Quantitative Aptitude PYQs

Shortcut Trick

Total distance for one round = 4 × 25 = 100 m.

Speed in m/s = 3 × (5/18) = 5/6 m/s.

Time = Distance ÷ Speed = 100 ÷ (5/6) = 100 × (6/5) = 120 seconds.

∴ The correct answer is 120 seconds.

Alternate Method

Given: Side of square field = 25 m, Speed = 3 km/hr.

Formula Used: Perimeter of square = 4 × Side; Time = Distance ÷ Speed; 1 km/hr = 5/18 m/s.

⇒ Total Distance = 4 × 25 = 100 m

⇒ Speed in m/s = 3 × (5/18) = 5/6 m/s

⇒ Time = 100 ÷ (5/6)

⇒ Time = (100 × 6) ÷ 5

⇒ Time = 600 ÷ 5 = 120 seconds

∴ The correct answer is 120 seconds.

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.

Square Properties

Perimeter of a square is 4a and Area is a², where 'a' is the length of the side.

Speed-Distance-Time

The basic relationship is Distance = Speed × Time. Ensure all units are in the same system (SI or Metric) before calculation.

Shortcut Trick

Extra distance covered = 25 − 20 = 5 km

Extra charge paid = ₹245 − ₹200 = ₹45

Charge per km = 45 ÷ 5 = ₹9/km

Fixed charge = Total charge (20 km) − (20 × 9) = 200 − 180 = ₹20

Charge for 1 km = Fixed charge + (1 × 9) = 20 + 9 = ₹29

∴ The correct answer is ₹29.

Alternate Method

Given:

Charge for 20 km = ₹200

Charge for 25 km = ₹245

Formula Used:

Total Charge = Fixed Charge + (Distance × Rate per km)

Calculations:

Let Fixed Charge be F and Rate per km be R.

⇒ F + 20R = 200 --- (i)

⇒ F + 25R = 245 --- (ii)

Subtracting (i) from (ii):

⇒ 5R = 45

⇒ R = 9

Substitute R in (i):

⇒ F + 20(9) = 200

⇒ F + 180 = 200

⇒ F = 20

Charge for 1 km:

⇒ Total = F + 1(R) = 20 + 9

⇒ Total = ₹29

∴ The correct answer is ₹29.

Additional Information

Linear Equation Modeling

Problems involving a constant base and a variable rate are solved using the linear form y = mx + c, where c is the fixed component and m is the variable rate.

Unitary Method in Charges

When the difference between two totals is divided by the difference in quantities, it directly yields the variable rate per unit.

Applications

This logic applies to electricity bills (Fixed meter rent + units consumed) and library fines (Fixed base + daily charges).

Shortcut Trick

Apply the Algebraic Identity: (a + b)² − (a − b)² = 4ab.

Here, a = 2x and b = 3.

The difference (2x + 3)² − (2x − 3)² = 4 × (2x) × (3) = 24x.

Substituting back into the expression: 24x − 24x = 0.

∴ The correct answer is 0.

Alternate Method

Given: (2x + 3)² − (2x − 3)² − 24x

Formula Used: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ Expand (2x + 3)² = (2x)² + 2 × (2x) × 3 + 3² = 4x² + 12x + 9

⇒ Expand (2x − 3)² = (2x)² − 2 × (2x) × 3 + 3² = 4x² − 12x + 9

⇒ Substitute these expansions: (4x² + 12x + 9) − (4x² − 12x + 9) − 24x

⇒ 4x² + 12x + 9 − 4x² + 12x − 9 − 24x

⇒ (4x² − 4x²) + (12x + 12x − 24x) + (9 − 9)

⇒ 0 + 0 + 0 = 0

∴ The correct answer is 0.

Additional Information

Difference of Two Squares Identity

The identity (a + b)² − (a − b)² = 4ab is useful for quickly solving algebraic simplifications involving binomial squares.

