A selling agent’s remuneration consists partly of a fixed monthly salary and partly of a fixed percentage on all sales made by him. During the first four months of 2011, he sold goods worth ₹26,272 and received a total remuneration of ₹5,284. During the rest of the year, he sold goods worth ₹43,008 and received a total remuneration of ₹9,376. Find his monthly salary and the percentage rate of commission.

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Points to Remember:

  • This is a mathematical word problem requiring the solution of simultaneous equations.
  • We need to identify the fixed monthly salary and the commission percentage.
  • The problem involves two sets of data representing different periods (first four months and remaining eight months).

Introduction:

This question is a classic example of a problem involving simultaneous equations. The remuneration of a selling agent is a common business scenario where compensation is structured as a combination of a fixed salary and a variable commission based on sales performance. Solving this problem requires setting up and solving a system of two linear equations with two unknowns: the monthly salary (let’s call it ‘S’) and the commission percentage (let’s call it ‘P’).

Body:

Setting up the Equations:

Let’s represent the given information as equations:

  • Equation 1 (First four months): 4S + 0.01P * 26272 = 5284
  • Equation 2 (Remaining eight months): 8S + 0.01P * 43008 = 9376

Solving the Simultaneous Equations:

We can solve these equations using either substitution or elimination methods. Let’s use the elimination method:

  1. Multiply Equation 1 by 2: 8S + 0.02P * 26272 = 10568
  2. Subtract Equation 2 from the modified Equation 1: (8S + 0.02P * 26272) – (8S + 0.01P * 43008) = 10568 – 9376
  3. Simplify: 0.02P * 26272 – 0.01P * 43008 = 1192

  4. Solve for P: 525.44P – 430.08P = 1192 => 95.36P = 1192 => P = 12.5%

  5. Substitute the value of P (12.5%) into Equation 1: 4S + 0.125 * 26272 = 5284

  6. Solve for S: 4S + 3284 = 5284 => 4S = 2000 => S = 500

Therefore, the monthly salary (S) is ₹500, and the commission percentage (P) is 12.5%.

Verification:

Let’s verify our solution by plugging the values of S and P into both equations:

  • Equation 1: 4(500) + 0.125 * 26272 = 2000 + 3284 = 5284 (Correct)
  • Equation 2: 8(500) + 0.125 * 43008 = 4000 + 5376 = 9376 (Correct)

Conclusion:

The selling agent’s monthly salary is ₹500, and his commission rate is 12.5% on all sales. This problem demonstrates the practical application of simultaneous equations in solving real-world business problems. Accurate calculation of remuneration is crucial for fair compensation and maintaining employee morale. Businesses should ensure transparent and clearly defined compensation structures to foster a positive and productive work environment. This approach promotes ethical business practices and contributes to a sustainable and equitable economic system.

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