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Shortcut Trick

In a partnership, the profit ratio is directly proportional to the product of Investment and Time.

Gautam's equivalent investment = 60000 × 24 months

Jatin's equivalent investment = 35000 × 16 months

Profit Ratio = (60000 × 24) : (35000 × 16) = (60 × 3) : (35 × 2) = 180 : 70 = 18 : 7

∴ The correct answer is 18:7.

Alternate Method

Given:

Gautam's investment (C₁) = ₹ 60,000

Jatin's investment (C₂) = ₹ 35,000

Total business duration = 2 years = 24 months

Jatin joins 8 months later.

Formula Used:

Profit Ratio = (C₁ × T₁) : (C₂ × T₂)

Gautam (₹ 60,000)

24 months

Jatin (₹ 35,000)

16 months

Joined 8 months later

Profit Ratio = (60k × 24) : (35k × 16)

Calculations:

⇒ Gautam's time period (T₁) = 24 months

⇒ Jatin's time period (T₂) = 24 − 8 = 16 months

⇒ Profit Ratio = (60000 × 24) : (35000 × 16)

⇒ Simplifying the capitals by removing common zeros:

⇒ Profit Ratio = (60 × 24) : (35 × 16)

⇒ Profit Ratio = 1440 : 560

⇒ Dividing both sides by 80:

⇒ Profit Ratio = 18 : 7

∴ The correct answer is 18:7.

Additional Information

Compound Partnership

When the capitals of partners are invested for different time periods, it is called a compound partnership. The share of profit for each partner is calculated by multiplying their respective capital with the time period.

Joining Mid-way

If a partner joins the business t months after the business started, and the total duration to calculate profit is T months, their capital is actively invested for only (T − t) months.

Active vs Sleeping Partner

An active partner manages the operations and usually receives a fixed percentage or salary from the total profit before the remainder is distributed. A sleeping partner only provides capital and shares the remaining profit based entirely on the investment ratio.

Shortcut Trick

Weight of the left person = Old Average + (Remaining People × Decrease in Average)

Old Average = 65 kg, Remaining People = 11

Decrease in Average = 65 − 64.5 = 0.5 kg

Weight = 65 + (11 × 0.5)

Weight = 65 + 5.5 = 70.5 kg

∴ The correct answer is 70.5 kg.

Alternate Method

Given:

Initial average weight of 12 people = 65 kg

New average weight after 1 person leaves = 64.5 kg

12 People | Average: 65 kg (Total: 780 kg)

11 People | Avg: 64.5 kg

1 Left

Total: 709.5 kg ? kg

Calculations:

⇒ Total initial weight of 12 people = 12 × 65 = 780 kg

⇒ Total weight of remaining 11 people = 11 × 64.5 = 709.5 kg

⇒ Weight of the person who left = (Total initial weight) − (Total new weight)

⇒ Weight of the person who left = 780 − 709.5

⇒ Weight of the person who left = 70.5 kg

∴ The correct answer is 70.5 kg.

Additional Information

When a Person Joins

If a new person joins and the average increases, New Weight = Old Average + (New total people × Increase in average).

When a Person is Replaced

If one person is replaced by another and the average changes, Weight of New Person = Weight of Replaced Person ± (Total number of people × Change in average).

Weighted Average

When combining two groups, Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Let the assumed average be 65.

Deviations of Q's sales (65, 68, 62, 64) from 65 are 0, +3, −3, and −1.

Sum of deviations = 0 + 3 − 3 − 1 = −1

Actual Average = Assumed Average + (Sum of deviations ÷ 4)

Actual Average = 65 − (1 ÷ 4) = 64.75

∴ The correct answer is ₹64.75.

Alternate Method

Given:

Sales of salesperson Q in different months:

Jan = 65, Feb = 68, March = 62, April = 64

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Total Sales of Q = 65 + 68 + 62 + 64

⇒ Total Sales of Q = 259

⇒ Total number of months = 4

⇒ Average Sales = 259 ÷ 4

⇒ Average Sales = 64.75

∴ The correct answer is ₹64.75.

Additional Information

Average Formula

Average is calculated by dividing the sum of all values by the total number of values. Average = Sum of observations ÷ Total number of observations.

Net Deviation Method

By selecting an assumed average, calculate the positive or negative deviation of each term. Net average = Assumed average + (Sum of deviations ÷ Total number of items).

Combined Average

When combining two distinct groups with quantities n₁ and n₂ and averages A₁ and A₂ respectively, the combined average is: (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂).