Sum of Two Squares Identity

Another common identity is (a + b)² + (a − b)² = 2(a² + b²), which cancels the middle product term (2ab).

BODMAS Rule

When simplifying polynomial expressions, always solve bracketed powers (exponents) first, then addition and subtraction from left to right.

Shortcut Trick

Ratio of the prices = 3 : 5 : 7.

Average of the ratio parts = (3 + 5 + 7) ÷ 3 = 5 parts.

Given value of 5 parts = ₹16,650 ⇒ 1 part = 16,650 ÷ 5 = ₹3,330.

Price of the costliest item (7 parts) = 7 × 3,330 = ₹23,310.

∴ The correct answer is ₹23,310.

Alternate Method

Given:

Average price of 3 items = ₹16,650

Ratio of their prices = 3 : 5 : 7

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Average (5 parts) = ₹16,650 3x Item 1 5x Item 2 7x Item 3 Costliest = ?

Calculations:

⇒ Let the prices of the three items be 3x, 5x, and 7x respectively.

⇒ Sum of their prices = 3x + 5x + 7x = 15x

⇒ Average price = 15x ÷ 3 = 5x

⇒ As per the given information, 5x = 16,650

⇒ x = 16,650 ÷ 5 = 3,330

⇒ The costliest item represents the largest ratio part (7x).

⇒ Price of the costliest item = 7 × 3,330 = 23,310

∴ The correct answer is ₹23,310.

Additional Information

Arithmetic Progression in Averages

When a given set of numbers are in an arithmetic sequence (like 3x, 5x, 7x having a common difference of 2x), their average is exactly equal to the middle term. Here, the average directly becomes 5x.

Weighted Average

When different groups have different averages, the overall average is given by the formula: A = (N₁A₁ + N₂A₂) ÷ (N₁ + N₂), where N represents the number of elements and A represents their respective averages.

Properties of Ratio

A ratio a : b remains unchanged if both terms are multiplied or divided by the same non-zero number. Thus, their actual absolute values can be represented as ax and bx by multiplying with a common variable x.

Shortcut Trick

Mode is defined as the observation that occurs most frequently in a data set.

Count the occurrences of each number: 16 repeats 4 times, while 17 repeats 3 times and 7 repeats 3 times.

Since 16 has the highest frequency (4), it is the mode of the given data.

∴ The correct answer is 16.

Alternate Method

Given Data: 22, 16, 17, 49, 7, 16, 45, 17, 7, 6, 16, 17, 22, 7, 16, 44

Formula Used: Mode = Value with the Maximum Frequency

To find the mode, let us calculate the frequency of each distinct value in the set:

⇒ Frequency of 6 = 1

⇒ Frequency of 7 = 3

⇒ Frequency of 16 = 4

⇒ Frequency of 17 = 3

⇒ Frequency of 22 = 2

⇒ Frequency of 44 = 1

⇒ Frequency of 45 = 1

⇒ Frequency of 49 = 1

⇒ Maximum Frequency is 4, which corresponds to the value 16.

∴ The correct answer is 16.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Bimodal and Multimodal Data

A data set with two values having the highest frequency is called bimodal. If there are more than two, it is multimodal.

Mean and Median

Mean is the average (Sum ÷ Count), and Median is the middle value when data is arranged in ascending order.

Shortcut Trick

Relative speed in same direction = 115 − 61 = 54 kmph.

Relative speed in m/s = 54 × (5/18) = 15 m/s.

Total distance covered = Speed × Time = 15 × 116 = 1740 m.

Length of one train = Total Distance ÷ 2 = 1740 ÷ 2 = 870 m.

∴ The correct answer is 870 m.

Alternate Method

Given:

Speed of Rajdhani Express (S₁) = 115 km/hr

Speed of Garib Rath Express (S₂) = 61 km/hr

Time taken to cross (T) = 116 seconds

Let the length of each train be 'L' meters.

