Understanding the Bike Sale Problem

This problem involves calculating the overall financial outcome (loss or profit) when two items are sold under different conditions. We are given the total selling price and the individual profit/loss percentages for each bike, along with a specific relationship between their cost and selling prices.

Defining Variables and Given Information

Let's define the variables:

• $CP_1$: Cost Price of the first bike
• $SP_1$: Selling Price of the first bike
• $CP_2$: Cost Price of the second bike
• $SP_2$: Selling Price of the second bike

From the problem statement, we have:

• Total Selling Price: $SP_1 + SP_2 = \text{₹ } 1,50,000$
• First bike sold at a loss of $33\frac{1}{3}\%$. This means $SP_1 = CP_1 \times (1 - 33\frac{1}{3}\%) = CP_1 \times (1 - \frac{1}{3}) = CP_1 \times \frac{2}{3}$.
• Second bike sold at a profit of $20\%$. This means $SP_2 = CP_2 \times (1 + 20\%) = CP_2 \times (1 + \frac{1}{5}) = CP_2 \times \frac{6}{5}$.
• The cost price of the first bike is equal to the selling price of the second bike: $CP_1 = SP_2$.

Step-by-Step Calculation of Selling Prices

We need to find the individual selling prices ($SP_1$ and $SP_2$). We can use the condition $CP_1 = SP_2$ to relate $SP_1$ and $SP_2$.

Substitute $CP_1 = SP_2$ into the equation for $SP_1$:

$SP_1 = CP_1 \times \frac{2}{3}$

Substituting $CP_1$ with $SP_2$, we get:

$SP_1 = SP_2 \times \frac{2}{3}$

Now we have a system of two equations with $SP_1$ and $SP_2$:

1. $SP_1 + SP_2 = 1,50,000$
2. $SP_1 = \frac{2}{3} SP_2$

Substitute the second equation into the first equation:

$\frac{2}{3} SP_2 + SP_2 = 1,50,000$

Combine the terms involving $SP_2$:

$(\frac{2}{3} + 1) SP_2 = 1,50,000$

$\frac{5}{3} SP_2 = 1,50,000$

Solve for $SP_2$:

$SP_2 = 1,50,000 \times \frac{3}{5}$

$SP_2 = 30,000 \times 3 = 90,000$

So, the selling price of the second bike is ₹ $90,000$.

Now, find $SP_1$ using the total selling price equation:

$SP_1 + 90,000 = 1,50,000$

$SP_1 = 1,50,000 - 90,000$

$SP_1 = 60,000$

The selling price of the first bike is ₹ $60,000$.

Calculating Cost Prices for Both Bikes

Now we calculate the cost prices ($CP_1$ and $CP_2$) using the selling prices and the given profit/loss percentages.

For the first bike (sold at $33\frac{1}{3}\%$ loss):

We know $SP_1 = CP_1 \times \frac{2}{3}$.

$60,000 = CP_1 \times \frac{2}{3}$

To find $CP_1$, rearrange the formula:

$CP_1 = 60,000 \times \frac{3}{2}$

$CP_1 = 30,000 \times 3 = 90,000$

The cost price of the first bike is ₹ $90,000$.

Check Condition: The problem states $CP_1 = SP_2$. We calculated $CP_1 = 90,000$ and $SP_2 = 90,000$. The condition is satisfied.

For the second bike (sold at $20\%$ profit):

We know $SP_2 = CP_2 \times \frac{6}{5}$.

$90,000 = CP_2 \times \frac{6}{5}$

To find $CP_2$, rearrange the formula:

$CP_2 = 90,000 \times \frac{5}{6}$

$CP_2 = 15,000 \times 5 = 75,000$

The cost price of the second bike is ₹ $75,000$.

Determining Overall Loss

To find the overall loss, we sum the cost prices and selling prices of both bikes and compare them.

Total Cost Price = ₹ $1,65,000$

Total Selling Price = ₹ $1,50,000$

Since the Total Cost Price (₹ $1,65,000$) is greater than the Total Selling Price (₹ $1,50,000$), there is an overall loss.

Calculate the overall loss:

Overall Loss = Total Cost Price - Total Selling Price

Overall Loss = $1,65,000 - 1,50,000$

Overall Loss = ₹ $15,000$

Conclusion

The overall loss incurred from the sale of the two bikes is ₹ $15,000$.