Quantitative Aptitude PYQs

Shortcut Trick

Apply the inverse proportion logic: M₁D₁ = M₂D₂.

20 × 13 = 65 × D₂

D₂ = (20 × 13) ÷ 65 = 260 ÷ 65 = 4.

∴ The correct answer is 4 days.

Alternate Method

Given:

Initial number of pupils (M₁) = 20

Initial days (D₁) = 13 days

Increased pupils (M₂) = 65

Formula Used:

In Time and Work (Provisions), Total Work = Number of People × Days.

Calculations:

⇒ Total provision = 20 × 13 = 260 units

⇒ Let the number of days for 65 pupils be x.

⇒ 65 × x = 260

⇒ x = 260 ÷ 65

⇒ x = 4

∴ The correct answer is 4 days.

Additional Information

Chain Rule (Inverse Proportion)

When the number of people increases, the number of days for which the food lasts decreases. This follows the formula M₁D₁H₁/W₁ = M₂D₂H₂/W₂.

Total Work Constant

If work or total provisions remain constant, then Man-Days (M × D) always remains a constant value.

Shortcut Trick

Age difference between Reena and Meena is always 5 years.

8 years ago, Reena was twice as old as Meena (Ratio 2 : 1).

Difference in ratio parts (2 − 1 = 1 unit) represents 5 years.

∴ 8 years ago, Reena age = 2 units × 5 = 10 years.

Present age of Reena = 10 + 8 = 18 years.

∴ The correct answer is 18 years.

Alternate Method

Given:

Age difference = 5 years

Condition: 8 years ago, Reena was twice as old as Meena.

Let Meena's present age be x years.

Then Reena's present age = (x + 5) years.

8 years ago:

⇒ Meena age = x − 8

⇒ Reena age = (x + 5) − 8 = x − 3

According to the question:

⇒ (x − 3) = 2 × (x − 8)

⇒ x − 3 = 2x − 16

⇒ 2x − x = 16 − 3

⇒ x = 13

Reena's present age = x + 5 = 13 + 5 = 18 years.

∴ The correct answer is 18 years.

Additional Information

Constant Age Difference

The difference between the ages of two persons remains constant at any point in time (past, present, or future).

Ratio Method for Ages

If the ratio of ages is given for two different time periods, equate the difference between the ratio parts to the constant age difference to find the value of 1 unit.

Future and Past Ages

To find age 'n' years ago, subtract 'n' from present age. To find age after 'n' years, add 'n' to present age.

Shortcut Trick
Scheme A: Buy 42 Get 47 means 5 items are free on 47. Discount% = 5 ÷ 47 × 100 ≈ 10.64%.
Scheme B: Successive 16% and 36%. Equivalent discount = 16 + 36 − (16 × 36 ÷ 100) = 52 − 5.76 = 46.24%.
Scheme C: Successive 46% and 46%. Equivalent discount = 46 + 46 − (46 × 46 ÷ 100) = 92 − 21.16 = 70.84%.
Comparing the three: 70.84% > 46.24% > 10.64%.
Therefore, Scheme C is the most beneficial to the customer.
∴ The correct answer is Only C.

Alternate Method
Given → Scheme A (Free items), Scheme B (16% & 36%), Scheme C (46% & 46%). Formulas: Discount% = (Free Items ÷ Total Items) × 100 | Successive Discount = a + b − ab ÷ 100.

⇒ For Scheme A: Discount = (47 − 42) ÷ 47 × 100 = 500 ÷ 47 = 10.638%
⇒ For Scheme B: Discount = 16 + 36 − (16 × 36) ÷ 100 = 52 − 5.76 = 46.24%
⇒ For Scheme C: Discount = 46 + 46 − (46 × 46) ÷ 100 = 92 − 21.16 = 70.84%
⇒ Higher discount percentage means more beneficial to the customer.
⇒ Scheme C provides the highest discount of 70.84%.
∴ The correct answer is Only C.

