Quantitative Aptitude PYQs

Shortcut Trick
Given: Selling Price (SP) of 10 watches = Cost Price (CP) of 8 watches.
Using the ratio method: 10 × SP = 8 × CP.
Ratio of SP/CP = 8/10 = 4/5.
Loss = CP − SP = 5 − 4 = 1 unit.
Loss% = (Loss ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%.
∴ The correct answer is 20%.

Alternate Method
Given: Total SP of 10 watches = ₹10,500. CP of 8 watches = ₹10,500.
Formulas: CP of 1 watch = Total CP ÷ Quantity | SP of 1 watch = Total SP ÷ Quantity | Loss% = ((CP − SP) ÷ CP) × 100.

⇒ Cost Price (CP) of 8 watches = ₹10,500.
⇒ CP of 1 watch = 10,500 ÷ 8 = ₹1312.5.
⇒ Selling Price (SP) of 10 watches = ₹10,500.
⇒ SP of 1 watch = 10,500 ÷ 10 = ₹1050.
⇒ Loss per watch = 1312.5 − 1050 = ₹262.5.
⇒ Loss% = (262.5 ÷ 1312.5) × 100 = 20%.
∴ The correct answer is 20%.

Additional Information
Loss and CP Relationship
When the SP of 'n' articles equals the CP of 'm' articles, if n > m, there is always a loss. Loss% = ((n − m) ÷ n) × 100.
Percentage Calculation
Loss and Profit percentages are always calculated on the Cost Price (CP) unless stated otherwise in the problem.

Shortcut Trick

Required percentage = (Value ÷ Base Value) × 100

Required percentage = (1/30 ÷ 2/20) × 100

⇒ (1/30 × 20/2 = 10/1) × 100 = 1/3 × 100

⇒ 33.33%

∴ The correct answer is 33.33%.

Alternate Method

Given:

First fraction (Part) = 1/30

Second fraction (Base) = 2/20 = 1/10

Formula Used:

Percentage = (Part ÷ Base) × 100

Calculations:

⇒ Percentage = ((1/30) / (1/10)) × 100

⇒ Percentage = (1/30 × 10/1) × 100

⇒ Percentage = (1/3) × 100

⇒ Percentage = 33.3333...

⇒ Percentage ≈ 33.33%

∴ The correct answer is 33.33%.

Additional Information

Fraction to Percentage

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

Percentage Change

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

Reciprocal Percentages

1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16.67%.

Shortcut Trick

To find the Mode, identify the observation that repeats most frequently in the given data.

Scanning the dataset: 53 appears 5 times, whereas 40 appears 3 times.

Values 50 and 51 appear only 2 times each, and others appear once.

Since 53 has the highest Frequency, it is the mode.

∴ The correct answer is 53.

Alternate Method

Given: Data = 53, 53, 40, 49, 50, 48, 53, 50, 41, 40, 53, 45, 40, 47, 43, 52, 51, 42, 51, 53

Formula Used: Mode is the value with the Maximum Frequency in a distribution.

Calculations:

⇒ Count the frequency of each distinct number in the set:

⇒ Frequency of 53 = 5

⇒ Frequency of 40 = 3

⇒ Frequency of 50 = 2

⇒ Frequency of 51 = 2

⇒ Frequency of {40, 41, 42, 43, 45, 47, 48, 49, 52} = 1 each

⇒ The value with the highest frequency (5) is 53.

∴ The correct answer is 53.

Additional Information

Mean

The Mean is the average of all observations, calculated as Σx ÷ n.

Median

The Median is the middle value when data is arranged in ascending or descending order. For even n, it is the average of the two middle terms.

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Bimodal and Multimodal

If two values have the same maximum frequency, the data is Bimodal. If more than two, it is Multimodal.

Shortcut Trick

Total work (LCM of 25 and 30) = 150 units.