Shortcut Trick

Rate = 10% = 1/10 ⇒ Multiplying factor per year = 11/10

Ratio of Amount to Principal = 66550 ÷ 50000 = 1331 ÷ 1000

We know that (11/10)³ = 1331 / 1000

Since the cube corresponds to 3 periods, the time taken is 3 years.

∴ The correct answer is 3 years.

Alternate Method

Given:

Principal (P) = ₹50,000

Amount (A) = ₹66,550

Rate of Interest (R) = 10% per annum

Formula Used:

A = P × (1 + R/100)ⁿ

Step-by-Step Compound Interest Growth

₹50,000

₹55,000

₹60,500

₹66,550

+10% 1 Yr

+10% 1 Yr

+10% 1 Yr

Total Time = 3 Years

Calculations:

⇒ 66,550 = 50,000 × (1 + 10/100)ⁿ

⇒ 66,550 ÷ 50,000 = (1 + 1/10)ⁿ

⇒ 6,655 ÷ 5,000 = (11/10)ⁿ

⇒ 1,331 ÷ 1,000 = (11/10)ⁿ

⇒ (11/10)³ = (11/10)ⁿ

⇒ n = 3

∴ The correct answer is 3 years.

Additional Information

Effective Rate of CI

For a 10% annual interest rate, the effective CI rate for 2 years is 21%, and for 3 years it is 33.1%. Here, 33.1% of ₹50,000 exactly equals the interest of ₹16,550.

Difference between CI and SI (2 Years)

The difference between compound interest and simple interest for a period of 2 years is calculated directly as: Difference = P × (R ÷ 100)².

Difference between CI and SI (3 Years)

For a period of 3 years, the formula extends to: Difference = P × (R ÷ 100)² × (3 + R ÷ 100).

Shortcut Trick

Given the ratio of Profit to Cost Price is 2 : 7.

Let Profit = 2 units and Cost Price = 7 units.

Selling Price = Cost Price + Profit = 7 + 2 = 9 units.

Required Ratio (Selling Price : Profit) = 9 : 2.

∴ The correct answer is 9:2.

Alternate Method

Given:

Ratio of Profit to Cost Price (CP) = 2 : 7

Cost Price (CP) = 7x Profit = 2x

Selling Price (SP) = 7x + 2x = 9x Profit & Loss Visualisation

Formula Used:

Selling Price (SP) = Cost Price (CP) + Profit

Calculation:

⇒ Let the Profit be 2x and the Cost Price be 7x.

⇒ SP = 7x + 2x

⇒ SP = 9x

⇒ The required ratio of Selling Price to Profit is SP : Profit

⇒ Ratio = 9x : 2x

⇒ Ratio = 9 : 2

∴ The correct answer is 9:2.

Additional Information

Profit and Loss Percentage

Both profit percentage and loss percentage are always calculated on the Cost Price (CP) unless specified otherwise. Formula: Profit % = (Profit ÷ CP) × 100.

Selling Price under Loss

When a product is sold at a loss, the Selling Price is less than the Cost Price. Formula: SP = CP − Loss.

Marked Price and Discount

Discount is always calculated on the Marked Price (MP) of the article. Selling Price = Marked Price − Discount.

Shortcut Trick

Let the Marked Price (MP) of 1 box = 100.

Cost Price (CP) of 1 box = (MP of 8 boxes) ÷ 10 = 800 ÷ 10 = 80.

Selling Price (SP) of 1 box = MP − 2% discount = 100 − 2 = 98.

Profit % = [(SP − CP) ÷ CP] × 100 = [(98 − 80) ÷ 80] × 100.

Profit % = (18 ÷ 80) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Alternate Method

Given:

Total number of boxes bought by the shop owner = 10

Price paid for 10 boxes = Price of 8 boxes

Discount allowed on each box during sale = 2% on Marked Price

Formula Used:

Profit = Selling Price (SP) − Cost Price (CP)

Profit % = (Profit ÷ CP) × 100

Cost Price (CP) of 1 box = 80

Marked Price (MP) of 1 box = 100

Selling Price (SP) of 1 box = 98 (-2% discount)

Profit = 18

Profit % = (18 ÷ 80) × 100 = 22.5%

Calculations:

⇒ Let the Marked Price (MP) of 1 box be 100.

⇒ The shopkeeper buys 10 boxes for the price of 8 boxes.

⇒ Cost Price (CP) of 10 boxes = 8 × 100 = 800.

⇒ Cost Price (CP) of 1 box = 800 ÷ 10 = 80.

⇒ He sells each box at a 2% discount on its Marked Price.

