Quantitative Aptitude PYQs

Shortcut Trick

Total work (LCM of 15, 12, 5) = 60 units.

Efficiency of X = 4, Y = 5, and X + Y + F = 12.

Efficiency of F = 12 − (4 + 5) = 3 units/day.

Time taken by F = 60 ÷ 3 = 20 days.

∴ The correct answer is 20 days.

Alternate Method

Given: Time taken by X = 15 days, Y = 12 days, and X + Y + F = 5 days.

Formula: Total Work = Days × Efficiency

Calculations:

⇒ Total efficiency of X and Y = 4 + 5 = 9 units/day.

⇒ Efficiency of F alone = Combined efficiency − (X + Y efficiency)

⇒ Efficiency of F = 12 − 9 = 3 units/day.

⇒ Time taken by F to complete work alone = Total Work ÷ Efficiency of F

⇒ Time = 60 ÷ 3 = 20 days.

∴ The correct answer is 20 days.

Additional Information

LCM Method for Time & Work

Total work is assumed as the Least Common Multiple (LCM) of the individual days to avoid working with complex fractions.

Work and Wages

Wages are distributed in the ratio of the efficiencies of the workers if they work for the same amount of time.

Combined Work Formula

If A and B work together, the time taken is (T_A × T_B) ÷ (T_A + T_B).

Shortcut Trick

Check options for 3 : 1 ratio: All options satisfy this (42:14, 45:15, etc.).

Subtract 5 years from both and check for 4 : 1 ratio.

Option (2): (45 − 5) = 40 and (15 − 5) = 10.

Ratio 40 : 10 = 4 : 1. This matches the condition.

∴ The correct answer is 45 years and 15 years.

Alternate Method

Given: Current Father : Son = 3 :

1. Five years ago, Father : Son = 4 :
2. Formula: Ratio of ages at different time intervals.

Let the present age of the son be x and father be 3x.

⇒ 5 years ago: Father = 3x − 5, Son = x − 5

⇒ According to the question: (3x − 5) = 4 × (x − 5)

⇒ 3x − 5 = 4x − 20

⇒ 4x − 3x = 20 − 5

⇒ x = 15

⇒ Father's present age = 3 × 15 = 45 years

⇒ Son's present age = 15 years

∴ The correct answer is 45 years and 15 years.

Additional Information

Age Ratio Difference Method

Make the difference between ratio parts equal for different time periods to find the value of 1 unit quickly.

Linear Equations in Ages

If 'n' years are added or subtracted, the difference between two people's ages always remains constant.

Shortcut Trick

Identify the common factor 5 in both terms.

Use the trigonometric identity: sec²θ − tan²θ = 1.

Expression = 5 × (sec²θ − tan²θ) = 5 × 1 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Expression: 5sec²θ − 5tan²θ

Formula Used:

Fundamental Trigonometric Identity: 1 + tan²θ = sec²θ

⇒ sec²θ − tan²θ = 1

Calculations:

⇒ 5sec²θ − 5tan²θ

⇒ 5(sec²θ − tan²θ)

⇒ 5 × (1)

⇒ 5

∴ The correct answer is 5.

Additional Information

Trigonometric Identities

Standard identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

secθ = 1/cosθ and tanθ = sinθ/cosθ, which also satisfy the identity when substituted.

Shortcut Trick

For the largest sphere inside a cube, the diameter of the sphere is always equal to the edge of the cube.

Direct formula for Volume = (1/6) × π × (side)³

Volume = (1/6) × π × 3³ = (27/6) π = 4.5 π cm³

∴ The correct answer is 4.5 π cm³.

Alternate Method

Given: Edge of the cube (a) = 3 cm

Formula Used: Volume of a sphere = (4/3) × π × r³

⇒ Diameter of the largest sphere = Side of the cube = 3 cm

⇒ Radius (r) of the sphere = 3 ÷ 2 = 1.5 cm

⇒ Volume = (4/3) × π × (1.5)³

⇒ Volume = (4/3) × π × 3.375

⇒ Volume = 4 × π × 1.125

⇒ Volume = 4.5 π cm³

∴ The correct answer is 4.5 π cm³.

