The cost of 8 pencils, 5 pens and 3 erasers is Rs 111. The cost of 9 pencils, 6 pens and 5 erasers is Rs 130. The cost of 16 pencils, 11 pens and 3 erasers is Rs 221. What is the total cost (in rupees) of buying 39 pencils, 26 pens and 13 erasers?

Introduction / Context:
This is a linear equations and arithmetic mixture question framed in a profit and loss or basic arithmetic context. The costs of pencils, pens, and erasers are hidden, but we are given total costs for three different combinations. We must determine the cost of a larger combination of items. The key idea is to treat each item cost as a variable and then either solve the system or cleverly combine equations to reach the required expression directly.

Given Data / Assumptions:

• 8 pencils + 5 pens + 3 erasers cost Rs 111.
• 9 pencils + 6 pens + 5 erasers cost Rs 130.
• 16 pencils + 11 pens + 3 erasers cost Rs 221.
• All pencils have the same cost, all pens have the same cost, and all erasers have the same cost.
• We need the cost of 39 pencils + 26 pens + 13 erasers.

Concept / Approach:
Let:

• Cost of one pencil = P.
• Cost of one pen = Q.
• Cost of one eraser = R.

We then write three linear equations by comparing coefficients. We could solve for P, Q, and R first, and then calculate the required combination. Another efficient approach is to notice that 39, 26, and 13 are linear combinations of the given sets and derive the cost directly once the unit costs are obtained.

Step-by-Step Solution:
Let cost of one pencil = P, one pen = Q, and one eraser = R. Equation 1: 8P + 5Q + 3R = 111. Equation 2: 9P + 6Q + 5R = 130. Equation 3: 16P + 11Q + 3R = 221. Solving these three equations simultaneously gives the unit costs. From the system, we obtain P = 10, Q = 5, and R = 2. Now compute the required cost: 39P + 26Q + 13R. 39P = 39 * 10 = 390. 26Q = 26 * 5 = 130. 13R = 13 * 2 = 26. Total cost = 390 + 130 + 26 = 546 rupees.

Verification / Alternative check:
Verify the unit costs against the original equations:

• For 8P + 5Q + 3R = 8 * 10 + 5 * 5 + 3 * 2 = 80 + 25 + 6 = 111, correct.
• For 9P + 6Q + 5R = 90 + 30 + 10 = 130, correct.
• For 16P + 11Q + 3R = 160 + 55 + 6 = 221, correct.

Since the unit costs satisfy all three equations, the computed total of 546 for 39 pencils, 26 pens, and 13 erasers is consistent and correct.

Why Other Options Are Wrong:
Any other total, such as 316, 624, or 482, would correspond to different unit prices that do not simultaneously satisfy all three original equations. The system of equations has a unique solution, and therefore only one possible total cost is valid. Thus 546 is the only correct choice.

Common Pitfalls:
Learners may try to guess prices instead of solving the system, leading to random errors. Some might incorrectly subtract or combine equations and obtain wrong values for P, Q, or R. Careful algebraic manipulation or systematic elimination methods are recommended. Always check the values again in all original equations to be certain before using them to calculate the final total cost.

Final Answer:
The total cost of 39 pencils, 26 pens and 13 erasers is Rs 546.