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Shortcut Trick

Calculate the Total Percentage Interest: 6% per month × 2 months = 12%.

Simple Interest = 12% of ₹1,500.

Simple Interest = (12/100) × 1500 = 12 × 15 = 180.

∴ The correct answer is ₹180.

Alternate Method

Given:

Principal (P) = ₹1,500

Rate of Interest (R) = 6% per month

Time (T) = 2 months

Formula Used:

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

Calculations:

⇒ SI = (1500 × 6 × 2) ÷ 100

⇒ SI = (1500 × 12) ÷ 100

⇒ SI = 15 × 12

⇒ SI = 180

∴ The correct answer is ₹180.

Additional Information

Consistency of Units

Ensure the Rate (R) and Time (T) are in the same time unit. If R is per annum, T must be in years. If R is per month, T must be in months.

Total Amount Formula

The total Amount (A) at the end of the period is the sum of the Principal and Simple Interest: A = P + SI.

Rate Conversion

To convert a monthly rate to an annual rate, multiply by 12. For example, 6% per month is equal to 6% × 12 = 72% per annum.

Shortcut Trick

Use the Successive Ratio method:

Product of profit multipliers = (125/100) × (160/100) × (125/100)

Net Multiplier = (5/4) × (8/5) × (5/4) = 5/2

Cost Price × (5/2) = ₹685

Cost Price = (685 × 2) ÷ 5 = 137 × 2 = ₹274

∴ The correct answer is ₹274.

Alternate Method

Given:

Manufacturer Profit = 25%, Wholesaler Profit = 60%, Retailer Profit = 25%

Final Selling Price (Customer pays) = ₹685

Let the Cost Price (CP) for the manufacturer be x.

⇒ Selling Price to Wholesaler = x × (125/100) = 1.25x

⇒ Selling Price to Retailer = 1.25x × (160/100) = 1.25x × 1.6 = 2x

⇒ Selling Price to Customer = 2x × (125/100) = 2x × 1.25 = 2.5x

⇒ 2.5x = 685

⇒ x = 685 ÷ 2.5

⇒ x = 274

∴ The correct answer is ₹274.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the net change is (a + b + ab/100)%.

Profit and Cost Price Relation

Selling Price (SP) = CP × (100 + Profit%)/100. This is used repeatedly in chain-sale problems.

Chain Rule in Profit/Loss

Final Price = Initial Price × (1 + p₁/100) × (1 + p₂/100) × (1 + p₃/100), where p_n are individual profit percentages.

Shortcut Trick

Ratio of Type A to Type B = (300 − 260) : (260 − 200) = 40 : 60 = 2 : 3.

Total units = 2 + 3 = 5 units.

According to question, 5 units = 10 kg ⇒ 1 unit = 2 kg.

Quantity of Type A = 2 × 2 = 4 kg.

Quantity of Type B = 3 × 2 = 6 kg.

∴ The correct answer is 4 kg of type A, 6 kg of type B.

Alternate Method

Given:

Cost Price of Type A tea = ₹200/kg

Cost Price of Type B tea = ₹300/kg

Price of the Mixture (Mean Price) = ₹260/kg

Total Quantity of mixture = 10 kg

Formula Used:

Alligation Rule: (Quantity of Cheaper / Quantity of Dearer) = (CP of Dearer − Mean Price) / (Mean Price − CP of Cheaper)

Calculations:

⇒ Ratio of Type A : Type B = (300 − 260) : (260 − 200)

⇒ Ratio = 40 : 60 = 2 : 3

⇒ Sum of ratio parts = 2 + 3 = 5

⇒ Quantity of Type A = (2 / 5) × 10 = 4 kg

⇒ Quantity of Type B = (3 / 5) × 10 = 6 kg

∴ The correct answer is 4 kg of type A, 6 kg of type B.

Additional Information

Alligation Rule

It is a modified form of the weighted average used to find the ratio in which two ingredients at given prices must be mixed to produce a mixture at a specific price.

Weighted Average Formula

Mean Price = (n₁c₁ + n₂c₂) / (n₁ + n₂), where n is quantity and c is cost per unit.

Three-Ingredient Mixture

For mixtures of more than two types, alligation is applied by pairing the cheapest with the dearest and checking the mean price.

Shortcut Trick

Net percentage growth for 2 years = x + y + (xy/100)

⇒ 3 + 3 + (3 × 3)/100 = 6 + 0.09 = 6.09%

Total population = 106.09% of 30,000

⇒ 30,000 × 1.0609 = 31,827

∴ The correct answer is 31,827.

