A shopkeeper bought 20 kg of rice at Rs. 55 per kg, 25 kg of rice at Rs. 50 per kg and 35 kg of rice at Rs. 60 per kg. He spent a sum of Rs. 150 on transportation. H mixed all the three types of rice and sold all the stock at Rs. 62.56 kg. His profit percent in the entire transaction is:
8.8
Understanding the Profit Calculation Problem
This problem asks us to calculate the overall profit percentage a shopkeeper makes after buying different types of rice at different prices, adding transportation costs, mixing them, and then selling the entire quantity at a single price per kilogram. To find the profit percentage, we first need to determine the total cost incurred by the shopkeeper and the total revenue generated from selling the rice.
Calculating the Total Cost Price
The total cost price includes the cost of buying all quantities of rice plus the transportation cost. The shopkeeper bought three different quantities of rice at different rates:
- 20 kg at Rs. 55 per kg
- 25 kg at Rs. 50 per kg
- 35 kg at Rs. 60 per kg
Let's calculate the cost for each type of rice:
- Cost of 20 kg rice: \(20 \times 55\) Rs.
- Cost of 25 kg rice: \(25 \times 50\) Rs.
- Cost of 35 kg rice: \(35 \times 60\) Rs.
Now, let's calculate the total cost of buying the rice:
\[ \text{Cost of 20 kg} = 20 \times 55 = 1100 \text{ Rs.} \] \[ \text{Cost of 25 kg} = 25 \times 50 = 1250 \text{ Rs.} \] \[ \text{Cost of 35 kg} = 35 \times 60 = 2100 \text{ Rs.} \]The total cost of the rice itself is the sum of these amounts:
\[ \text{Total rice cost} = 1100 + 1250 + 2100 = 4450 \text{ Rs.} \]In addition to the cost of the rice, the shopkeeper also spent Rs. 150 on transportation. This transportation cost is also part of the total cost price.
\[ \text{Total Cost Price (CP)} = \text{Total rice cost} + \text{Transportation cost} \] \[ \text{Total CP} = 4450 + 150 = 4600 \text{ Rs.} \]So, the total cost incurred by the shopkeeper is Rs. 4600.
Calculating the Total Selling Price
The shopkeeper mixed all the rice and sold the entire stock at Rs. 62.56 per kg. First, we need to find the total quantity of rice sold.
\[ \text{Total quantity of rice} = \text{Quantity 1} + \text{Quantity 2} + \text{Quantity 3} \] \[ \text{Total quantity} = 20 \text{ kg} + 25 \text{ kg} + 35 \text{ kg} = 80 \text{ kg} \]The total quantity of rice sold is 80 kg. The selling price per kg is Rs. 62.56. The total selling price is the total quantity multiplied by the selling price per kg.
\[ \text{Total Selling Price (SP)} = \text{Total quantity} \times \text{Selling price per kg} \] \[ \text{Total SP} = 80 \times 62.56 \text{ Rs.} \]Let's calculate the total selling price:
\[ 80 \times 62.56 = 8 \times 625.6 = 5004.8 \text{ Rs.} \]So, the total selling price of the rice is Rs. 5004.80.
Calculating the Profit
Profit is the difference between the Total Selling Price (SP) and the Total Cost Price (CP).
\[ \text{Profit} = \text{Total SP} - \text{Total CP} \] \[ \text{Profit} = 5004.80 - 4600 \text{ Rs.} \] \[ \text{Profit} = 404.80 \text{ Rs.} \]The shopkeeper made a profit of Rs. 404.80 from the entire transaction.
Calculating the Profit Percentage
The profit percentage is calculated with respect to the Total Cost Price (CP).
