Quantitative Aptitude PYQs

Shortcut Trick

Apply the Successive Discount formula: a + b − (a × b)/100.

Given: a = 30% and b = 10%.

Net Discount % = 30 + 10 − (30 × 10)/100

Net Discount % = 40 − 3 = 37%

∴ The correct answer is 37%.

Alternate Method

Given: Original Price = 1,500, First Discount = 30%, Second Discount = 10%.

⇒ First discount amount = 30% of 1,500 = (30/100) × 1500 = 450

⇒ Price after 1^st discount = 1500 − 450 = 1050

⇒ Second discount amount = 10% of 1,050 = (10/100) × 1050 = 105

⇒ Final Selling Price = 1050 − 105 = 945

⇒ Total Discount = 1500 − 945 = 555

⇒ Equivalent Discount % = (Total Discount / Original Price) × 100

⇒ Equivalent Discount % = (555 / 1500) × 100 = 37%

∴ The correct answer is 37%.

Additional Information

Net Discount for Three Rates

For three successive discounts a%, b%, and c%, the net discount is calculated by first finding the net of a and b, then applying the formula to that result and c.

Marked Price (MP) and Selling Price (SP)

Discount is always computed on the Marked Price. SP = MP × (100 − D₁%)/100 × (100 − D₂%)/100.

Multiplying Factor Method

Final Price = Initial Price × 0.70 × 0.90. The overall multiplier is 0.63, which implies a (1 − 0.63) × 100 = 37% discount.

Shortcut Trick

Interest earned in 1 year = 3600 ÷ 4 = ₹900

Rate of Interest % = (Annual Interest ÷ Principal) × 100

Rate % = (900 ÷ 15000) × 100 = 6%

∴ The correct answer is 6%.

Alternate Method

Given:

Principal (P) = ₹15,000

Time (T) = 4 years

Simple Interest (SI) = ₹3,600

Formula Used:

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

Rate (R) = (SI × 100) ÷ (P × T)

Calculations:

⇒ R = (3600 × 100) ÷ (15000 × 4)

⇒ R = 360000 ÷ 60000

⇒ R = 6

∴ The correct answer is 6%.

Additional Information

Amount Calculation

The total sum to be repaid after the time period is the Amount, calculated as Amount = Principal + Simple Interest.

Time and Rate Relationship

For a fixed interest amount and principal, the Rate of Interest is inversely proportional to the Time (R ∝ 1/T).

Constant Annual Interest

In Simple Interest, the interest earned remains the same for each year of the time period.

Shortcut Trick

Pick Assumed Mean (A) = 28. Find the mid-point (x) for each class interval: 4, 12, 20, 28, 36, 44.

Calculate deviations d = x − A: −24, −16, −8, 0, 8, 16.

Calculate Σf×d = 4(−24) + 6(−16) + 7(−8) + 3(0) + 12(8) + 8(16) = −96 − 96 − 56 + 0 + 96 + 128 = −24.

Mean = A + (Σfd ÷ Σf) = 28 + (−24 ÷ 40) = 28 − 0.6 = 27.4.

∴ The correct answer is 27.4.

Alternate Method

Given: Frequencies and class intervals for a dataset.

Formula Used: Direct Method Mean (μ) = Σ(f_i × x_i) ÷ Σf_i

Step 1: Calculate Mid-points (x_i) and Products (f_ix_i):

⇒ (0-8): x₁ = 4, f₁x₁ = 4 × 4 = 16

⇒ (8-16): x₂ = 12, f₂x₂ = 6 × 12 = 72

⇒ (16-24): x₃ = 20, f₃x₃ = 7 × 20 = 140

⇒ (24-32): x₄ = 28, f₄x₄ = 3 × 28 = 84

⇒ (32-40): x₅ = 36, f₅x₅ = 12 × 36 = 432

⇒ (40-48): x₆ = 44, f₆x₆ = 8 × 44 = 352

Step 2: Calculate Total Frequency and Sum of Products:

⇒ Σf_i = 4 + 6 + 7 + 3 + 12 + 8 = 40

⇒ Σf_ix_i = 16 + 72 + 140 + 84 + 432 + 352 = 1096

Step 3: Calculate the Mean:

⇒ Mean = 1096 ÷ 40

⇒ Mean = 27.4

∴ The correct answer is 27.4.