Additional Information
Buy X Get Y Free
In a 'Buy X get Y' scheme, the discount percentage is calculated as [Y ÷ (X + Y)] × 100. In this problem, 'get 47' implies 47 is the total items, so free = 5.
Successive Discount Formula
For two successive discounts of a% and b%, the effective discount is (a + b − ab/100)%. A higher effective discount is always better for the buyer.

Shortcut Trick

In a triangle with an angle of 120°, the side opposite to it is calculated as: XZ² = XY² + YZ² + (XY × YZ).

⇒ XZ² = 12² + 15² + (12 × 15)

⇒ XZ² = 144 + 225 + 180 = 549

⇒ XZ = √549 = 3√61 cm

∴ The correct answer is 3√61 cm.

Alternate Method

Given: Side XY = 12 cm, Side YZ = 15 cm, and Angle ∠Y = 120°.

Formula Used: Cosine Rule: b² = a² + c² − 2ac cos(B)

⇒ Applying Cosine Rule for side XZ:

⇒ XZ² = XY² + YZ² − 2(XY)(YZ) cos(∠Y)

⇒ XZ² = (12)² + (15)² − 2 × 12 × 15 × cos(120°)

⇒ XZ² = 144 + 225 − 360 × (−0.5) [Since cos 120° = −1/2]

⇒ XZ² = 369 + 180

⇒ XZ² = 549

⇒ XZ = √549 = √(9 × 61)

⇒ XZ = 3√61 cm

∴ The correct answer is 3√61 cm.

Additional Information

Cosine Rule

It relates the lengths of the sides of a triangle to the cosine of one of its angles: c² = a² + b² − 2ab cos(C).

Special Case: 120 Degrees

When the angle is 120°, the term −2ab cos(120°) becomes +ab because cos(120°) = −1/2.

Sine Rule

In any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C.

Shortcut Trick

For four numbers a, b, c, and d in proportion, the product of extremes equals the product of means: a × d = b × c.

Given a = 8, b = X, c = 15, and d = 135.

8 × 135 = X × 15

X = (8 × 135) ÷ 15 ⇒ X = 8 × 9 = 72.

∴ The correct answer is 72.

Alternate Method

Given: First term (a) = 8, Second term (b) = X, Third term (c) = 15, Fourth term (d) = 135.

Formula Used: a/b = c/d

⇒ 8 / X = 15 / 135

⇒ 8 / X = 1 / 9

⇒ X = 8 × 9

⇒ X = 72

∴ The correct answer is 72.

Additional Information

Fourth Proportional

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

Third Proportional

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

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Shortcut Trick

Ratio CSA : TSA = 2 : 3 ⇒ 2πrh : 2πr(h + r) = 2 : 3 ⇒ h : (h + r) = 2 : 3 ⇒ h = 2r.

CSA = 2 × (22/7) × r × (2r) = 55.44 ⇒ r² = (55.44 × 7) / 88 = 4.41 ⇒ r = 2.1 m.

Height h = 2 × 2.1 = 4.2 m.

Volume = πr²h = (22/7) × 2.1 × 2.1 × 4.2 = 13.86 × 4.2 = 58.212 m³.

∴ The correct answer is 58.21 m³.

Alternate Method

Given:

Curved Surface Area (CSA) = 55.44 m²

CSA = (2/3) × Total Surface Area (TSA)

Formula Used:

CSA of cylinder = 2πrh

TSA of cylinder = 2πr(h + r)

Volume of cylinder = πr²h

Calculations:

⇒ 2πrh = (2/3) × 2πr(h + r)

⇒ h = (2/3)(h + r)

⇒ 3h = 2h + 2r

⇒ h = 2r

⇒ Now, 2 × (22/7) × r × h = 55.44

⇒ 2 × (22/7) × r × 2r = 55.44

⇒ (88/7) × r² = 55.44

⇒ r² = (55.44 × 7) / 88

⇒ r² = 0.63 × 7 = 4.41

⇒ r = 2.1 m

⇒ h = 2 × 2.1 = 4.2 m

⇒ Volume = (22/7) × (2.1)² × 4.2

⇒ Volume = (22/7) × 4.41 × 4.2

⇒ Volume = 22 × 0.63 × 4.2 = 58.212 m³

∴ The correct answer is 58.21 m³.