Additional Information

Sphere Inscribed in a Cube

When a sphere is inscribed in a cube of side 'a', the radius r = a/2 and the Volume = (πa³)/6.

Cube Inscribed in a Sphere

When a cube of side 'a' is inscribed in a sphere, the diameter of the sphere is equal to the body diagonal of the cube (a√3).

Surface Area of Sphere

The total surface area of a sphere with radius 'r' is given by the formula 4πr².

Volume of a Cube

The volume of a cube with edge 'a' is a³ and its total surface area is 6a².

Shortcut Trick

Expenditure is the product of Price and Quantity.

Ratio of Price (Paddy : Jowar) = 1 : 3

Ratio of Quantity (Paddy : Jowar) = 5 : 7

Ratio of Expenditure = (1 × 5) : (3 × 7) = 5 : 21

∴ The correct answer is 5 : 21.

Alternate Method

Given:

Price ratio of Paddy to Jowar = 1 : 3

Quantity ratio of Paddy to Jowar = 5 : 7

Formula Used:

Expenditure = Price × Quantity

Calculations:

⇒ Let price of Paddy = 1x and Jowar = 3x

⇒ Let quantity of Paddy = 5y and Jowar = 7y

⇒ Expenditure on Paddy = 1x × 5y = 5xy

⇒ Expenditure on Jowar = 3x × 7y = 21xy

⇒ Ratio of Expenditure = 5xy : 21xy

⇒ Ratio = 5 : 21

∴ The correct answer is 5 : 21.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (antecedent with antecedent and consequent with consequent), it is called a compound ratio.

Price-Consumption Relation

If expenditure is constant, Price is inversely proportional to Quantity (Consumption). P₁/P₂ = Q₂/Q₁.

Proportion Rule

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

Shortcut Trick

Income ratio (Vipul : Vijay) = 5 : 8 | Expenditure ratio = 1 : 4.

Difference in income units = 8 − 5 = 3 | Difference in expenditure units = 4 − 1 = 3.

Since savings are equal and the difference between units is same, we map income to expenditure.

Saving units for Vipul = 5 − 1 = 4 units | Saving units for Vijay = 8 − 4 = 4 units.

4 units = ₹84,000 → 1 unit = ₹21,000.

Vipul's income = 5 units = 5 × ₹21,000 = ₹1,05,000.

∴ The correct answer is ₹1,05,000.

Alternate Method

Given:

Income ratio = 5 : 8 | Expenditure ratio = 1 : 4 | Savings = ₹84,000 each.

Formula Used:

Income − Savings = Expenditure

Calculations:

⇒ (5x − 84000) / (8x − 84000) = 1 / 4

⇒ 4 × (5x − 84000) = 1 × (8x − 84000)

⇒ 20x − 3,36,000 = 8x − 84,000

⇒ 20x − 8x = 3,36,000 − 84,000

⇒ 12x = 2,52,000

⇒ x = 21,000

⇒ Vipul's Income = 5x = 5 × 21,000 = 1,05,000

∴ The correct answer is ₹1,05,000.

Additional Information

Income-Expenditure Relation

Income = Expenditure + Savings. This is the fundamental equation for all personal finance problems in quantitative aptitude.

Ratio Balancing Method

If the difference between two ratios (Income and Expenditure) is to be equated to a constant saving, ensure the difference between the terms of each ratio is the same by multiplying with suitable factors.

Cross Multiplication Method

For unequal savings, use: (I₁ × E₂ − I₂ × E₁) units = (S₁ × E₂ − S₂ × E₁) to find the value of one unit directly.

Shortcut Trick

Find the difference between the two numbers: 378 − 252 = 126.

The HCF must be a factor of this difference.

Check if 126 divides both numbers: 252 ÷ 126 = 2 and 378 ÷ 126 = 3.

Since 126 divides both, it is the Highest Common Factor.

∴ The correct answer is 126.

Alternate Method

Given: Numbers are 252 and 378.

