

Simple Interest (SI) = Principal × Rate × Time

(where Rate is taken as a decimal fraction r)

Lent to B₹5000 × 2 yrsLent to C₹3000 × 4 yrsTotal SI = ₹2200

Calculations:

⇒ Interest from B = 5000 × r × 2 = 10000r

⇒ Interest from C = 3000 × r × 4 = 12000r

⇒ Total Interest = 10000r + 12000r

⇒ 22000r = 2200

⇒ r = (2200 ÷ 22000) × 100

⇒ r = 10%

∴ The correct answer is 0.1 per annum.

Additional Information

Simple Interest (SI) Formula

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

Decimal vs Percentage Rate

When solving equations, the percentage rate R can be expressed directly as a decimal r (where r = R ÷ 100) making the SI calculation simply P × r × T.

Equivalent Principal

When multiple investments are made at the same simple interest rate, the overall principal equivalent to a 1-year term is the sum of the products of each Principal and its respective Time period.

Shortcut Trick

Volume of any prism is directly proportional to its height when the base area is constant.

Original volume = 25 × 10 = 250 cm³

Since the height increases by 20%, the volume also increases by exactly 20%.

New volume = 250 × 1.2 = 300 cm³

∴ The correct answer is 300 cm³.

Alternate Method

Given:

Base area of the triangular prism = 25 cm²

Original height = 10 cm

Increase in height = 20%

Formula Used:

Volume of a prism = Base Area × Height

        Original h = 10 cm New h = 12 cm   Base Area = 25 cm² New Volume = 25 × 12 = 300 cm³

Calculations:

⇒ New height = 10 + (20% of 10)

⇒ New height = 10 + 2 = 12 cm

⇒ New volume = Base Area × New height

⇒ New volume = 25 × 12 = 300 cm³

∴ The correct answer is 300 cm³.

Additional Information

Volume of a Prism

The volume of any regular prism (triangular, rectangular, hexagonal, etc.) is always the product of its uniform base area and its vertical height: V = Base Area × h.

Proportionality in 3D Shapes

If only a single dimension (like height) of a 3D figure changes by x%, the total volume also undergoes an exact x% change, provided all other dimensions remain constant.

Total Surface Area of Prism

The Total Surface Area (TSA) is calculated as: TSA = Lateral Surface Area + (2 × Base Area) = (Perimeter of Base × Height) + (2 × Base Area).

Shortcut Trick

For a circular disc, Area ∝ (Radius)².

The net percentage change can be calculated using the successive change formula: x + y + (x × y) ÷ 100.

Since the radius is decreased by 10%, we substitute x = −10 and y = −10.

Net Change = −10 − 10 + [(−10) × (−10)] ÷ 100 = −20 + 1 = −19%.

The negative sign indicates a 19% decrease in the overall area.

∴ The correct answer is 19%.

Alternate Method

Given:

The radius of the circular disc is decreased by 10%.

Formula Used:

Area of a circle = π × r²

Percentage Decrease = [(Initial Area − Final Area) ÷ Initial Area] × 100

r Initial Area = 100%

− 10% Radius

0.9r Final Area = 81%

Net Decrease in Area = 19%

Calculation:

⇒ Let the original radius of the disc be 10 units.

⇒ Initial Area = π × 10² = 100π sq. units.

⇒ New radius = 10 − (10% of 10) = 10 − 1 = 9 units.

⇒ New Area = π × 9² = 81π sq. units.

⇒ Decrease in Area = 100π − 81π = 19π sq. units.

⇒ Percentage Decrease = (19π ÷ 100π) × 100 = 19%.

∴ The correct answer is 19%.

Additional Information

Successive Percentage Change

When a value is subjected to two successive percentage changes of x% and y%, the overall net change is calculated using the formula: Net Change = x + y + (xy ÷ 100). Use negative values for any decrease.

Volumetric Percentage Changes

For 3D shapes like spheres or cubes, if the radius or side changes by x%, the volume change is determined by applying the successive percentage formula twice consecutively to account for all three dimensions.

