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Shortcut Trick

Numerical value of CSA₁ = 2 × π × 4 × 7 = 56π.

Given radius of second cylinder (r₂) = 2 cm and its TSA₂ = 56π.

Relation: TSA = CSA + 2 × Base Area.

56π = CSA₂ + 2 × (π × 2²) ⇒ 56π = CSA₂ + 8π.

CSA₂ = 48π cm².

∴ The correct answer is 48π cm².

Alternate Method

Given:

First Cylinder: Radius (r₁) = 4 cm, Height (h₁) = 7 cm.

Second Cylinder: Radius (r₂) = 4 ÷ 2 = 2 cm.

Formula Used:

Curved Surface Area (CSA) of cylinder = 2πrh

Total Surface Area (TSA) of cylinder = 2πr(r + h)

⇒ CSA of first cylinder = 2 × π × 4 × 7 = 56π cm².

⇒ TSA of second cylinder = 2 × π × r₂ × (r₂ + h₂) = 56π.

⇒ 2 × π × 2 × (2 + h₂) = 56π.

⇒ 4π × (2 + h₂) = 56π.

⇒ 2 + h₂ = 56 ÷ 4 = 14.

⇒ h₂ = 14 − 2 = 12 cm.

⇒ CSA of second cylinder = 2 × π × r₂ × h₂.

⇒ CSA₂ = 2 × π × 2 × 12 = 48π cm².

∴ The correct answer is 48π cm².

Additional Information

Volume of a Cylinder

The volume represents the space occupied and is calculated as Base Area × Height: V = πr²h.

Total Surface Area (TSA)

TSA includes both the curved part and the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Area of the Circular Base

Since the base of a cylinder is a circle, its area is given by πr².

Shortcut Trick

The net percentage change can be calculated using the formula: a + b + (a × b)/100.

Here, a = −25 (discount) and b = +20 (increase).

Net change = −25 + 20 + (−25 × 20)/100 = −5 − 5 = −10%.

Final Selling Price = 4000 × (100 − 10)% = 4000 × 0.9 = ₹3,600.

∴ The correct answer is ₹3,600.

Alternate Method

Given:

Marked Price (MP) = ₹4,000

First Discount = 25%

Price Increase = 20%

Calculations:

⇒ Price after first discount = MP − 25% of MP

⇒ Price after discount = 4000 − (25/100 × 4000) = 4000 − 1000 = ₹3,000

⇒ Price after increase = Price after discount + 20% of Price after discount

⇒ Final Selling Price = 3000 + (20/100 × 3000) = 3000 + 600 = ₹3,600

∴ The correct answer is ₹3,600.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the effective change is (x + y + xy/100)%. Use positive for increase and negative for discount/decrease.

Selling Price Formula

Selling Price (SP) = Marked Price (MP) × (100 − Discount%)/100. If there is a further markup, multiply by (100 + Increase%)/100.

Ratio Method for Successive Changes

Convert percentages to fractions: 25% discount = 4 to 3 ratio; 20% increase = 5 to 6 ratio. Multiply ratios: (4×5) to (3×6) = 20:18 = 10:9.

Shortcut Trick

New sum of resulting numbers = 72 + 8 − 6 = 74.

Total ratio parts = 3 + 2 = 5 parts.

Value of 1 part = 74 ÷ 5 = 14.8.

New smaller number = 2 parts × 14.8 = 29.6.

Original smaller number = 29.6 + 6 = 35.6.

∴ The correct answer is 35.6.

Alternate Method

Given: Sum of two numbers = 72. Let the larger number be x and the smaller number be y.

Formula Used: Linear equation in two variables and Ratio proportion properties.

⇒ x + y = 72 → (Equation 1)

⇒ According to the condition: (x + 8) ÷ (y − 6) = 3 ÷ 2

⇒ 2 × (x + 8) = 3 × (y − 6)

⇒ 2x + 16 = 3y − 18

⇒ 2x − 3y = −34 → (Equation 2)

⇒ Substitute x = 72 − y in Equation 2:

⇒ 2(72 − y) − 3y = −34

⇒ 144 − 2y − 3y = −34

⇒ 144 − 5y = −34

⇒ 5y = 144 + 34 = 178

⇒ y = 178 ÷ 5 = 35.6

∴ The correct answer is 35.6.

Additional Information

Ratio Division Rule

To divide a total sum S into a ratio a : b, the parts are [a/(a+b)] × S and [b/(a+b)] × S.

Linear Equations

For two variables, substitution or elimination methods are used to find the values of x and y.

Net Change Property

If values in a sum change, the new sum = original sum + total increase − total decrease.

Shortcut Trick

Factor out the smallest power: 9^x−1(9¹ − 1) = 72

8 × 9^x−1 = 72 ⇒ 9^x−1 = 9 ⇒ x − 1 = 1 ⇒ x = 2

Substitute x = 2 in the expression: (2 × 2 − 1) / (2 × 2 + 3) = 3 / 7

∴ The correct answer is 3/7.

