The cost of 4 pens, 6 notebooks and 9 files is Rs 305, and the cost of 3 pens, 4 notebooks and 2 files is Rs 145. Based on this information, what is the total cost in rupees of buying 5 pens, 8 notebooks and 16 files together?

Introduction / Context:
This is a linear equations word problem about the cost of pens, notebooks and files. Instead of asking for the individual prices of each item, the question directly asks for the cost of a particular combination. This allows us to use linear combination of equations without fully solving for every variable.

Given Data / Assumptions:
- Cost of 4 pens, 6 notebooks and 9 files is Rs 305.
- Cost of 3 pens, 4 notebooks and 2 files is Rs 145.
- We want the cost of 5 pens, 8 notebooks and 16 files.
- All prices are in rupees and are assumed to be constant per item.

Concept / Approach:
Let the cost of one pen be p rupees, one notebook be n rupees and one file be f rupees. We translate the statements into linear equations in p, n and f. Then we try to express the target combination (5p + 8n + 16f) as a linear combination of the given equations, which is more efficient than solving for p, n and f individually.

Step-by-Step Solution:
Step 1: From the first sentence we get 4p + 6n + 9f = 305.Step 2: From the second sentence we get 3p + 4n + 2f = 145.Step 3: We want S = 5p + 8n + 16f. Try to write S as a(4p + 6n + 9f) + b(3p + 4n + 2f).Step 4: This gives a(4, 6, 9) + b(3, 4, 2) = (5, 8, 16).Step 5: Solve 4a + 3b = 5 and 6a + 4b = 8. Subtracting gives a + b = 1, so b = 1 − a.Step 6: Substitute into 4a + 3(1 − a) = 5, which gives a = 2 and hence b = −1.Step 7: Therefore S = 2(305) − 1(145) = 610 − 145 = 465.

Verification / Alternative check:
If desired, we could solve for p, n and f explicitly and then compute 5p + 8n + 16f, but the linear combination method is faster and less error prone. The final total of Rs 465 is consistent with both original equations when back-substituted through the derived coefficients a = 2 and b = −1.

Why Other Options Are Wrong:
Values like 415 or 440 arise from incorrect algebra such as choosing wrong coefficients a and b, or making arithmetic mistakes while subtracting or adding the equations. The option “Cannot be determined” is incorrect because the specific combination we need lies in the span of the two given combinations, so it is uniquely determined even though we have three unknown prices.

Common Pitfalls:
Students often think that three unknowns always require three equations. However, when we only need a specific linear combination, two equations may be enough. Another common pitfall is to attempt full substitution, which is longer and increases chances of arithmetic errors, instead of directly expressing the target combination as a linear combination of the given ones.

Final Answer:
The total cost of 5 pens, 8 notebooks and 16 files is Rs 465.