Additional Information

Direct Method for Mean

It is the simplest method where Mean = Σf_ix_i ÷ Σf_i, used when values of x_i and f_i are small.

Assumed Mean Method

Used to simplify calculations by choosing a central value A. Mean = A + (Σf_id_i ÷ Σf_i), where d_i = x_i − A.

Step-Deviation Method

Used when class sizes (h) are uniform. Mean = A + [(Σf_iu_i ÷ Σf_i) × h], where u_i = (x_i − A) ÷ h.

Empirical Relationship

The connection between the three measures of central tendency: 3 × Median = Mode + 2 × Mean.

Given:

((³√4096) ÷ (⁴√10000)) × (5/48) × 186

Formula used:

Cube root: ³√x = x^1/3

Fourth root: ⁴√x = x^1/4

Calculations:

Evaluate the roots:

³√4096 = 16

⁴√10000 = 10

Substitute values and simplify the expression:

⇒ (16 ÷ 10) × (5 ÷ 48) × 186

⇒ (8/5) × (5/48) × 186

⇒ (8/48) × 186

⇒ (1/6) × 186

⇒ 31

∴ The correct answer is option (4).

Shortcut Trick

Ratio of Marked Price to Reduced Price = 500 : 450 = 10 : 9.

Decrease in price = 10 − 9 = 1 unit.

Percentage decrease = (Decrease ÷ Original Price) × 100 = (1 ÷ 10) × 100 = 10%.

∴ The correct answer is 10%.

Alternate Method

Given:

Marked Price (MP) = ₹500

Reduced Price (SP) = ₹450

Formula Used:

Percentage Decrease = [(Original Value − New Value) ÷ Original Value] × 100

Calculations:

⇒ Total reduction in price = 500 − 450 = ₹50

⇒ Percentage decrease = (50 ÷ 500) × 100

⇒ Percentage decrease = (1 ÷ 10) × 100

⇒ Percentage decrease = 10%

∴ The correct answer is 10%.

Additional Information

Discount and Marked Price

The amount deducted from the Marked Price is called the Discount. Discount % is always calculated on the Marked Price: Discount % = (Discount ÷ MP) × 100.

Successive Discounts

When two or more discounts are applied one after another, the effective discount is calculated using the formula: x + y − (x × y ÷ 100).

Selling Price Calculation

The final price after the discount is the Selling Price (SP). SP = MP × (100 − Discount %) ÷ 100.

Shortcut Trick

Co-prime numbers are pairs that have no common factor other than 1 (HCF = 1).

Analyze options: 27 and 36 are divisible by 9; 44 and 99 are divisible by 11; 42 and 56 are divisible by 14.

For 25 and 36, their factors do not overlap: 25 (5 × 5) and 36 (2 × 2 × 3 × 3).

∴ The correct answer is 25 and 36.

Alternate Method

Given: A set of number pairs to identify the co-prime pair.

Definition: Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

Calculation for all options:

⇒ Option 1: HCF(27, 36) → Factors of 27 = {1, 3, 9, 27}; Factors of 36 = {1, 2, 3, 4, 6, 9, ...} ⇒ HCF = 9

⇒ Option 2: HCF(44, 99) → Factors of 44 = {1, 2, 4, 11, 22, 44}; Factors of 99 = {1, 3, 9, 11, 33, 99} ⇒ HCF = 11

⇒ Option 3: HCF(25, 36) → Factors of 25 = {1, 5, 25}; Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36} ⇒ HCF = 1

⇒ Option 4: HCF(42, 56) → Factors of 42 = {1, 2, 3, 6, 7, 14, ...}; Factors of 56 = {1, 2, 4, 7, 8, 14, ...} ⇒ HCF = 14

∴ The correct answer is 25 and 36.

Additional Information

Properties of Co-prime Numbers

Any two prime numbers are always co-prime to each other. Also, any two consecutive numbers are always co-prime (e.g., 9 and 10).

HCF and LCM Relation

For a pair of co-prime numbers a and b, their HCF = 1 and their LCM = a × b.

