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Shortcut Trick

Convert all animals into equivalent cows: 1 horse = 2.5 cows, and 3 sheep = 1 cow.

A's usage = (20 × 2.5) cows × 5 months = 250 cow-months.

B's usage = (30 cows × 4 months) + (45 ÷ 3) cows × 5 months = 120 + 75 = 195 cow-months.

Total equivalent usage = 250 + 195 = 445 cow-months.

A's rent portion = A's usage ÷ Total usage = 250 ÷ 445 = 50/89.

∴ The correct answer is 50/89.

Alternate Method

Given:

⇒ A uses 20 horses for 5 months.

⇒ B uses 30 cows for 4 months and 45 sheep for 5 months.

⇒ Conversion rates: 2 horses = 5 cows, and 3 cows = 9 sheep.

Formula Used:

⇒ Portion of Rent = (Individual Equivalent Usage) ÷ (Total Equivalent Usage)

Equivalent Cow-Months Usage

A: 250 units

B: 195 units Share: 250 / 445 = 50 / 89 Share: 195 / 445 = 39 / 89

Calculations:

⇒ Determine equivalent cow units for A:

⇒ 2 horses = 5 cows ⇒ 20 horses = (5 ÷ 2) × 20 = 50 cows.

⇒ A's total usage = 50 cows × 5 months = 250 cow-months.

⇒ Determine equivalent cow units for B:

⇒ 9 sheep = 3 cows ⇒ 45 sheep = (3 ÷ 9) × 45 = 15 cows.

⇒ B's usage from cows = 30 cows × 4 months = 120 cow-months.

⇒ B's usage from sheep = 15 cows × 5 months = 75 cow-months.

⇒ B's total usage = 120 + 75 = 195 cow-months.

⇒ Total usage by A and B = 250 + 195 = 445 cow-months.

⇒ A's portion = 250 ÷ 445.

⇒ Simplify the fraction by dividing numerator and denominator by 5:

⇒ 250 ÷ 5 = 50

⇒ 445 ÷ 5 = 89

⇒ A's Portion = 50/89

∴ The correct answer is 50/89.

Additional Information

Partnership in Renting

When individuals share a rented property or field, the rent must be divided proportionally based on both the amount of resources used and the time duration (Usage × Time).

Animal Equivalence

When multiple entity types are involved (horses, cows, sheep), convert all of them into a single common base unit before calculating the total usage to ensure a uniform scale.

Compound Ratio

The final ratio of share is a compound ratio formed by multiplying the equivalent units ratio and the time ratio for each individual.

Shortcut Trick

A invests 2 times more than B ⇒ Initial capital ratio A : B = 3 : 1.

A's profit units = (3 × 8) + (1.5 × 4) = 30; B's profit units = (1 × 8) + (3 × 4) = 20.

Profit ratio A : B = 30 : 20 = 3 : 2.

A's share = [3 ÷ (3 + 2)] × 75000 = ₹45,000.

∴ The correct answer is ₹45,000.

Alternate Method

Given:

⇒ Total profit after one year (12 months) = ₹75,000

⇒ A invests 2 times more than B initially.

Formula Used:

⇒ Profit Ratio = (Capital × Time)

A 8 mo × 3x 4 mo × 1.5x Sum = 30x B 8 mo × 1x 4 mo × 3x Sum = 20x Profit Ratio (A : B) = 30x : 20x = 3 : 2

Calculations:

⇒ Let B's initial investment be x.

⇒ A invests 2 times more than B, so A's initial investment = x + 2x = 3x.

⇒ For the first 8 months, A's capital is 3x and B's capital is x.

⇒ For the remaining 4 months, A withdraws half of his capital, so A's new capital = 3x ÷ 2 = 1.5x.

⇒ For the remaining 4 months, B triples his capital, so B's new capital = 3 × x = 3x.

⇒ A's profit equivalent = (3x × 8) + (1.5x × 4) = 24x + 6x = 30x.

⇒ B's profit equivalent = (x × 8) + (3x × 4) = 8x + 12x = 20x.

⇒ Profit Ratio of A : B = 30x : 20x = 3 : 2.

⇒ Total ratio parts = 3 + 2 = 5 parts.

⇒ A's share = (3 ÷ 5) × 75000 = 3 × 15000

⇒ A's share = ₹45,000.

∴ The correct answer is ₹45,000.

Additional Information

Partnership Core Principle

Profit is exclusively distributed among partners in the ratio of their effective investments (the product of capital and duration). Key formula: Profit Ratio = C₁T₁ : C₂T₂.

"More Than" Phrasing Concept

When a question states "A is n times more than B", the expression evaluates to A = B + nB. It must not be confused with "A is n times of B" which simply means A = nB.

Active vs Sleeping Partner

An active partner actively works for the business and typically claims a salary or fixed percentage of total profit before the remainder is split purely based on the initial capital-time ratio among all partners.

