Calculating the Selling Price to Gain 10%:

Given:
- Selling price at a loss of 20% = $800

Step 1: Calculate the Cost Price:
To find the selling price to gain 10%, we first need to calculate the cost price of the cycle.

Formula:
Selling Price = Cost Price + Profit

Given that the selling price at a loss of 20% is $800, we can write the equation as:
$800 = Cost Price - (20/100) * Cost Price
Simplifying the equation:
$800 = Cost Price * (1 - 20/100)
$800 = Cost Price * (1 - 0.2)
$800 = Cost Price * 0.8

Now, we can solve for the cost price:

Cost Price = $800 / 0.8
Cost Price = $1000

Therefore, the cost price of the cycle is $1000.

Step 2: Calculate the Selling Price to Gain 10%:
Now that we have the cost price, we can calculate the selling price to gain 10%.

Formula:
Selling Price = Cost Price + Profit

Given that we want to gain 10% profit, we can write the equation as:
Selling Price = $1000 + (10/100) * $1000
Selling Price = $1000 + $100

Simplifying the equation:
Selling Price = $1100

Therefore, the cycle should be sold for $1100 to gain a 10% profit.

Explanation:
- The given information states that the cycle is sold at a loss of 20% for $800. This means the selling price is 80% of the cost price.
- To calculate the cost price, we use the formula: Selling Price = Cost Price * (1 - Loss/100).
- By substituting the given values, we can find the cost price of the cycle, which is $1000.
- To find the selling price to gain 10% profit, we use the formula: Selling Price = Cost Price + Profit.
- By substituting the cost price and profit percentage, we can calculate the selling price, which is $1100.
- Therefore, to gain a 10% profit, the cycle should be sold for $1100.

₹ 1100