Quantitative Aptitude PYQs

Shortcut Trick

Teacher's Age = New Average + (Initial Strength × Increase in Average)

Initial students = 48, Initial average = 19 years

New average = 20 years, Increase = 20 − 19 = 1 year

Teacher's Age = 20 + (48 × 1) = 68 years

∴ The correct answer is 68 years.

Alternate Method

Given:

Number of students = 48

Average age of students = 19 years

Average age including teacher = 20 years

Formula Used:

Total Age = Average × Number of persons

Calculations:

⇒ Total age of 48 students = 48 × 19

⇒ Total age of 48 students = 912 years

⇒ Total number of people including teacher = 48 + 1 = 49

⇒ Total age of 49 people = 49 × 20

⇒ Total age of 49 people = 980 years

⇒ Age of the teacher = Total age of group − Total age of students

⇒ Age of the teacher = 980 − 912

⇒ Age of the teacher = 68 years

∴ The correct answer is 68 years.

Additional Information

Inclusion of a New Member

When a new person joins a group and the average increases, the new person's value = Old Average + (New Total Number × Increase in Average).

Exclusion of a Member

When a person leaves and the average changes, the value of the person who left = Old Average ± (New Total Number × Change in Average).

Weighted Average Formula

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the quantity and x is the average of each group.

Shortcut Trick

When equal distances (d) are covered at speeds a, b, and c, use the Harmonic Mean formula.

Average Speed = 3abc ÷ (ab + bc + ca)

Average Speed = (3 × 10 × 20 × 40) ÷ (200 + 800 + 400) = 24000 ÷ 1400

Average Speed = 240 ÷ 14 = 120 ÷ 7 = 17(1/7) km/hr.

∴ The correct answer is 17(1/7) km/hr.

Alternate Method

Given: Three successive distances = 20 km each. Speeds = 10 km/hr, 20 km/hr, 40 km/hr.

Formula Used: Average Speed = Total Distance ÷ Total Time

⇒ Total Distance = 20 + 20 + 20 = 60 km

⇒ Time 1 = 20 ÷ 10 = 2 hours

⇒ Time 2 = 20 ÷ 20 = 1 hour

⇒ Time 3 = 20 ÷ 40 = 0.5 hours

⇒ Total Time = 2 + 1 + 0.5 = 3.5 hours = 7/2 hours

⇒ Average Speed = 60 ÷ (7/2) = (60 × 2) ÷ 7 = 120/7

⇒ Average Speed = 17(1/7) km/hr

∴ The correct answer is 17(1/7) km/hr.

Additional Information

Average Speed for Two Equal Distances

When two equal distances are covered at speeds u and v, the average speed is given by (2uv) ÷ (u + v).

Relative Speed

For two objects moving in the opposite direction, relative speed is the sum of their speeds (s₁ + s₂). For the same direction, it is the difference (s₁ − s₂).

Conversion Units

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

Shortcut Trick

First value = 60% of 45 = 0.6 × 45 = 27

Second value = 2/5 of 15 = 2 × 3 = 6

Difference = 27 − 6 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Expression 1: 60% of 45

Expression 2: 2/5 of 15

Calculations:

⇒ Value 1 = (60/100) × 45

⇒ Value 1 = (3/5) × 45 = 3 × 9 = 27

⇒ Value 2 = (2/5) × 15 = 2 × 3 = 6

⇒ Difference = Value 1 − Value 2

⇒ Difference = 27 − 6 = 21

∴ The correct answer is 21.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: x% = x/100. For example, 60% = 3/5.

Concept of 'Of' in Mathematics

In arithmetic word problems, the word 'of' represents the multiplication (×) operation between two numbers or expressions.

Percentage Change

Percentage change = [(New Value − Original Value) / Original Value] × 100.

Shortcut Trick

Average speed for equal distances = (2 × S₁ × S₂) ÷ (S₁ + S₂)

Average speed = (2 × 24 × 28) ÷ (24 + 28) = 1344 ÷ 52

Average speed = 25.846... ≈ 25.85 km/hr

∴ The correct answer is 25.85 km/hr.

Alternate Method

Given:

Speed going (S₁) = 24 km/hr

Speed returning (S₂) = 28 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Let the distance between the two places be d km.

