A man sells two cows for Rs.15,640 each, gaining 15% on one and losing 15% on the other. Find his total gain or loss.
Rs.720 loss
Solving Profit and Loss Problems
This problem involves calculating the total gain or loss when two items are sold at the same price, but one results in a percentage gain and the other in an equal percentage loss. This is a classic scenario in profit and loss calculations.
To find the total gain or loss, we first need to determine the original cost price (CP) of each cow. We are given the selling price (SP) for each cow and the percentage gain or loss for each transaction.
Calculating the Cost Price (CP)
The formula relating Selling Price (SP), Cost Price (CP), and percentage gain or loss is:
- If there is a gain: $\text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain \%}}{100}\right)$
- If there is a loss: $\text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss \%}}{100}\right)$
We can rearrange these formulas to find the CP:
- If there is a gain: $\text{CP} = \frac{\text{SP}}{\left(1 + \frac{\text{Gain \%}}{100}\right)} = \text{SP} \times \frac{100}{(100 + \text{Gain \%})}$
- If there is a loss: $\text{CP} = \frac{\text{SP}}{\left(1 - \frac{\text{Loss \%}}{100}\right)} = \text{SP} \times \frac{100}{(100 - \text{Loss \%})}$
Step-by-Step Calculation for Each Cow
Cow 1: Sold with 15% Gain
Selling Price (SP) of Cow 1 = Rs. 15,640
Gain % = 15%
Cost Price (CP) of Cow 1 (let's call it CP1) can be calculated as:
$\text{CP1} = 15640 \times \frac{100}{(100 + 15)}$
$\text{CP1} = 15640 \times \frac{100}{115}$
$\text{CP1} = 15640 \times \frac{20}{23}$
Dividing 15640 by 23:
$15640 \div 23 = 680$
So,
$\text{CP1} = 680 \times 20 = 13600$
The cost price of the first cow was Rs. 13,600.
Cow 2: Sold with 15% Loss
Selling Price (SP) of Cow 2 = Rs. 15,640
Loss % = 15%
Cost Price (CP) of Cow 2 (let's call it CP2) can be calculated as:
$\text{CP2} = 15640 \times \frac{100}{(100 - 15)}$
$\text{CP2} = 15640 \times \frac{100}{85}$
$\text{CP2} = 15640 \times \frac{20}{17}$
Dividing 15640 by 17:
$15640 \div 17 = 920$
So,
$\text{CP2} = 920 \times 20 = 18400$
The cost price of the second cow was Rs. 18,400.
Calculating Total Cost Price and Total Selling Price
Total Selling Price (Total SP) for both cows:
Total SP = SP of Cow 1 + SP of Cow 2
Total SP = $15640 + 15640 = 31280$
The total selling price is Rs. 31,280.
Total Cost Price (Total CP) for both cows:
Total CP = CP1 + CP2
Total CP = $13600 + 18400 = 32000$
The total cost price is Rs. 32,000.
Determining Total Gain or Loss
To find the total gain or loss, we compare the Total SP and Total CP:
- If Total SP > Total CP, there is a total gain.
- If Total SP < Total CP, there is a total loss.
In this case, Total SP (31280) is less than Total CP (32000).
$31280 < 32000$
This indicates a total loss in the transaction.
Total Loss = Total CP - Total SP
Total Loss = $32000 - 31280 = 720$
The total loss is Rs. 720.
Here is a summary of the transaction:
| Cost Price (CP) | Selling Price (SP) | Gain/Loss | |
|---|---|---|---|
| Cow 1 (+15% Gain) | Rs. 13,600 | Rs. 15,640 | Rs. 2,040 Gain |
| Cow 2 (-15% Loss) | Rs. 18,400 | Rs. 15,640 | Rs. 2,760 Loss |
| Total | Rs. 32,000 | Rs. 31,280 | Rs. 720 Loss |
The total loss on selling the two cows is Rs. 720.
Revision Table: Profit and Loss Concepts
| Concept | Formula (in terms of CP and %) | Formula (in terms of SP and %) |
|---|---|---|
| Profit (Gain) | $SP = CP \times (1 + \text{Gain\%}/100)$ | $CP = SP \times 100/(100 + \text{Gain\%})$ |
| Loss | $SP = CP \times (1 - \text{Loss\%}/100)$ | $CP = SP \times 100/(100 - \text{Loss\%})$ |
| Profit Amount | $SP - CP$ | |
| Loss Amount | $CP - SP$ |
Additional Information: Selling Two Items at Same Price with Equal % Gain/Loss
When two items are sold at the same selling price (SP), and one is sold at a gain of x% and the other at a loss of x%, there is always a net loss in the transaction. The percentage loss in such a case can be directly calculated using the formula:
Net Loss % $= \left(\frac{\text{x}}{10}\right)^2$
Where 'x' is the percentage of gain or loss (which is the same for both items). In this problem, x = 15.
Net Loss % $= \left(\frac{15}{10}\right)^2 = (1.5)^2 = 2.25\%$
This means the total transaction resulted in a 2.25% loss on the total cost price.
Total Loss Amount = Total CP $\times$ Net Loss % / 100
Total Loss Amount = $32000 \times 2.25 / 100$
Total Loss Amount = $32000 \times 2.25 / 100 = 320 \times 2.25 = 720$
This confirms the total loss amount of Rs. 720, which matches our earlier calculation and the problem solution.
This direct formula is useful for quickly determining the percentage loss in such specific scenarios, but calculating individual CPs and summing them up is a more general approach applicable to all profit and loss problems.
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