Alternate Method

Given:

Current Population (P) = 30,000

Growth Rate (R) = 3% per year

Time (n) = 2 years

Formula Used:

Population after n years = P × (1 + R/100)ⁿ

Calculations:

⇒ Population after 1 year = 30,000 × (1 + 3/100)

⇒ Population after 1 year = 30,000 × (103/100) = 30,900

⇒ Population after 2 years = 30,900 × (1 + 3/100)

⇒ Population after 2 years = 30,900 × (1.03)

⇒ Population after 2 years = 31,827

∴ The correct answer is 31,827.

Additional Information

Successive Growth Formula

When a value increases by R% successively for n years, the final amount is P(1 + R/100)ⁿ. This is identical to the Compound Interest formula.

Population Decrease/Depreciation

If the population decreases or an asset depreciates at a rate R%, the value after n years is P(1 − R/100)ⁿ.

Effective Percentage Change

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

Shortcut Trick

Effective Rate of Interest for 2 years at 6% compounded annually = 6 + 6 + (6 × 6)/100 = 12.36%.

Total Compound Interest = 12.36% of ₹22,500.

⇒ CI = (12.36 ÷ 100) × 22,500 = 12.36 × 225 = ₹2781.

∴ The correct answer is ₹2781.

Alternate Method

Given:

Principal (P) = ₹22,500

Time (T) = 2 years

Rate (R) = 6% per annum

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Compound Interest (CI) = A − P

Calculations:

⇒ Year 1 Interest = 6% of 22,500 = (6 ÷ 100) × 22,500 = ₹1350

⇒ Amount after 1 year = 22,500 + 1350 = ₹23,850

⇒ Year 2 Interest = 6% of 23,850 = (6 ÷ 100) × 23,850 = ₹1431

⇒ Total Interest = 1350 + 1431 = ₹2781

∴ The correct answer is ₹2781.

Additional Information

Compound Interest Formula

CI = P[(1 + R/100)^t − 1], where P is principal, R is rate, and t is time.

Successive Percentage Change

For two years, the effective rate is given by x + y + (xy/100). For CI, x = y = R.

Difference between CI and SI

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

Shortcut Trick

Initial Ratio Spirit : Water = 5 : 2 (Total 7 units → 28 litres, 1 unit = 4 litres).

Current Spirit = 5 × 4 = 20 litres and Water = 2 × 4 = 8 litres.

In new mixture, Water = 60%, so Spirit = 40%.

Since Spirit remains constant, 40% = 20 litres ⇒ 100% = 50 litres.

Water to be added = New total − Old total = 50 − 28 = 22 litres.

∴ The correct answer is 22 litres.

Alternate Method

Given:

Total initial mixture = 28 litres.

Ratio of Spirit to Water = 5 : 2.

Target concentration of Water = 60%.

Calculations:

⇒ Quantity of Spirit = (5 ÷ 7) × 28 = 20 litres.

⇒ Quantity of Water = (2 ÷ 7) × 28 = 8 litres.

⇒ Let 'x' litres of water be added.

⇒ Total mixture becomes (28 + x) litres.

⇒ Total water becomes (8 + x) litres.

⇒ (8 + x) ÷ (28 + x) = 60 ÷ 100

⇒ (8 + x) ÷ (28 + x) = 3 ÷ 5

⇒ 5 × (8 + x) = 3 × (28 + x)

⇒ 40 + 5x = 84 + 3x

⇒ 2x = 44

⇒ x = 22 litres.

∴ The correct answer is 22 litres.

Additional Information

Alligation Method

It is used to find the ratio in which two ingredients at different prices/concentrations are mixed to produce a mixture of a desired price/concentration.

Concentration Rule

When one component (like spirit) is not added, its absolute quantity remains constant. Use this constant value to find the new total volume quickly.

Successive Dilution

Final quantity = Initial quantity × (1 − x/V)ⁿ, where x is quantity removed/added and V is total volume.

Shortcut Trick

Successive increase ratio: 112% × 124% = 1.12 × 1.24 = 1.3888

Current Sales = 1.3888 × Initial Sales

₹173600 = 1.3888 × Initial Sales

Initial Sales = 173600 ÷ 1.3888 = 125000

∴ The correct answer is ₹125000.

Alternate Method

Given:

Current Sales = ₹1,73,600

Increase in 1^st year (r₁) = 12%

Increase in 2^nd year (r₂) = 24%

Formula Used:

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

Let the sales two years ago be S.