Alternate Method

Given: 9^x − 9^x−1 = 72

Formula Used: Laws of Indices: a^m − a^m−n = a^m(1 − a⁻ⁿ)

⇒ 9^x − (9^x / 9) = 72

⇒ 9^x (1 − 1/9) = 72

⇒ 9^x (8/9) = 72

⇒ 9^x = (72 × 9) / 8

⇒ 9^x = 9 × 9 = 81

⇒ 9^x = 9²

Comparing the exponents as the bases are equal:

⇒ x = 2

Now, substitute the variable x into the required expression:

⇒ Value = (2x − 1) / (2x + 3)

⇒ Value = (2 × 2 − 1) / (2 × 2 + 3)

⇒ Value = (4 − 1) / (4 + 3) = 3/7

∴ The correct answer is 3/7.

Additional Information

Laws of Indices

For any non-zero base, the quotient rule states that a^m ÷ aⁿ = a^m−n.

Comparison Property

If a^x = a^y and a ≠ 1, 0, or −1, then the powers must be equal, such that x = y.

Negative Exponents

A term with a negative exponent can be written as its reciprocal with a positive exponent: a⁻ⁿ = 1 / aⁿ.

Shortcut Trick

Let the Cost Price (CP) of 1 pen be 1 unit.

Total CP of 60 pens = 60 units.

Selling Price (SP) of 60 pens = CP of 72 pens = 72 units.

Profit = 72 − 60 = 12 units.

Profit % = (12 ÷ 60) × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given:

Quantity of pens bought = 60

Selling Price (SP) of 60 pens = Cost Price (CP) of 72 pens

Formula Used:

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

Calculations:

⇒ Let the CP of 1 pen = ₹x

⇒ Total CP of 60 pens = ₹60x

⇒ Total SP of 60 pens = CP of 72 pens = ₹72x

⇒ Profit = ₹72x − ₹60x = ₹12x

⇒ Profit % = (12x ÷ 60x) × 100

⇒ Profit % = 1/5 × 100 = 20%

∴ The correct answer is 20%.

Additional Information

Profit and Loss Formulas

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

Quantity Based Trick

If CP of 'a' articles = SP of 'b' articles, then Profit/Loss % = [(a − b) ÷ b] × 100. A positive result indicates profit, while negative indicates loss.

Marked Price (MP) and Discount

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

Shortcut Trick

Convert speed of Train X: 72 × (5/18) = 20 m/s.

Relative Speed (Opposite) = Total Length ÷ Time = 500 ÷ 15 = 100/3 m/s.

Speed of Train Y = (100/3) − 20 = 40/3 m/s.

Relative Speed (Same) = 20 − (40/3) = 20/3 m/s.

Time to overtake = 500 ÷ (20/3) = 75 sec.

∴ The correct answer is 75 sec.

Alternate Method

Given:

Length of Train X (L_X) = 200 m

Length of Train Y (L_Y) = 300 m

Speed of Train X (S_X) = 72 km/hr = 72 × (5/18) = 20 m/s

Time in opposite direction (T_opp) = 15 sec

Formula Used:

Relative Speed (Opposite) = S₁ + S₂

Relative Speed (Same) = |S₁ − S₂|

Time = (L₁ + L₂) ÷ Relative Speed

Calculations:

⇒ Total distance to cover = 200 + 300 = 500 m

⇒ In opposite direction: S_X + S_Y = 500 ÷ 15

⇒ 20 + S_Y = 100/3

⇒ S_Y = 100/3 − 20 = (100 − 60)/3 = 40/3 m/s

⇒ In same direction: Relative speed = S_X − S_Y = 20 − 40/3

⇒ Relative speed (same) = (60 − 40)/3 = 20/3 m/s

⇒ Time to overtake = Total Distance ÷ Relative Speed (same)

⇒ Time = 500 ÷ (20/3) = 500 × (3/20)

⇒ Time = 25 × 3 = 75 sec

∴ The correct answer is 75 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). When they move in the same direction, their speeds are subtracted (S₁ − S₂).

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert from m/s to km/hr, multiply by 18/5.

Total Distance for Trains

Whether crossing each other or overtaking, the total distance to be covered is always the sum of the lengths of the two trains (L₁ + L₂).

Shortcut Trick

The Third Proportional (c) of two numbers a and b is calculated as: c = b² ÷ a.

Given values: a = 5 and b = 10.

Applying the trick: c = (10 × 10) ÷ 5

c = 100 ÷ 5 = 20.

∴ The correct answer is 20.

Alternate Method

Given: First number (a) = 5, Second number (b) = 10.

Formula Used: If three numbers a, b, and c are in continued proportion, then a : b :: b : c, which means a/b = b/c.

Let the third proportional be x.