Additional Information

Surface Area of Cylinder

The Curved Surface Area (CSA) excludes the top and bottom circles, while the Total Surface Area (TSA) includes them: TSA = 2πrh + 2πr².

Volume Calculation

The volume of any right circular cylinder is the product of the base area and its height: Volume = πr²h.

Hollow Cylinder

For a hollow cylinder with outer radius R and inner radius r, the Volume of material = π(R² − r²)h.

Shortcut Trick

Substitute the given values x = 2 and y = 1 directly into the length and breadth expressions.

Length = 2(2) + 1² = 4 + 1 = 5 m

Breadth = 2(2) − 1² = 4 − 1 = 3 m

Area = Length × Breadth = 5 × 3 = 15 m²

∴ The correct answer is 15 m².

Alternate Method

Given:

Length (L) = (2x + y²) m

Breadth (B) = (2x − y²) m

Values: x = 2, y = 1

Formula Used:

Area of a Rectangle = Length × Breadth

Algebraic Identity: (a + b)(a − b) = a² − b²

Calculations:

⇒ Area = (2x + y²) × (2x − y²) ⇒ Area = (2x)² − (y²)² ⇒ Area = 4x² − y⁴ ⇒ Substituting x = 2 and y = 1: ⇒ Area = 4(2)² − (1)⁴ ⇒ Area = 4 × 4 − 1 ⇒ Area = 16 − 1 = 15 m²

∴ The correct answer is 15 m².

Additional Information

Area of Rectangle

The total space occupied by a rectangle in a 2D plane is calculated as the product of its length and breadth.

Perimeter of Rectangle

The total distance around the boundary of a rectangle, given by 2 × (Length + Breadth).

Algebraic Identity

The identity (a + b)(a − b) = a² − b² is frequently used to simplify products of binomials in geometry problems.

Shortcut Trick

Convert Speed from km/hr to m/min: 84.6 × (1000 ÷ 60) = 1,410 m/min.

Distance = Speed × Time = 1,410 × 20 = 28,200 m.

∴ The correct answer is 28,200 m.

Alternate Method

Given:

Speed (S) = 84.6 km/hr

Time (T) = 20 minutes

Formula Used:

Distance = Speed × Time

1 km = 1000 m | 1 hour = 60 minutes

Calculations:

⇒ Speed in m/s = 84.6 × (5/18) = 4.7 × 5 = 23.5 m/s

⇒ Time in seconds = 20 × 60 = 1200 seconds

⇒ Distance = 23.5 × 1200

⇒ Distance = 28,200 m

∴ The correct answer is 28,200 m.

Additional Information

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Speed-Time Relationship

Distance is directly proportional to time when speed is constant (D = S × T).

Average Speed

Average Speed = Total Distance traveled ÷ Total Time taken.

Shortcut Trick

The angle formed by the bisectors of two adjacent angles in a quadrilateral is equal to half the sum of the other two angles.

⇒ ∠STR = (∠P + ∠Q) ÷ 2

⇒ 18° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 36°

Given ratio ∠P : ∠Q = 2 : 7. Total parts = 2 + 7 = 9 units.

⇒ 9 units = 36° ⇒ 1 unit = 4°

Difference = (7 − 2) units = 5 units = 5 × 4° = 20°.

∴ The correct answer is 20°.

Alternate Method

Given: ∠STR = 18°, ∠P : ∠Q = 2 : 7, ST and RT are angle bisectors of ∠S and ∠R.

Formula Used: Sum of interior angles of a quadrilateral = 360°; Sum of angles in a triangle = 180°.