Shortcut Trick

Ratio of Senior Staff to Other Employees = 8 : 16 = 1 : 2

Deviation of senior staff average = ₹60,000 − ₹40,000 = +20,000

Total deviation of senior staff (1 unit) = 1 × 20,000 = +20,000

To balance the average, the other employees (2 units) must have a deviation of −20,000 ÷ 2 = −10,000

Average salary of other employees = ₹40,000 − 10,000 = ₹30,000

∴ The correct answer is ₹30,000.

Alternate Method

Given:

Total number of employees = 24

Average salary of all employees = ₹40,000

Number of senior staff = 8

Average salary of senior staff = ₹60,000

8 Senior Staff Avg: ₹60,000

16 Other Employees Avg: ₹x

Total: 24 Employees (Avg: ₹40,000)

Formula Used:

Total Salary = Average Salary × Number of Employees

Calculations:

⇒ Total salary of all 24 employees = 24 × 40,000 = ₹9,60,000

⇒ Total salary of 8 senior staff = 8 × 60,000 = ₹4,80,000

⇒ Number of other employees = 24 − 8 = 16

⇒ Total salary of 16 other employees = 9,60,000 − 4,80,000 = ₹4,80,000

⇒ Average salary of other employees = 4,80,000 ÷ 16 = ₹30,000

∴ The correct answer is ₹30,000.

Additional Information

Combined Average

When two groups of sizes n₁ and n₂ have averages A₁ and A₂, their combined average is given by (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method in Averages

The sum of deviations of all individual values from their true average is always zero. If one group has a positive deviation, the remaining group must have an equal negative deviation to balance the overall average.

Weighted Average

Instead of calculating the total sum with large numbers, the simplified ratios of the group sizes can be used as weights to find the combined average much faster.

Shortcut Trick

When an item is excluded, its value can be calculated directly by the average deviation method:

Excluded value = Original Average + (Drop in Average × Remaining number of items)

Here, Drop in Average = 58 − 54 = 4

Excluded highest score = 58 + (4 × 9) = 58 + 36 = 94

∴ The correct answer is 94 runs.

Alternate Method

Average Score Partition

10 Innings (Average: 58)

Total Score = 10 × 58 = 580

9 Innings (Average: 54)

Total = 9 × 54 = 486

Highest Score

94

Given:

⇒ Average score of a cricketer in 10 innings = 58

⇒ Average score in 9 innings (after highest score is excluded) = 54

Formula Used:

⇒ Total Value = Average × Number of Observations

Calculations:

⇒ Total score of the cricketer in 10 innings = 10 × 58 = 580

⇒ Total score of the cricketer in 9 innings = 9 × 54 = 486

⇒ Highest score = Total score in 10 innings − Total score in 9 innings

⇒ Highest score = 580 − 486 = 94

∴ The correct answer is 94 runs.

Additional Information

Averages (Basic Concept)

Average is the central value of a set of data, calculated as the sum of all observations divided by the total number of observations.

Effect of Excluding an Observation

When an item is removed and the overall average decreases, it indicates that the value of the removed item was strictly greater than the initial average.

Combined Average

If two sets of data with n₁ and n₂ observations have averages A₁ and A₂ respectively, their combined average is (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick Since the number of rounds played by both teams is the same (3 rounds), the team with the higher total score will naturally have the higher average. Total score of Team A = 42 + 38 + 40 = 120 Total score of Team B = 45 + 46 + 44 = 135 Since 135 > 120, Team B has the higher total and thus the higher average. ∴ The correct answer is Team B.

Alternate Method Given: Scores of Team A in Round 1, 2, and 3 = 42, 38, and 40 respectively. Scores of Team B in Round 1, 2, and 3 = 45, 46, and 44 respectively. Formula Used: Average = (Sum of all observations) ÷ (Total number of observations) Calculations: ⇒ Average score of Team A = (42 + 38 + 40) ÷ 3 ⇒ Average score of Team A = 120 ÷ 3 = 40 ⇒ Average score of Team B = (45 + 46 + 44) ÷ 3 ⇒ Average score of Team B = 135 ÷ 3 = 45 ⇒ Comparing both averages, we can see that 45 > 40. ⇒ Hence, Team B has a higher average score than Team A. ∴ The correct answer is Team B.

Additional Information Average of a Set of Numbers The average (or arithmetic mean) of a set of data is found by adding all the numbers in the data set and then dividing the sum by the total number of values in the set. Direct Comparison Method When comparing the averages of two or more groups that have the exact same number of observations, calculating the total sum alone is sufficient. The group with the highest total sum will always have the highest average. Weighted Average When observations appear with varying frequencies (or weights), the weighted average formula is used: Average = (w1x1 + w2x2 + ... + wnxn) ÷ (w1 + w2 + ... + wn).

