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Steel bottle is manufactured by A is 122 and copper bottle is =122+21 = 143

Copper A and Steel B is in ratio 13: 17 

⇒ 13 unit = 143 then 17 is equal to 187

Copper bottle by B is 25 more than the copper by A = 143+25 = 168

Copper bottle manufactured by C is 164 then Steel bottle is 216

Then average no of steels bottles manufactured by all 

⇒ 122+187+216 = 525

⇒ 525/3

⇒ 175

Hence 175 is the correct answer.

Total no of copper bottles in shop C is 164

Total no of steel bottles in all three shops is 525 

⇒( 164 / 525 ) ×100

⇒ 31.23% ~ 32%

Hence 32% is the correct answer.

Steel bottles in shop C is 216

Shop D of copper bottles  is 25% more than steel shop C

⇒ 216 × 125 / 100

⇒ 216× 5 /4

⇒ 270 unit 

Copper bottles in shop A is 143 

Then Ratio of these respective entities is 143 : 270

Hence 143 : 270 is the right answer.

Total no of steel bottles in all three shops is 525 units 

Total no of copper bottles in all three shops is 475 units 

Required difference is ,

⇒ 525 - 475 

⇒ 50 units 

Hence 50 is the correct answer.

Cost price of  copper bottles is Rs 250

Cost price of silver bottles is Rs 300

Profit earned by Shop selling all bottles 

⇒ 187× 300 + 250 × 168

⇒ 56100 + 42000

⇒ Rs 98100

Hence Rs 98100 is the correct answer.

Shortcut Trick

Ratio change in P and Q from 3:4 to 6:7 involves an increase of 3 units each (6−3 = 3 and 7−4 = 3).

This 3 units increase corresponds to 27 years.

∴ 1 unit = 27 ÷ 3 = 9 years.

Present age of Q = 4 units = 4 × 9 = 36 years.

∴ The correct answer is 36 years.

Alternate Method

Given:

Ratio of present ages of P and Q = 3:4

Difference between P and Q = 9 years

Calculations:

Let the present age of P be 3x and Q be 4x.

⇒ 4x − 3x = 9

⇒ x = 9

⇒ Present age of Q = 4x

⇒ 4 × 9 = 36 years

Verification with future ratio:

⇒ (27 + 27) / (36 + 27) = 54 / 63 = 6 / 7 (Correct)

∴ The correct answer is 36 years.

Additional Information

Age Difference Property

The difference between the ages of two persons remains constant at any point in time (past, present, or future).

Ratio Balancing Method

If the difference in ratio parts between two periods is not the same, multiply the ratios by the difference of the other ratio to balance the gaps.

Shortcut Trick

Let Efficiency of P = 2 units/day → Q = 3 units/day (50% more).

Total Work = 30 days × 2 = 60 units.

Days taken by Q = 60 ÷ 3 = 20 days. Days for R = 20 − 10 = 10 days.

Efficiency of R = 60 ÷ 10 = 6 units/day.

Equation: (Efficiency_R+Q × X) + (Efficiency_P × (X + 8)) = 60

⇒ (6 + 3)X + 2(X + 8) = 60 ⇒ 9X + 2X + 16 = 60 ⇒ 11X = 44 ⇒ X = 4.

∴ The correct answer is 4.

Alternate Method

Given: P completes task in 30 days. Q is 50% more efficient than P. R takes 10 days less than Q.

Calculations:

⇒ Efficiency of Q = 1.5 × Efficiency of P. If P = 2, then Q = 3.

⇒ Total Work = Days × Efficiency = 30 × 2 = 60 units.

⇒ Days taken by Q = 60 ÷ 3 = 20 days.

⇒ Days taken by R = 20 − 10 = 10 days.

⇒ Efficiency of R = 60 ÷ 10 = 6 units/day.

⇒ Combined efficiency of R and Q = 6 + 3 = 9 units/day.

⇒ Work done by R and Q in X days = 9X.

⇒ Work done by P in (X + 8) days = 2(X + 8).

⇒ 9X + 2(X + 8) = 60

⇒ 9X + 2X + 16 = 60

⇒ 11X = 44

⇒ X = 4.

∴ The correct answer is 4.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is a:b, the ratio of time taken is b:a.

Work and Wages

Wages are generally distributed in the ratio of the work done by each individual, which is (Efficiency × Time).

Combined Work Formula

If A and B work together, their combined efficiency is the sum of their individual efficiencies. Total Work = (E_A + E_B) × Time.

Shortcut Trick

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

Given Difference (D) = ₹15 and Rate (r) = 5%.