Constant Expenditure Rule

If the price of an item increases by R%, the consumption must be reduced by [R ÷ (100 + R)] × 100% to keep the total expenditure constant.

Shortcut Trick

For any given perimeter, the ratio of the area of a circle to a square is the constant 4 : π.

Ratio = 4 / π ≈ 4 ÷ 3.1416

Ratio ≈ 1.27 : 1

The total wire length of 100 m is not needed to find this ratio.

∴ The correct answer is 1.27:1.

Alternate Method

Given:

Total length of wire = Perimeter of circle = Perimeter of square = 100 m

Formula Used:

Circumference of circle = 2×π×r, Area of circle = π×r²

Perimeter of square = 4×a, Area of square = a²

r Circle Perimeter = 100 m

a Square Perimeter = 100 m

Calculations:

⇒ Circumference of the circle = 2×π×r = 100

⇒ Radius (r) = 50/π m

⇒ Area of circle = π×(50/π)² = 2500/π m²

⇒ Perimeter of the square = 4×a = 100

⇒ Side (a) = 25 m

⇒ Area of square = 25² = 625 m²

⇒ Ratio of areas = (2500/π) ÷ 625

⇒ Ratio = 4 / π

⇒ Ratio ≈ 4 ÷ 3.1416 = 1.2732...

⇒ Approximate ratio = 1.27 : 1

∴ The correct answer is 1.27:1.

Additional Information

Isoperimetric Theorem

For any given perimeter, the circle is the 2D shape that encloses the maximum possible area among all geometric figures.

Area Ratio for Same Perimeter

If a circle, a square, and an equilateral triangle all have the exact same perimeter, their respective areas are in the approximate ratio of 1 : 0.785 : 0.604.

Minimum Perimeter for Fixed Area

Conversely, if a fixed area must be enclosed, the circle is the shape that requires the absolute minimum perimeter (or boundary length) to do so.

Shortcut Trick

By the principle of displacement, Volume of oil raised = Volume of immersed hemisphere.

⇒ π × R² × h = ²⁄₃ × π × r³

⇒ R² × 3 = ²⁄₃ × 7³

⇒ R² = 686 ÷ 9

⇒ R = √686 ÷ 3 ≈ 26.19 ÷ 3 ≈ 8.73 cm

∴ The correct answer is 8.73 cm.

Alternate Method

Given:

Radius of the hemispherical stone (r) = 7 cm

Rise in the oil level (h) = 3 cm

Formula Used:

Volume of a cylinder = πR²h

Volume of a hemisphere = ²⁄₃πr³

R = ?

r = 7 cm

h = 3 cm

Rise in Level Original Level

Volume Displaced = Volume of Hemisphere

Calculations:

When the stone is immersed completely, the volume of the oil displaced equals the volume of the stone.

⇒ Volume of displaced oil = Volume of hemisphere

⇒ π × R² × h = ²⁄₃ × π × r³

⇒ π × R² × 3 = ²⁄₃ × π × 7³

Cancelling π from both sides:

⇒ 3 × R² = ²⁄₃ × 343

⇒ R² = 686 ÷ 9

⇒ R = √(686 ÷ 9) = √686 ÷ 3

⇒ R ≈ 26.19 ÷ 3

⇒ R ≈ 8.73 cm

∴ The correct answer is 8.73 cm.

Additional Information

Displacement Principle

When a solid object is fully immersed in a liquid, the volume of the displaced liquid is exactly equal to the volume of the object.

Volume of Cylinder

The total space occupied by a right circular cylinder is calculated as V = πR²h.

Volume of Hemisphere

A hemisphere is half of a sphere. Its volume is given by V = ²⁄₃πr³.

Shortcut Trick

Convert the central angle of the second sector from radians to degrees to easily compare them.

Angle in degrees = (2π ÷ 3) × (180° ÷ π) = 120°.

Both sectors belong to the same circle (radius = 7 cm) and have equal central angles of 120°.

Since both the radius and the central angles are identical, their areas must be exactly equal.

