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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).