Formula Used:

Distance = Relative Speed × Time

Calculations:

⇒ Relative Speed (V_rel) = S₁ − S₂ = 115 − 61 = 54 km/hr

⇒ V_rel in m/s = 54 × (5/18) = 15 m/s

⇒ Total Distance = L + L = 2L

⇒ 2L = 15 × 116

⇒ 2L = 1740

⇒ L = 870 m

∴ The correct answer is 870 m.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed = S₁ − S₂. When they move in opposite directions, relative speed = S₁ + S₂.

Unit Conversion

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

Train Crossing Distance

When a train crosses another train, the total distance covered is the sum of the lengths of both trains (L₁ + L₂).

Shortcut Trick

Let Savings = 1 unit. Expenditure (100% more) = 2 units. Income = 1 + 2 = 3 units.

Net % change in Income = (Expenditure % change × Expenditure + Savings % change × Savings) ÷ Total Income

Net change = (−2% × 2 + 22% × 1) ÷ 3 = (−4 + 22) ÷ 3 = 18% ÷ 3 = 6%

∴ The correct answer is 6%.

Alternate Method

Given: Expenditure is 100% more than Savings. Expenditure decrease = 2%, Savings increase = 22%.

Formula: Income = Expenditure + Savings

⇒ Let Savings = 100

⇒ Expenditure = 100 + 100% of 100 = 200

⇒ Initial Income = 100 + 200 = 300

⇒ New Savings = 100 + 22% of 100 = 122

⇒ New Expenditure = 200 − 2% of 200 = 200 − 4 = 196

⇒ New Income = 122 + 196 = 318

⇒ Increase in Income = 318 − 300 = 18

⇒ % Increase = (18 ÷ 300) × 100 = 6%

∴ The correct answer is 6%.

Additional Information

Income-Expenditure Concept

Income is always the sum of Expenditure and Savings. Any change in Income is the weighted average of changes in Expenditure and Savings.

Percentage Change

Percentage change = (New Value − Old Value) ÷ Old Value × 100. Always ensure the base is the original value.

Alligation in Income

Alligation can be used where Expenditure % change and Savings % change are the components and Income % change is the mean value.

Shortcut Trick

Using the fundamental trigonometric identity: cos²θ = 1 − sin²θ

Given sinθ = 3/5, we get:

cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25

∴ The correct answer is 16/25.

Alternate Method

Given: sinθ = 3/5

Formula Used:

In a right-angled triangle, sinθ = Perpendicular / Hypotenuse

By Pythagoras theorem: Base² = Hypotenuse² − Perpendicular²

cosθ = Base / Hypotenuse

Calculations:

⇒ Perpendicular (P) = 3

⇒ Hypotenuse (H) = 5

⇒ Base (B) = √(5² − 3²) = √(25 − 9) = √16 = 4

⇒ cosθ = Base / Hypotenuse = 4/5

⇒ cos²θ = (4/5)² = 16/25

∴ The correct answer is 16/25.

Additional Information

Trigonometric Identities

The three basic Pythagorean identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Pythagorean Triplets

Common integer sets that satisfy the Pythagoras theorem are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Shortcut Trick

Find semi-perimeter (s): s = (41 + 84 + 85) ÷ 2 = 105 cm.

Calculate Area using Heron's formula: √[105 × (105 − 41) × (105 − 84) × (105 − 85)] = √[105 × 64 × 21 × 20] = 1680 cm².

Altitude to base 84 cm = (2 × Area) ÷ Base = (2 × 1680) ÷ 84 = 40 cm.

∴ The correct answer is 40 cm.

Alternate Method

Given: Sides of the triangle are a = 41 cm, b = 84 cm, and c = 85 cm.

Formula Used: Heron's Formula for Area (A) = √[s(s − a)(s − b)(s − c)] and Area = (1/2) × base × altitude.