In ΔSTR:

⇒ ∠TSR + ∠TRS + ∠STR = 180°

⇒ (∠S ÷ 2) + (∠R ÷

Shortcut Trick

The relationship between Cost Price (CP) and Marked Price (MP) is given by the ratio:

CP : MP = (100 − Discount%) : (100 + Profit%)

CP : MP = (100 − 23) : (100 + 44)

CP : MP = 77 : 144

∴ The correct answer is 77 : 144.

Alternate Method

Given: Discount = 23%, Profit = 44%

Formulas: SP = MP × (100 − D%) ÷ 100 | SP = CP × (100 + P%) ÷ 100

⇒ Let Marked Price (MP) be 100.

⇒ Selling Price (SP) = 100 − 23% of 100 = 77.

⇒ Since Profit is 44%, SP = CP × 144 ÷ 100.

⇒ 77 = CP × 144 ÷ 100 ⇒ CP = (77 × 100) ÷ 144.

⇒ Ratio CP : MP = [(77 × 100) ÷ 144] : 100.

⇒ CP : MP = 77 : 144.

∴ The correct answer is 77 : 144.

Additional Information

Marked Price & Discount

The Marked Price (Listed Price) is the price printed on the label. Discount is always calculated on the MP to arrive at the Selling Price.

Markup Percentage

Markup% = (MP − CP) ÷ CP × 100. It represents how much higher the marked price is compared to the cost price.

Shortcut Trick

Given Marked Price (MP) = ₹6,370 and Discount% = 50%.

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

SP = 6,370 × (100 − 50) ÷ 100 = 6,370 × 50 ÷ 100.

SP = 6,370 ÷ 2 = 3,185.

∴ The correct answer is ₹3,185.

Alternate Method

Given: Marked Price = ₹6,370, Discount Percentage = 50%.

Formulas: Discount = (Discount% × MP) ÷ 100 | Selling Price (SP) = MP − Discount.

⇒ Discount Amount = 50% of 6,370

⇒ Discount = (50 ÷ 100) × 6,370 = 3,185

⇒ Selling Price = MP − Discount = 6,370 − 3,185

⇒ Selling Price = 3,185

∴ The correct answer is ₹3,185.

Additional Information

Marked Price & Discount

Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP unless specified otherwise. Formula: SP = MP × (100 − D)%.

50% Discount Rule

A 50% discount implies the customer pays exactly half of the Marked Price. It is equivalent to a 1:1 ratio between the discount and the selling price.

Shortcut Trick

Total Capacity (LCM of 15 and 40) = 120 units

Efficiency of P + Q = 120 ÷ 15 = 8 units/hr

Efficiency of P = 120 ÷ 40 = 3 units/hr

Efficiency of Q = 8 − 3 = 5 units/hr

Time for Q to fill 1/4^th = (120 × 1/4) ÷ 5 = 30 ÷ 5 = 6 hours

∴ The correct answer is 6 hours.

Alternate Method

Given: P + Q can fill in 15 hours. P alone can fill in 40 hours.

Formula Used: Total Work = Efficiency × Time | 1/Q = 1/(P+Q) − 1/P

⇒ Efficiency of Q = Efficiency of (P + Q) − Efficiency of P

⇒ Efficiency of Q = 8 − 3 = 5 units/hr

⇒ Total capacity of cistern = 120 units

⇒ One-fourth of the cistern = 120 ÷ 4 = 30 units

⇒ Time taken by Q = 30 ÷ 5 = 6 hours

∴ The correct answer is 6 hours.

Additional Information

Efficiency Concept

Efficiency is inversely proportional to the time taken to complete a work. Total Work = LCM of individual time durations.

Combined Work Formula

If two pipes A and B fill a tank in x and y hours, then together they fill it in (xy) ÷ (x + y) hours.

Pipes and Cisterns

Inlet pipes have positive efficiency while outlet/leakage pipes have negative efficiency.

Shortcut Trick

Use the algebraic identity: if x + y + z = 0, then x³ + y³ + z³ = 3xyz.

