Cost Calculation Puzzle: Solving for Pencils, Pens, and Erasers

In this guide, we will explore a problem-solving method to determine the costs of pencils, pens, and erasers. We will use linear algebra and systems of equations to find the exact amounts. If you're a student, a math enthusiast, or even a teacher looking for an interesting problem, this article is for you.

Introduction to the Problem

We are given the following information:

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.

We need to find the cost in Rs. of 39 pencils, 26 pens, and 13 erasers. Let's denote the cost of one pencil as PC, one pen as PN, and one eraser as ER.

Step-by-Step Solution

We start by setting up our system of equations based on the given information:

8PC 5PN 3ER 111 … (a) 9PC 6PN 5ER 130 … (b) 16PC 11PN 3ER 221 … (c)

Step 1: Eliminate Common Terms

First, let's eliminate the term involving erasers (ER) using equations (c) and (a).

16PC 11PN 3ER - (8PC 5PN 3ER) 221 - 111

8PC 6PN 110 … (d)

Step 2: Solve Another Pair of Equations

Next, we solve (5a - 3b) to eliminate ER again.

5a - 3b:

5(8PC 5PN 3ER) - 3(9PC 6PN 5ER) 5(111) - 3(130)

40PC 25PN 15ER - 27PC - 18PN - 15ER 555 - 390

13PC 7PN 165 … (e)

Step 3: Further Simplification

Now, we use (3e - 7d) to eliminate PN.

3e - 7d:

3(13PC 7PN) - 7(8PC 6PN) 3(165) - 7(110)

39PC 21PN - 56PC - 42PN 495 - 770

-17PC - 21PN -275

17PC 110

PC 10

From equation (4d), we can now solve for PN.

4PC 3PN 55

4(10) 3PN 55

40 3PN 55

3PN 15

PN 5

Final Step: Determine the Cost of Erasers

Using the value of PC and PN in equation (a), we can solve for ER.

8PC 5PN 3ER 111

8(10) 5(5) 3ER 111

80 25 3ER 111

105 3ER 111

3ER 6

ER 2

Final Calculation

Now, we use the values of PC, PN, and ER to find the cost of 39 pencils, 26 pens, and 13 erasers.

39PC 26PN 13ER 39(10) 26(5) 13(2)

390 130 26

Rs. 546

Conclusion

The cost in Rs. of 39 pencils, 26 pens, and 13 erasers is Rs. 546. This exercise not only enhances our problem-solving skills but also reinforces the application of linear algebra in everyday scenarios.

Related Keywords: cost calculation, algebraic equations, system of equations