Efficiency of Pipe F = 150 ÷ 25 = 6 units/min; Efficiency of Pipe M = 150 ÷ 30 = 5 units/min.

Pipe M runs for the full 15 minutes: Work by M = 15 × 5 = 75 units.

Remaining work for F = 150 − 75 = 75 units.

Time for pipe F = 75 ÷ 6 = 12.5 minutes.

∴ The correct answer is 12.5 min.

Alternate Method

Given: Time taken by Pipe F = 25 min, Time taken by Pipe M = 30 min, Total time to fill = 15 min.

Formula Used: Work = Efficiency × Time; Total Work = LCM of individual times.

⇒ Let Pipe F be closed after t minutes.

⇒ Pipe F works for t minutes and Pipe M works for the entire 15 minutes.

⇒ (Efficiency of F × t) + (Efficiency of M × 15) = Total Work

⇒ (6 × t) + (5 × 15) = 150

⇒ 6t + 75 = 150

⇒ 6t = 75

⇒ t = 75 ÷ 6 = 12.5 minutes

∴ The correct answer is 12.5 min.

Additional Information

Pipes and Cisterns Efficiency

Efficiency is inversely proportional to the time taken to fill the tank. If a pipe fills a tank in x hours, its efficiency is 1/x part per hour.

Negative Work (Inlet vs Outlet)

Inlet pipes do positive work (filling), while outlet pipes (leaks) do negative work (emptying). Net work = Σ(Inlet efficiencies) − Σ(Outlet efficiencies).

Combined Work Formula

If two pipes fill a tank in x and y mins respectively, then together they fill it in (x × y) ÷ (x + y) mins.

Shortcut Trick

For an angle of elevation of 45°, the height of the tower is always equal to the distance from the foot.

Since the distance is 29.6 m, the height must also be 29.6 m.

∴ The correct answer is 29.6 m.

Alternate Method

Given:

Distance from the foot of the tower (BC) = 29.6 m

Angle of elevation (θ) = 45°

Formula Used:

tan θ = Perpendicular / Base = AB / BC

Calculations:

⇒ tan 45° = Height (h) / 29.6

⇒ 1 = h / 29.6

⇒ h = 29.6 × 1

⇒ h = 29.6 m

∴ The correct answer is 29.6 m.

Additional Information

Trigonometric Ratios for 45°

In a right-angled triangle with a 45° angle, the Perpendicular and Base are equal because tan 45° = 1.

Special Right Triangle (45-45-90)

The sides of a 45°−45°−90° triangle are in the ratio 1 : 1 : √2. If height is 'a', the hypotenuse is a√2.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when looking up at an object.

Shortcut Trick

For a number to be divisible by 9, the sum of its digits must be a multiple of 9.

Sum of digits of A6B59 = A + 6 + B + 5 + 9 = 20 + A + B.

Next multiples of 9 after 20 are 27, 36... ⇒ A + B = 7 (for sum 27) or 16 (for sum 36).

To minimize \( (A^2+2B) \) with A + B = 7, test pairs: (1,6) → 1+12=13; (2,5) → 4+10=14; (3,4) → 9+8=17.

∴ The correct answer is 13.

Alternate Method

Given: The number A6B59 is divisible by 9. We need to find the least value of \( (A^2+2B) \).

Divisibility Rule of 9: Sum of digits is divisible by 9.

⇒ A + 6 + B + 5 + 9 = 20 + A + B

⇒ 20 + A + B must be 27, 36, ...

Case 1: A + B = 7 (Since 20 + 7 = 27)

⇒ Possible (A, B) pairs: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0). (

Note:

A cannot be 0 as it is the first digit).