⇒ Time taken going (T₁) = d ÷ 24

⇒ Time taken returning (T₂) = d ÷ 28

⇒ Average Speed = (d + d) ÷ (d/24 + d/28)

⇒ Average Speed = 2d ÷ [d(28 + 24) ÷ (24 × 28)]

⇒ Average Speed = (2 × 24 × 28) ÷ 52

⇒ Average Speed = 1344 ÷ 52 = 25.84615...

⇒ Average Speed ≈ 25.85 km/hr

∴ The correct answer is 25.85 km/hr.

Additional Information

Average Speed (Equal Distances)

When a person covers two equal distances at speeds x and y, the average speed is given by (2xy) ÷ (x + y).

General Average Speed Formula

Average Speed = Total Distance Covered ÷ Total Time Taken. This applies even when distances or time intervals are unequal.

Relative Speed

If two objects move in the same direction, relative speed is |s₁ − s₂|. In opposite directions, it is s₁ + s₂.

Shortcut Trick

Effective Compound Interest rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%

Total Amount = 100% (Principal) + 21% (Interest) = 121% of Principal

Amount = 4200 × 121/100 = 42 × 121

Amount = ₹5,082

∴ The correct answer is ₹5,082.

Alternate Method

Given:

Principal (P) = ₹4,200

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4200 × (1 + 10/100)²

⇒ A = 4200 × (1 + 0.1)²

⇒ A = 4200 × (1.1)²

⇒ A = 4200 × 1.21

⇒ A = 42 × 121

⇒ A = ₹5,082

∴ The correct answer is ₹5,082.

Additional Information

Effective Interest Rate

For two successive years with rate R%, the effective rate is R + R + (R²/100)%.

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is P(R/100)².

Compound Interest Formula

Interest (CI) = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Shortcut Trick

Net Multiplier = (100 − 18)% × (100 − 6)% × (100 − 25)%

Net Multiplier = 0.82 × 0.94 × 0.75 = 0.5781

Equivalent Discount = (1 − 0.5781) × 100 = 42.19%

∴ The correct answer is 42.19%.

Alternate Method

Given → Discounts: 18%, 6%, and 25%.
Formulas: Successive Discount = a + b − (ab ÷ 100)

⇒ First combine 18% and 6%: 18 + 6 − (18 × 6 ÷ 100) = 24 − 1.08 = 22.92%
⇒ Now combine 22.92% and 25%: 22.92 + 25 − (22.92 × 25 ÷ 100)
⇒ 47.92 − (573 ÷ 100) = 47.92 − 5.73 = 42.19%

∴ The correct answer is 42.19%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). SP = MP × (100 − D)% ÷ 100.

Equivalent Discount Order

The order in which successive discounts are applied does not change the final single equivalent discount.

Shortcut Trick

Let the dimensions be l : b : h = 5x : 4x : 4x.

The three adjacent face areas are lb = 160 cm², bh = 64 cm², and lh = 40 cm².

Multiplying these: (l × b × h)² = 160 × 64 × 40 ⇒ Volume (V) = √(409600) = 640 cm³.

Using V = l × b × h ⇒ 5x × 4x × 4x = 640 ⇒ 80x³ = 640 ⇒ x = 2.

Thus, dimensions are l = 10 cm, b = 8 cm, and h = 8 cm.

Lateral Surface Area = 2h(l + b) = 2 × 8 × (10 + 8) = 16 × 18 = 288 cm².

∴ The correct answer is 288 cm².

Alternate Method

Given ratio of dimensions of the cuboid: l : b : h = 5 : 4 : 4

Let l = 5x, b = 4x, and h = 4x

Areas of adjacent faces: A₁ = 160 cm², A₂ = 64 cm², A₃ = 40 cm²

Formula for Volume (V) of a cuboid from its adjacent face areas:

⇒ V = √(A₁ × A₂ × A₃)

⇒ V = √(160 × 64 × 40)

⇒ V = √(409600) = 640 cm³

Also, Volume of a cuboid:

⇒ V = l × b × h

⇒ 640 = 5x × 4x × 4x

⇒ 640 = 80x³

⇒ x³ = 8

⇒ x = 2

Substitute the value of x to find the actual dimensions:

⇒ l = 5 × 2 = 10 cm

⇒ b = 4 × 2 = 8 cm

⇒ h = 4 × 2 = 8 cm

Formula for Lateral Surface Area of a cuboid:

⇒ LSA = 2h(l + b)

⇒ LSA = 2 × 8 × (10 + 8)

⇒ LSA = 16 × 18

⇒ LSA = 288 cm²

∴ The correct answer is 288 cm².

