Solving the Mixture Ratio Puzzle: A Can Contains Two Liquids in Specific Ratios

Imagine a scenario where a can holds a mixture of two liquids, A and B, in a specific ratio. In this case, the ratio is 9:7. When a portion of this mixture is removed and replaced with a different liquid, the ratio changes. This article explores the solution to the problem using a step-by-step algebraic approach.

Problem Description

The problem can be described as follows: A can contains a mixture of two liquids, A and B, in the ratio 9:7. When 9 litres of this mixture are drawn off, the can is then filled with liquid B, and the new ratio of A to B becomes 7:9. The challenge is to determine how many litres of liquid A the can initially contained.

Initial Assumptions and Calculations

To solve this problem, we begin by making some assumptions and setting up the initial conditions. Let's denote the total volume of the mixture in the can as V litres. Given the initial ratio 9:7, we can express the quantities of A and B as:

Quantity of A: ( frac{9}{16}V )

Quantity of B: ( frac{7}{16}V )

When 9 litres of the mixture is drawn off, the same ratio of A to B is maintained. Therefore, the amount of A and B in the 9 litres removed can be calculated as:

Amount of A removed: ( frac{9}{16} times 9 frac{81}{16} ) litres

Amount of B removed: ( frac{7}{16} times 9 frac{63}{16} ) litres

After removing 9 litres, the remaining quantities of A and B in the can are:

Remaining A: ( frac{9}{16}V - frac{81}{16} frac{9V - 81}{16} )

Remaining B: ( frac{7}{16}V - frac{63}{16} frac{7V - 63}{16} )

Then, we fill the can back up with 9 litres of liquid B. The new quantity of B becomes:

New B: ( frac{7V - 63}{16} 9 frac{7V - 63 144}{16} frac{7V 81}{16} )

Setting Up and Solving the Equation

The new ratio of A to B is given as 7:9. Therefore, we can set up the following equation:

( frac{frac{9V - 81}{16}}{frac{7V 81}{16}} frac{7}{9} )

By cross-multiplying, we get:

9 (9V - 81) 7 (7V 81)

Expanding both sides:

81V - 729 49V 567

Now, we can isolate V:

81V - 49V 567 729

32V 1296

V ( frac{1296}{32} 40.5 ) litres

Now, to find the initial amount of liquid A:

( Quantity , of , A frac{9}{16} times 40.5 frac{364.5}{16} 22.828125 ) litres

Therefore, the initial quantity of liquid A in the can is approximately 22.83 litres.

Conclusion

In conclusion, using a systematic approach involving algebraic manipulation and ratio determination, we have solved the mixture ratio puzzle. This problem not only tests one's understanding of ratios but also reinforces the application of algebraic equations to real-world scenarios.