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Shortcut Trick

Effective capital ratio (Investment × Time):

A : B : C = (1,21,500 × 12) : (1,02,060 × 9) : (97,200 × 6 + 72,700 × 6)

A : B : C = 14,58,000 : 9,18,540 : 10,19,400 = 24,300 : 15,309 : 16,990

Sum of ratio parts = 24,300 + 15,309 + 16,990 = 56,599 units

Total profit = ₹1,69,797 ⇒ 1 unit = ₹3

Difference (A − B) = (24,300 − 15,309) × 3 = 8,991 × 3 = ₹26,973.

∴ The correct answer is ₹26,973.

Alternate Method

Given:

Investment of A = ₹1,21,500 for 12 months

Investment of B = ₹1,02,060 for 9 months

Investment of C = ₹97,200 for 6 months and (₹97,200 − ₹24,500) = ₹72,700 for 6 months

Total Profit = ₹1,69,797

Formula: Profit Share Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Calculations:

⇒ Equivalent Capital of A = 1,21,500 × 12 = 14,58,000

⇒ Equivalent Capital of B = 1,02,060 × 9 = 9,18,540

⇒ Equivalent Capital of C = (97,200 × 6) + (72,700 × 6) = 5,83,200 + 4,36,200 = 10,19,400

⇒ Ratio A : B : C = 14,58,000 : 9,18,540 : 10,19,400

⇒ Dividing by common factor 60: 24,300 : 15,309 : 16,990

⇒ Sum of ratios = 24,300 + 15,309 + 16,990 = 56,599

⇒ Total Profit = ₹1,69,797

⇒ Value of 1 unit = 1,69,797 ÷ 56,599 = ₹3

⇒ Difference between share of A and B = (24,300 − 15,309) units

⇒ Difference = 8,991 × 3 = ₹26,973

∴ The correct answer is ₹26,973.

Additional Information

Basic Partnership Law

Profit is always distributed in the ratio of the product of Capital (C) and Time (T), i.e., P₁ : P₂ = C₁×T₁ : C₂×T₂.

Effective Capital for Multiple Transactions

When a partner changes investment mid-term, effective capital = (C₁×T₁ + C₂×T₂ + ...), where T is the duration for each capital amount.

Calculating Unknown Shares

Individual Profit = (Individual Ratio Part ÷ Total Ratio Sum) × Total Profit.

Shortcut Trick

Total Simple Interest for 3 years = 20% × 3 = 60%.

Total Amount = 100% (Principal) + 60% (Interest) = 160% = ₹5,280.

Simple Interest for 2 years = 20% × 2 = 40%.

Required SI = (5,280 ÷ 160%) × 40% = 5,280 ÷ 4 = ₹1,320.

∴ The correct answer is ₹1,320.

Alternate Method

Given:

Amount (A) = ₹5,280, Rate (R) = 20% p.a., Time (T₁) = 3 years.

Formula Used:

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

Amount (A) = Principal (P) + SI = P × (1 + (R × T) ÷ 100)

Calculations:

⇒ 5,280 = P × (1 + (20 × 3) ÷ 100)

⇒ 5,280 = P × (1 + 0.6)

⇒ 5,280 = 1.6P

⇒ Principal (P) = 5,280 ÷ 1.6 = ₹3,300

⇒ Simple Interest for 2 years (T₂ = 2) = (3,300 × 20 × 2) ÷ 100

⇒ SI = 33 × 40 = ₹1,320

∴ The correct answer is ₹1,320.

Additional Information

Simple Interest Properties

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

Principal Formula

The Principal can be calculated if the Amount, Rate, and Time are known: P = (A × 100) ÷ (100 + RT).

SI vs CI

For the first year (if compounded annually), Simple Interest and Compound Interest are equal for the same principal and rate.

Shortcut Trick

The largest 6-digit number is 999999.

Divide 999999 by 96 to find the remainder.

999999 ÷ 96 gives a remainder of 63.

Subtract this remainder from 999999 to get the largest divisible number.

999999 − 63 = 999936.