In the numerator, (a² − b²) + (b² − c²) + (c² − a²) = 0, so it becomes 3(a² − b²)(b² − c²)(c² − a²).

In the denominator, (a − b) + (b − c) + (c − a) = 0, so it becomes 3(a − b)(b − c)(c − a).

Dividing these gives [(a−b)(a+b)(b−c)(b+c)(c−a)(c+a)] ÷ [(a−b)(b−c)(c−a)].

Canceling the common terms leaves (a + b)(b + c)(c + a).

∴ The correct answer is (a + b)(b + c)(c + a).

Alternate Method

Given: ((a² − b²)³ + (b² − c²)³ + (c² − a²)³) ÷ ((a − b)³ + (b − c)³ + (c − a)³)

Formula Used: If x + y + z = 0, then x³ + y³ + z³ = 3xyz; and x² − y² = (x − y)(x + y).

⇒ Let X = a² − b², Y = b² − c², and Z = c² − a².

⇒ X + Y + Z = a² − b² + b² − c² + c² − a² = 0.

⇒ Therefore, Numerator = 3(a² − b²)(b² − c²)(c² − a²).

⇒ Now, let p = a − b, q = b − c, and r = c − a.

⇒ p + q + r = a − b + b − c + c − a = 0.

⇒ Therefore, Denominator = 3(a − b)(b − c)(c − a).

⇒ Expression = [3(a² − b²)(b² − c²)(c² − a²)] ÷ [3(a − b)(b − c)(c − a)]

⇒ Expression = [3(a − b)(a + b)(b − c)(b + c)(c − a)(c + a)] ÷ [3(a − b)(b − c)(c − a)]

⇒ Expression = (a + b)(b + c)(c + a).

∴ The correct answer is (a + b)(b + c)(c + a).

Additional Information

Conditional Identity

The condition a + b + c = 0 implies a³ + b³ + c³ = 3abc, which is a specific case of the expansion of a³ + b³ + c³ − 3abc.

Sum of Three Cubes Formula

The general identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Difference of Squares

The property a² − b² = (a − b)(a + b) allows for the factorization and cancellation of terms in complex rational expressions.

Shortcut Trick

Solve the brackets using the BODMAS rule directly.

Inner bracket: (−8 − 20 − 18) = −46

Square bracket: [1 × {10 + (−2) × (−3)}] = [1 × {10 + 6}] = 16

Remaining expression: 86 − (−16) × (−46) ÷ 16

⇒ 86 − (736 ÷ 16) = 86 − 46 = 40

∴ The correct answer is 40.

Alternate Method

Given:

Expression: 86 − (−16) × (−8 − 20 − 18) ÷ [1 × {10 + (−2) × (−3)}]

Method Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ 86 − (−16) × (−8 − 20 − 18) ÷ [1 × {10 + (−2) × (−3)}]

⇒ 86 − (−16) × (−46) ÷ [1 × {10 + 6}]

⇒ 86 − (−16) × (−46) ÷ [1 × 16]

⇒ 86 − (−16) × (−46) ÷ 16

⇒ 86 − 736 ÷ 16

⇒ 86 − 46

⇒ 40

∴ The correct answer is 40.

Additional Information

BODMAS Rule

It stands for Bracket, Of, Division, Multiplication, Addition, and Subtraction. Expressions are simplified in this strict sequential order.

Order of Brackets

Always solve brackets in the order: Line bracket (bar), Simple brackets (), Curly brackets {}, and finally Square brackets [].

Sign Multiplication Rules

When multiplying numbers with the same signs, the result is positive (− × − = +). When multiplying numbers with different signs, the result is negative (+ × − = −).

Shortcut Trick

To find 31³, focus on the unit digit. Since 31 ends in 1, its cube must end in 1 (1 × 1 × 1 = 1).

Check the given options:

(1) ends in 9, (2) ends in 8, (3) ends in 1, and (4) ends in 5.

Since only 29791 ends in 1, it is the correct answer.

∴ The correct answer is 29791 m³.