Additional Information

Total Surface Area (TSA) of Cuboid

The total area of all six faces of a cuboid is given by: TSA = 2(lb + bh + lh).

Volume and Adjacent Face Area Relation

If the areas of three adjacent faces are A, B, and C, then the volume V of the cuboid is V = √(A × B × C).

Diagonal of a Cuboid

The length of the longest rod that can be placed inside a cuboid is: d = √(l² + b² + h²).

4

Shortcut Trick

When one person's age is the arithmetic mean of two others, the middle person's age is simply the average of the other two.

Age of Akhil = (Sum of ages of Vijay and Karthik) ÷ 2

Age of Akhil = 84 ÷ 2 = 42 years

∴ The correct answer is 42 years.

Alternate Method

Given:

Sum of ages of Vijay (V) and Karthik (K) = 84 years

Akhil (A) is as much older than Vijay as he is younger than Karthik.

Calculations:

⇒ Age difference 1: Akhil − Vijay = A − V

⇒ Age difference 2: Karthik − Akhil = K − A

⇒ According to question: A − V = K − A

⇒ A + A = K + V

⇒ 2A = K + V

⇒ 2A = 84

⇒ A = 84 ÷ 2 = 42

∴ The correct answer is 42 years.

Additional Information

Arithmetic Progression in Ages

When a person is equally older and younger than two others, their ages form an Arithmetic Progression (AP) where the middle person is the average.

Average of Ages

The average of n people is the Total Sum of Ages divided by n. For two people, it is (Age₁ + Age₂) ÷ 2.

Ratio & Proportion in Ages

If ratios are given for different time periods (past/future), use the property that the age difference between any two people always remains constant over time.

Shortcut Trick

Height of a cuboid = Volume ÷ (Length × Breadth)

Height = 9198 ÷ (21 × 2)

Height = 9198 ÷ 42

Height = 219 cm

∴ The correct answer is 219 cm.

Alternate Method

Given:

Length (l) = 21 cm

Breadth (b) = 2 cm

Volume (V) = 9198 cm³

Formula Used:

Volume of a Cuboid = Length × Breadth × Height

Calculations:

⇒ Volume = l × b × h

⇒ 9198 = 21 × 2 × Height

⇒ 9198 = 42 × Height

⇒ Height = 9198 ÷ 42

⇒ Height = 219 cm

∴ The correct answer is 219 cm.

Additional Information

Total Surface Area of Cuboid

The total area of all six faces: TSA = 2(lb + bh + hl).

Lateral Surface Area of Cuboid

The area of the four side faces (excluding top and bottom): LSA = 2h(l + b).

Diagonal of a Cuboid

The longest rod that can be placed inside: Diagonal = √(l² + b² + h²).

Shortcut Trick

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

CP ÷ 45,000 = (100 − 44) ÷ (100 + 80).

CP ÷ 45,000 = 56 ÷ 180.

CP = (56 ÷ 180) × 45,000.

CP = 56 × 250 = 14,000.

∴ The correct answer is ₹14,000.

Alternate Method

Given → Marked Price (MP) = ₹45,000, Discount = 44%, Profit = 80%.
Formulas: SP = MP × (100 − D%) ÷ 100 | CP = SP × 100 ÷ (100 + P%).

⇒ Step 1: Calculate Selling Price (SP) = 45,000 × (100 − 44) ÷ 100.
⇒ SP = 45,000 × 0.56 = 25,200.
⇒ Step 2: Calculate Cost Price (CP) using SP and Profit.
⇒ CP = 25,200 × 100 ÷ (100 + 80).
⇒ CP = 25,200 × 100 ÷ 180 = 25,200 ÷ 1.8.
⇒ CP = 14,000.

∴ The correct answer is ₹14,000.