⇒ Semi-perimeter s = (41 + 84 + 85) ÷ 2 = 210 ÷ 2 = 105 cm

⇒ Area = √[105 × (105 − 41) × (105 − 84) × (105 − 85)]

⇒ Area = √[105 × 64 × 21 × 20]

⇒ Area = √[21 × 5 × 64 × 21 × 5 × 4]

⇒ Area = 21 × 5 × 8 × 2 = 1680 cm²

⇒ Let the altitude corresponding to side 84 cm be h.

⇒ 1680 = (1/2) × 84 × h

⇒ 1680 = 42 × h

⇒ h = 1680 ÷ 42 = 40 cm

∴ The correct answer is 40 cm.

Additional Information

Heron's Formula

Used to find the area of a triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter.

Altitude of a Triangle

The perpendicular segment from a vertex to the opposite side. Area = (1/2) × base × altitude.

Pythagorean Property

For a right-angled triangle with sides a, b and hypotenuse c, a² + b² = c². The triangle in this problem is not a right triangle as 41² + 84² ≠ 85².

Shortcut Trick

The volume ratio depends only on the multiplicative increase of each dimension.

Length increase: 25% → Factor = 1 + 1/4 = 5/4

Breadth increase: 33 1/3% → Factor = 1 + 1/3 = 4/3

Height increase: 40% → Factor = 1 + 2/5 = 7/5

Net Volume Factor = (5/4) × (4/3) × (7/5) = 7/3

∴ The correct answer is 7/3 times.

Alternate Method

Given:

Ratio of Length (L) : Breadth (B) : Height (H) = 2/3 : 1/4 : 5/6

Percent changes: L (+25%), B (+33 1/3%), H (+40%)

Calculations:

⇒ Simplify ratio L : B : H by multiplying by LCM(3, 4, 6) = 12:

⇒ L = (2/3) × 12 = 8, B = (1/4) × 12 = 3, H = (5/6) × 12 = 10

⇒ Original Volume (V₁) = 8 × 3 × 10 = 240 units³

⇒ New Length (L') = 8 + 25% of 8 = 8 + 2 = 10

⇒ New Breadth (B') = 3 + 33 1/3% of 3 = 3 + 1 = 4

⇒ New Height (H') = 10 + 40% of 10 = 10 + 4 = 14

⇒ New Volume (V₂) = 10 × 4 × 14 = 560 units³

⇒ Ratio V₂ / V₁ = 560 / 240 = 56 / 24 = 7 / 3

∴ The correct answer is 7/3 times.

Additional Information

Total Surface Area of Cuboid

The total surface area is given by 2(LB + BH + HL). It represents the sum of areas of all 6 faces.

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is its diagonal, calculated as √(L² + B² + H²).

Successive Percentage Change

When three dimensions change by x%, y%, and z%, the net volume change factor is (1 + x/100)(1 + y/100)(1 + z/100).

Shortcut Trick

Profit = Selling Price (SP) − Cost Price (CP) = 45,000 − 40,000 = 5,000.

Profit% = (Profit ÷ CP) × 100.

Profit% = (5,000 ÷ 40,000) × 100 = (1 ÷ 8) × 100.

We know the fraction 1/8 is equivalent to 12.5%.

∴ The correct answer is 12.5%.

Alternate Method

Given:

Cost Price (CP) of the laptop = ₹40,000

Selling Price (SP) of the laptop = ₹45,000

Formulas:

Profit = SP − CP

Profit% = (Profit ÷ CP) × 100

Steps:

⇒ Total Profit = 45,000 − 40,000 = 5,000

⇒ Profit Percentage = (5,000 ÷ 40,000) × 100

⇒ Profit Percentage = (5 ÷ 40) × 100

⇒ Profit Percentage = 1/8 × 100 = 12.5%

∴ The correct answer is 12.5%.

Additional Information

Cost Price (CP)

The price at which an article is purchased. Overhead expenses like transport are added to it.

