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Shortcut Trick

Investment ratio of A : B : C = 5 : 7 : 5

Time period ratio of A : B : C = 12 : 12 : 8 = 3 : 3 : 2

Profit sharing ratio = (5 × 3) : (7 × 3) : (5 × 2) = 15 : 21 : 10

Total ratio units = 15 + 21 + 10 = 46 units

Sum of B and C shares = 21 + 10 = 31 units

Value = (31 ÷ 46) × 92,000 = 31 × 2,000 = ₹62,000

∴ The correct answer is ₹62,000.

Alternate Method

Given: Investment ratio A : B = 5 : 7. After 4 months, C joins with investment = A. Total profit = ₹92,000.

⇒ Let the initial investments of A and B be 5x and 7x.

⇒ Investment of C = Investment of A = 5x.

⇒ Time duration: A = 12 months, B = 12 months, C = (12 − 4) = 8 months.

⇒ Profit Ratio = (Capital_A × Time_A) : (Capital_B × Time_B) : (Capital_C × Time_C)

⇒ Profit Ratio = (5x × 12) : (7x × 12) : (5x × 8)

⇒ Profit Ratio = 60 : 84 : 40 = 15 : 21 : 10

⇒ Total profit units = 15 + 21 + 10 = 46 units.

⇒ Given, 46 units = ₹92,000.

⇒ 1 unit = 92,000 ÷ 46 = ₹2,000.

⇒ Sum of shares of B and C = (21 + 10) units = 31 units.

⇒ Sum = 31 × 2,000 = ₹62,000.

∴ The correct answer is ₹62,000.

Additional Information

Partnership Ratio Formula

The profit sharing ratio in a business is always calculated as P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂), where C is capital and T is time.

Compound Partnership

When partners invest their capitals for different time periods, it is called a compound partnership.

Active vs Sleeping Partner

An active partner manages the business and may receive a salary or extra share, while a sleeping partner only invests money.

Given:

{ (12/33) ÷ (12/2) } ÷ ( (3/7) × (14/9) + (3/11) ) + (2/8) ÷ (31/25) of (25/8)

Formula used:

The expression is solved using the BODMAS rule, which stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

Solving the first Bracket and the Of operation:

⇒ { (12/33) × (2/12) } ÷ ( (3/7 × 14/9) + 3/11 ) + (1/4) ÷ (31/25 × 25/8)

⇒ (2/33) ÷ ( 2/3 + 3/11 ) + (1/4) ÷ (31/8)

Solving the second Bracket and performing Division:

⇒ (2/33) ÷ ( (22 + 9)/33 ) + (1/4) × (8/31)

⇒ (2/33) ÷ (31/33) + 2/31

Continuing with the Division operation:

⇒ (2/33) × (33/31) + 2/31

⇒ 2/31 + 2/31

Performing the final Addition:

⇒ 4/31

∴ The correct answer is option (3).

Shortcut Trick

Apply the successive percentage change formula: a + b + (ab/100).

Here, increase a = +10% and discount b = −15%.

Net Change = 10 − 15 + (10 × (−15))/100 = −5 − 1.5 = −6.5%.

Final Price = 600 × (100 − 6.5)% = 600 × 93.5% = 6 × 93.5 = 561.

∴ The correct answer is ₹561.

Alternate Method

Given: Initial Price = ₹600, Price Increase = 10%, Discount = 15%.

Calculations:

⇒ Increased Price = 600 + (10% of 600)

⇒ Increased Price = 600 + (10/100) × 600 = 600 + 60 = ₹660

⇒ Discount Amount = 15% of ₹660

⇒ Discount Amount = (15/100) × 660 = 0.15 × 660 = ₹99

⇒ Final Selling Price = Increased Price − Discount

⇒ Final Selling Price = 660 − 99 = ₹561

∴ The correct answer is ₹561.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the net percentage change is given by [x + y + (xy/100)]%.

Multiplying Factor Method

Final Value = Initial Value × (1 + r₁/100) × (1 + r₂/100). For a 10% increase, factor is 1.10; for a 15% discount, factor is 0.85.

Profit and Loss Relation

If the final price is greater than the initial cost, there is a profit; if less, there is a loss. Here, 561 < 600, so the net result is a 6.5% loss.

Shortcut Trick

For a number to be divisible by 24, it must be divisible by both 3 and 8.

Divisibility by 8: Last three digits 84B ÷ 8 ⇒ B can be 0 or 8.

Divisibility by 3: Sum of digits (4 + 6 + A + 4 + 8 + 4 + B) = 26 + A + B must be a multiple of 3.

To find the minimum (A + B), we test B = 0: 26 + A + 0 = 27 (nearest multiple) ⇒ A = 1.

Then, A + B = 1 + 0 = 1.

∴ The correct answer is 1.