∴ The correct answer is 999936.

Alternate Method

Given: Divisor = 96, Largest 6-digit number = 999999.

Formula: Required Number = (Largest n-digit number) − (Remainder when largest n-digit number is divided by divisor).

⇒ Largest 6-digit number = 999999

⇒ Dividing 999999 by 96:

⇒ 999999 = 10416 × 96 + 63

⇒ Remainder = 63

⇒ Required number = 999999 − 63

⇒ Required number = 999936

∴ The correct answer is 999936.

Additional Information

Divisibility Rule for 96

A number is divisible by 96 if it is divisible by both 3 (sum of digits) and 32 (last 5 digits divisible by 32).

Smallest n-digit number divisible by x

Required number = Smallest n-digit number + (x − remainder), where remainder is obtained by dividing the smallest n-digit number by x.

Division Algorithm

Dividend = (Divisor × Quotient) + Remainder. For perfect divisibility, the remainder must be zero.

Shortcut Trick

Assume the initial price of fuel is 100.

Final Price = 100 × (0.55) × (0.75) × (0.85) × (1.45)

Final Price = 100 × 0.50840625 = 50.840625

Net percentage decrease = 100 − 50.840625 = 49.159375%

Rounding to 2 decimal places, we get 49.16%.

∴ The correct answer is 49.16%.

Alternate Method

Given:

Initial changes: 45% decrease, 25% decrease, 15% decrease.

Fourth month change: 45% increase.

Formula Used:

Final Value = Initial Value × (1 − r₁/100) × (1 − r₂/100) × (1 − r₃/100) × (1 + r₄/100)

Calculations:

⇒ Let Initial Price = 100

⇒ Price after 1^st month = 100 − 45% of 100 = 55

⇒ Price after 2^nd month = 55 − 25% of 55 = 55 − 13.75 = 41.25

⇒ Price after 3^rd month = 41.25 − 15% of 41.25 = 41.25 − 6.1875 = 35.0625

⇒ Price after 4^th month = 35.0625 + 45% of 35.0625 = 35.0625 + 15.778125 = 50.840625

⇒ Net Percentage Decrease = 100 − 50.840625 = 49.159375%

⇒ Rounding to two decimal places = 49.16%

∴ The correct answer is 49.16%.

Additional Information

Successive Percentage Change

For two successive changes of a% and b%, the net change is given by (a + b + ab/100)%. Use negative values for decrease.

Multiplying Factor Concept

To find the final value after an x% change, multiply the base by (1 ± x/100). For multiple changes, multiply all factors sequentially.

Rounding Rules

When rounding to 2 decimal places, if the third decimal digit is 5 or greater, round up the second digit.

Shortcut Trick

Let Cost Price (CP) = 100.

Marked Price (MP) = 100 + 120 = 220.

To gain 43%, Selling Price (SP) must be 143.

Discount needed = MP − SP = 220 − 143 = 77.

Discount % = (77 ÷ 220) × 100 = 35%.

∴ The correct answer is 35%.

Alternate Method

Given:

Markup = 120% above Cost Price (CP)

Required Gain (Profit) = 43%

Formula Used:

1. Marked Price (MP) = CP × (100 + Markup%) ÷ 100
2. Selling Price (SP) = CP × (100 + Profit%) ÷ 100
3. Discount % = [(MP − SP) ÷ MP] × 100

Calculations:

⇒ Let CP = 100

⇒ MP = 100 + (120% of 100) = 100 + 120 = 220

⇒ SP = 100 + (43% of 100) = 100 + 43 = 143

⇒ Discount = MP − SP = 220 − 143 = 77

⇒ Discount % = (77 ÷ 220) × 100

⇒ Discount % = (7 ÷ 20) × 100 = 7 × 5 = 35%

∴ The correct answer is 35%.

Additional Information

Relationship between CP and MP

The ratio of Cost Price to Marked Price is directly related to the profit and discount: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Net Percentage Change Formula

The overall profit or loss can be calculated using the successive percentage change formula: Net % = Markup − Discount − (Markup × Discount ÷ 100).