Perfect Squares

Two perfect squares can be co-prime if their square roots do not share any common factors (e.g., 25 = 5² and 36 = 6²).

Shortcut Trick

The overall profit is the product of individual multipliers for markup, discount, and weight fraud.

Net Multiplier = (1 + 24/100) × (1 − 18/100) × (1000 ÷ 800)

Net Multiplier = 1.24 × 0.82 × 1.25

Net Multiplier = 1.0168 × 1.25 = 1.271

Profit % = (1.271 − 1) × 100 = 27.1%

∴ The correct answer is 27.1%.

Alternate Method

Given:

Markup = 24% above CP | Discount = 18% | Actual weight = 800 g | Declared weight = 1 kg (1000 g).

Formula Used:

Overall Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Calculations:

⇒ Let the Cost Price (CP) of 1000 g of sugar be ₹1000 (so ₹1 per gram).

⇒ Marked Price (MP) = 1000 + 24% of 1000 = ₹1240.

⇒ Selling Price (SP) = 1240 − 18% of 1240 = 1240 − 223.2 = ₹1016.8.

⇒ Since the shopkeeper gives only 800 g, his actual CP = ₹800.

⇒ Total Profit = SP − Actual CP = 1016.8 − 800 = ₹216.8.

⇒ Overall Profit % = (216.8 ÷ 800) × 100 = 27.1%.

∴ The correct answer is 27.1%.

Additional Information

Successive Percentage Change

When multiple changes occur, the net effect is calculated using multipliers: Net Factor = (1 ± r₁/100) × (1 ± r₂/100) × ...

Dishonest Dealer Rule

Profit % due to weight fraud = [(True Weight − False Weight) ÷ False Weight] × 100, assuming sale is at CP.

CP, MP, and SP Relationship

The direct ratio is given by (100 + Profit%) ÷ (100 − Discount%) = Marked Price ÷ Cost Price.

Shortcut Trick

Net Rate Difference per year = 6.5% − 3.5% = 3%.

Total Interest Difference for 2 years = 3% × 2 = 6%.

Required Difference = 6% of ₹2,60,000.

⇒ 2,60,000 × (6 ÷ 100) = 2,600 × 6 = 15,600.

∴ The correct answer is ₹15,600.

Alternate Method

Given:

Principal (P) for Bank A and Bank B = ₹2,60,000 each.

Interest Rate (R₁) for Bank A = 3.5% per annum.

Interest Rate (R₂) for Bank B = 6.5% per annum.

Time period (T) = 2 years.

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Calculations:

⇒ Interest paid to Bank A (SI_A) = (2,60,000 × 3.5 × 2) ÷ 100

⇒ SI_A = 2,600 × 7 = 18,200

⇒ Interest paid to Bank B (SI_B) = (2,60,000 × 6.5 × 2) ÷ 100

⇒ SI_B = 2,600 × 13 = 33,800

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = 33,800 − 18,200 = 15,600

According to the calculation, the interest difference is ₹15,600 which corresponds to option (2).

∴ The correct answer is ₹15,600.

Additional Information

Simple Interest Principle

Simple interest is calculated only on the initial Principal amount. The formula is SI = (P × R × T) ÷ 100, where P is Principal, R is annual rate, and T is time in years.

Interest Difference Shortcut

When Principal and Time are constant, the difference in interest is directly proportional to the difference in rates: Difference = [P × T × (R₂ − R₁)] ÷ 100.

Total Amount

The total amount (A) to be repaid after the time period is the sum of the Principal and the Interest: A = P + SI.

Shortcut Trick

Area of rectangular field = 55 × 70 = 3850 m².

As per the problem, Area of circular field = πr² = 3850 m².

⇒ (22/7) × r² = 3850 ⇒ r² = 175 × 7 = 1225 ⇒ r = 35 m.

Perimeter (Circumference) = 2πr = 2 × (22/7) × 35 = 220 m.

∴ The correct answer is 220 m.