⇒ If A = 1, B = 6: \( 1^2 + 2(6) \) = 1 + 12 = 13

⇒ If A = 2, B = 5: \( 2^2 + 2(5) \) = 4 + 10 = 14

⇒ If A = 3, B = 4: \( 3^2 + 2(4) \) = 9 + 8 = 17

⇒ If A = 7, B = 0: \( 7^2 + 2(0) \) = 49

Case 2: A + B = 16 (Since 20 + 16 = 36)

⇒ Min A is 7, B = 9: \( 7^2 + 2(9) \) = 49 + 18 = 67 (Higher than Case 1 values)

⇒ Comparing all results, the minimum value is 13.

∴ The correct answer is 13.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility Rule of 11

Difference between the sum of digits at odd places and even places must be 0 or a multiple of 11.

Minimization Logic

To minimize a sum of squares and linear terms like \( A^2 + 2B \), generally, the variable with the higher power (A) should be as small as possible.

Shortcut Trick

Original Volume = Final Volume ÷ 0.8 = 10,752 ÷ 0.8 = 13,440 cm³

Volume Formula: 4/3 × 3.14 × R³ = 13,440

R³ = 13,440 × 3 ÷ 12.56 ≈ 3210.19

Since 14³ = 2744 and 15³ = 3375, R ≈ 14.75 cm.

Rounding to the nearest integer, R = 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given:

⇒ Total volume of 64 small spheres = 10,752 cm³

⇒ Metal waste percentage = 20%

⇒ Value of π = 3.14

Formula Used:

⇒ Volume of a sphere = 4/3 × π × R³

Calculations:

⇒ Let the radius of the original sphere be R.

⇒ Effective metal percentage after loss = 100% − 20% = 80%

⇒ Total volume of original sphere × 80% = Total volume of 64 spheres

⇒ Original Volume × 0.8 = 10,752

⇒ Original Volume = 10,752 ÷ 0.8 = 13,440 cm³

⇒ 4/3 × 3.14 × R³ = 13,440

⇒ 12.56 ÷ 3 × R³ = 13,440

⇒ R³ = (13,440 × 3) ÷ 12.56

⇒ R³ = 40,320 ÷ 12.56 = 3210.19

⇒ R = √³3210.19 ≈ 14.75 cm

⇒ Nearest integer to 14.75 is 15.

∴ The correct answer is 15 cm.

Additional Information

Volume Conservation in Recasting

When a solid is melted and recast into other shapes, the total volume remains constant if there is no loss. If loss occurs, Final Volume = Initial Volume − Waste.

Sphere Properties

Volume = (4/3)πr³ and Surface Area = 4πr². If the radius is scaled by factor k, the volume is scaled by k³.

Approximate Cube Roots

Memorizing basic cubes like 10³ = 1000, 15³ = 3375, and 20³ = 8000 helps in estimating the radius during exams without full calculation.

Shortcut Trick

Incomes ratio = 5 : 9 and Expenditures ratio = 1 : 3.

Since savings are equal (��51,000), the difference between income and expenditure parts must be equal.

Difference in income parts (9 − 5) = 4; Difference in expenditure parts (3 − 1) = 2.

Make differences equal: Multiply expenditure ratio by 2 → 2 : 6.

Now, Savings = 5 − 2 = 3 units. ⇒ 3 units = ₹51,000 ⇒ 1 unit = ₹17,000.

Chetan monthly income = 5 units = 5 × ₹17,000 = ₹85,000.

∴ The correct answer is ₹85,000.

Alternate Method

Given: Ratio of Monthly Incomes = 5 : 9 | Monthly Savings = ₹51,000 each | Ratio of Expenditures = 1 : 3.

Formula: Income − Savings = Expenditure

Let the incomes be 5x and 9x.

⇒ (5x − 51,000) / (9x − 51,000) = 1 / 3

⇒ 3 × (5x − 51,000) = 1 × (9x − 51,000)

⇒ 15x − 1,53,000 = 9x − 51,000

⇒ 6x = 1,02,000

⇒ x = 17,000

⇒ Chetan monthly income = 5x = 5 × 17,000 = 85,000

∴ The correct answer is ₹85,000.