Additional Information

Relation between CP and MP

When both discount percentage and profit percentage are given, the ratio of Cost Price to Marked Price is always (100 − D%) : (100 + P%).

Marked Price & Discount

Marked Price is the price printed on the product. Discount is always calculated on the Marked Price to arrive at the Selling Price.

Shortcut Trick

Simplify ratios: Second : First = 2:12 = 1:6 and Third : Second = 5:10 = 1:2.

Combine: First : Second : Third = (6×2) : (1×2) : (1×1) = 12 : 2 :

1. Total units = 12 + 2 + 1 = 15 units → 270.

1 unit = 270 ÷ 15 = 18.

Second number = 2 units = 2 × 18 = 36.

∴ The correct answer is 36.

Alternate Method

Given: Sum of numbers = 270. Ratio N₂ : N₁ = 2:12. Ratio N₃ : N₂ = 5:10.

Calculations:

⇒ Let numbers be N₁, N₂, N₃.

⇒ N₁ : N₂ = 12 : 2 = 6 :

1. ⇒ N₂ : N₃ = 10 : 5 = 2 :
2. ⇒ To equalize N₂, multiply first ratio by 2: N₁ : N₂ = 12 : 2.

⇒ Combined ratio N₁ : N₂ : N₃ = 12 : 2 :

1. ⇒ Sum of ratios = 12 + 2 + 1 = 15 parts.

⇒ 15 parts = 270.

⇒ 1 part = 270 ÷ 15 = 18.

⇒ Second number (N₂) = 2 parts = 2 × 18 = 36.

∴ The correct answer is 36.

Additional Information

Compound Ratio

When ratios A:B and B:C are given, the combined ratio A:B:C is found by making the common term B equal in both sets.

Proportion Rule

If a:b = c:d, then product of extremes (a×d) equals product of means (b×c).

Ratio Scaling

A ratio remains unchanged if both terms are multiplied or divided by the same non-zero number.

Shortcut Trick

Simplify both fractions first: 1/30 and 2/60 = 1/30.

Since both values are identical, the percentage is 100%.

∴ The correct answer is 100%.

Alternate Method

Given:

First value = 1/30

Second value (Base) = 2/60

Formula Used:

Percentage = (Value / Base) × 100

Calculations:

⇒ Simplify the base value: 2/60 = 1/30

⇒ Percentage = [(1/30) / (1/30)] × 100

⇒ Percentage = 1 × 100

⇒ Percentage = 100%

∴ The correct answer is 100%.

Additional Information

Percentage Change

Percentage change = [(New Value − Old Value) / Old Value] × 100.

Fraction to Percentage

To convert any fraction a/b to a percentage, multiply the fraction by 100 and add the % symbol.

Base Change Property

If Value A is x% of Value B, then Value B is (10000/x)% of Value A.

Shortcut Trick

√48 = 4√3 and √75 = 5√3

⇒ (4√3 + 5√3) ÷ √3 = 9√3 ÷ √3

⇒ = 9

∴ Answer = 9

Alternate Method

Given:

(√48 + √75) ÷ √3

Concept Used:

Surd simplification and BODMAS rule

Calculation:

⇒ (√48 + √75) ÷ √3

⇒ (4√3 + 5√3) ÷ √3

⇒ 9√3 ÷ √3

⇒ 9

∴ The correct answer is 9.

Additional Information

Surd Simplification

√48 = √(16×3) = 4√3 and √75 = √(25×3) = 5√3

BODMAS Rule

Follow order: Brackets → Division → Addition

Like Terms

Combine terms with same root: a√x + b√x = (a+b)√x

Division Rule

√a ÷ √a = 1

Shortcut Trick

Ratio of Girls to Boys = 1 : 7

Given, 1 unit (Girls) = 300

Number of boys = 7 units = 7 × 300

∴ The correct answer is 2100.

Alternate Method

Given:

Ratio of Girls : Boys = 1 : 7

Total number of Girls = 300

Calculations:

⇒ Let the number of girls be 1x and boys be 7x.

⇒ 1x = 300

⇒ x = 300

⇒ Number of boys = 7x

⇒ Number of boys = 7 × 300

⇒ Number of boys = 2100

∴ The correct answer is 2100.