Discount Calculation

Discount is always calculated on the Marked Price. Discount = MP − SP. Discount % = (Discount ÷ MP) × 100.

Shortcut Trick

The effective price after successive discounts is: MP × 70% × 88% = SP.

MP × 0.7 × 0.88 = ₹2,464

MP × 0.616 = ₹2,464

MP = 2,464 ÷ 0.616 = ₹4,000.

∴ The correct answer is ₹4,000.

Alternate Method

Given:

Trade Discount = 30%

Scheme Discount = 12%

Selling Price (SP) = ₹2,464

Formula Used:

Final SP = MP × (100 − d₁)/100 × (100 − d₂)/100

Calculations:

⇒ Let the Marked Price (MP) of the electric kettle be M.

⇒ Price after 30% trade discount = M × (100 − 30) ÷ 100 = 0.70M.

⇒ Final Selling Price after 12% scheme discount = 0.70M × (100 − 12) ÷ 100.

⇒ 2,464 = 0.70M × 0.88

⇒ 2,464 = 0.616 × M

⇒ M = 2,464 ÷ 0.616

⇒ M = 4,000

∴ The correct answer is ₹4,000.

Additional Information

Single Equivalent Discount

For two successive discounts of a% and b%, the single equivalent discount is [a + b − (a × b)/100]%.

Relation between MP and CP

The ratio between Marked Price and Cost Price is given by: MP/CP = (100 + Profit%)/(100 − Discount%).

Basis of Discount

A discount (whether trade or scheme) is always calculated on the current Marked Price or the preceding balance price in a chain.

Shortcut Trick

Break the expression into three parts: Part 1 is √36 = 6.

Part 2 is √(324 ÷ 81) = √4 = 2.

Part 3: Count decimal places. Top has 6, Bottom has 8. Net adjustment is 10² in the numerator.

Simplify: √[(95 ÷ 19) × (85 ÷ 17) × (189 ÷ 21) × 100] = √(5 × 5 × 9 × 100) = 150.

Total Sum = 6 + 2 + 150 = 158.

∴ The correct answer is 158.

Alternate Method

Given Expression: √36 + √(324 ÷ 81) + √((9.5 × 0.0085 × 18.9) ÷ (0.021 × 0.0017 × 1.9))

The expression involves square roots and decimal simplification. We solve each term step-by-step:

⇒ Term 1: √36 = 6

⇒ Term 2: √(324 ÷ 81) = √4 = 2

⇒ Term 3: To simplify √((9.5 × 0.0085 × 18.9) ÷ (0.021 × 0.0017 × 1.9)), remove decimals by multiplying both numerator and denominator by 10⁸.

⇒ √((95 × 85 × 189 × 100) ÷ (21 × 17 × 19))

⇒ √((95 ÷ 19) × (85 ÷ 17) × (189 ÷ 21) × 100)

⇒ √(5 × 5 × 9 × 100)

⇒ √(22500) = 150

Now, add all the simplified values obtained from the steps above:

⇒ 6 + 2 + 150 = 158

∴ The correct answer is 158.

Additional Information

BODMAS Rule

Always prioritize Orders (roots and exponents) before addition and subtraction to ensure correct arithmetic evaluation.

Decimal Shift in Fractions

When dividing decimals, shift the decimal point to the right in both the numerator and denominator by the same number of places to simplify into whole numbers.

Square Root of Products

The property √(a × b × c) = √a × √b × √c allows for the factorization of large numbers under a root for easier calculation.

Shortcut Trick

Ratio 5 years ago (M:S) = 7 : 1 (Difference = 6 units)

Current Ratio (M:S) = 4 : 1 (Difference = 3 units)

Equalize differences by multiplying current ratio by 2 → 8 : 2

Change in son's age units from 5 years ago to now = 2 − 1 = 1 unit → 5 years

Son's present age = 2 units = 2 × 5 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given:

Current age of man is four times the age of his son.

Five years ago, the man was seven times older than his son.

