A seller professes to sell his fruits at cost price but still gains \(5\frac{5}{19}\)%. How much does he give for 1 kg?
950 gm
Understanding False Weight Profit Problems
This problem deals with a dishonest seller who claims to sell goods at the cost price but uses a false weight to make a profit. The profit is earned because the seller charges the price of a larger quantity (like 1 kg) while actually selling a smaller quantity.
Let's assume the cost price (CP) of 1 kg (1000 gm) of fruits is \(C\). The seller professes to sell at this cost price.
However, the seller gives a smaller weight, let's say \(W\) grams, instead of 1000 grams.
The seller charges the price of 1000 gm, which is \(C\), for only \(W\) grams of fruits. So, the selling price (SP) of \(W\) grams is \(C\).
The actual cost price of \(W\) grams of fruits is \(C \times \frac{W}{1000}\).
The gain is the difference between the selling price and the actual cost price for the quantity sold:
Gain = SP of \(W\) gm - CP of \(W\) gm
Gain = \(C - C \times \frac{W}{1000} = C \left(1 - \frac{W}{1000}\right)\)
The gain percentage is calculated on the actual cost price of the quantity sold (which is \(W\) gm):
Gain% = \(\frac{\text{Gain}}{\text{CP of W gm}} \times 100\)
Gain% = \(\frac{C \left(1 - \frac{W}{1000}\right)}{C \times \frac{W}{1000}} \times 100\)
We can cancel \(C\) from the numerator and denominator:
Gain% = \(\frac{1 - \frac{W}{1000}}{\frac{W}{1000}} \times 100\)
Gain% = \(\frac{\frac{1000 - W}{1000}}{\frac{W}{1000}} \times 100\)
Gain% = \(\frac{1000 - W}{W} \times 100\)
Calculating the Given Gain Percentage
The problem states the seller gains \(5\frac{5}{19}\)%. Let's convert this mixed fraction percentage into an improper fraction:
\(5\frac{5}{19}\)% = \(\frac{5 \times 19 + 5}{19}\)% = \(\frac{95 + 5}{19}\)% = \(\frac{100}{19}\)%
Solving for the Weight Given
Now we set up the equation using the gain percentage formula and the given value:
\(\frac{1000 - W}{W} \times 100 = \frac{100}{19}\)
Divide both sides by 100:
\(\frac{1000 - W}{W} = \frac{1}{19}\)
Cross-multiply:
\(19 \times (1000 - W) = 1 \times W\)
\(19000 - 19W = W\)
Add \(19W\) to both sides:
\(19000 = W + 19W\)
\(19000 = 20W\)
Solve for \(W\):
\(W = \frac{19000}{20}\)
\(W = \frac{1900}{2}\)
\(W = 950\)
So, the seller gives 950 grams for 1 kg.
Verification and Conclusion
The calculation shows that the seller gives 950 gm while charging for 1000 gm. This matches one of the options provided.
The calculated weight is 950 gm.
| Concept | Details |
|---|---|
| Claimed Sale Price | Cost price of 1000 gm |
| Actual Weight Sold | \(W\) gm |
| Actual Cost Price of Sold Weight | CP of \(W\) gm |
| Gain Source | Difference between price of 1000 gm and cost of \(W\) gm |
| Gain Percentage Calculation Base | Actual cost price of \(W\) gm |
Revision Table: Key Concepts in False Weight Problems
| Term | Explanation |
|---|---|
| False Weight | Using a weight less than what is indicated or charged for. |
| Gain % (False Weight) | \(\frac{\text{Error}}{\text{Actual Weight Used}} \times 100\) |
| Error | True Weight - Actual Weight Used (e.g., 1000 gm - W gm) |
| Selling at Cost Price Claim | Charging the buyer the cost price of the advertised weight (e.g., cost of 1000 gm). |
Additional Information: Profit Calculation Variations
In profit and loss problems, the gain or loss percentage is usually calculated on the cost price. In false weight scenarios where the seller claims to sell at cost price, the gain comes purely from the difference in weight supplied versus weight charged for. The formula used above, \(\frac{\text{Error}}{\text{Actual Weight}} \times 100\), is a direct application of the gain percentage formula where the 'gain' is proportional to the 'error' in weight, and the base for percentage calculation is the 'actual weight' supplied (as this is the quantity whose cost is effectively recovered plus a profit).
If a seller claims to sell at a profit/loss percentage while using a false weight, the calculation becomes a bit more complex, combining both the declared profit/loss and the gain/loss from the weight difference.
Understanding the base on which the percentage is calculated (usually CP, but effectively the cost of the actual quantity sold in false weight cases at CP) is crucial for solving these problems.
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