15 = P × (5/100) × (5/100)

15 = P × (1/20) × (1/20) ⇒ 15 = P ÷ 400

P = 15 × 400 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given: Difference (D) = ₹15, Time (T) = 2 years, Rate (R) = 5%

Formula Used: D = P(R/100)²

⇒ 15 = P × (5/100)²

⇒ 15 = P × (25/10000)

⇒ 15 = P × (1/400)

⇒ P = 15 × 400

⇒ P = 6,000

∴ The correct answer is ₹6,000.

Additional Information

CI vs SI Difference for 3 Years

The formula for the difference for 3 years is D = P(r/100)² × (300 + r)/100.

Effective Rate Method

For 2 years, SI effective rate = 2r%. CI effective rate = (2r + r²/100)%. Difference = (r²/100)% of Principal.

Compound Interest Formula

Amount (A) = P(1 + r/100)ⁿ, where n is the number of years and r is the annual interest rate.

Shortcut Trick

Initial Ratio of Water : Alcohol = 2 : 1. After removing 12 liters, the ratio remains 2 : 1.

Let quantities be 2x and 1x. Replacing 12 liters of mixture with 12 liters of water gives:

(2x + 12) / 1x = 4 / 1 ⇒ 2x + 12 = 4x ⇒ 2x = 12 ⇒ x = 6 liters.

Quantity of Alcohol after removal = 6 liters. Initial Alcohol = 6 + (1/3 × 12) = 10 liters.

∴ The correct answer is 10 litres.

Alternate Method

Given: Initial ratio Water : Alcohol = 2 : 1. Final ratio after replacing 12L mixture with 12L water = 4 : 1.

Let the initial total mixture be T liters.

⇒ Initial Water = 2T/3, Initial Alcohol = T/3.

⇒ Alcohol removed in 12L = (1/3) × 12 = 4 liters.

⇒ Water removed in 12L = (2/3) × 12 = 8 liters.

⇒ New Alcohol = (T/3) − 4.

⇒ New Water = (2T/3) − 8 + 12 = (2T/3) + 4.

⇒ Final Ratio: ((2T/3) + 4) / ((T/3) − 4) = 4 / 1.

⇒ (2T/3) + 4 = 4 × ((T/3) − 4) ⇒ 2T/3 + 4 = 4T/3 − 16.

⇒ 2T/3 = 20 ⇒ T = 30 liters.

⇒ Initial Alcohol = T/3 = 30 / 3 = 10 liters.

∴ The correct answer is 10 litres.

Additional Information

Ratio in Mixtures

When a portion of a mixture is removed, the ratio of components in the remaining mixture stays the same as the initial ratio.

Replacement Rule

Final Quantity = Initial Quantity × (1 − Replacement Volume / Total Volume)ⁿ. This applies when the same component is repeatedly replaced.

Shortcut Trick

Investment ratio = (Profit ratio ÷ Time period ratio).

Time periods: A = 12 months, B = 9 months, C = 6 months. Ratio = 4 : 3 : 2.

Profit ratio = 2 : 3 : 5.

Investment ratio = (2/4) : (3/3) : (5/2) = 0.5 : 1 : 2.5 = 1 : 2 : 5.

Given A's investment (1 unit) = Rs. 36000.

Difference between B and C = 5 − 2 = 3 units = 3 × 36000 = Rs. 108000.

∴ The correct answer is Rs. 108000.

Alternate Method

Given: A's investment = Rs. 36000. Time of A = 12 months, B = 9 months, C = 6 months. Profit ratio A : B : C = 2 : 3 : 5.

Formulas: Profit = Investment × Time | Investment = Profit ÷ Time

⇒ Let B's investment be I_B and C's investment be I_C.

⇒ Ratio of profits = (36000 × 12) : (I_B × 9) : (I_C × 6) = 2 : 3 : 5

⇒ For A and B: (36000 × 12) / (I_B × 9) = 2 / 3

⇒ (4000 × 12) / I_B = 2 / 3 ⇒ 48000 / I_B = 2 / 3 ⇒ I_B = (48000 × 3) / 2 = Rs. 72000

⇒ For A and C: (36000 × 12) / (I_C × 6) = 2 / 5

⇒ (6000 × 12) / I_C = 2 / 5 ⇒ 72000 / I_C = 2 / 5 ⇒ I_C = (72000 × 5) / 2 = Rs. 180000

⇒ Difference = I_C − I_B = 180000 − 72000 = Rs. 108000

∴ The correct answer is Rs. 108000.

Additional Information

Partnership Basics

In a partnership, the profit is shared in the ratio of the product of Capital (Investment) and Time. If time is constant, profit ratio is the same as investment ratio.

Ratio of Investments

If profits P₁, P₂ are earned in times T₁, T₂, then the ratio of investments I₁ : I₂ = (P₁/T₁) : (P₂/T₂).