Formula Used:

Age difference and ratio calculation using linear equations.

Calculations:

⇒ Let the son's present age be S years.

⇒ Therefore, the man's present age is 4S years.

⇒ Five years ago, son's age = (S − 5) years.

⇒ Five years ago, man's age = (4S − 5) years.

⇒ According to the condition: (4S − 5) = 7 × (S − 5)

⇒ 4S − 5 = 7S − 35

⇒ 35 − 5 = 7S − 4S

⇒ 30 = 3S

⇒ S = 10

∴ The correct answer is 10 years.

Additional Information

Age Difference Constant Rule

The numerical difference between the ages of two people always remains the same, regardless of how many years pass.

Ratio Scaling

When solving age problems using ratios, always ensure the unit difference between the two ratios is equalized relative to the time gap.

General Age Formula

If the present age is X, then age 'n' years ago is (X − n) and age after 'n' years is (X + n).

Shortcut Trick

Ratio of Cost Price (CP) for P to Q = 100 : 125 = 4 : 5

Ratio of CP for Q to R = 100 : 40 = 5 : 2

Combined ratio P : Q : R = 4 : 5 : 2

Given R paid 2 units = ₹5,166

1 unit = ₹2,583 ⇒ P purchased for 4 units = 4 × 2,583 = ₹10,332

∴ The correct answer is ₹10,332.

Alternate Method

Given:

Price paid by R = ₹5,166

Profit percentage of P = 25%

Loss percentage of Q = 60%

Formula Used:

Selling Price (SP) = Cost Price (CP) × (100 ± Profit/Loss%)/100

Calculations:

⇒ Let the purchase price of the item for P be x.

⇒ P sells it to Q at 25% profit, so CP of Q = x × (125/100) = 1.25x

⇒ Q sells it to R at 60% loss, so CP of R = 1.25x × (1 − 60/100)

⇒ CP of R = 1.25x × 0.40 = 0.5x

⇒ Given that R paid ₹5,166, therefore: 0.5x = 5,166

⇒ x = 5,166 ÷ 0.5 = 5,166 × 2

⇒ x = 10,332

∴ The correct answer is ₹10,332.

Additional Information

Successive Percentage Change

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

Fractional Conversion

Profit of 25% is equivalent to a multiplier of 5/4 (1 + 1/4), and a loss of 60% is equivalent to a multiplier of 2/5 (1 − 3/5).

Final Price Formula

Final Price = Initial Price × (100 ± r₁)/100 × (100 ± r₂)/100 × ... × (100 ± r_n)/100.

Shortcut Trick

In a standard 30°−60°−90° right-angled triangle, the ratio of sides opposite to the angles is 1 : √3 : 2.

The side opposite to 30° (building height) corresponds to 1 unit = 100√3 m.

The side opposite to 60° (horizontal distance) corresponds to √3 units.

⇒ Horizontal distance = 100√3 × √3 = 300 m.

∴ The correct answer is 300 m.

Alternate Method

Given:

Height of the building (AB) = 100√3 m

Angle of depression from the top of the building to the bicycle = 30°

Since the horizontal ground is parallel to the horizontal line of sight from the top, the angle of elevation of the top of the building from the bicycle (Angle ACB) = 30° (alternate interior angles).

Let the horizontal distance from the base of the building to the bicycle (BC) = d

Formula Used:

In a right-angled triangle ABC:

tan(θ) = Opposite Side ÷ Adjacent Side

Calculations:

In right-angled triangle ABC:

⇒ tan(30°) = AB ÷ BC

⇒ 1 ÷ √3 = 100√3 ÷ d

⇒ d = 100√3 × √3

⇒ d = 100 × 3

⇒ d = 300 m

∴ The correct answer is 300 m.

Additional Information

Angle of Elevation and Depression

The angle of elevation is measured upwards from the horizontal line, while the angle of depression is measured downwards. They are equal in magnitude because of alternate interior angles.

Standard Trigonometric Ratios for Height & Distance

Common ratio values to remember: tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) =