Alternate Method

Given:

Length of the rectangular field (L) = 70 m

Breadth of the rectangular field (B) = 55 m

Formula Used:

Area of rectangle = Length × Breadth

Area of circular field = π × r²

Perimeter (Circumference) of circle = 2 × π × r

Calculations:

⇒ Area of rectangular field = 70 × 55 = 3850 m²

⇒ Area of circular field = 3850 m²

⇒ πr² = 3850

⇒ (22/7) × r² = 3850

⇒ r² = (3850 × 7) ÷ 22

⇒ r² = 175 × 7 = 1225

⇒ Radius (r) = √1225 = 35 m

⇒ Perimeter = 2 × (22/7) × 35

⇒ Perimeter = 2 × 22 × 5 = 220 m

∴ The correct answer is 220 m.

Additional Information

Circle Properties

The ratio of the circumference to the diameter of any circle is always constant, denoted by π (≈ 22/7 or 3.14).

Comparison of Perimeter

For a given area, a circle has the shortest perimeter compared to any other plane figure like a rectangle or square.

Unit Conversion

Ensure all dimensions are in the same units (e.g., meters) before equating areas or calculating perimeters.

Shortcut Trick

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

Sum of digits for 7KM93 = 7 + K + M + 9 + 3 = 19 + K + M.

Possible sums are 27 or 36, giving K + M = 8 or K + M = 17.

To minimize 4K² + 2M, keep K as small as possible because it has a higher power and coefficient.

Let K = 0, then M = 8 ⇒ 4(0)² + 2(8) = 16.

∴ The correct answer is 16.

Alternate Method

Given: A five-digit number 7KM93 is divisible by 9. We need to find the least value of the expression 4K² + 2M.

Formula Used: Sum of digits for divisibility by 9.

⇒ Sum of digits = 7 + K + M + 9 + 3 = 19 + K + M

⇒ Since the number is divisible by 9, 19 + K + M = 27 or 36

⇒ Case 1: K + M = 27 − 19 = 8. Possible (K, M) are (0, 8), (1, 7), (2, 6), ..., (8, 0).

⇒ For K = 0, M = 8: 4(0)² + 2(8) = 16.

⇒ For K = 1, M = 7: 4(1)² + 2(7) = 4 + 14 = 18.

⇒ Case 2: K + M = 36 − 19 = 17. Possible (K, M) are (8, 9) or (9, 8).

⇒ For K = 8, M = 9: 4(8)² + 2(9) = 4(64) + 18 = 256 + 18 = 274.

⇒ Comparing all values, the least value is 16.

∴ The correct answer is 16.

Additional Information

Divisibility by 3

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

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is either 0 or a multiple of 11.

Divisibility by 8

A number is divisible by 8 if its last three digits form a number that is divisible by 8.

Shortcut Trick

Use the successive percentage formula: Net Profit % = x + y + (xy ÷ 100)

Here, markup (x) = +30 and discount (y) = −10.

Net Profit % = 30 − 10 + [30 × (−10)] ÷ 100

Net Profit % = 20 − 3 = 17%

∴ The correct answer is 17%.

Alternate Method

Let the Cost Price (CP) of the product be 100 units.

Given: Markup % = 30%, Discount % = 10%.

⇒ Marked Price (MP) = CP + 30% of CP

⇒ MP = 100 + (30 ÷ 100 × 100) = 130 units.

⇒ Discount = 10% of MP = 10 ÷ 100 × 130 = 13 units.

⇒ Selling Price (SP) = MP − Discount

⇒ SP = 130 − 13 = 117 units.

⇒ Profit = SP − CP = 117 − 100 = 17 units.

⇒ Profit % = (Profit ÷ CP) × 100 = (17 ÷ 100) × 100 = 17%.

∴ The correct answer is 17%.

Additional Information

Relationship between MP and CP

The ratio of Marked Price to Cost Price is given by: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Successive Discounts

If two successive discounts of d₁% and d₂% are given, the equivalent single discount is (d₁ + d₂ − d₁d₂ ÷ 100)%.

Selling Price Formula

Selling Price can be calculated directly from the Marked Price as SP = MP × (100 − Discount%) ÷ 100.

Shortcut Trick

Total Work (LCM of 12 and 24) = 24 units.

Efficiency: A = 24 ÷ 12 = 2; B = 24 ÷ 24 = 1.

Work done by (A + B) in 2 days = (2 + 1) × 2 = 6 units.

Remaining work = 24 − 6 = 18 units.