⇒ 5 : 10 :: 10 : x

⇒ 5 ÷ 10 = 10 ÷ x

⇒ 1 ÷ 2 = 10 ÷ x

⇒ x = 10 × 2

⇒ x = 20

∴ The correct answer is 20.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Fourth Proportional

If four numbers a, b, c, d are in proportion (a:b = c:d), then the fourth proportional d = (b × c) ÷ a.

Continued Proportion

Three numbers a, b, c are in continued proportion if the ratio of the first to the second is the same as the ratio of the second to the third (a/b = b/c).

Shortcut Trick

To find the next time trains from Delhi, Lucknow, and Jaipur depart together, calculate the LCM of their departure intervals.

LCM(12, 18, 27) = 108 minutes.

Convert 108 minutes to hours: 108 ÷ 60 = 1 hour 48 minutes.

Add this duration to the initial departure time: 6:00 am + 1:48 = 7:48 am.

∴ The correct answer is 7:48 am.

Alternate Method

Given:

Interval for Delhi train = 12 min

Interval for Lucknow train = 18 min

Interval for Jaipur train = 27 min

Initial simultaneous departure = 6:00 am

Formula Used:

Next simultaneous event = Initial time + LCM of individual intervals.

Calculations:

⇒ Prime factorization of 12 = 2 × 2 × 3 = 2² × 3¹

⇒ Prime factorization of 18 = 2 × 3 × 3 = 2¹ × 3²

⇒ Prime factorization of 27 = 3 × 3 × 3 = 3³

⇒ LCM = 2^max(2,1,0) × 3^max(1,2,3) = 2² × 3³

⇒ LCM = 4 × 27 = 108 minutes

⇒ Converting minutes to hours: 108 minutes = 60 minutes + 48 minutes = 1 hour 48 minutes

⇒ Next meeting time = 6:00 am + 1 hour 48 minutes

⇒ Next meeting time = 7:48 am

∴ The correct answer is 7:48 am.

Additional Information

Least Common Multiple (LCM)

The smallest number that is a multiple of two or more numbers. It is used to find the first time periodic events coincide.

Relationship between LCM and HCF

For any two numbers A and B, LCM(A, B) × HCF(A, B) = A × B. This is a fundamental property in number theory.

Periodic Events

For problems involving traffic lights, bells, or circular tracks, the next time of coincidence is always the LCM of their individual time periods.

Shortcut Trick

Ratio of the two parts of the Marked Price = 70% : 30% = 7 : 3.

Total Percentage Discount = [(Part₁ × Rate₁) + (Part₂ × Rate₂)] ÷ (Total Parts)

Total Discount % = [(7 × 15) + (3 × 5)] ÷ (7 + 3)

Total Discount % = (105 + 15) ÷ 10 = 120 ÷ 10 = 12%

∴ The correct answer is 12%.

Alternate Method

Given: Marked Price (MP) = ₹50,000; Discount₁ = 15% on 70% MP; Discount₂ = 5% on remaining MP.

⇒ First part of MP = 70% of 50,000 = (70 ÷ 100) × 50,000 = ₹35,000

⇒ Second part of MP = 30% of 50,000 = (30 ÷ 100) × 50,000 = ₹15,000

⇒ Discount 1 = 15% of 35,000 = 0.15 × 35,000 = ₹5,250

⇒ Discount 2 = 5% of 15,000 = 0.05 × 15,000 = ₹750

⇒ Total Discount Amount = 5,250 + 750 = ₹6,000

⇒ Total Percentage Discount = (Total Discount ÷ Total MP) × 100

⇒ Total Discount % = (6,000 ÷ 50,000) × 100 = 12%

∴ The correct answer is 12%.

Additional Information

Weighted Average Discount

When different portions of a Marked Price are sold at different discount rates, the overall discount is the weighted average: D_total = (w₁d₁ + w₂d₂) ÷ (w₁ + w₂).

Successive Discounts

If two discounts d₁ and d₂ are applied one after another, the single equivalent discount is [d₁ + d₂ − (d₁ × d₂) ÷ 100]%.

Net Selling Price

The Selling Price is calculated as MP × [1 − (Discount % ÷ 100)]. In this problem, SP = ₹50,000 − ₹6,000 = ₹44,000.

Shortcut Trick

Effective Compound Interest rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount percentage = 100% + 21% = 121%.

Total Amount = 121% of ₹4000 = 1.21 × 4000.

Total Amount = ₹4840.

∴ The correct answer is ₹4840.

Alternate Method

Given:

Principal (P) = ₹4000

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4000 × (1 + 10/100)²

⇒ A = 4000 × (110/100) × (110/100)

⇒ A = 4000 × (1.1) × (1.1)

⇒ A = 4000 × 1.21

⇒ A = 4840

∴ The correct answer is ₹4840.

Additional Information

Compound Interest Formula

Compound Interest (CI) = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Difference between CI and SI

For 2 years, the difference (D) between CI and SI is given by D = P(R/100)².

Effective Rate for 2 Years

If the rate is r%, the effective compound interest rate for 2 years is (2r + r²/100)%.