Additional Information

Income, Expenditure, and Saving Relation

The basic governing formula for these problems is Income = Expenditure + Savings.

Ratio Equalization Method

When a constant value is added or subtracted from both terms of a ratio, the difference between the ratio parts should be made equal to maintain proportionality.

Cross Multiplication Method

For two ratios a:b and c:d with a constant change k: x = |(bk − dk) / (ad − bc)| helps in finding the variable value quickly.

Shortcut Trick

Ratio of numbers = 12 : 14 : 4

25% of 1^st number (12 units) = 3 units

Given: 3 units = 18 ⇒ 1 unit = 6

Difference between 2^nd and 3^rd = 14 − 4 = 10 units = 60

30% of 60 = 18

∴ The correct answer is 18.

Alternate Method

Given: Ratio of numbers = 12 : 14 : 4, 25% of the first number = 18.

Formula Used: Percentage = (Value/Total) × 100

⇒ Let the numbers be 12x, 14x, and 4x.

⇒ According to question: 25% of 12x = 18

⇒ (25/100) × 12x = 18

⇒ 3x = 18 ⇒ x = 6

⇒ Second number = 14 × 6 = 84

⇒ Third number = 4 × 6 = 24

⇒ Difference = 84 − 24 = 60

⇒ Required value = 30% of 60 = (30/100) × 60 = 18

∴ The correct answer is 18.

Additional Information

Ratio and Proportion

A ratio compares two quantities; if a:b = c:d, then product of extremes equals product of means (a × d = b × c).

Percentage Conversion

To find x% of a number N, use the formula: (x/100) × N. Common fractions: 25% = 1/4, 50% = 1/2.

Shortcut Trick

Sum of ages after 20 years = 92 years.

Present sum of ages = 92 − (2 × 20) = 52 years.

Ratio of Wife : Husband = 6 : 7 (Total parts = 13).

Value of 1 part = 52 ÷ 13 = 4 years.

Present ages = 6 × 4 and 7 × 4 = 24 and 28 years.

∴ The correct answer is 24, 28 years.

Alternate Method

Given: Present ratio = 6 : 7; Sum after 20 years = 92.

Formula: Let Present Age of Wife = 6x and Husband = 7x.

⇒ Age after 20 years: (6x + 20) and (7x + 20)

⇒ Sum: (6x + 20) + (7x + 20) = 92

⇒ 13x + 40 = 92

⇒ 13x = 52

⇒ x = 4

⇒ Wife's present age = 6 × 4 = 24 years

⇒ Husband's present age = 7 × 4 = 28 years

∴ The correct answer is 24, 28 years.

Additional Information

Age Sum Property

If the sum of ages of 'n' persons is 'S' now, then after 't' years, the sum will be S + (n × t).

Ratio & Proportion

If ages are in ratio a:b and sum is S, individual ages are S × [a/(a+b)] and S × [b/(a+b)].

Age Difference Property

The difference between the ages of two people remains constant regardless of the number of years passed.

Shortcut Trick

Direct formula for Median: Median = (Mode + 2 × Mean) ÷ 3

Substitute values: Median = (88.7 + 2 × 20) ÷ 3

Median = (88.7 + 40) ÷ 3 = 128.7 ÷ 3

Median = 42.9

∴ The correct answer is 42.9.

Alternate Method

Given:

Mean of data = 20

Mode of data = 88.7

Formula Used:

The empirical relationship between central tendencies is:

Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 88.7 = 3 × Median − 2 × 20

⇒ 88.7 = 3 × Median − 40

⇒ 3 × Median = 88.7 + 40

⇒ 3 × Median = 128.7

⇒ Median = 128.7 ÷ 3

⇒ Median = 42.9

∴ The correct answer is 42.9.

Additional Information

Empirical Relationship

This formula is applicable for moderately asymmetrical distributions. It was suggested by Karl Pearson.