Formula Used: Prime Factorization Method

⇒ Prime factorization of 252 = 2 × 2 × 3 × 3 × 7 = 2² × 3² × 7

⇒ Prime factorization of 378 = 2 × 3 × 3 × 3 × 7 = 2 × 3³ × 7

⇒ Common factors with lowest powers = 2¹ × 3² × 7¹

⇒ HCF = 2 × 9 × 7 = 126

∴ The correct answer is 126.

Additional Information

HCF vs LCM Relation

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Euclidean Algorithm

HCF can be found by repeated division: HCF(a, b) = HCF(b, a mod b). Continue until the remainder is zero; the last non-zero divisor is the HCF.

HCF of Fractions

To find the HCF of fractions, use the formula: HCF = (HCF of Numerators) ÷ (LCM of Denominators).

Shortcut Trick

Net percentage of successive discounts can be calculated using the formula: a + b − (a × b) ÷ 100.

First two discounts (23% and 11%): 23 + 11 − (23 × 11) ÷ 100 = 34 − 2.53 = 31.47%.

Combining 31.47% with the third discount (29%): 31.47 + 29 − (31.47 × 29) ÷ 100.

Sum: 60.47 − (912.63 ÷ 100) = 60.47 − 9.1263 = 51.3437%.

∴ The correct answer is 51.34%.

Alternate Method

Given: Three successive discounts: d₁ = 23%, d₂ = 11%, d₃ = 29%.

Formulas: Equivalent Discount % = [1 − (1 − d₁/100) × (1 − d₂/100) × (1 − d₃/100)] × 100

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 1^st discount = 100 × (77 ÷ 100) = 77.

⇒ Price after 2^nd discount = 77 × (89 ÷ 100) = 68.53.

⇒ Price after 3^rd discount (Selling Price) = 68.53 × (71 ÷ 100) = 48.6563.

⇒ Total Discount = MP − SP = 100 − 48.6563 = 51.3437.

⇒ Discount % = (51.3437 ÷ 100) × 100 = 51.3437%.

∴ The correct answer is 51.34%.

Additional Information

Successive Discounts

When multiple discounts are applied one after another, the base for the subsequent discount is the value obtained after the previous discount.

Effective Discount for Multiple Values

The order of applying successive discounts does not change the final effective discount percentage.

Shortcut Trick

Ratio of Ramesh : Mahesh at present = 3 : 1 (Difference = 2 units)

Ratio 5 years ago = 4 : 1 (Difference = 3 units)

To equate the age gap, multiply the present ratio by 3 and the past ratio by 2:

New Present Ratio = 9 : 3 | New Past Ratio = 8 : 2

Difference (9 − 8) = 1 unit → 5 years. Therefore, Ramesh's present age (9 units) = 9 × 5 = 45 years.

∴ The correct answer is 45 years.

Alternate Method

Given: Present age of Ramesh is 3 times Mahesh. 5 years ago, the ratio was 4 :

1. Let Mahesh's present age = x years.

⇒ Ramesh's present age = 3x years.

According to the question (5 years ago):

⇒ (3x − 5) / (x − 5) = 4 / 1

⇒ 3x − 5 = 4(x − 5)

⇒ 3x − 5 = 4x − 20

⇒ 4x − 3x = 20 − 5

⇒ x = 15 years.

⇒ Ramesh's present age = 3 × 15 = 45 years.

∴ The correct answer is 45 years.

Additional Information

Age Difference Property

The difference between the ages of two people remains constant at any point in time.

Ratio Balancing

To solve age problems quickly, ensure the numerical difference between the ratio terms is the same by multiplying with a suitable factor.

Variable Assumption

Always try to assume the present age as the variable to simplify the final calculation for the required answer.

Shortcut Trick

For any twin prime numbers, let the middle even number be x.

The sum of the twin primes is 2x = 36 ⇒ x = 18.

The twin primes are (x − 1) and (x + 1), which are 17 and 19.

Product = x² − 1 = 18² − 1 = 324 − 1 = 323.

∴ The correct answer is 323.

Alternate Method

Given: Sum of twin prime numbers = 36.

Definition: Twin primes are prime numbers that have a difference of 2.

Let the two twin prime numbers be p and (p + 2).

