We can solve this problem using a system of linear equations. Let's define:

• $x$ = the cost of one bag (in Rs.)
• $y$ = the cost of one box (in Rs.)

From the problem statement, we have two pieces of information:

1. The cost of 4 bags and 3 boxes is Rs. 555: $4x + 3y = 555 \quad \text{(Equation 1)}$

2. The cost of 3 bags and 4 boxes is Rs. 460: $3x + 4y = 460 \quad \text{(Equation 2)}$

Step 1: Solve for one variable

Let's solve the two equations simultaneously. First, we will multiply each equation to make the coefficients of one variable (either $x$ or $y$) the same.

Multiply Equation 1 by 3: $3(4x + 3y) = 3(555)$ $12x + 9y = 1665 \quad \text{(Equation 3)}$

Now, multiply Equation 2 by 4: $4(3x + 4y) = 4(460)$ $12x + 16y = 1840 \quad \text{(Equation 4)}$

Step 2: Subtract the equations

Now subtract Equation 3 from Equation 4 to eliminate $x$:

$(12x + 16y) - (12x + 9y) = 1840 - 1665$ $7y = 175$ $y = \frac{175}{7} = 25$

Step 3: Substitute the value of $y$ into one of the original equations

Now that we know $y = 25$, substitute this value into Equation 1 to solve for $x$:

$4x + 3(25) = 555$ $4x + 75 = 555$ $4x = 555 - 75$ $4x = 480$ $x = \frac{480}{4} = 120$

Conclusion:

The cost of one bag is Rs. 120.

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Would you like further details on how we solved this, or do you have any other questions?

Here are some related questions to explore further:

1. How would the solution change if the cost of a box was Rs. 40 instead of Rs. 25?
2. Can we solve this system using matrices or determinants? How?
3. How would you approach this problem if there were more than two variables involved?
4. What happens if the total cost of the bags and boxes changes (e.g., Rs. 600 and Rs. 450)?
5. Can this approach be applied to problems involving more than two items (e.g., bags, boxes, and bottles)?

Tip: When solving systems of equations, always check if one variable can be eliminated by adding or subtracting equations. This helps simplify the process.