This problem involves calculating the cost price of items based on total cost, individual profit/loss percentages, and overall profit.

Define Variables

Let the cost price (CP) of one pen be represented by P and the cost price of one pencil be represented by C.

Formulate Equations

From the information given:

• Rohit buys 8 pens and 4 pencils for Rs. 2400. This gives us the first equation:

  $$ 8P + 4C = 2400 $$

  We can simplify this equation by dividing all terms by 4:

  $$ 2P + C = 600 \quad (Equation \, 1) $$

• He sells pencils at a profit of 20 percent and pens at a loss of 10 percent.
  • Selling Price (SP) of 8 pens = CP of 8 pens $\times$ (1 - Loss Percentage)

    $$ SP_{pens} = 8P \times (1 - \frac{10}{100}) = 8P \times (1 - 0.10) = 8P \times 0.90 = 7.2P $$

  • Selling Price (SP) of 4 pencils = CP of 4 pencils $\times$ (1 + Profit Percentage)

    $$ SP_{pencils} = 4C \times (1 + \frac{20}{100}) = 4C \times (1 + 0.20) = 4C \times 1.20 = 4.8C $$

• His overall profit is Rs. 240. The overall profit is the difference between the total selling price and the total cost price.

  Total SP = $ SP_{pens} + SP_{pencils} = 7.2P + 4.8C $

  Total CP = $ 8P + 4C = 2400 $

  Overall Profit = Total SP - Total CP

  $$ (7.2P + 4.8C) - (8P + 4C) = 240 $$

  Simplify the equation:

  $$ 7.2P + 4.8C - 8P - 4C = 240 $$

  $$ -0.8P + 0.8C = 240 $$

  Divide the entire equation by 0.8:

  $$ -P + C = 300 $$

  Rearranging this gives us:

  $$ C - P = 300 \quad (Equation \, 2) $$

Solve the System of Equations

Now we have two equations:

1. $$ 2P + C = 600 $$
2. $$ C - P = 300 $$

From Equation 2, we can express C in terms of P:

$$ C = P + 300 $$

Substitute this expression for C into Equation 1:

$$ 2P + (P + 300) = 600 $$

Combine like terms:

$$ 3P + 300 = 600 $$

Subtract 300 from both sides:

$$ 3P = 600 - 300 $$

$$ 3P = 300 $$

Divide by 3 to find the value of P:

$$ P = \frac{300}{3} $$

$$ P = 100 $$

So, the cost price of one pen is Rs. 100.

Calculate Cost Price of Pencil

Now substitute the value of P back into the expression for C (from Equation 2):

$$ C = P + 300 $$

$$ C = 100 + 300 $$

$$ C = 400 $$

So, the cost price of one pencil is Rs. 400.

Calculate the Sum of Cost Prices

The question asks for the sum of the cost price of one pen and one pencil:

Sum = $ P + C $

Sum = $ 100 + 400 $

Sum = $ 500 $

Therefore, the sum of the cost price of one pen and one pencil is Rs. 500.