∴ The correct answer is 1:1.

Alternate Method

Given:

Radius of the circle (r) = 7 cm

Central angle of the first sector (θ₁) = 120°

Central angle of the second sector (θ₂) = 2π/3 radians

120° 7 cm Sector 1

2π/3 rad 7 cm Sector 2 :

Formulas Used:

Area of sector (degrees) = (θ ÷ 360°) × πr²

Area of sector (radians) = (1 ÷ 2) × θ × r²

Calculations:

Calculating the area of the first sector (A₁):

⇒ A₁ = (120° ÷ 360°) × π × (7)²

⇒ A₁ = (1 ÷ 3) × 49π

⇒ A₁ = 49π ÷ 3 cm²

Calculating the area of the second sector (A₂):

⇒ A₂ = (1 ÷ 2) × (2π ÷ 3) × (7)²

⇒ A₂ = (π ÷ 3) × 49

⇒ A₂ = 49π ÷ 3 cm²

Finding the ratio of their areas:

⇒ A₁ : A₂ = (49π ÷ 3) : (49π ÷ 3)

⇒ A₁ : A₂ = 1 : 1

∴ The correct answer is 1:1.

Additional Information

Area of a Sector

The area of a sector is directly proportional to its central angle. It represents a specific fraction of the entire area of the circle. Formula: Area = (θ ÷ 360°) × πr²

Radians and Degrees

Degrees and radians are two distinct units for measuring angles. A complete circle equals 360° or 2π radians. Conversion: Angle in Degrees = Angle in Radians × (180° ÷ π).

Arc Length

The length of the boundary arc of a sector relies on the central angle. The formula is: L = (θ ÷ 360°) × 2πr (when in degrees) or simply L = rθ (when in radians).

Shortcut Trick Substitute approximate decimal values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 ⇒ x ≈ 1.414 + 1.732 = 3.146 ⇒ y ≈ 2.236 + 1 = 3.236 ⇒ z ≈ 2 × 1.414 + 1 = 3.828 ⇒ Comparing values: 3.146 < 3.236 < 3.828 ⇒ x < y < z ∴ The correct answer is x < y < z. Alternate Method Given: x = √2 + √3 y = √5 + √1 z = 2√2 + 1 Calculations: ⇒ Squaring all the given terms to compare their magnitudes: ⇒ x2 = (√2 + √3)2 = 2 + 3 + 2√6 = 5 + √24 ⇒ y2 = (√5 + 1)2 = 5 + 1 + 2√5 = 6 + √20 ⇒ z2 = (2√2 + 1)2 = 8 + 1 + 4√2 = 9 + √32 ⇒ Using approximations for easier comparison: √20 ≈ 4.47, √24 ≈ 4.89, √32 ≈ 5.65 ⇒ x2 ≈ 5 + 4.89 = 9.89 ⇒ y2 ≈ 6 + 4.47 = 10.47 ⇒ z2 ≈ 9 + 5.65 = 14.65 ⇒ Comparing the calculated squares gives: 9.89 < 10.47 < 14.65 ⇒ x2 < y2 < z2 ⇒ Since x, y, and z are all positive numerical values, their order remains the same: x < y < z. ∴ The correct answer is x < y < z. Additional Information Comparing Surds To compare irrational numbers (surds), squaring both sides is a standard method. It removes the square roots and helps compare the resulting rational and irrational parts more easily. Important Square Roots Memorising common square roots helps quickly estimate and simplify values in exams: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449. Inequality Property of Squares For any positive real numbers a and b, if a2 > b2, it directly implies a > b. This fundamental property allows us to determine the order of numbers by ordering their squares.

Shortcut Trick

Net percentage change can be calculated directly using the weighted average of the given ratios.

Net % Change = [(Ratio₁ × %₁) + (Ratio₂ × %₂) + (Ratio₃ × %₃)] ÷ Total Ratio parts

Net % Change = [5 × 5 + 3 × 10 + 2 × (−20)] ÷ (5 + 3 + 2)

Net % Change = (25 + 30 − 40) ÷ 10 = 15 ÷ 10

Net % Change = +1.5% (A positive value denotes an increase)

∴ The correct answer is 1.5% increase.

