Some fruits are bought at a rate of 11 for Rs. 100 and an equal number at a rate of 9 for Rs. 100. If all the fruits are sold at a rate of 10 for Rs. 100, then what is the gain or loss percent in the entire transaction?
Loss, 1%
Let's break down this problem step-by-step to calculate the overall gain or loss percentage in the entire transaction involving buying and selling fruits.
The problem states that fruits are bought in two lots of equal numbers, but at different rates, and then all are sold at a single rate. We need to find the total cost price, the total selling price, and then the percentage gain or loss.
Understanding the Fruit Buying and Selling Rates
- First lot: 11 fruits are bought for Rs. 100.
- Second lot: An equal number of fruits are bought at the rate of 9 for Rs. 100.
- Selling rate: All the fruits are sold at the rate of 10 for Rs. 100.
Calculating the Cost Price (CP) of Fruits
Since an equal number of fruits are bought in each lot, let's assume a convenient number of fruits for each lot. A good number to choose would be the Least Common Multiple (LCM) of 11 and 9, which is $11 \times 9 = 99$.
Let's assume 99 fruits were bought in the first lot and 99 fruits were bought in the second lot. The total number of fruits is $99 + 99 = 198$ fruits.
Cost Price of the First Lot (99 Fruits)
Rate is 11 fruits for Rs. 100.
Cost of 1 fruit = Rs. $\frac{100}{11}$
Cost of 99 fruits = Rs. $\frac{100}{11} \times 99 = 100 \times 9 = \text{Rs. } 900$.
Cost Price of the Second Lot (99 Fruits)
Rate is 9 fruits for Rs. 100.
Cost of 1 fruit = Rs. $\frac{100}{9}$
Cost of 99 fruits = Rs. $\frac{100}{9} \times 99 = 100 \times 11 = \text{Rs. } 1100$.
Total Cost Price (CP)
Total CP = Cost of First Lot + Cost of Second Lot
Total CP = Rs. $900 + \text{Rs. } 1100 = \text{Rs. } 2000$.
So, the total cost price for 198 fruits is Rs. 2000.
Calculating the Selling Price (SP) of Fruits
All 198 fruits are sold at the rate of 10 fruits for Rs. 100.
Cost of 1 fruit = Rs. $\frac{100}{10} = \text{Rs. } 10$.
Selling Price of 198 fruits = $10 \times 198 = \text{Rs. } 1980$.
So, the total selling price for 198 fruits is Rs. 1980.
Determining Gain or Loss
Total Cost Price (CP) = Rs. 2000
Total Selling Price (SP) = Rs. 1980
Since Total SP < Total CP, there is a Loss in the transaction.
Loss = Total CP - Total SP
Loss = Rs. $2000 - \text{Rs. } 1980 = \text{Rs. } 20$.
Calculating the Loss Percentage
Loss Percentage is calculated on the Cost Price.
Loss % = $\left( \frac{\text{Loss}}{\text{Total CP}} \right) \times 100\%$
Loss % = $\left( \frac{20}{2000} \right) \times 100\%$
Loss % = $\left( \frac{1}{100} \right) \times 100\%$
Loss % = $1\%$.
Thus, there is a loss of 1% in the entire transaction.
| Transaction | Rate | Number of Fruits (Assumed) | Cost/Selling Price |
|---|---|---|---|
| Buying (Lot 1) | 11 for Rs. 100 | 99 | $\frac{100}{11} \times 99 = 900$ |
| Buying (Lot 2) | 9 for Rs. 100 | 99 | $\frac{100}{9} \times 99 = 1100$ |
| Total Buying | 198 | $900 + 1100 = 2000$ (Total CP) | |
| Selling | 10 for Rs. 100 | 198 | $\frac{100}{10} \times 198 = 10 \times 198 = 1980$ (Total SP) |
Comparing Total CP (Rs. 2000) and Total SP (Rs. 1980), we see a loss of Rs. 20. The loss percentage is calculated as $(20/2000) \times 100\% = 1\%$.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Cost Price (CP) | The price at which an article is bought. |
| Selling Price (SP) | The price at which an article is sold. |
| Profit | When SP > CP (SP - CP). |
| Loss | When CP > SP (CP - SP). |
| Profit % | $\left( \frac{\text{Profit}}{\text{CP}} \right) \times 100\%$ |
| Loss % | $\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100\%$ |
Additional Information on Profit and Loss Calculations
When dealing with different rates for buying or selling, it's often helpful to calculate the price per unit item (in this case, per fruit) or work with a total number of items that is easy to manage (like using the LCM for the number of fruits). In problems where equal numbers are bought at different rates, choosing a multiple of the numbers of items in the rates simplifies the calculations significantly.
The overall gain or loss percentage is always calculated on the total cost price of the entire transaction, unless otherwise specified.
Important Questions from Mixture Problems
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