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In a vessel, there are two types of liquids A and B in the ratio of 5:9. 28 litres of the mixture is taken out and 2 litres of types B liquid is poured into the vessel. The new ratio (A:B) thus formed is 1:2. Find the initial quantity of mixture in the vessel?

A

84 litres

B

42 litres

C

50 litres

D

56 litres

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will follow these steps: ### Step 1: Define the initial quantities of liquids A and B Let the initial quantities of liquids A and B be represented as: - Quantity of A = 5x - Quantity of B = 9x ### Step 2: Calculate the total initial quantity The total initial quantity of the mixture is: \[ \text{Total initial quantity} = 5x + 9x = 14x \] ### Step 3: Determine the quantities of A and B removed when 28 liters of the mixture is taken out Since the ratio of A to B is 5:9, the total parts of the mixture is 14 parts. Therefore, the quantities of A and B removed when 28 liters are taken out can be calculated as follows: - Quantity of A removed = \( \frac{5}{14} \times 28 = 10 \) liters - Quantity of B removed = \( \frac{9}{14} \times 28 = 18 \) liters ### Step 4: Calculate the remaining quantities of A and B after removal After removing 28 liters of the mixture: - Remaining quantity of A = \( 5x - 10 \) - Remaining quantity of B = \( 9x - 18 \) ### Step 5: Add 2 liters of liquid B to the vessel After adding 2 liters of liquid B, the new quantity of B becomes: \[ \text{New quantity of B} = (9x - 18) + 2 = 9x - 16 \] ### Step 6: Set up the new ratio of A to B According to the problem, the new ratio of A to B is given as 1:2. Therefore, we can set up the equation: \[ \frac{5x - 10}{9x - 16} = \frac{1}{2} \] ### Step 7: Cross-multiply to solve for x Cross-multiplying gives us: \[ 2(5x - 10) = 1(9x - 16) \] Expanding both sides: \[ 10x - 20 = 9x - 16 \] ### Step 8: Rearrange to find the value of x Rearranging the equation: \[ 10x - 9x = -16 + 20 \] \[ x = 4 \] ### Step 9: Calculate the initial quantity of the mixture Now that we have the value of x, we can find the initial quantity: \[ \text{Initial quantity} = 14x = 14 \times 4 = 56 \text{ liters} \] ### Final Answer The initial quantity of the mixture in the vessel is **56 liters**. ---
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