Selling Price (SP)

The price at which an article is sold to a customer.

Gain or Loss

If SP > CP, it is a Gain. If CP > SP, it is a Loss. Both are always calculated on the Cost Price unless specified otherwise.

Shortcut Trick

Total deduction = 92 + 9 + 48 = ₹149.

Remaining sum = 7085 − 149 = ₹6936.

Ratio = 3 : 15 : 16 → Total units = 3 + 15 + 16 = 34.

1 unit = 6936 ÷ 34 = ₹204.

Difference between Y and Z shares = (16 − 15) units + (48 − 9) = 1 × 204 + 39 = ₹243.

∴ The correct answer is ₹243.

Alternate Method

Given: Total amount = ₹7,085. Deductions: X = ₹92, Y = ₹9, Z = ₹48. New ratio = 3 : 15 : 16.

Calculations:

⇒ Net Amount = 7085 − (92 + 9 + 48) = 7085 − 149 = ₹6936

⇒ Total Ratio Parts = 3 + 15 + 16 = 34 units

⇒ Value of 1 unit = 6936 ÷ 34 = ₹204

⇒ Y's share after deduction = 15 × 204 = ₹3060

⇒ Z's share after deduction = 16 × 204 = ₹3264

⇒ Y's original share = 3060 + 9 = ₹3069

⇒ Z's original share = 3264 + 48 = ₹3312

⇒ Difference = 3312 − 3069 = ₹243

∴ The correct answer is ₹243.

Additional Information

Ratio and Proportion

Ratio represents the relative size of two or more values. To find the actual value, multiply each part of the ratio by a common constant k.

Deduction in Ratios

When specific amounts are deducted from shares, the total sum is reduced by the sum of all deductions before applying the ratio division.

Difference between Original Shares

Difference = (New share difference) ± (Difference in deductions). Always calculate original shares individually or adjust the unit difference carefully.

Shortcut Trick

List the first 21 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73.

Group them for faster addition: (2+3+5+7+11+13+17+19+23+29) + (31+37+41+43+47+53+59+61+67+71) + 73.

Sum = 129 + 510 + 73 = 712.

∴ The correct answer is 712.

Alternate Method

Given: Find the sum of the first 21 prime numbers.

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73.

2 + 3 + ... + 71 + 712

⇒ Sum of first 5: 2 + 3 + 5 + 7 + 11 = 28

⇒ Sum of next 5: 13 + 17 + 19 + 23 + 29 = 101

⇒ Sum of next 5: 31 + 37 + 41 + 43 + 47 = 199

⇒ Sum of next 5: 53 + 59 + 61 + 67 + 71 = 311

⇒ Total Sum = 28 + 101 + 199 + 311 + 73 = 712

∴ The correct answer is 712.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The only even prime number is 2.

Sum of Primes up to N

There is no simple algebraic formula for the sum of the first n prime numbers; it must be calculated by listing and summing the terms.

Primality Test

To check if a number n is prime, test for divisibility by all primes less than or equal to √n.

Shortcut Trick

In a proportion a : b :: c : d, the product of means equals the product of extremes.

Product of extremes = P × 21

Product of means = 14 × 57

P = (14 × 57) ÷ 21 = (2 × 57) ÷ 3 = 2 × 19 = 38

∴ The correct answer is 38.

Alternate Method

Given: P : 14 :: 57 : 21

Formula Used: a/b = c/d

⇒ P / 14 = 57 / 21

⇒ P = (57 / 21) × 14

⇒ P = (19 / 7) × 14

⇒ P = 19 × 2

⇒ P = 38

∴ The correct answer is 38.

Additional Information

Fourth Proportional

If a, b, c, d are in proportion, then d is called the fourth proportional, where d = (b × c) ÷ a.

Mean Proportional

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

Third Proportional

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

Shortcut Trick

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

CP : MP = (100 − 58) : (100 + 63) = 42 : 163.