Shortcut Trick

Total Capacity = 0.005 kl = 0.005 × 1,000,000 ml = 5,000 ml.

Empty volume = 5,000 − 3,500 = 1,500 ml.

Empty Percentage = (1,500 ÷ 5,000) × 100 = 30%.

∴ The correct answer is 30%.

Alternate Method

Given:

Filled volume of the tank = 3,500 ml

Total capacity of the tank = 0.005 kl

Formula Used:

1 kiloliter (kl) = 1,000 liters (L)

1 liter (L) = 1,000 milliliters (ml)

Percentage = (Part value ÷ Total value) × 100

Total Capacity 5000 ml

Empty = 1500 ml ( ? % )

Filled = 3500 ml

Calculations:

⇒ Total capacity = 0.005 kl = 0.005 × 1,000 × 1,000 ml

⇒ Total capacity = 5,000 ml

⇒ Empty volume = Total capacity − Filled volume

⇒ Empty volume = 5,000 − 3,500 = 1,500 ml

⇒ Percentage of empty tank = (1,500 ÷ 5,000) × 100

⇒ Percentage of empty tank = 0.3 × 100 = 30%

∴ The correct answer is 30%.

Additional Information

Unit Conversions of Volume

1 Kiloliter (kl) = 10³ Liters = 10⁶ Milliliters (ml). Always ensure all given quantities are in the exact same measurement unit before performing any arithmetic calculations.

Complementary Percentages

The sum of the percentage of the filled portion and the empty portion of a container is always exactly 100%. If a tank is 70% full, it is fundamentally 100% − 70% = 30% empty.

Percentage Calculation Rule

To calculate what percentage quantity X is of quantity Y, use the standard formula: (X ÷ Y) × 100. The base reference value (Total Capacity) must always reside in the denominator.

Shortcut Trick

Let the price of standard and premium pens be 1 and 2 units respectively, and standard boxes ordered be y.

Original Cost = (8 × 2) + (y × 1) = 16 + y

New Cost (after interchange) = (y × 2) + (8 × 1) = 2y + 8

Given a 25% increase in the bill, the multiplier is 5/4, giving: (2y + 8) / (16 + y) = 5 / 4

⇒ 8y + 32 = 5y + 80 ⇒ 3y = 48 ⇒ y = 16

Ratio of Premium to Standard = 8 : 16 = 1 : 2.

∴ The correct answer is 1 : 2.

Alternate Method

Given:

Original number of premium boxes = 8

Price of a premium box is twice the price of a standard box.

New total bill is 25% more than the original bill due to interchanged quantities.

Original Bill (100%)

16 + y

New Bill (125%)

2y + 8

• 25% Increase

Calculations:

⇒ Let the original number of standard pen boxes be y and price per standard box be P.

⇒ Price per premium box = 2P.

⇒ Original Bill = (8 × 2P) + (y × P) = P(16 + y)

⇒ When quantities are interchanged, there are y premium boxes and 8 standard boxes.

⇒ New Bill = (y × 2P) + (8 × P) = P(2y + 8)

⇒ The new bill is 25% more than the original, so New Bill = 1.25 × Original Bill

⇒ P(2y + 8) = 1.25 × P(16 + y)

⇒ 2y + 8 = 1.25 × (16 + y)

⇒ 2y + 8 = 20 + 1.25y

⇒ 2y − 1.25y = 20 − 8

⇒ 0.75y = 12

⇒ y = 12 ÷ 0.75 = 16

⇒ Original number of standard pen boxes = 16.

⇒ Required ratio (Premium boxes : Standard boxes) = 8 : 16 = 1 : 2.

∴ The correct answer is 1 : 2.

Additional Information

Ratio & Proportion

A ratio compares values, showing how much of one there is compared to another. If two quantities are in the ratio a:b, they can be represented directly as ax and bx for calculation purposes to quickly establish a relation.

Percentage Increase

When a quantity increases by R%, its new value becomes (100 + R)% of the original. An increase of 25% corresponds to a multiplying factor of 125% or 5/4.

Interchange of Quantities

When two items with original quantities x and y and respective unit prices a and b are interchanged, the net change in the total bill follows the shortcut formula: Change = (a − b)(y − x). A positive result directly indicates a percentage increase.

Shortcut Trick

Multiplying factor for 10% compound interest = 1 + 10/100 = 11/10.

Ratio of final Amount to Principal = 133100 ÷ 100000 = 1331/1000.

We know that (11/10)ⁿ = 1331/1000.

Since 11³ = 1331 and 10³ = 1000, the value of n must be 3.