⇒ p + (p + 2) = 36

⇒ 2p + 2 = 36

⇒ 2p = 34

⇒ p = 17

⇒ The second prime number = p + 2 = 17 + 2 = 19.

⇒ Product = 17 × 19

⇒ Product = 323.

∴ The correct answer is 323.

Additional Information

Twin Prime Numbers

Pairs of prime numbers that differ by exactly 2 (e.g., 3 and 5, 5 and 7, 11 and 13).

Prime Number Property

Except for (3, 5), the number between twin primes is always divisible by 6.

Co-prime Numbers

Two numbers whose HCF is 1. All twin primes are also co-prime.

Shortcut Trick

Convert ratio 1/2 : 1/3 : 1/6 to integers by multiplying with LCM (6) → 3 : 2 :

1. Total units = 3 + 2 + 1 = 6 units.

Value of 6 units = 602 bananas.

First monkey (3 units) = (602 ÷ 6) × 3 = 602 ÷ 2 = 301.

∴ The correct answer is 301 bananas.

Alternate Method

Given: Total bananas = 602. Ratio = 1/2 : 1/3 : 1/6.

⇒ Simplify Ratio: LCM of 2, 3, 6 = 6.

⇒ Ratio = (1/2 × 6) : (1/3 × 6) : (1/6 × 6) = 3 : 2 :

1. ⇒ Let parts be 3x, 2x, and 1x.

⇒ 3x + 2x + 1x = 602

⇒ 6x = 602

⇒ x = 602 ÷ 6 = 301/3

⇒ Share of first monkey = 3x = 3 × (301/3) = 301.

∴ The correct answer is 301 bananas.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d to integers, multiply both terms by the LCM of denominators b and d.

Inverse Ratio

The inverse ratio of a : b is 1/a : 1/b, which simplifies to b : a for two terms.

Proportional Division

Share of a part = (Specific Ratio Part ÷ Sum of Ratio Parts) × Total Value.

Shortcut Trick

Number of 4-legged animals (goats) = (Total feet − 2 × Total heads) ÷ 2

⇒ Number of goats = (82 − 2 × 33) ÷ 2

⇒ Number of goats = (82 − 66) ÷ 2 = 16 ÷ 2 = 8

∴ The correct answer is 8 goats.

Alternate Method

Given:

Total animal heads = 33

Total animal feet = 82

Number of feet for one hen = 2

Number of feet for one goat = 4

Formula Used:

Total feet = (2 × Number of hens) + (4 × Number of goats)

Calculations:

Let the number of goats be G and the number of hens be H.

⇒ G + H = 33 → (Equation 1)

⇒ 4G + 2H = 82 → (Equation 2)

Divide Equation 2 by 2:

⇒ 2G + H = 41 → (Equation 3)

Subtract Equation 1 from Equation 3:

⇒ (2G + H) − (G + H) = 41 − 33

⇒ G = 8

∴ The correct answer is 8 goats.

Additional Information

Alligation Method

You can find the ratio of hens to goats by placing 2 and 4 at the ends and the average feet per head (82/33) in the center.

Assumption Method

If we assume all animals are hens, total feet = heads × 2. The extra feet (Total − Assumed) are due to goats, where each goat contributes 2 extra feet.

Linear Variables

For any problem with two types of items having different counts (like 2-legged/4-legged or ₹2/₹5 coins), use x + y = Total Quantity and ax + by = Total Value.

Shortcut Trick

Apply the direct Algebraic Identity: a³ + b³ + c³ = (a + b + c)[(a + b + c)² − 3(ab + bc + ca)] + 3abc.

Substitute the given values: 14 × [14² − 3(47)] + 3(15).

Simplify: 14 × [196 − 141] + 45 = 14 × 55 + 45.

Calculate: 770 + 45 = 815.

∴ The correct answer is 815.

Alternate Method

Given: a + b + c = 14, ab + bc + ca = 47, and abc = 15.

