Question

The monthly sales of a product from January to April were 120, 135, 150 and 165 units, respectively. The cost price of the product was Rs. 240 per unit, and a fixed marked price was used for the product in all the four months. Discounts of 20%, 10% and 5% were given on the marked price per unit in January, February and March, respectively, while no discounts were given in April. If the total profit from January to April was Rs. 138825, then the marked price per unit, in rupees, was

• A. \(525\)
• B. \(510\)
• C. \(520\)
• D. \(515\)

Hint:

In profit and loss problems over multiple periods:
Keep the marked price as a single variable \(M\).
Express each month's profit as \((\text{SP} - \text{CP}) \times \text{quantity}\).
Add all monthly profits and equate to the given total profit. The coefficients usually simplify nicely.

Answer 1

The Correct Option is A

To solve the problem, we need to determine the marked price per unit of the product, given the total profit over four months and the discounts applied during the first three months. Let's break down the calculations step by step.

1.  Identify the information given and what needs to be determined: 
   • Cost price (CP) per unit = Rs. 240
   • Monthly sales units = 120 (January), 135 (February), 150 (March), 165 (April)
   • Discounts given = 20% (January), 10% (February), 5% (March), 0% (April)
   • Total profit from January to April = Rs. 138825
   • We need to find the marked price (MP) per unit.
2. Calculate total selling price for each month:
   • January:
     Discount = 20%, So, Selling Price (SP) = MP × (1 - 0.20)
     Total SP = 120 units × MP × 0.80
   • February:
     Discount = 10%, So, SP = MP × (1 - 0.10)
     Total SP = 135 units × MP × 0.90
   • March:
     Discount = 5%, So, SP = MP × (1 - 0.05)
     Total SP = 150 units × MP × 0.95
   • April:
     No discount, SP = MP × 1
     Total SP = 165 units × MP
3. Calculate Total Cost Price (TCP) and Total Selling Price (TSP):
   • Total Cost Price (TCP):
     = Total units sold × CP per unit
     = (120 + 135 + 150 + 165) units × Rs. 240
     = 570 units × Rs. 240
     = Rs. 136800
   • Total Selling Price (TSP):
     = 120 × 0.80 × MP + 135 × 0.90 × MP + 150 × 0.95 × MP + 165 × MP
     = MP × (96 + 121.5 + 142.5 + 165)
     = MP × 525
4. Calculate Total Profit:
   • Profit = Total Selling Price - Total Cost Price
   • Given Profit = Rs. 138825, TCP = Rs. 136800
   • So, TSP - Rs. 136800 = Rs. 138825
   • TSP = Rs. 138825 + Rs. 136800 = Rs. 275625
5. Set up the equation to find MP:
   • MP × 525 = Rs. 275625
   • MP = Rs. 275625 / 525 = Rs. 525

Therefore, the marked price per unit of the product is Rs. 525.

Answer 2

Let the marked price per unit be \(M\) rupees. Cost price per unit is Rs. \(240\).
Step 1: Compute selling price and profit per unit for each month.
January: Discount \(= 20%\) Selling price \(= 0.8M\) Profit per unit \(= 0.8M - 240\) Units sold \(= 120\) \[ \text{Profit in Jan} = 120(0.8M - 240) \]
February: Discount \(= 10%\) Selling price \(= 0.9M\) Profit per unit \(= 0.9M - 240\) Units sold \(= 135\) \[ \text{Profit in Feb} = 135(0.9M - 240) \]
March: Discount \(= 5%\) Selling price \(= 0.95M\) Profit per unit \(= 0.95M - 240\) Units sold \(= 150\) \[ \text{Profit in Mar} = 150(0.95M - 240) \]
April: No discount Selling price \(= M\) Profit per unit \(= M - 240\) Units sold \(= 165\) \[ \text{Profit in Apr} = 165(M - 240) \]
Step 2: Use total profit to form the equation. Total profit: \[ 120(0.8M - 240) + 135(0.9M - 240) + 150(0.95M - 240) + 165(M - 240) = 138825. \] Simplify each term: \[ 120 \cdot 0.8M = 96M,\quad 135 \cdot 0.9M = 121.5M,\quad 150 \cdot 0.95M = 142.5M,\quad 165M. \] Sum of coefficients of \(M\): \[ 96M + 121.5M + 142.5M + 165M = 525M. \] Constant terms: \[ 120(-240) + 135(-240) + 150(-240) + 165(-240) = -240(120 + 135 + 150 + 165) = -240 \cdot 570 = -136800. \] So the equation becomes: \[ 525M - 136800 = 138825. \] \[ 525M = 138825 + 136800 = 275625 \quad \Rightarrow \quad M = \frac{275625}{525} = 525. \] Hence, the marked price per unit is: \[ \boxed{525}. \]