Given Cost Price (CP) = 42 units = 1,848.

1 unit = 1,848 ÷ 42 = 44.

Marked Price (MP) = 163 units = 163 × 44 = 7,172.

∴ The correct answer is ₹7,172.

Alternate Method

Given: CP = ₹1,848, Discount = 58%, Profit = 63%.

Formulas: SP = CP × (100 + Profit%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%)

⇒ Selling Price (SP) = 1,848 × (100 + 63) ÷ 100 = 1,848 × 1.63 = 3,012.24

⇒ Marked Price (MP) = SP × 100 ÷ (100 − 58) = 3,012.24 × 100 ÷ 42

⇒ MP = 301,224 ÷ 42 = 7,172.

∴ The correct answer is ₹7,172.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP) to arrive at the Selling Price (SP). Formula: SP = MP − Discount.

Relation between CP and MP

When both profit and discount percentages are known, the direct relation is: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

In a 30°−60°−90° triangle, the sides are in the ratio 1 : √3 : 2.

The side opposite to 30° (base) = 10√3 m.

The side opposite to 60° (height) = Base × √3.

Height = 10√3 × √3 = 10 × 3 = 30 m.

∴ The correct answer is 30 m.

Alternate Method

Given:

Angle of depression = 60° ⇒ Angle of elevation at object = 60°.

Distance from foot of tower to object (Base) = 10√3 m.

Formula Used:

tan(θ) = Perpendicular / Base

Calculations:

⇒ tan(60°) = Height / 10√3

⇒ √3 = Height / 10√3

⇒ Height = 10√3 × √3

⇒ Height = 10 × 3 = 30 m.

∴ The correct answer is 30 m.

Additional Information

Angle of Depression

The angle formed by the line of sight with the horizontal when the object is below the horizontal level. It is numerically equal to the angle of elevation.

Standard Trigonometric Ratios

tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3. These are essential for solving heights and distances problems.

Right Triangle Properties

In any right-angled triangle, the relation between sides and angles is given by SOH-CAH-TOA (Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj).

Shortcut Trick

Total Voters = 1,20,000

Polled Votes = 72% of 1,20,000

Winning Margin = (27% − 15%) = 12% of Polled Votes

Required Votes = 1,20,000 × 0.72 × 0.12 = 10,368

∴ The correct answer is 10,368 votes.

Alternate Method

Given:

Total number of voters = 1,20,000

Voting percentage = 72%

Winner's share of polled votes = 27%

Runner-up's share of polled votes = 15%

Calculations:

⇒ Total polled votes = 72% of 1,20,000 = (72 ÷ 100) × 1,20,000 = 86,400

⇒ Winning margin percentage = 27% − 15% = 12%

⇒ Winning margin in votes = 12% of 86,400

⇒ Margin = (12 ÷ 100) × 86,400

⇒ Margin = 12 × 864 = 10,368

∴ The correct answer is 10,368 votes.

Additional Information

Percentage Change and Base

Always identify the base value (Total Voters vs. Polled Votes). Here, percentages were of polled votes, not total voters.

Effective Percentage

Effective % = (Polled %) × (Margin %). In this case, 72% × 12% = 8.64% of the total voter list.

Successive Percentage Application

Used when one percentage is applied on the result of another. Formula: Final Value = Initial Value × (p1/100) × (p2/100).

Shortcut Trick

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

Here, the number of terms n = 9.

Average = 9 + 1 = 10.

∴ The correct answer is 10.

Alternate Method

Given:

First 9 even natural numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18.

Number of terms (n) = 9

Formula Used:

Sum of first n even numbers = n(n + 1)

Average = Sum of numbers ÷ Total numbers

Calculations:

⇒ Sum of first 9 even numbers = 9 × (9 + 1)

⇒ Sum = 9 × 10 = 90

⇒ Average = 90 ÷ 9

⇒ Average = 10

∴ The correct answer is 10.