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹1,00,000

Amount (A) = ₹1,33,100

Rate of Interest (R) = 10% compounded annually

Formula Used:

A = P × (1 + R/100)ⁿ

Start (P) 100k

Year 1 110k

Year 2 121k

Year 3 (A) 133.1k

+10%

+10%

+10%

Total Time Required = 3 Years

Calculations:

⇒ 1,33,100 = 1,00,000 × (1 + 10/100)ⁿ

⇒ 133100 ÷ 100000 = (11/10)ⁿ

⇒ 1331/1000 = (11/10)ⁿ

⇒ (11/10)³ = (11/10)ⁿ

⇒ By equating the powers on both sides, we get n = 3.

∴ The correct answer is 3 years.

Additional Information

Effective Rate of Interest

For a 10% annual rate compounded annually over 3 years, the effective percentage increase directly becomes 33.1%.

Simple Interest vs Compound Interest

For the first year, SI and CI are completely identical. From the second year onward, CI starts to grow faster because interest is also applied to the accumulated interest of the preceding years.

Half-Yearly Compounding

If the interest was compounded half-yearly instead of annually, the applied rate would be halved (R/2) and the number of calculation cycles would double (2n).

Shortcut Trick

In percentage terms, the Cost Price (CP) is always considered 100%.

The given loss is 5% of the Cost Price, which is equal to ₹125.

5% → ₹125

100% → (125 ÷ 5) × 100 = ₹2500

∴ The correct answer is ₹2500.

Alternate Method

Given:

Loss incurred = ₹125

Loss percentage = 5%

Selling Price (95%) Loss (5%)

Total Cost Price (100%)

₹125

Formula Used:

Loss % = (Loss ÷ Cost Price) × 100

Calculation:

⇒ 5 = (125 ÷ CP) × 100

⇒ 5 × CP = 12500

⇒ CP = 12500 ÷ 5

⇒ CP = 2500

∴ The correct answer is ₹2500.

Additional Information

Profit and Loss Base

Profit or Loss is always calculated on the base of Cost Price (CP) unless stated otherwise in the question.

Key Formulas

Profit % = (Profit ÷ CP) × 100 | Loss % = (Loss ÷ CP) × 100

Selling Price (SP) Evaluation

When there is a profit: SP = CP × [(100 + Profit %) ÷ 100]

When there is a loss: SP = CP × [(100 − Loss %) ÷ 100]

Shortcut Trick

Let the Cost Price (CP) be 100. With a 10% loss, the initial Selling Price Y = 90.

New Marked Price (MP) = 0.8 × Y = 0.8 × 90 = 72.

New Selling Price after 5% discount = 72 × 0.95 = 68.4.

Overall Loss = CP − New SP = 100 − 68.4 = 31.6.

Percentage Loss = (31.6 ÷ 100) × 100 = 31.6%.

∴ The correct answer is 31.6%.

Alternate Method

Given:

Initial Selling Price (SP) = Y

Loss = 10%

Marked Price (MP) = 0.8Y

Discount = 5%

Formula Used:

Cost Price (CP) = SP × [ 100 ÷ (100 − Loss%) ]

Selling Price after discount = MP × [ (100 − Discount%) ÷ 100 ]

Percentage Loss = [ (CP − SP) ÷ CP ] × 100

CP = 100 Y = 90 MP = 72 SP₂ = 68.4 −10% × 0.8 −5% Overall Loss = 100 − 68.4 = 31.6%

Calculations:

Let the Cost Price (CP) of the novel be ₹100.

⇒ Loss = 10% of CP = ₹10

⇒ Initial SP (Y) = CP − Loss = 100 − 10 = ₹90

Now, the new Marked Price (MP) is set to 0.8Y to clear old stock.

⇒ MP = 0.8 × 90 = ₹72

The bookseller gives a 5% discount on this new MP.

⇒ Discount = 5% of 72 = ₹3.6

⇒ New SP = MP − Discount = 72 − 3.6 = ₹68.4

Now, calculating the overall loss on the clearance sale with respect to the original CP:

⇒ Overall Loss = CP − New SP = 100 − 68.4 = ₹31.6

⇒ Percentage Loss = (Overall Loss ÷ CP) × 100 = (31.6 ÷ 100) × 100 = 31.6%

(Note: The exact result is 31.6%. The option 31.59% is generated if intermediate fractional steps like 10/9 are rounded early.)

∴ The correct answer is 31.6%.

Additional Information

Marked Price and Successive Discount

When discounts are applied sequentially, the effective selling price is calculated directly as MP × (1 − d₁/100) × (1 − d₂/100).

Cost Price vs Selling Price

Profit or loss is always calculated on the base Cost Price (CP) unless the question specifically directs otherwise. The general relation is given by CP = SP × [ 100 ÷ (100 ± Profit/Loss%) ].

Discount vs Loss

Discount is an amount deduced solely from the Marked Price (MP), whereas Loss is the deficit calculated against the Cost Price (CP). Offering deep discounts can frequently push the Selling Price below the Cost Price, leading to an overall loss.