Symmetrical Distribution

In a perfectly symmetrical distribution (like the Normal Distribution), the values of Mean, Median, and Mode are all equal (Mean = Median = Mode).

Skewness Property

If Mode > Median > Mean, the distribution is negatively skewed (skewed to the left). If Mean > Median > Mode, it is positively skewed.

Shortcut Trick

Given HCF = 23 and Ratio = 1 : 2 : 3.

Numbers = Ratio Parts × HCF.

First number = 1 × 23 = 23.

Second number = 2 × 23 = 46.

Third number = 3 × 23 = 69.

∴ The correct answer is 23, 46, 69.

Alternate Method

Given: Ratio of three numbers = 1 : 2 : 3 and Highest Common Factor (HCF) = 23.

Let the three numbers be 1x, 2x, and 3x.

The HCF of 1x, 2x, and 3x is the common variable x.

⇒ x = 23

⇒ First number = 1 × 23 = 23

⇒ Second number = 2 × 23 = 46

⇒ Third number = 3 × 23 = 69

∴ The correct answer is 23, 46, 69.

Additional Information

HCF and LCM Relation

For two numbers a and b, Product of numbers = HCF × LCM.

Ratio and Multiples

If numbers are in ratio a:b:c, then the actual numbers are a×H, b×H, and c×H where H is the HCF.

LCM Formula

LCM of numbers in ratio a:b:c is equal to (a × b × c × HCF) if a, b, c are co-prime.

Shortcut Trick

Successive rate for 2 years at 20% = 20 + 20 + (20 × 20)/100 = 44%.

Total Amount (%) = 100% (Principal) + 44% (Interest) = 144%.

Amount = 144% of ₹5,200 = 1.44 × 5200 = ₹7,488.

∴ The correct answer is ₹7,488.

Alternate Method

Given:

Principal (P) = ₹5,200

Rate of Interest (R) = 20% per annum

Time (n) = 2 years (Compounded Annually)

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 5200(1 + 20/100)²

⇒ A = 5200(1 + 1/5)²

⇒ A = 5200 × (6/5) × (6/5)

⇒ A = 5200 × 36 / 25

⇒ A = 208 × 36

⇒ A = ₹7,488

∴ The correct answer is ₹7,488.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate is calculated as R + R + (R × R)/100.

Compound Interest (CI)

CI is the difference between the total Amount and the Principal: CI = A − P.

Difference between CI and SI

For a period of 2 years, the difference (D) between Compound Interest and Simple Interest is D = P × (R/100)².

Shortcut Trick

Total distance = Length of A + Length of B = 150 + 100 = 250 m.

Relative speed (opposite direction) = Speed of A + Speed of B = 72 + 54 = 126 km/h.

Convert speed to m/s: 126 × (5/18) = 35 m/s.

Time = Distance ÷ Relative Speed = 250 ÷ 35 ≈ 7.14 sec.

∴ The correct answer is 7.14 sec.

Alternate Method

Given:

Length of Train A (L₁) = 150 m | Length of Train B (L₂) = 100 m

Speed of Train A (S₁) = 72 km/h | Speed of Train B (S₂) = 54 km/h

Formula Used:

Time taken to cross = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 150 + 100 = 250 m

⇒ Relative Speed = 72 + 54 = 126 km/h

⇒ Relative Speed in m/s = 126 × (5/18) = 7 × 5 = 35 m/s

⇒ Time taken = 250 ÷ 35

⇒ Time taken = 50 ÷ 7 ≈ 7.1428... sec

∴ The correct answer is 7.14 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). If they move in the same direction, the speeds are subtracted (|S₁ − S₂|).

Unit Conversion

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

Crossing Distance

When a train crosses another train or a bridge/platform, the total distance covered is always the sum of the lengths of both objects.

Shortcut Trick

Using the Multiplying Factor method: Let original price be 100.

New Price = 100 × 0.85 × 0.70 × 0.40 × 1.60

New Price = 100 × 0.595 × 0.64 = 38.08

Net change = 100 − 38.08 = 61.92% decrease.

