Calculating the Selling Price for 5% Gain

The problem asks for the selling price required to achieve a 5% gain, given the current selling price results in a 5% loss.

Understanding the Initial Situation

When the tradesman sells the horse for Rs. 670, he incurs a 5% loss. This means that Rs. 670 represents 95% of the original cost price (CP) of the horse.

We can write this relationship mathematically:

\( \text{Selling Price (Loss)} = \text{Cost Price} \times (1 - \text{Loss Percentage}) \)

\( 670 = \text{CP} \times (1 - 0.05) \)

\( 670 = \text{CP} \times 0.95 \)

Finding the Cost Price (CP)

To find the cost price, we can rearrange the equation:

\( \text{CP} = \frac{670}{0.95} \)

To simplify the division, we can multiply the numerator and denominator by 100:

\( \text{CP} = \frac{670 \times 100}{0.95 \times 100} = \frac{67000}{95} \)

Dividing 67000 by 95 gives the cost price:

\( \text{CP} = 705.263... \)

We'll keep this value precise for the next step.

Calculating the Selling Price for 5% Gain

Now, the tradesman wants to sell the horse to gain 5%. This means the new selling price (SP) should be 105% of the cost price (CP).

\( \text{Selling Price (Gain)} = \text{Cost Price} \times (1 + \text{Gain Percentage}) \)

\( \text{SP} = \text{CP} \times (1 + 0.05) \)

\( \text{SP} = \text{CP} \times 1.05 \)

Substitute the calculated value of CP into this equation:

\( \text{SP} = \frac{67000}{95} \times 1.05 \)

\( \text{SP} = \frac{67000}{95} \times \frac{105}{100} \)

We can simplify this calculation:

\( \text{SP} = \frac{670 \times 100}{95} \times \frac{105}{100} \)

The '100' in the numerator and denominator cancel out:

\( \text{SP} = 670 \times \frac{105}{95} \)

Simplify the fraction \(\frac{105}{95}\) by dividing both numerator and denominator by 5:

\( \frac{105 \div 5}{95 \div 5} = \frac{21}{19} \)

So, the selling price for a 5% gain is:

\( \text{SP} = 670 \times \frac{21}{19} \)

\( \text{SP} = \frac{670 \times 21}{19} = \frac{14070}{19} \)

Performing the Final Calculation

Now, we divide 14070 by 19:

\( \frac{14070}{19} \approx 740.5263... \)

Rounding this value to two decimal places gives approximately Rs. 740.53. However, matching the provided options, we see that Rs. 740.52 is an option.

Let's check if dividing 14070 by 19 gives exactly 740.52 when considered to two decimal places:

\( 14070 \div 19 = 740 \text{ with remainder } 10 \)

\( \frac{10}{19} \approx 0.5263... \)

Thus, \(\frac{14070}{19} \approx 740.5263\), which rounds to 740.53. The option provided, Rs. 740.52, is very close to this value, potentially resulting from slight rounding differences in a practical context or specific calculation method used to derive the options.

Based on the provided options, the closest value obtained from the calculation \(\frac{14070}{19}\) is Rs. 740.52.

Therefore, the tradesman must sell the horse for approximately Rs. 740.52 to gain 5%.