Alternate Method

Given: The edge of the cube (a) = 31 m.

Formula Used: Volume of a cube = Side³ = a³

⇒ Volume = (31 m)³

⇒ Volume = 31 × 31 × 31

⇒ Volume = 961 × 31

⇒ Volume = 29791 m³

∴ The correct answer is 29791 m³.

Additional Information

Total Surface Area of Cube

The total surface area is the sum of the areas of all six faces: TSA = 6a².

Lateral Surface Area of Cube

The area of the four side faces (excluding top and bottom): LSA = 4a².

Diagonal of a Cube

The longest distance between two opposite corners: d = a√3.

Shortcut Trick

Ratio of Profit = Ratio of Investment (when time is constant).

Sunita : Javed : Rani = 139 : 47 : 154.

Total ratio parts = 139 + 47 + 154 = 340 units.

340 units = ₹4,50,500 ⇒ 1 unit = 4,50,500 ÷ 340 = ₹1,325.

Rani's share (154 units) = 154 × 1,325 = ₹2,04,050.

∴ The correct answer is ₹2,04,050.

Alternate Method

Given → Sunita's Investment = ₹13,90,000 | Javed's Investment = ₹4,70,000 | Rani's Investment = ₹15,40,000 | Total Profit = ₹4,50,500.
Formulas: Profit Share = (Individual Investment ÷ Total Investment) × Total Profit.

⇒ Ratio of investments = 13,90,000 : 4,70,000 : 1540,000 = 139 : 47 : 154.
⇒ Sum of ratios = 139 + 47 + 154 = 340.
⇒ Rani's share = (Rani's ratio ÷ Total ratio) × Total Profit.
⇒ Rani's share = (154 ÷ 340) × 4,50,500.
⇒ Rani's share = 154 × 1,325 = 2,04,050.

∴ The correct answer is ₹2,04,050.

Additional Information

Partnership Concept

When multiple people invest in a business, the profit is distributed in the ratio of (Investment × Time Period).

Simple vs. Compound Partnership

In a simple partnership, all partners invest for the same duration. In a compound partnership, durations differ.

Shortcut Trick

When the distance for both journeys is the same, use the harmonic mean formula.

Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

⇒ (2 × 131 × 24) ÷ (131 + 24) = 6288 ÷ 155 ≈ 40.57 km/hr

∴ The correct answer is 40.57 km/hr.

Alternate Method

Given:

Distance (A to B) = 663 km

Speed from A to B (S₁) = 131 km/hr

Speed from B to A (S₂) = 24 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 663 + 663 = 1326 km

⇒ Time taken (A to B) = 663 ÷ 131 ≈ 5.061 hours

⇒ Time taken (B to A) = 663 ÷ 24 = 27.625 hours

⇒ Total Time = 5.061 + 27.625 = 32.686 hours

⇒ Average Speed = 1326 ÷ 32.686 ≈ 40.57 km/hr

∴ The correct answer is 40.57 km/hr.

Additional Information

Average Speed (Different Distances)

If a person covers distances d₁, d₂, d₃ with speeds s₁, s₂, s₃, then Average Speed = (d₁ + d₂ + d₃) ÷ (d₁/s₁ + d₂/s₂ + d₃/s₃).

Relative Speed

When two objects move in opposite directions, relative speed is (u + v). When they move in the same direction, relative speed is |u − v|.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Shortcut Trick

Larger number = (Sum + Difference) ÷ 2 = (74 + 12) ÷ 2 = 43

Smaller number = (Sum − Difference) ÷ 2 = (74 − 12) ÷ 2 = 31

Product = 43 × 31 = 1333

∴ The correct answer is 1333.

Alternate Method

Given: Sum of two numbers = 74 and Difference of two numbers = 12.

Formula Used: Product (xy) = [(x + y)² − (x − y)²] ÷ 4

⇒ Let the numbers be x and y.