Time taken by B alone = 18 ÷ 1 = 18 days.

∴ The correct answer is 18 days.

Alternate Method

Given: A takes 12 days and B takes 24 days to finish the work.

⇒ Total work to be completed = 24 units.

⇒ Efficiency of A = 24 ÷ 12 = 2 units/day.

⇒ Efficiency of B = 24 ÷ 24 = 1 unit/day.

⇒ Work done by A and B together in 2 days = (2 + 1) × 2 = 6 units.

⇒ Remaining work after A leaves = Total work − Work done = 24 − 6 = 18 units.

⇒ Time taken by B to complete the remaining work = Remaining work ÷ Efficiency of B.

⇒ Time = 18 ÷ 1 = 18 days.

∴ The correct answer is 18 days.

Additional Information

Work and Efficiency Relation

Efficiency is inversely proportional to time. If time ratio is x : y, then efficiency ratio is y : x.

Total Work Calculation

Total Work = Efficiency × Time. Using the LCM of individual days is the fastest way to find a common work unit.

Work and Wages

Wages are distributed in the ratio of the total work done by each person, which is (Efficiency × Days worked).

Shortcut Trick

If a, b, c, d are in proportion, then a × d = b × c.

Given: 5, 15, 25, k are in proportion.

5 × k = 15 × 25

k = 3 × 25 = 75 (Since 15 ÷ 5 = 3)

∴ The correct answer is k = 75.

Alternate Method

Given:

First term (a) = 5

Second term (b) = 15

Third term (c) = 25

Fourth proportional = k

Formula Used:

For four numbers to be in proportion: a ÷ b = c ÷ d

Calculations:

⇒ 5 ÷ 15 = 25 ÷ k

⇒ 1 ÷ 3 = 25 ÷ k

⇒ k = 25 × 3

⇒ k = 75

∴ The correct answer is k = 75.

Additional Information

Mean Proportional

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

Third Proportional

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

Product Property

In any proportion, the product of extremes (outer terms) is always equal to the product of means (inner terms).

Shortcut Trick

Net percentage change = a + b + (ab/100)

Here, a = +30 and b = −20

Net change = 30 − 20 + (30 × (−20) / 100) = 10 − 6 = 4%

Since 4% change corresponds to ₹40 increase, then 100% (original price) = (40 ÷ 4) × 100 = ₹1,000.

∴ The correct answer is ₹1,000.

Alternate Method

Given:

First increase = 30%

Second decrease = 20%

Difference between final price and original price = ₹40

Let the original price of the article be ₹ x.

⇒ Price after 30% increase = x + (30 ÷ 100) × x = 1.3x

⇒ Price after 20% decrease = 1.3x − (20 ÷ 100) × 1.3x

⇒ Final Price = 1.3x − 0.26x = 1.04x

⇒ Given: Final Price − Original Price = ₹40

⇒ 1.04x − x = 40

⇒ 0.04x = 40

⇒ x = 40 ÷ 0.04

⇒ x = 4000 ÷ 4 = 1000

∴ The correct answer is ₹1,000.

Additional Information

Successive Percentage Change

When a value is changed by a% and then by b%, the effective change is given by [a + b + (ab/100)]%.

Multiplying Factor Method

An increase of x% is equivalent to multiplying the value by (1 + x/100), and a decrease of y% is equivalent to multiplying by (1 − y/100).

Ratio Method for Successive Changes

Convert percentages to fractions (30% = 13/10 and −20% = 4/5). Multiply the ratios: (13/10) × (4/5) = 52/50 = 104/100, which shows a 4% increase.

Given:

Expression = $\dfrac{27^{2/3} \times 9^{-1/2}}{3^{-1} \times 81^{1/4}}$

Formula used:

(a^m)ⁿ = a^{m × n}

a⁻ⁿ = 1 / aⁿ

Calculation:

Simplify the Numerator:

⇒ 27^2/3 × 9^−1/2

⇒ (3³)^2/3 × (3²)^−1/2

⇒ 3² × 3⁻¹

⇒ 9 × (1/3) = 3

Simplify the Denominator:

⇒ 3⁻¹ × 81^1/4

⇒ (1/3) × (3⁴)^1/4

⇒ (1/3) × 3 = 1

Evaluate the final Value:

⇒ Result = 3 / 1

⇒ Result = 3

∴ The correct answer is option 3.