⇒ Selling Price (SP) of 1 box = 100 − (2% of 100) = 98.

⇒ Profit earned on 1 box = SP − CP = 98 − 80 = 18.

⇒ Profit % = (18 ÷ 80) × 100

⇒ Profit % = (9 ÷ 40) × 100 = 22.5%.

∴ The correct answer is 22.5%.

Additional Information

Marked Price and Discount

Discount is always calculated strictly on the Marked Price (MP) unless specified otherwise. The Selling Price is derived as SP = MP − Discount.

Buy X Get Y Free Offers

In promotional schemes, the discount percentage is based on the total number of items taken. Discount % = [Free Items ÷ Total Items (X + Y)] × 100.

Successive Discounts

When two consecutive discounts of a% and b% are applied on an item, the equivalent single discount percentage is given by the formula: (a + b − (a × b) ÷ 100)%.

Dishonest Dealer Profit

When a dealer claims to sell at cost price but uses a false weight, the actual profit percentage is calculated as: [(True Weight − False Weight) ÷ False Weight] × 100.

Shortcut Trick

Let the Cover Price be 100 units.

Discounted Price = 100 − 30% = 70 units.

Total Cost Price = 70 + (5% of 70) = 73.5 units.

Profit = 40% of Cost Price = 0.40 × 73.5 = 29.4 units.

Since 29.4 units = ₹1620, Cover Price (100 units) = (1620 ÷ 29.4) × 100 ≈ ₹5510.20

∴ The correct answer is ₹5,510.20.

Alternate Method

Given:

Discount on Cover Price = 30%

Shipping charge on discounted price = 5%

Profit percentage = 40%

Profit amount (sold for ₹1620 more) = ₹1620

Cover Price(x)-30% disc+5% shipCost Price(0.735x)+40% Profit= ₹1620Selling Price(CP + 1620)Equation: 40% of 0.735x = 1620

Formula Used:

Cost Price (CP) = Discounted Price × (1 + Shipping Rate)

Profit = Profit % × CP

Calculations:

⇒ Let the original Cover Price of the novel be x.

⇒ Discounted Price = x − 30% of x = 0.70x

⇒ Cost Price (CP) = 0.70x + 5% of 0.70x

⇒ CP = 0.70x + 0.035x = 0.735x

⇒ Profit = 40% of CP = 0.40 × 0.735x = 0.294x

⇒ We are given that the Profit earned is ₹1620.

⇒ 0.294x = 1620

⇒ x = 1620 ÷ 0.294

⇒ x = 1620000 ÷ 294

⇒ x ≈ 5510.20

∴ The correct answer is ₹5,510.20.

Additional Information

Marked Price (Cover Price)

It is the original price printed on an item. Successive discounts or flat discounts are always calculated on the Marked Price before deciding the Selling Price or Purchase Cost.

Effective Cost Price with Overheads

When extra charges like shipping, repairs, or transportation are incurred, they are added to the initial discounted purchase price to evaluate the total effective Cost Price.

Profit and Loss Application

All Profit and Loss calculations and percentage margins are applied purely on the final Cost Price, not on the original marked price. Formula: Profit % = (Profit ÷ CP) × 100.

Shortcut Trick

Fraction of syrup replaced each time = 10 ÷ 100 = 1/10.

Fraction of syrup left after 2 replacements = (1 − 1/10)² = (9/10)² = 81/100.

Final syrup = 81/100 × 100 L = 81 L.

Final water = Total volume − Final syrup = 100 L − 81 L = 19 L.

Ratio of syrup to water = 81 : 19.

∴ The correct answer is 81:19.

Alternate Method

Given:

Initial volume of pure syrup (V) = 100 L

Volume drawn and replaced by water (x) = 10 L

Number of operations (n) = 2

Formula Used:

Final amount of pure liquid = V × (1 − x/V)ⁿ

Initial 100L Pure Syrup −10L Mixture +10L Water (2 Times) Final 81L Syrup 19L Water Ratio 81:19

Calculations:

⇒ Final amount of syrup = 100 × (1 − 10/100)²

⇒ Final amount of syrup = 100 × (1 − 1/10)²

⇒ Final amount of syrup = 100 × (9/10)²

⇒ Final amount of syrup = 100 × 81/100 = 81 L

⇒ Volume of water in the final mixture = Total Volume − Final amount of syrup

⇒ Volume of water = 100 − 81 = 19 L

⇒ Ratio of syrup to water = 81 : 19

∴ The correct answer is 81:19.