Additional Information

Average of first n natural numbers

The average of the first n natural numbers starting from 1 is (n + 1) ÷ 2.

Average of first n odd natural numbers

The average of the first n odd natural numbers (1, 3, 5, ...) is always equal to n.

Average of numbers in A.P.

For any sequence in Arithmetic Progression, Average = (First term + Last term) ÷ 2.

Shortcut Trick

For any regular polygon with n sides:

Exterior Angle = 360° ÷ n

For n = 18: Exterior Angle = 360° ÷ 18 = 20°

Since Interior Angle + Exterior Angle = 180°:

Interior Angle = 180° − 20° = 160°

∴ The correct answer is 160°.

Alternate Method

Given:

Number of sides of the regular polygon (n) = 18

Formula Used:

Measure of each interior angle of a regular polygon with n sides = ((n − 2) × 180°) ÷ n

Calculations:

Substitute n = 18 into the formula:

⇒ Interior Angle = ((18 − 2) × 180°) ÷ 18

⇒ Interior Angle = (16 × 180°) ÷ 18

⇒ Interior Angle = 16 × 10°

⇒ Interior Angle = 160°

Let us verify the other options to confirm they are incorrect:

⇒ For Option (2) 150°: The exterior angle would be 180° − 150° = 30°. The number of sides would be 360° ÷ 30° = 12, which does not match the given 18 sides.

⇒ For Option (3) 140°: The exterior angle would be 180° − 140° = 40°. The number of sides would be 360° ÷ 40° = 9, which does not match the given 18 sides.

⇒ For Option (4) 120°: The exterior angle would be 180° − 120° = 60°. The number of sides would be 360° ÷ 60° = 6, which does not match the given 18 sides.

∴ The correct answer is 160°.

Additional Information

Sum of Interior Angles of a Polygon

The sum of all interior angles of any n-sided polygon is given by the formula: Sum = (n − 2) × 180°.

Sum of Exterior Angles of a Polygon

For any convex polygon, the sum of all exterior angles is always constant and equal to 360°.

Number of Diagonals in a Polygon

The total number of diagonals in a polygon with n sides is given by the formula: Diagonals = (n × (n − 3)) ÷ 2.

Shortcut Trick

Nisha should choose the option that gives the highest equivalent discount percentage.

Option 1: 23 + 23 − (23 × 23) ÷ 100 = 46 − 5.29 = 40.71%.

Option 2: Single discount = 64%.

Option 3: 7 + 45 − (7 × 45) ÷ 100 = 52 − 3.15 = 48.85%.

Option 4: 10 + 39 − (10 × 39) ÷ 100 = 49 − 3.9 = 45.1%.

Comparing all values, 64% is significantly higher than the others.

∴ The correct answer is A single discount of 64%.

Alternate Method

Given: Marked Price (MP) = ₹5,000.
Formula: Successive Discount = a + b − (a × b) ÷ 100.

⇒ Option 1: Successive 23% and 23% ⇒ 23 + 23 − (529 ÷ 100) = 46 − 5.29 = 40.71%.

⇒ Option 2: Single discount = 64%.

⇒ Option 3: Successive 7% and 45% ⇒ 7 + 45 − (315 ÷ 100) = 52 − 3.15 = 48.85%.

⇒ Option 4: Successive 10% and 39% ⇒ 10 + 39 − (390 ÷ 100) = 49 − 3.9 = 45.1%.

⇒ Maximum advantage occurs at the highest discount percentage, which is 64%.

∴ The correct answer is A single discount of 64%.

Additional Information

Successive Discount Logic

A single discount is always better for the buyer than two successive discounts that add up to the same numerical value, because in successive discounts, the second percentage is calculated on a reduced amount.

Maximum Advantage

To gain the maximum advantage, the customer seeks the lowest Selling Price (SP), which corresponds to the highest Discount Percentage.