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Total CP}} \right) \times 100\% \]Using the values we calculated:
\[ \text{Profit Percentage} = \left( \frac{404.80}{4600} \right) \times 100\% \] \[ \text{Profit Percentage} = \left( \frac{404.8}{4600} \right) \times 100 \] \[ \text{Profit Percentage} = \frac{404.8 \times 100}{4600} \] \[ \text{Profit Percentage} = \frac{40480}{4600} \] \[ \text{Profit Percentage} = \frac{4048}{460} \]Let's simplify the fraction:
\[ \frac{4048}{460} = \frac{2024}{230} = \frac{1012}{115} \]Now, perform the division:
\[ 1012 \div 115 \]We can estimate: \(115 \times 8 = 920\). \(1012 - 920 = 92\). \(920 / 115 = 8\). So \(1012/115 = 8\) with a remainder. Add a decimal and a zero to 92, making it 920. \(920 \div 115 = 8\). So the result is 8.8.
\[ \frac{1012}{115} = 8.8 \]So, the profit percentage is 8.8%.
Summary of Calculations for Profit Percentage
| Item | Calculation | Amount (Rs.) |
|---|---|---|
| Cost of 20 kg Rice | \(20 \times 55\) | 1100 |
| Cost of 25 kg Rice | \(25 \times 50\) | 1250 |
| Cost of 35 kg Rice | \(35 \times 60\) | 2100 |
| Total Rice Cost | \(1100 + 1250 + 2100\) | 4450 |
| Transportation Cost | 150 | |
| Total Cost Price (CP) | \(4450 + 150\) | 4600 |
| Total Quantity of Rice | \(20 + 25 + 35\) | 80 kg |
| Selling Price per kg | 62.56 | |
| Total Selling Price (SP) | \(80 \times 62.56\) | 5004.80 |
| Profit | \(5004.80 - 4600\) | 404.80 |
| Profit Percentage | \(\left( \frac{404.80}{4600} \right) \times 100\) | 8.8% |
The profit percentage in the entire transaction is 8.8%.
Revision Table: Profit and Loss Concepts
| Term | Definition | Formula |
|---|---|---|
| Cost Price (CP) | The price at which an item is bought. Can include additional costs like transportation. | |
| Selling Price (SP) | The price at which an item is sold. | |
| Profit | Occurs when SP > CP. | Profit = SP - CP |
| Loss | Occurs when SP < CP. | Loss = CP - SP |
| Profit Percentage | Profit expressed as a percentage of the Cost Price. | \(\left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\) |
| Loss Percentage | Loss expressed as a percentage of the Cost Price. | \(\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100\) |
Additional Information: Weighted Average Cost
In problems like this, where different quantities of the same item are bought at different prices and then mixed, we could also calculate a weighted average cost per kilogram before adding transportation. However, directly calculating the total cost price is often simpler and less prone to rounding errors if the total cost is needed for profit percentage calculation.
Let's quickly see the weighted average cost of the rice before transportation:
- Total cost of rice = Rs. 4450
- Total quantity of rice = 80 kg
- Weighted average cost per kg (excluding transportation) = \(\frac{4450}{80}\) Rs./kg
- \(\frac{4450}{80} = \frac{445}{8} = 55.625\) Rs./kg
The total cost price per kg including transportation is the Total CP divided by the Total Quantity:
- Total CP per kg = \(\frac{4600}{80}\) Rs./kg
- \(\frac{4600}{80} = \frac{460}{8} = \frac{230}{4} = \frac{115}{2} = 57.50\) Rs./kg
The selling price per kg is Rs. 62.56. Since 62.56 > 57.50, there is a profit. The profit per kg is \(62.56 - 57.50 = 5.06\) Rs.
Profit percentage per kg = \(\left( \frac{\text{Profit per kg}}{\text{Total CP per kg}} \right) \times 100\)
\[ \left( \frac{5.06}{57.50} \right) \times 100 = \frac{5.06 \times 100}{57.50} = \frac{506}{57.5} = \frac{5060}{575} \]Simplifying the fraction:
\[ \frac{5060}{575} = \frac{1012}{115} = 8.8 \]This confirms the profit percentage calculation using the total cost and total selling price method. Both approaches yield the same result, but calculating total costs and total selling prices is generally more straightforward for profit percentage calculations on the entire transaction.
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