Additional Information

Ratio and Proportion

A ratio compares two quantities. If a : b is the ratio, then the quantities can be expressed as ax and bx where x is the common multiplier.

Total Quantity

To find the total number of participants, add the parts: (1 + 7) units = 8 units. Total = 8 × 300 = 2400.

Direct Proportion

When the number of girls increases, the number of boys increases to maintain the constant ratio 1/7.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) = 388 : 485.

Dividing both by their HCF (97): 388 ÷ 97 : 485 ÷ 97 = 4 : 5.

Profit = 5 − 4 = 1 unit.

Profit% = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given → Cost Price (CP) = ₹388, Selling Price (SP) = ₹485.
Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100.

Steps:
⇒ Profit = 485 − 388 = ₹97.
⇒ Profit% = (97 ÷ 388) × 100.
⇒ Profit% = (1 ÷ 4) × 100.
⇒ Profit% = 25%.

∴ The correct answer is 25%.

Additional Information

Profit and Loss Basics

Profit occurs when the selling price is greater than the cost price. It is calculated on the Cost Price (CP) unless otherwise stated.

Percentage Calculation

To convert a fraction to a percentage, multiply by 100. For example, 1/4 × 100 = 25% and 1/5 × 100 = 20%.

Shortcut Trick

The average of the first n odd natural numbers is always equal to n.

Here, the question asks for the average of the first 8 odd natural numbers.

⇒ n = 8

⇒ Average = 8

∴ The correct answer is 8.

Alternate Method

Given: First 8 odd natural numbers are 1, 3, 5, 7, 9, 11, 13, and 15.

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15

⇒ Sum = 64

⇒ Average = 64 ÷ 8

⇒ Average = 8

∴ The correct answer is 8.

Additional Information

Sum of First n Odd Numbers

The sum of the first n odd natural numbers is always calculated as n².

Average of First n Even Numbers

The average of the first n even natural numbers is given by the formula (n + 1).

Average of Arithmetic Progression (AP)

For any sequence in AP (like consecutive odd/even numbers), Average = (First term + Last term) ÷ 2.

Shortcut Trick

Let total voters = 100%.

Non-voters = 10%, so voted = 90%.

Winner = 49% of total = 49%.

Loser = 90% − 49% = 41%.

Difference (8%) = 120 votes ⇒ 1% = 15 votes ⇒ 100% = 1500 votes.

∴ The correct answer is 1500.

Alternate Method

Given:

Winner gets 49% of total electoral list.

10% voters did not vote.

Winner defeats loser by 120 votes.

⇒ Let total voters = x.

⇒ Winner's votes = 0.49x.

⇒ Total votes polled = 0.90x.

⇒ Loser's votes = 0.90x − 0.49x = 0.41x.

⇒ Difference = 0.49x − 0.41x = 120.

⇒ 0.08x = 120.

⇒ x = 120 ÷ 0.08 = 1500.

∴ The correct answer is 1500.

Additional Information

Percentage Change

Percentage change = (Difference ÷ Original Value) × 100.

Valid Votes vs Total Votes

Always distinguish between % of total voters and % of valid votes cast. If winner gets % of valid votes, subtract invalid/non-voters from total first.

Unitary Method in Elections

Find the value of 1% (or 1 unit) by dividing the vote margin by the percentage difference between candidates.

Shortcut Trick

In a 30°−60°−90° triangle, the ratio of sides opposite to these angles is 1 : √3 : 2.

Side opposite to 30° (Height) = 1 unit = 60 m.

Side opposite to 60° (Distance) = √3 units = 60 × √3 m.

Distance = 60 × 1.7 = 102 m.

∴ The correct answer is 102 m.

Alternate Method

Given:

Height of tower (AB) = 60 m

Angle of depression = 30° ⇒ Angle of elevation at car (C) = 30°

Formula Used:

tan(θ) = Perpendicular ÷ Base

Calculations:

⇒ tan(30°) = AB ÷ BC

⇒ 1/√3 = 60 ÷ BC

⇒ BC = 60 × √3

⇒ BC = 60 × 1.7

⇒ BC = 102 m

∴ The correct answer is 102 m.

Additional Information

Angle of Depression

The angle formed by the line of sight with the horizontal when looking down at an object. It is numerically equal to the angle of elevation due to alternate interior angles.