Formula Used:

1. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

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

Calculations:

⇒ 14² = a² + b² + c² + 2(47)

⇒ 196 = a² + b² + c² + 94

⇒ a² + b² + c² = 196 − 94 = 102

⇒ a³ + b³ + c³ − 3(15) = 14 × (102 − 47)

⇒ a³ + b³ + c³ − 45 = 14 × 55

⇒ a³ + b³ + c³ − 45 = 770

⇒ a³ + b³ + c³ = 770 + 45

⇒ a³ + b³ + c³ = 815

∴ The correct answer is 815.

Additional Information

Special Case of the Cubic Identity

If a + b + c = 0, then the identity simplifies significantly to a³ + b³ + c³ = 3abc.

Alternative Form of Identity

The expression a² + b² + c² − ab − bc − ca can also be written as ½[(a − b)² + (b − c)² + (c − a)²], which is useful when differences between variables are known.

Square of a Trinomial

The basic expansion used to find the sum of squares is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

For a number N with n distinct prime factors, the number of pairs of coprime factors (a, b) such that a × b = N is 2ⁿ⁻¹.

Prime factorization of 180 = 2² × 3² × 5¹.

Number of distinct prime factors (n) = 3 (which are 2, 3, and 5).

Number of coprime pairs = 2³⁻¹ = 2² = 4.

∴ The correct answer is 4.

Alternate Method

Given: Product of two coprime integers is 180.

Formula Used: Two numbers are coprime if their Highest Common Factor (HCF) is 1.

⇒ Prime factorization of 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5¹.

⇒ Let the two coprime numbers be x and y. Since they are coprime, any prime power (like 2²) must stay together in one of the numbers.

⇒ The distinct prime power blocks are (2²), (3²), and (5¹), i.e., 4, 9, and 5.

⇒ Possible pairs (x, y):

⇒ 1) (1, 180)

⇒ 2) (4, 9 × 5) = (4, 45)

⇒ 3) (9, 4 × 5) = (9, 20)

⇒ 4) (5, 4 × 9) = (5, 36)

∴ The correct answer is 4.

Additional Information

Coprime Numbers

Two numbers a and b are coprime or relatively prime if HCF(a, b) = 1. They do not need to be prime themselves.

Number of Coprime Pairs Formula

For a number N = p₁^a × p₂^b × ... × p_n^k, the number of ways to write N as a product of two coprime integers is 2ⁿ⁻¹.

Total Number of Factors

The total number of factors of N is (a+1)(b+1)...(k+1). For 180, total factors = (2+1)(2+1)(1+1) = 18.

Shortcut Trick

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

Here, a = 16 and b = 7.

Equivalent discount = (16 + 7 − (16 × 7) ÷ 100)%

= (23 − 112 ÷ 100)%

= 23 − 1.12 = 21.88%.

∴ The correct answer is 21.88%.

Alternate Method

Given: First discount = 16%, Second discount = 7%.

Formula: Single Equivalent Discount = 100 − [100 × (100 − d₁)% × (100 − d₂)%]

⇒ Let the Marked Price (MP) be 100.

⇒ Price after first discount = 100 − (16% of 100) = 84.

⇒ Price after second discount = 84 − (7% of 84) = 84 − 5.88 = 78.12.

⇒ Total discount = 100 − 78.12 = 21.88.

⇒ Equivalent discount % = (21.88 ÷ 100) × 100 = 21.88%.

∴ The correct answer is 21.88%.

Additional Information

Successive Discount

When a series of discounts are applied one after another on the reducing balance, they are called successive discounts. The order of applying discounts does not change the final result.

Three Successive Discounts

For three discounts a, b, and c, first find the equivalent of a and b (say x), then find the equivalent of x and c.

Shortcut Trick

For any quadrilateral WXYZ, the angle formed by the bisectors of two adjacent angles (∠Y and ∠Z) is equal to half the sum of the other two angles (∠W and ∠X).

∠ZOY = (∠W + ∠X) ÷ 2

∠ZOY = (74° + 58°) ÷ 2

∠ZOY = 132° ÷ 2 = 66°

∴ The correct answer is 66°.

Alternate Method

Given:

In quadrilateral WXYZ:

∠W = 74°

∠X = 58°

The bisectors of ∠Y and ∠Z meet at O.