⇒ S × (1 + 12/100) × (1 + 24/100) = 173600

⇒ S × (112/100) × (124/100) = 173600

⇒ S × 1.12 × 1.24 = 173600

⇒ S × 1.3888 = 173600

⇒ S = 173600 ÷ 1.3888

⇒ S = 125000

∴ The correct answer is ₹125000.

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)]%.

Multiplying Factor Method

For an increase of r%, the multiplying factor is (1 + r/100). For a decrease, it is (1 − r/100).

Reverse Percentage Calculation

To find the initial value after a percentage increase, divide the final value by the multiplying factor.

Shortcut Trick

For a finite series of the form √(a √(a √(a ... n times))), the value is a^{(2ⁿ − 1) / 2ⁿ}.

Here, the base a = 6 and the square root is repeated n = 5 times.

Value = 6^{(2⁵ − 1) / 2⁵} = 6^{(32 − 1) / 32} = 6^31/32.

∴ The correct answer is 6^31/32.

Alternate Method

Given: √(6 × √(6 × √(6 × √(6 × √6))))

Formula Used: a^m × aⁿ = a^m+n and (a^m)ⁿ = a^m×n

⇒ Let X = √(6 × √(6 × √(6 × √(6 × √6))))

⇒ Converting the surds into fractional indices: X = 6^1/2 × 6^1/4 × 6^1/8 × 6^1/16 × 6^1/32

⇒ Since the bases are same, add the powers: X = 6^{(1/2 + 1/4 + 1/8 + 1/16 + 1/32)}

⇒ Taking LCM of denominators (32): X = 6^{(16 + 8 + 4 + 2 + 1) / 32}

⇒ X = 6^31/32

∴ The correct answer is 6^31/32.

Additional Information

Infinite Surds Series (Multiplication)

For an infinite series √(a √(a √(a ... ∞))), the value is always equal to a.

Infinite Surds Series (Addition)

For √(a + √(a + √(a ... ∞))), if a = n(n + 1), the answer is the larger factor (n + 1). For subtraction, it is n.

General Surd Formula

For m^th root repeated n times: √_m(a √_m(a ...)) = a^{(mⁿ − 1) / (mⁿ(m − 1))}.

Shortcut Trick

Identify all unique bases in the given expressions: 5, 8, 14, and 17.

To find the LCM, select the highest power of each base present in any of the expressions.

Highest power of 5 = 5³ | Highest power of 8 = 8³.

Highest power of 14 = 14² | Highest power of 17 = 17².

LCM = 5³ × 8³ × 14² × 17².

∴ The correct answer is 5³ × 8³ × 14² × 17².

Alternate Method

Given:

Expression 1 = 5³ × 8² × 14¹

Expression 2 = 5² × 14² × 17¹

Expression 3 = 8³ × 14² × 17²

Method Used: To find the Least Common Multiple (LCM) of numbers in exponential form, we take the product of the highest powers of all factors involved.

Calculations:

⇒ Compare powers of 5: max(3, 2, 0) = 3 → Term is 5³

⇒ Compare powers of 8: max(2, 0, 3) = 3 → Term is 8³

⇒ Compare powers of 14: max(1, 2, 2) = 2 → Term is 14²

⇒ Compare powers of 17: max(0, 1, 2) = 2 → Term is 17²

⇒ LCM = 5³ × 8³ × 14² × 17²

∴ The correct answer is 5³ × 8³ × 14² × 17².

Additional Information

HCF of Exponential Numbers

To find the Highest Common Factor (HCF), take the lowest power of only the common bases present in all expressions.

Relationship between LCM and HCF

For any two numbers 'a' and 'b', LCM(a, b) × HCF(a, b) = a × b.

LCM of Fractions

LCM of Fractions = LCM of Numerators ÷ HCF of Denominators.

Given:

Total Number of tourists to all places = 10.61 + 8.48 + 9.86 + 7.58 + 8.47 = 45.00 million

Formula Used:

Central Angle (in degrees) = (Individual value ÷ Total value) × 360

Calculation:

According to the question,

⇒ Total number of tourists = 45 million

⇒ Tourists visiting Manali = 7.58 million

⇒ Central angle for Manali = (7.58 ÷ 45) × 360

⇒ Central angle = 7.58 × (360 ÷ 45)

⇒ Central angle = 7.58 × 8

⇒ Central angle = 60.64

∴ The correct answer is 60.64.