Trigonometric Ratios for Standard Angles

Common values include tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) = √3.

Height and Distance Relationship

Distance = Height × cot(θ). In this case, 60 × cot(30°) = 60 × √3.

Shortcut Trick

The expression is based on the algebraic identity: (a³ + b³ + c³ − 3abc) ÷ (a² + b² + c² − ab − bc − ca) = (a + b + c).

Identifying the values: a = 6.9, b = 2.8, and c = 6.3.

The result is simply the sum of these three numbers.

Sum = 6.9 + 2.8 + 6.3 = 16.0.

∴ The correct answer is 16.

Alternate Method

Given:

Numerator = (6.9)³ + (2.8)³ + (6.3)³ − 3 × 6.9 × 2.8 × 6.3

Denominator = (6.9)² + (2.8)² + (6.3)² − (6.9 × 2.8) − (2.8 × 6.3) − (6.9 × 6.3)

Formula Used:

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

Calculations:

Let a = 6.9, b = 2.8, and c = 6.3.

⇒ Expression = [(a + b + c)(a² + b² + c² − ab − bc − ca)] ÷ (a² + b² + c² − ab − bc − ca)

⇒ Expression = a + b + c

⇒ Expression = 6.9 + 2.8 + 6.3

⇒ Expression = 16.0

∴ The correct answer is 16.

Additional Information

Standard Algebraic Identity

The cubic expansion a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca) is a fundamental factorization tool in algebra.

Special Case (Conditional Identity)

If (a + b + c) = 0, then the expression a³ + b³ + c³ − 3abc becomes 0, which implies a³ + b³ + c³ = 3abc.

Related Form

Another version of the second bracket is ½[(a−b)² + (b−c)² + (c−a)²], often used in quantitative aptitude problems.

Shortcut Trick

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

∴ ∠LZK = (∠I + ∠J) ÷ 2

⇒ ∠LZK = (11° + 79°) ÷ 2

⇒ ∠LZK = 90° ÷ 2 = 45°

∴ The correct answer is 45°.

Alternate Method

Given:

In quadrilateral IJKL:

∠I = 11°

∠J = 79°

Formula Used:

• Sum of all interior angles of a quadrilateral = 360°
• Sum of angles of a triangle = 180°

Calculations:

In quadrilateral IJKL, the sum of all interior angles is 360°:

⇒ ∠I + ∠J + ∠K + ∠L = 360°

Substitute the given values of ∠I and ∠J:

⇒ 11° + 79° + ∠K + ∠L = 360°

⇒ 90° + ∠K + ∠L = 360°

⇒ ∠K + ∠L = 360° − 90°

⇒ ∠K + ∠L = 270°

Since KZ and LZ are the angle bisectors of ∠K and ∠L respectively:

⇒ ∠ZKL = ∠K ÷ 2

⇒ ∠ZLK = ∠L ÷ 2

In ▵LZK, the sum of angles is 180°:

⇒ ∠LZK + ∠ZKL + ∠ZLK = 180°

⇒ ∠LZK + (∠K ÷ 2) + (∠L ÷ 2) = 180°

⇒ ∠LZK + (∠K + ∠L) ÷ 2 = 180°

Substitute the value of (∠K + ∠L = 270°):

⇒ ∠LZK + (270° ÷ 2) = 180°

⇒ ∠LZK + 135° = 180°

⇒ ∠LZK = 180° − 135°

⇒ ∠LZK = 45°

∴ The correct answer is 45°.

Option Analysis:

• Option (1) 31°: Incorrect, as the calculated value of ∠LZK is 45°, not 31°.
• Option (2) 45°: Correct, as verified by the angle sum properties of quadrilaterals and triangles.
• Option (3) 55°: Incorrect, as the calculated value of ∠LZK is 45°, not 55°.
• Option (4) 51°: Incorrect, as the calculated value of ∠LZK is 45°, not 51°.

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 remaining two angles. For quadrilateral IJKL, ∠LZK = (∠I + ∠J) ÷ 2.

Sum of Interior Angles of a Polygon

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

Angle Bisectors of Opposite Angles

The angle between the bisectors of two opposite angles of a quadrilateral is equal to half the difference of the other two angles.