∴ The correct answer is Decreases by 61.92%.

Alternate Method

Given: Successive changes are −15%, −30%, −60% and +60%.

Formula Used: Final Price = Initial Price × (1 ± r₁/100) × (1 ± r₂/100) × ...

⇒ Let original price = 100

⇒ Price after 1^st month = 100 − 15% of 100 = 85

⇒ Price after 2^nd month = 85 − 30% of 85 = 85 − 25.5 = 59.5

⇒ Price after 3^rd month = 59.5 − 60% of 59.5 = 59.5 − 35.7 = 23.8

⇒ Price after 4^th month = 23.8 + 60% of 23.8 = 23.8 + 14.28 = 38.08

⇒ Total Decrease = 100 − 38.08 = 61.92

⇒ Percentage Decrease = (61.92 / 100) × 100 = 61.92%

∴ The correct answer is Decreases by 61.92%.

Additional Information

Successive Percentage Change Formula

For two successive changes of a% and b%, the net change is (a + b + ab/100)%.

Base Value Stability

In successive changes, the base for the next percentage calculation is the resultant value of the previous calculation.

Multiplying Factor (MF)

For x% increase, MF = (1 + x/100); for x% decrease, MF = (1 − x/100). Net Result = Initial × MF₁ × MF₂...

Shortcut Trick

Effective rate for 2 years at 20% Compound Interest = 20 + 20 + (20 × 20)/100 = 44%.

Total Amount = 100% (Principal) + 44% (Interest) = 144%.

Amount = 1.44 × 5,000.

Amount = ₹7,200.

∴ The correct answer is ₹7,200.

Alternate Method

Given:

⇒ Principal (P) = ₹5,000

⇒ Annual Rate (R) = 20%

⇒ Time (T) = 2 years

Formula Used:

⇒ Amount (A) = P(1 + R/100)^T

Calculations:

⇒ A = 5000 × (1 + 20/100)²

⇒ A = 5000 × (1 + 1/5)²

⇒ A = 5000 × (6/5) × (6/5)

⇒ A = 5000 × 36/25

⇒ A = 200 × 36

⇒ A = ₹7,200

∴ The correct answer is ₹7,200.

Additional Information

Compound Interest Formula

Interest alone is calculated as CI = P[(1 + R/100)ⁿ − 1].

CI vs SI for 2 Years

The difference (D) between CI and SI for 2 years is given by D = P(R/100)².

Half-Yearly Compounding

If compounded half-yearly, the rate is halved (R/2) and the time is doubled (2T).

Shortcut Trick

Age ratio (Father : Son) 7 years ago = 5 : 1 (Difference = 4)

Age ratio (Father : Son) 7 years hence = 3 : 1 (Difference = 2)

Equalize difference: New ratio 7 years hence = 6 : 2 (Difference = 4)

Ratio change in 14 years (7 ago to 7 hence) for son: 2 − 1 = 1 unit → 14 years

Son's age 7 years ago = 1 unit = 14 years ⇒ Present age = 14 + 7 = 21 years

∴ The correct answer is 21 years.

Alternate Method

Given: Ratio 7 years ago = 5 : 1, Ratio 7 years later = 3 :

1. Formula Used: Age calculation based on linear equations.

⇒ Let the age of Son and Father 7 years ago be x and 5x respectively.

⇒ Present age of Son = x + 7 and Father = 5x + 7.

⇒ 7 years from now: Father's age = (5x + 7) + 7 = 5x + 14.

⇒ 7 years from now: Son's age = (x + 7) + 7 = x + 14.

⇒ According to question: 5x + 14 = 3 × (x + 14)

⇒ 5x + 14 = 3x + 42 ⇒ 2x = 28 ⇒ x = 14.

⇒ Son's present age = x + 7 = 14 + 7 = 21 years.