Alternate Method

Given:

Ratio of cost of land acquisition : building materials : labor = 5 : 3 : 2

Change in land acquisition cost = +5%

Change in building materials cost = +10%

Change in labor cost = −20%

Formula Used:

Percentage change = (Net Change ÷ Initial Total) × 100

Land (500)+5% (→ +25)Materials (300)+10% (→ +30)Labor (200)−20% (→ −40)Total Base = 1000Net Change = +25 + 30 − 40 = +15

Calculation:

⇒ Let the initial cost of land acquisition, building materials, and labor be 500, 300, and 200 respectively.

⇒ Total initial project cost = 500 + 300 + 200 = 1000

⇒ Change in land acquisition cost = 5% of 500 = +25

⇒ Change in building materials cost = 10% of 300 = +30

⇒ Change in labor cost = −20% of 200 = −40

⇒ Net change in total cost = 25 + 30 − 40 = +15

⇒ Percentage change = (15 ÷ 1000) × 100

⇒ Percentage change = 1.5%

⇒ Since the net change is positive, there is an overall increase in the total cost.

∴ The correct answer is 1.5% increase.

Additional Information

Percentage Change

Percentage change is calculated as [(New Value − Old Value) ÷ Old Value] × 100. A positive result denotes an increase, and a negative result denotes a decrease.

Weighted Average Method

When components of a total are subjected to different percentage changes, the net percentage change can be found using the weighted average formula: Net % = (w₁x₁ + w₂x₂ + ... + w_nx_n) ÷ (w₁ + w₂ + ... + w_n).

Successive Percentage Change

If a single quantity changes by a% and then further changes by b%, the effective percentage change is given by: Net Change = a + b + (a × b)÷100 %.

Shortcut Trick

According to BODMAS, solve inner brackets first: (1 − 1/3) = 2/3

Multiply inner terms: 1/2 × 2/3 = 1/3

Solve curly brackets: 1/6 + 1/3 = 1/2

Solve square brackets: 2/3 ÷ 1/2 = 2/3 × 2 = 4/3

Final subtraction: 6 − 4/3 = 14/3

∴ The correct answer is 14/3.

Alternate Method

Given:

Expression: 6 − [2/3 ÷ {1/6 + (1/2 × (1 − 1/3))}]

Formula Used:

BODMAS Rule: Brackets → Of → Division → Multiplication → Addition → Subtraction

Calculations:

⇒ Step 1: Solve the innermost round bracket ( )

⇒ 1 − 1/3 = (3 − 1)/3 = 2/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + (1/2 × 2/3)}]

⇒ Step 2: Solve the multiplication inside the remaining round bracket

⇒ 1/2 × 2/3 = 2/6 = 1/3

⇒ The expression becomes: 6 − [2/3 ÷ {1/6 + 1/3}]

⇒ Step 3: Solve the addition inside the curly bracket { }

⇒ 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2

⇒ The expression becomes: 6 − [2/3 ÷ 1/2]

⇒ Step 4: Solve the division inside the square bracket [ ]

⇒ 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3

⇒ The expression becomes: 6 − 4/3

⇒ Step 5: Perform the final subtraction

⇒ 6 − 4/3 = (18 − 4)/3 = 14/3

∴ The correct answer is 14/3.

Additional Information

BODMAS Rule

BODMAS stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction. It sets the standard order of operations to solve mathematical expressions correctly.

Types of Brackets

Brackets must be solved in a specific order: Line bracket or Vinculum first, followed by Round brackets ( ), Curly brackets { }, and finally Square brackets [ ].

Fraction Division Rule

When dividing two fractions, you must multiply the first fraction by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Shortcut Trick

Since the time period is identical for all, the profit ratio equals the investment ratio.

Investment Ratio of X : Y : Z = 25000 : 35000 : 40000 = 5 : 7 : 8

Total profit parts = 5 + 7 + 8 = 20 parts

Y's share = (7 ÷ 20) × 30000 = 7 × 1500 = 10500

∴ The correct answer is ₹10,500.