Alternate Method

Given: The 7-digit number 46A484B is divisible by 24.

Formula Used: Divisibility rules for 3 and 8 (since 24 = 3 × 8).

Calculations:

⇒ For divisibility by 8, the last three digits 84B must be divisible by 8.

⇒ 840 ÷ 8 = 105 (Remainder 0), so B = 0 is possible.

⇒ 848 ÷ 8 = 106 (Remainder 0), so B = 8 is possible.

⇒ For divisibility by 3, sum of digits = 4 + 6 + A + 4 + 8 + 4 + B = 26 + A + B.

⇒ If B = 0, Sum = 26 + A. The smallest A making this divisible by 3 is A = 1 (as 26 + 1 = 27).

⇒ In this case, A + B = 1 + 0 = 1.

⇒ If B = 8, Sum = 26 + A + 8 = 34 + A. The smallest A making this divisible by 3 is A = 2 (as 34 + 2 = 36).

⇒ In this case, A + B = 2 + 8 = 10.

⇒ Comparing both cases, the minimum value of (A + B) is 1.

∴ The correct answer is 1.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits form a number that is divisible by 8 or if the last three digits are 000.

Divisibility Rule of 3

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

Composite Divisibility

To check divisibility by a composite number like 24, check divisibility by its co-prime factors (e.g., 3 and 8, where HCF is 1).

Shortcut Trick

Total Investment (CP) = 7000 + 400 + 600 = ₹8000

Selling Price (SP) = ₹9800

Profit = 9800 − 8000 = ₹1800

Profit % = (1800 ÷ 8000) × 100 = 22.5%

∴ The correct answer is 22.5%.

Alternate Method

Given:

Purchase Price of microwave oven = ₹7,000

Transportation cost = ₹400

Repair cost = ₹600

Selling Price (SP) = ₹9,800

Formula Used:

Total CP = Purchase Price + Overhead Expenses

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

Calculations:

⇒ Total Investment (Total CP) = 7000 + 400 + 600

⇒ Total CP = ₹8,000

⇒ Profit = SP − Total CP

⇒ Profit = 9800 − 8000 = ₹1,800

⇒ Profit % = (1800 ÷ 8000) × 100

⇒ Profit % = 180 ÷ 8 = 22.5%

∴ The correct answer is 22.5%.

Additional Information

Overhead Charges

Expenses like transportation, repairs, and labor are always added to the purchase price to calculate the effective Cost Price.

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100 and Loss % = (Loss ÷ CP) × 100. Both are always calculated on the Cost Price unless specified otherwise.

Selling Price Calculation

SP = CP × (100 + Profit%) ÷ 100 or SP = CP × (100 − Loss%) ÷ 100.

Shortcut Trick

Ratio of number of boys to girls = 8 : 4 = 2 : 1.

Using Weighted Average method: Combined Average = (Ratio₁ × Avg₁ + Ratio₂ × Avg₂) ÷ (Total Ratio)

Combined Average = (2 × 56 + 1 × 80) ÷ (2 + 1)

Combined Average = (112 + 80) ÷ 3 = 192 ÷ 3 = 64.

∴ The correct answer is 64.

Alternate Method

Given:

Number of boys (n₁) = 8, Average score (a₁) = 56

Number of girls (n₂) = 4, Average score (a₂) = 80

Formula Used:

Combined Average = (Total sum of scores) ÷ (Total number of students)

Calculations:

⇒ Total score of boys = 8 × 56 = 448

⇒ Total score of girls = 4 × 80 = 320

⇒ Total score of all students = 448 + 320 = 768

⇒ Total number of students = 8 + 4 = 12

⇒ Average score of all students = 768 ÷ 12 = 64

∴ The correct answer is 64.

Additional Information

Weighted Average Formula

It is used when groups of different sizes are combined: Avg = (n₁×a₁ + n₂×a₂ + ...) ÷ (n₁ + n₂ + ...).

Deviation Method

Pick an assumed mean (A), calculate deviations (d), then find the true average using: Average = A + (Σf_id_i ÷ Σf_i).

Combined Mean Property

The combined average always lies between the lowest and the highest individual group averages.

Shortcut Trick

For any two-digit number, the difference between its digits is equal to the change in value upon reversing digits divided by 9.

Difference between digits = 27 ÷ 9 = 3

Given digit sum = 9

Unit digit = (Sum + Difference) ÷ 2 = (9 + 3) ÷ 2 = 6

∴ The correct answer is 6.

Alternate Method

Given:

Sum of digits of a two-digit number = 9

Number + 27 = Reversed Number

Method: Algebraic representation where tens digit is x and unit digit is y.

Let the tens digit be x and the unit digit be y.

⇒ x + y = 9 ... (i)

⇒ The original number is 10x + y.