∴ The correct answer is 21 years.

Additional Information

Ratio Difference Concept

The age difference between two people always remains constant. If the difference in parts is not equal, multiply the ratios by the difference of the other ratio.

Present Age Conversion

If age 'n' years ago is x, present age is x + n. If age 'n' years later is y, present age is y − n.

Shortcut Trick

Mode is simply the value that appears most frequently in the given data set.

Scan the list for repeating numbers: 20 appears 3 times, 32 appears 2 times, and 34 appears 5 times.

Since 34 has the highest count, it is the mode.

∴ The correct answer is 34.

Alternate Method

Given observations: 21, 32, 34, 34, 34, 20, 32, 24, 34, 34, 20, 25, 29, 26, 20

To find the mode, we calculate the frequency (number of occurrences) of each observation:

⇒ Frequency of 20: Found at positions 6, 11, 15 → Count = 3

⇒ Frequency of 21: Found at position 1 → Count = 1

⇒ Frequency of 24: Found at position 8 → Count = 1

⇒ Frequency of 25: Found at position 12 → Count = 1

⇒ Frequency of 26: Found at position 14 → Count = 1

⇒ Frequency of 29: Found at position 13 → Count = 1

⇒ Frequency of 32: Found at positions 2, 7 → Count = 2

⇒ Frequency of 34: Found at positions 3, 4, 5, 9, 10 → Count = 5

⇒ Since the value 34 has the maximum frequency of 5, it is the Mode.

∴ The correct answer is 34.

Additional Information

Empirical Formula

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

Types of Mode

Data with one mode is unimodal, two modes is bimodal, and more than two is multimodal.

Median Calculation

To find the Median, arrange data in ascending order. If N is odd, Median = ((N+1)/2)^th term.

Shortcut Trick

Successive multiplier: MP × (100 − d₁)% × (100 − d₂)% = SP.

34,400 × (87 ÷ 100) × ((100 − Z) ÷ 100) = 22,446.

29,928 × (100 − Z) ÷ 100 = 22,446.

100 − Z = (22,446 × 100) ÷ 29,928 ⇒ 100 − Z = 75.

Z = 100 − 75 = 25.

∴ The correct answer is 25.

Alternate Method

Formulas: Price after discount = MP × (100 − D%) ÷ 100 | Second Discount % = (Intermediate SP − Final SP) ÷ Intermediate SP × 100

⇒ Marked Price (MP) = ₹34,400.
⇒ Price after 1^st discount of 13% = 34,400 − (13% of 34,400) = 34,400 − 4,472 = ₹29,928.
⇒ Final Selling Price (SP) after 2^nd discount = ₹22,446.
⇒ Amount of 2^nd discount = 29,928 − 22,446 = ₹7,482.
⇒ Z% = (7,482 ÷ 29,928) × 100 = 0.25 × 100 = 25%.

∴ The correct answer is 25.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The total effective discount is always less than the sum of individual percentages.

Effective Discount Formula

For two successive discounts a% and b%, the single equivalent discount is given by: (a + b − (a × b ÷ 100))%.

Shortcut Trick

Since the denominators (30) are the same, compare only the numerators.

Numerator 1 = 1, Numerator 2 = 2.

Required percentage = (1 ÷ 2) × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: First value = 1/30, Second value = 2/30.

Formula: Percentage = (Value / Base) × 100

⇒ Percentage = [(1/30) / (2/30)] × 100

⇒ Percentage = (1/30) × (30/2) × 100

⇒ Percentage = (1/2) × 100

⇒ Percentage = 50%

∴ The correct answer is 50%.

Additional Information

Percentage Change

To find the percentage change between two numbers: [(New − Old) ÷ Old] × 100.

Fraction to Percentage

To convert any fraction x/y into a percentage, multiply it by 100 and add the % symbol.

Base Comparison

In questions like 'A is what % of B', B is always the denominator (base) and A is the numerator.