Alternate Method

Given:

Investment of X = ₹25,000

Investment of Y = ₹35,000

Investment of Z = ₹40,000

Total profit at the end of the year = ₹30,000

Formula Used:

Profit Ratio = Capital Investment × Time Period

Partnership Investment & Profit Ratio

₹25,000 5 parts Partner X

₹35,000 7 parts Partner Y

₹40,000 8 parts Partner Z

Total Profit = ₹30,000 (20 parts)

Calculations:

⇒ Profit Ratio of X : Y : Z = (25000 × 1) : (35000 × 1) : (40000 × 1)

⇒ X : Y : Z = 25 : 35 : 40

⇒ X : Y : Z = 5 : 7 : 8

⇒ Sum of profit ratios = 5 + 7 + 8 = 20 parts

⇒ 20 parts = ₹30,000

⇒ 1 part = 30000 ÷ 20 = ₹1,500

⇒ Y's share = 7 parts = 7 × 1500 = ₹10,500

∴ The correct answer is ₹10,500.

Additional Information

Simple Partnership

When all partners invest their money for the exact same duration, the profit or loss is distributed purely in the ratio of their original investment amounts.

Compound Partnership

When partners invest their capital for different time periods, the profit ratio is calculated as the product of their investment and their respective time duration (Profit = Capital × Time).

Active vs Sleeping Partner

An active partner manages the business operations and generally receives a fixed percentage or salary from the profit before the remaining profit is distributed among all partners (including sleeping partners) based on their capital ratio.

Shortcut Trick

Ratio of Profit = (Capital_A × Time_A) : (Capital_B × Time_B)

Profit Ratio = [(48 × 6) + (24 × 6)] : [(72 × 6) + (96 × 6)]

Profit Ratio = [6 × (48 + 24)] : [6 × (72 + 96)] = 72 : 168 = 3 : 7

B's share = (7 ÷ 10) × 55000 = ₹38,500

∴ The correct answer is ₹38,500.

Alternate Method

Given:

Initial capital of A = ₹48,000

Initial capital of B = ₹72,000

Total time of investment = 1 year = 12 months

Total profit earned at the end of the year = ₹55,000

A

₹48k × 6m

₹24k × 6m = 432k

B

₹72k × 6m

₹96k × 6m = 1008k

Profit Ratio (A : B) = 432 : 1008 = 3 : 7

Formula Used:

Profit Ratio = Product of Capital and Time Period = C₁T₁ : C₂T₂

Calculations:

⇒ Equivalent capital of A = (₹48,000 × 6 months) + (₹24,000 × 6 months)

⇒ Equivalent capital of A = ₹2,88,000 + ₹1,44,000 = ₹4,32,000

⇒ Equivalent capital of B = (₹72,000 × 6 months) + (₹96,000 × 6 months)

⇒ Equivalent capital of B = ₹4,32,000 + ₹5,76,000 = ₹10,08,000

⇒ Ratio of profit (A : B) = 4,32,000 : 10,08,000

⇒ Ratio of profit (A : B) = 432 : 1008 = 3 : 7

⇒ Sum of ratio parts = 3 + 7 = 10 parts

⇒ B's share in profit = (7 ÷ 10) × Total Profit

⇒ B's share = (7 ÷ 10) × 55,000

⇒ B's share = 7 × 5,500 = ₹38,500

∴ The correct answer is ₹38,500.

Additional Information

Partnership Profit Ratio

When investments are made for unequal time periods, the profit is distributed in the ratio of the product of capital and corresponding time period: C₁T₁ : C₂T₂ : C₃T₃.

Sleeping vs Working Partner

A working partner manages the business and usually receives a fixed salary or a certain percentage of the total profit before the remaining profit is distributed according to capital ratios.

Joining Mid-Way

If a partner joins x months after the start of a year, their investment time period is taken as (12 − x) months when calculating the effective capital for profit distribution.