⇒ According to the problem: (10x + y) + 27 = 10y + x

⇒ 10x − x + y − 10y = −27

⇒ 9x − 9y = −27

⇒ x − y = −3 or y − x = 3 ... (ii)

⇒ Adding equation (i) and (ii): (x + y) + (y − x) = 9 + 3

⇒ 2y = 12

⇒ y = 6

⇒ x = 9 − 6 = 3

The number is 36 and its unit digit is 6.

∴ The correct answer is 6.

Additional Information

Number Representation

A two-digit number with tens digit x and unit digit y is expressed as 10x + y. Its reversed form is 10y + x.

Difference Property

The difference between any two-digit number and its reverse is always a multiple of 9, specifically 9 × |x − y|.

Divisibility by 9

A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Shortcut Trick

Total Interest Percentage = Rate × Time = 20% × 3 = 60%

Given: 60% of Principal = ₹1,320

Principal (100%) = (1,320 ÷ 60) × 100

⇒ 22 × 100 = ₹2,200

∴ The correct answer is ₹2,200.

Alternate Method

Given: Simple Interest (SI) = ₹1,320, Rate (R) = 20% per year, Time (T) = 3 years.

Formula Used: SI = (P × R × T) ÷ 100

⇒ 1,320 = (P × 20 × 3) ÷ 100

⇒ 1,320 = (P × 60) ÷ 100

⇒ 1,320 = 0.6 × P

⇒ P = 1,320 ÷ 0.6

⇒ P = (1,320 × 10) ÷ 6

⇒ P = 220 × 10

⇒ P = ₹2,200

∴ The correct answer is ₹2,200.

Additional Information

Total Amount in Simple Interest

The total amount (A) is the sum of Principal and Interest. Formula: A = P + SI or A = P(1 + RT/100).

Constant Nature of SI

Unlike Compound Interest, Simple Interest remains constant for every year as it is always calculated on the original Principal amount.

Relationship between P, R, T and SI

If SI, R, and T are known, Principal can be found using P = (SI × 100) ÷ (R × T).

Shortcut Trick

The effective interest rate for 2 years at 10% per annum compounded annually is 10 + 10 + (10 × 10 ÷ 100) = 21%.

Given that the interest of 21% is equal to ₹462.

Value of 1% = 462 ÷ 21 = ₹22.

The total amount payable is Principal (100%) + Interest (21%) = 121%.

Total amount = 121 × 22 = ₹2,662.

∴ The correct answer is ₹2,662.

Alternate Method

Given:

Principal (P) = ₹y

Rate of Interest (R) = 10% per annum

Time (T) = 2 years

Compound Interest (CI) = ₹462

Formula Used:

CI = P × [(1 + R/100)^T − 1]

Amount (A) = Principal (P) + Compound Interest (CI)

Calculations:

⇒ 462 = y × [(1 + 10/100)² − 1]

⇒ 462 = y × [(1.1)² − 1]

⇒ 462 = y × [1.21 − 1]

⇒ 462 = 0.21 × y

⇒ y = 462 ÷ 0.21 = 2200

⇒ Total amount payable = y + 462 = 2200 + 462 = ₹2,662

∴ The correct answer is ₹2,662.

Additional Information

Effective Interest Rate Formula

For a rate of R% for 2 years, the effective compound interest rate is calculated as R + R + (R × R ÷ 100).

Compound Interest Amount Formula

The total amount A is given by A = P × (1 + R/100)ⁿ, where n is the number of years.

Difference between CI and SI

The difference between Compound Interest and Simple Interest for 2 years is P × (R ÷ 100)².

Shortcut Trick

Excluded Number = Old Average + (Remaining Count × Decrease in Average)

Remaining numbers = 15 − 1 = 14

Decrease in average = 6

Excluded Number = 82 + (14 × 6) = 82 + 84 = 166

∴ The correct answer is 166.

Alternate Method

Given:

Total numbers (n₁) = 15

Initial arithmetic mean (M₁) = 82

Mean reduction = 6

Formula Used:

Sum of numbers = Average × Total count

Calculations:

⇒ Total sum of 15 numbers = 15 × 82 = 1230

⇒ Remaining numbers after exclusion = 15 − 1 = 14

⇒ New mean = 82 − 6 = 76

⇒ Total sum of 14 numbers = 14 × 76 = 1064

⇒ Excluded number = Sum of 15 numbers − Sum of 14 numbers

⇒ Excluded number = 1230 − 1064 = 166

∴ The correct answer is 166.

Additional Information

Arithmetic Mean

The average of a set of data calculated by dividing the sum of all values by the total number of values.

Effect of Excluding an Item

If the excluded number is greater than the average, the mean of the remaining items decreases. If it is smaller, the mean increases.

Net Deviation Rule

The sum of deviations of all observations from their arithmetic mean is always zero. This property is used to find missing or excluded values quickly.