⇒ x + y = 74 --- (1)

⇒ x − y = 12 --- (2)

⇒ Adding (1) and (2): 2x = 86 ⇒ x = 43

⇒ Substituting x in (1): 43 + y = 74 ⇒ y = 31

⇒ Product = 43 × 31

⇒ Product = 1333

∴ The correct answer is 1333.

Additional Information

Algebraic Identity for Product

The relationship between sum, difference, and product is given by: 4xy = (x + y)² − (x − y)². This allows finding the product without calculating individual numbers.

Finding Individual Numbers

If S is the sum and D is the difference, then the numbers are (S + D) ÷ 2 and (S − D) ÷ 2.

Sum and Product Relationship

For a quadratic equation t² − (sum)t + (product) = 0, the roots are the two numbers themselves.

Shortcut Trick

For any quadrilateral EFGH, the angle formed by the angle bisectors of two adjacent angles (say, ∠G and ∠H) meeting at a point X is equal to half the sum of the other two angles (∠E and ∠F):

∠HXG = (∠E + ∠F) ÷ 2

Given: ∠E = 52° and ∠F = 37°

⇒ ∠HXG = (52° + 37°) ÷ 2

⇒ ∠HXG = 89° ÷ 2 = 44.5°

∴ The correct answer is 44.5°.

Alternate Method

Given:

In quadrilateral EFGH:

∠E = 52°

∠F = 37°

The bisectors of ∠G and ∠H meet at X.

Formula Used:

1. Sum of interior angles of a quadrilateral = 360°
2. Sum of angles of a triangle = 180°

Calculations:

In quadrilateral EFGH:

⇒ ∠E + ∠F + ∠G + ∠H = 360°

⇒ 52° + 37° + ∠G + ∠H = 360°

⇒ 89° + ∠G + ∠H = 360°

⇒ ∠G + ∠H = 360° − 89° = 271°

Since GX and HX are the angle bisectors of ∠G and ∠H respectively:

⇒ ∠XGH = ∠G ÷ 2

⇒ ∠XHG = ∠H ÷ 2

In ΔHXG:

⇒ ∠HXG + ∠XGH + ∠XHG = 180°

⇒ ∠HXG + (∠G ÷ 2) + (∠H ÷ 2) = 180°

⇒ ∠HXG + (∠G + ∠H) ÷ 2 = 180°

⇒ ∠HXG + 271° ÷ 2 = 180°

⇒ ∠HXG + 135.5° = 180°

⇒ ∠HXG = 180° − 135.5° = 44.5°

∴ The correct answer is 44.5°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is always equal to half the sum of the other two remaining angles.

Sum of Interior Angles of a Polygon

For any n-sided polygon, the sum of interior angles is given by (n − 2) × 180°. For a quadrilateral (n = 4), the sum is 360°.

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral meet, the angle between them is equal to half the difference of the other two angles.

Shortcut Trick

Use the direct relationship formula: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

MP ÷ 1,600 = (100 + 20) ÷ (100 − 25) = 120 ÷ 75.

MP = 1,600 × (120 ÷ 75) = 1,600 × (8 ÷ 5).

MP = 320 × 8 = 2,560.

∴ The correct answer is ₹2,560.

Alternate Method

Given: Cost Price (CP) = ₹1,600, Profit = 20%, Discount = 25%.
Formulas: SP = CP × (100 + P%) ÷ 100 | MP = (SP × 100) ÷ (100 − D%)

⇒ Step 1: Calculate Selling Price (SP) = 1,600 × (120 ÷ 100) = ₹1,920.

⇒ Step 2: Marked Price (MP) × (100 − 25) ÷ 100 = 1,920.

⇒ Step 3: MP × (75 ÷ 100) = 1,920.

⇒ Step 4: MP = 1,920 × (4 ÷ 3) = 640 × 4 = ₹2,560.

∴ The correct answer is ₹2,560.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP: SP = MP − Discount.

CP-MP Relationship

For problems involving both Profit% and Discount%, the ratio MP/CP = (100 + P%)/(100 − D%) is the most efficient solving method.