Shortcut Trick

Investment Ratio (P : Q) = 2300 : 4600 = 1 : 2

Since time is constant, Profit Ratio = Investment Ratio = 1 : 2

Total units = 1 + 2 = 3 units → ₹37,272

P's share (1 unit) = 37,272 ÷ 3 = ₹12,424

∴ The correct answer is ₹12,424.

Alternate Method

Given:

Investment of P = ₹2,300

Investment of Q = ₹4,600

Total Profit earned at the end of 1 year = ₹37,272

Formula Used:

Profit Share of a Partner = (Individual Investment × Time Period) ÷ (Total Investment × Time Period) × Total Profit

Calculations:

⇒ Ratio of profits of P and Q = (2300 × 1) : (4600 × 1)

⇒ Ratio (P : Q) = 23 : 46 = 1 : 2

⇒ Sum of ratio components = 1 + 2 = 3

⇒ P's share in profit = (1 ÷ 3) × 37272

⇒ P's share = ₹12,424

∴ The correct answer is ₹12,424.

Additional Information

Simple Partnership

If all partners invest their capital for the same time period, then the profits or losses are distributed in the ratio of their investments.

Compound Partnership

If partners invest for different time periods, the profit is shared in the ratio of the product of their capital and the time (Capital₁ × Time₁ : Capital₂ × Time₂).

Working and Sleeping Partners

A working partner manages the business and may receive a salary or extra commission, while a sleeping partner only provides the capital.

Shortcut Trick

When two articles are sold at the same Selling Price, one at a gain of x% and the other at a loss of x%, the transaction always results in an overall loss.

Net Loss % = (x / 10)²

Given x = 25%

Net Loss % = (25 / 10)² = (2.5)² = 6.25%

∴ The correct answer is 6.25%.

Alternate Method

Given: Selling Price (SP) is same for both articles. 1^st article profit = 25%, 2^nd article loss = 25%.

⇒ For 1^st article (25% profit): 25% = 1/4 → CP₁ : SP₁ = 4 : 5

⇒ For 2^nd article (25% loss): 25% = 1/4 → CP₂ : SP₂ = 4 : 3

⇒ To make Selling Prices equal, take LCM(5, 3) = 15 units.

⇒ Article 1: CP₁ = 4 × 3 = 12 units | SP₁ = 5 × 3 = 15 units

⇒ Article 2: CP₂ = 4 × 5 = 20 units | SP₂ = 3 × 5 = 15 units

⇒ Total Cost Price = 12 + 20 = 32 units

⇒ Total Selling Price = 15 + 15 = 30 units

⇒ Overall Loss % = [(32 − 30) / 32] × 100 = (2 / 32) × 100 = 6.25%

∴ The correct answer is 6.25%.

Additional Information

Net Percentage Change Rule

For two successive percentage changes of a% and b%, the net change is given by (a + b + ab/100)%. However, when SP is same, the shortcut (x/10)² is derived from specific CP adjustments.

Same Cost Price Case

If two articles are bought at the same Cost Price (CP) and sold at x% profit and x% loss respectively, there is No Profit No Loss overall.

Profit and Loss Ratios

Profit % = (Profit / CP) × 100 and Loss % = (Loss / CP) × 100. Always ensure the denominator is Cost Price unless specified otherwise.

Shortcut Trick

Prime factorization of 360 = 2³ × 3² × 5 = 8 × 9 × 5.

To minimize the sum, a, b, and c should be as close to √³(360) ≈ 7.1 as possible.

Condition (iii) requires a and c to share a factor, so we split the 2³ into 2² for a and 2¹ for c.

Assigning factors: a = 2² = 4, b = 3² = 9, c = 2 × 5 = 10.

Check: gcd(4, 9)=1, gcd(9, 10)=1, gcd(4, 10)=2. All conditions satisfied.

Minimum sum = 4 + 9 + 10 = 23.

∴ The correct answer is 23.

Alternate Method

Given: abc = 360, gcd(a, b) = 1, gcd(b, c) = 1, gcd(a, c) > 1.