Formula Used:

Sum of interior angles of a quadrilateral = 360°

Sum of angles of a triangle = 180°

Calculations:

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

⇒ ∠W + ∠X + ∠Y + ∠Z = 360°

Substitute the given values of ∠W and ∠X:

⇒ 74° + 58° + ∠Y + ∠Z = 360°

⇒ 132° + ∠Y + ∠Z = 360°

⇒ ∠Y + ∠Z = 360° − 132°

⇒ ∠Y + ∠Z = 228°

Since ZO and YO are the angle bisectors of ∠Z and ∠Y respectively:

⇒ ∠OZY = ∠Z ÷ 2

⇒ ∠OYZ = ∠Y ÷ 2

In triangle ZOY, the sum of angles is 180°:

⇒ ∠ZOY + ∠OZY + ∠OYZ = 180°

⇒ ∠ZOY + (∠Z ÷ 2) + (∠Y ÷ 2) = 180°

⇒ ∠ZOY + (∠Z + ∠Y) ÷ 2 = 180°

Substitute the value of (∠Z + ∠Y) = 228°:

⇒ ∠ZOY + 228° ÷ 2 = 180°

⇒ ∠ZOY + 114° = 180°

⇒ ∠ZOY = 180° − 114°

⇒ ∠ZOY = 66°

Option Analysis:

• Option (1) 68°: Incorrect. If ∠ZOY = 68°, then the sum of ∠Y and ∠Z would be 2 × (180° − 68°) = 224°, which implies ∠W + ∠X = 360° − 224° = 136°. This contradicts the given sum of 132°.
• Option (2) 60°: Incorrect. If ∠ZOY = 60°, then the sum of ∠Y and ∠Z would be 2 × (180° − 60°) = 240°, which implies ∠W + ∠X = 360° − 240° = 120°. This contradicts the given sum of 132°.
• Option (3) 66°: Correct. As mathematically proven, ∠ZOY = 180° − (∠Y + ∠Z) ÷ 2 = 66°.
• Option (4) 59°: Incorrect. If ∠ZOY = 59°, then the sum of ∠Y and ∠Z would be 2 × (180° − 59°) = 242°, which implies ∠W + ∠X = 360° − 242° = 118°. This contradicts the given sum of 132°.

∴ The correct answer is 66°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining two angles. Formula: ∠ZOY = (∠W + ∠X) ÷ 2.

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 always 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 opposite angles.

Shortcut Trick

Convert percentages and fractions into a single multiplying factor.

80% × 50% × 3/8 → 4/5 × 1/2 × 3/8 = 3/20

If 3 units = 537, then 1 unit = 179.

Total value (20 units) = 179 × 20 = 3580.

∴ The correct answer is 3580.

Alternate Method

Given: 80% of 50% of 3/8 of a number is 537.

Formula: Result = (P₁/100) × (P₂/100) × Fraction × Number

Let the number be x.

⇒ (80/100) × (50/100) × (3/8) × x = 537

⇒ (4/5) × (1/2) × (3/8) × x = 537

⇒ (12/80) × x = 537

⇒ (3/20) × x = 537

⇒ x = (537 × 20) / 3

⇒ x = 179 × 20

⇒ x = 3580

∴ The correct answer is 3580.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, remember common fractions: 80% = 4/5, 50% = 1/2, 25% = 1/4, and 12.5% = 1/8.

Successive Percentage

When applying multiple percentages to a single base, the net percentage is the product of their decimal equivalents.

Base Value Concept

Always identify the 'of' term as the denominator or base value in ratio-based solving.

Shortcut Trick

Since Mode = 22.75, the Modal Class is 21−28.

Using Mode formula: 22.75 = 21 + [(11 − 9) / (2 × 11 − 9 − X)] × 7

⇒ 1.75 = [2 / (13 − X)] × 7 ⇒ 13 − X = 14 / 1.75 = 8

⇒ X = 5

Total Frequency = 1 + 7 + 9 + 11 + 5 + 3 = 36.

∴ The correct answer is 36.