Additional Information

Successive Replacement Formula

When a constant volume 'x' is replaced 'n' times from an initial volume 'V', Final amount = V × (1 − x/V)ⁿ.

Multiple Different Replacements

If replaced amounts are different (x, y, z) from initial volume V, Final amount = V × (1 − x/V) × (1 − y/V) × (1 − z/V).

Alligation Cross Method

Used to find the ratio in which two mixtures should be mixed. The ratio of quantities is inversely proportional to the differences in their unit concentrations.

Shortcut Trick

Volume is proportional to the cube of the radius, while surface area is proportional to the square of the radius.

Since 1 large sphere is melted into 8 identical smaller spheres, the volume ratio of 1 Large : 1 Small = 8 : 1.

Therefore, the radius ratio of Large : Small = 8^1/3 : 1^1/3 = 2 : 1.

The surface area ratio of 1 Large : 1 Small = 2² : 1² = 4 : 1.

Ratio of the surface area of 1 Large sphere to 8 Small spheres = 4 : (8 × 1) = 4 : 8 = 1 : 2.

∴ The correct answer is 1:2.

Alternate Method

Given:

Radius of original metal sphere (R) = 10 cm

Number of identical smaller spheres (n) = 8

Formula Used:

Volume of a sphere = (4/3) × π × r³

Surface Area of a sphere = 4 × π × r²

R = 10 Original Sphere

Melted into 8

r × 8 Smaller Spheres

Calculations:

⇒ Total volume remains constant when melting and recasting.

⇒ Volume of original sphere = 8 × Volume of one smaller sphere

⇒ (4/3) × π × R³ = 8 × (4/3) × π × r³

⇒ R³ = 8r³

⇒ 10³ = 8r³

⇒ 1000 = 8r³

⇒ r³ = 125 ⇒ r = 5 cm

⇒ Surface Area of Original Sphere (S₁) = 4 × π × 10²

⇒ S₁ = 400π cm²

⇒ Total Surface Area of 8 Smaller Spheres (S₂) = 8 × (4 × π × 5²)

⇒ S₂ = 8 × 100π = 800π cm²

⇒ Required Ratio = S₁ : S₂ = 400π : 800π

⇒ Ratio = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Melting and Recasting Law

When a solid object is melted and recast into new shapes, the total volume of the material is always conserved, though the total surface area will change.

Dimensional Scaling Factors

If the linear dimension (radius, side) of a 3D figure changes by a factor of k, then its surface area scales by k² and its volume scales by k³.

Sphere vs Hemisphere Properties

For a full sphere, Total Surface Area = 4πr². For a solid hemisphere, the Curved Surface Area is 2πr² while the Total Surface Area is 3πr² because of the flat circular base.

Shortcut Trick

When a large shape is melted into smaller identical shapes, the number of objects is the ratio of their volumes.

For identical shapes like hemispheres, constants cancel out, leaving only the ratio of the radii cubed.

Number of small hemispheres = (Radius of large ÷ Radius of small)³

⇒ Number = (12 ÷ 3)³ = 4³

⇒ Number = 64

∴ The correct answer is 64.

Alternate Method

12 cm Large Hemisphere

Melted

3 cm

...

n = ? n Small Hemispheres

Given:

Radius of large hemisphere (R) = 12 cm

Radius of small hemisphere (r) = 3 cm

Formula Used:

Volume of a hemisphere = (2 ÷ 3) × π × radius³

Calculation:

Let the total number of small hemispheres produced be n.

Since the large hemisphere is melted to form smaller ones, the total volume of the material remains constant.

⇒ Volume of large hemisphere = n × Volume of small hemisphere

⇒ (2 ÷ 3) × π × R³ = n × (2 ÷ 3) × π × r³

Canceling out (2 ÷ 3) × π from both sides:

⇒ R³ = n × r³

⇒ 12³ = n × 3³

⇒ n = 12³ ÷ 3³

⇒ n = (12 ÷ 3)³

⇒ n = 4³

⇒ n = 64

∴ The correct answer is 64.

Additional Information

Melting and Recasting

When a solid object is melted and recast into new shapes, the total volume remains constant regardless of the new shape. Number of shapes = Total initial volume ÷ Volume of one new shape.

Volume of a Hemisphere

The space occupied by a hemisphere is exactly half the volume of a full sphere. It is given by V = (2 ÷ 3) × π × r³, where r is the radius.

Surface Areas of a Hemisphere

The Curved Surface Area (CSA) of a hemisphere is 2 × π × r². The Total Surface Area (TSA) includes the flat circular top, making it 3 × π × r².