Prime Factorization: 360 = 2³ × 3² × 5¹.

⇒ Total prime factors available are {2, 2, 2}, {3, 3}, {5}.

⇒ For gcd(a, c) > 1, both a and c must share at least one prime factor (2, 3, or 5).

⇒ If we share prime 2: a contains 2^x and c contains 2^y. Since b is coprime to both, b cannot have any factor of 2.

⇒ Trial 1: b = 3² = 9. Then ac = 360 ÷ 9 = 40. Since a, c are even, possible pairs (a, c) are (4, 10), (2, 20). Sums: 4+9+10 = 23; 2+9+20 = 31.

⇒ Trial 2: b = 5. Then ac = 360 ÷ 5 = 72. To share a factor and keep sum low, try (a, c) = (6, 12). Sum: 6+5+12 = 23.

⇒ Trial 3: b = 8. Then ac = 360 ÷ 8 = 45. To share a factor, try (a, c) = (3, 15). Sum: 3+8+15 = 26.

⇒ Comparing all valid combinations, the minimum sum is 23.

∴ The correct answer is 23.

Additional Information

Coprime Numbers

Two numbers are coprime if their Greatest Common Divisor (GCD) is 1. They do not share any prime factors.

Prime Factorization

Every composite number can be expressed as a unique product of prime powers. For 360, it is 2³ × 3² × 5¹.

AM-GM Inequality Concept

For a fixed product, the sum of positive numbers is minimized when the numbers are as close to each other as possible (ideally equal to the n-th root).

Shortcut Trick

The ratio of the first two numbers is 240 : 120 = 2 : 1.

For the third proportional (c), the ratio 120 : c must also be 2 : 1.

Since the ratio is halved, c = 120 ÷ 2 = 60.

∴ The correct answer is 60.

Alternate Method

Given:

First term (a) = 240

Second term (b) = 120

Formula Used:

If three numbers a, b, and c are in continued proportion, then:

a : b :: b : c ⇒ c = b² ÷ a

Calculations:

⇒ Let the third proportional be c.

⇒ 240 : 120 = 120 : c

⇒ 240 ÷ 120 = 120 ÷ c

⇒ 2 = 120 ÷ c

⇒ c = 120 ÷ 2

⇒ c = 60

∴ The correct answer is 60.

Additional Information

Mean Proportional

The Mean Proportional between two numbers x and y is given by the formula √(x × y).

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b = c : d), then d is the Fourth Proportional, calculated as (b × c) ÷ a.

Product Rule

In any proportion, the Product of Extremes (first and last terms) is always equal to the Product of Means (middle terms).

Shortcut Trick

Simple Interest (SI) for 2 years = ₹6,400 − ₹5,200 = ₹1,200

Principal (P) = Amount after 2 years − SI for 2 years = ₹5,200 − ₹1,200 = ₹4,000

Rate (R) = (SI for 2 years × 100) ÷ (P × Time)

Rate = (1,200 × 100) ÷ (4,000 × 2) = 15%

∴ The correct answer is 15 percent.

Alternate Method

Given:

Amount after 2 years (A₁) = ₹5,200

Amount after 4 years (A₂) = ₹6,400

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Amount (A) = Principal (P) + Simple Interest (SI)

Calculations:

⇒ Simple Interest for 2 years = A₂ − A₁

⇒ SI for 2 years = 6,400 − 5,200 = 1,200

⇒ Principal (P) = A₁ − SI for 2 years

⇒ P = 5,200 − 1,200 = 4,000

⇒ We know, SI = (P × R × T) ÷ 100

⇒ 1,200 = (4,000 × R × 2) ÷ 100

⇒ 1,200 = 80 × R

⇒ R = 1,200 ÷ 80 = 15

∴ The correct answer is 15 percent.

Additional Information

Simple Interest Constant Property

In simple interest, the interest earned remains constant every year because it is always calculated on the original principal.

Relationship between Amount and Principal

The amount (A) is the sum of the principal (P) and the accumulated interest (I) over a period of time T: A = P + (P × R × T)/100.

Finding Principal from Two Amounts

If a sum amounts to A₁ in T₁ years and A₂ in T₂ years, then SI for (T₂ − T₁) years is (A₂ − A₁).