Alternate Method

Given:

Mode = 22.75

Modal Class = 21−28 (since 22.75 lies in this range)

Lower limit (L) = 21

Frequency of modal class (f₁) = 11

Frequency of class preceding modal class (f₀) = 9

Frequency of class succeeding modal class (f₂) = X

Class width (h) = 7

Formula Used:

Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Calculations:

⇒ 22.75 = 21 + [(11 − 9) / (2 × 11 − 9 − X)] × 7

⇒ 22.75 − 21 = [2 / (22 − 9 − X)] × 7

⇒ 1.75 = [2 / (13 − X)] × 7

⇒ 1.75 = 14 / (13 − X)

⇒ 13 − X = 14 / 1.75

⇒ 13 − X = 8

⇒ X = 13 − 8 = 5

Total Frequency (Sum of all frequencies):

⇒ N = 1 + 7 + 9 + 11 + 5 + 3

⇒ N = 36

∴ The correct answer is 36.

Additional Information

Empirical Relationship

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

Modal Class Identification

The class interval with the highest frequency is called the Modal Class.

Median Formula

Median = L + [((N/2) − cf) / f] × h, where cf is the cumulative frequency of the preceding class.

Shortcut Trick
Scheme i: Total Discount = 37%.
Scheme ii: Successive Discount = a+b−(ab÷100) = 33+43−(33×43÷100) = 76−14.19 = 61.81%.
Scheme iii: Successive Discount = 46+22−(46×22÷100) = 68−10.12 = 57.88%.
Compare the discounts: 37% (i) < 57.88% (iii) < 61.81% (ii).
The least discount is the least beneficial to the customer.
∴ The correct answer is Scheme i.

Alternate Method
Given: Three different discount schemes. To find the least beneficial (minimum discount percentage).
Formulas: Successive Discount = a + b − (ab ÷ 100)%

⇒ Scheme i: Single discount = 37%.
⇒ Scheme ii: Single equivalent of 33% and 43% = [33 + 43 − (33 × 43)/100]% = [76 − 14.19]% = 61.81%.
⇒ Scheme iii: Single equivalent of 46% and 22% = [46 + 22 − (46 × 22)/100]% = [68 − 10.12]% = 57.88%.
⇒ Comparing all three: 37% is the minimum value among the available options.
∴ The correct answer is Scheme i.

Additional Information
Customer Perspective vs. Seller Perspective
For a customer, the most beneficial scheme is the one offering the highest discount. Conversely, the least beneficial is the one offering the lowest discount.
Successive Discount Calculation
When two successive discounts x% and y% are given, the net discount is always less than the sum (x + y)% due to the reduction in the base amount for the second discount.

Shortcut Trick

Difference in annual interest rate = 7% − 3.5% = 3.5%

Total interest percentage difference for 4 years = 3.5% × 4 = 14%

Positive difference in ₹ = 14% of ₹4,80,000

Calculation: 14/100 × 4,80,000 = 14 × 4,800 = 67,200

∴ The correct answer is ₹67,200.

Alternate Method

Given:

Principal (P) = ₹4,80,000

Time (T) = 4 years

Rate of Bank A (R_A) = 3.5% p.a.

Rate of Bank B (R_B) = 7% p.a.

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ SI paid to Bank A = (4,80,000 × 3.5 × 4) ÷ 100

⇒ SI paid to Bank A = 4,800 × 14 = ₹67,200

⇒ SI paid to Bank B = (4,80,000 × 7 × 4) ÷ 100

⇒ SI paid to Bank B = 4,800 × 28 = ₹1,34,400

⇒ Positive Difference = ₹1,34,400 − ₹67,200

⇒ Positive Difference = ₹67,200

∴ The correct answer is ₹67,200.

Additional Information

Total Amount Formula

Amount (A) = Principal (P) + Simple Interest (SI). It represents the total money repaid to the lender.

Direct Difference Rule

When Principal and Time are constant, the difference in SI is directly proportional to the difference in rates: ΔSI = (P × ΔR × T) ÷ 100.

Simple Interest Nature

Simple Interest remains constant every year if the principal and rate do not change, unlike Compound Interest which increases annually.