A merchant purchases a cloth at a rate of Rs. 100 per meter and receives 5 cm length of cloth free for every 100 cm length of cloth purchased by him. He sells the same cloth at a rate of Rs.110 per meter but cheats his customers by giving 95 cm length of cloth for every 100 cm length of cloth purchased by the customers. If the merchant provides a 5% discount, the resulting profit earned by him is

To find the profit percentage earned by the merchant, let us analyze the transactions based on costs and revenues for a specific quantity of cloth. Let us trace the actual length of cloth bought, the cost incurred, the actual length sold, and the revenue generated.

Step 1: Analyze the Purchase (Cost Price)
The merchant purchases cloth at a nominal rate of Rs. 100 per meter (which is 100 cm).
However, for every 100 cm of cloth purchased, he receives an additional 5 cm free.
This means for every Rs. 100 he pays, he actually receives:
$100 \text{cm}+5 \text{cm}=105 \text{cm}$ of cloth.
Therefore, the Cost Price (CP) of 105 cm of cloth to the merchant is Rs. 100.
From this, the Cost Price per cm of cloth is:
$\text{CP per cm}=\frac{100}{105}=\text{Rs.} \frac{20}{21}$

Step 2: Analyze the Sale (Selling Price with Cheating and Discount)
The merchant sells the cloth at a nominal rate of Rs. 110 per meter (100 cm).
He offers a 5% discount on this selling rate.
The discounted nominal selling price for 100 cm is:
$110\times (1-0.05)=110\times 0.95=\text{Rs.} 104.5$
However, when a customer pays for 100 cm of cloth, the merchant only gives them 95 cm of cloth.
This means the merchant receives Rs. 104.5 by selling actually only 95 cm of cloth.
Therefore, the Selling Price (SP) of 95 cm of cloth is Rs. 104.5.
From this, the Selling Price per cm of cloth is:
$\text{SP per cm}=\frac{104.5}{95}=\text{Rs.} 1.1$

Step 3: Calculate the Profit Percentage
Now we have both the cost price and the selling price per centimeter of cloth:
$\text{CP per cm}=\text{Rs.} \frac{20}{21}\approx \text{Rs.} 0.95238$
$\text{SP per cm}=\text{Rs.} 1.1$
The profit percentage is calculated using the formula:
$\text{Profit \%}=\left(\frac{\text{SP per cm}-\text{CP per cm}}{\text{CP per cm}}\right)\times 100$
Substituting the values:
$\text{Profit \%}=\left(\frac{1.1-\frac{20}{21}}{\frac{20}{21}}\right)\times 100$
Multiply the numerator and denominator of the fraction inside the parenthesis by 21:
$\text{Profit \%}=\left(\frac{1.1\times 21-20}{20}\right)\times 100$
$\text{Profit \%}=\left(\frac{23.1-20}{20}\right)\times 100$
$\text{Profit \%}=\left(\frac{3.1}{20}\right)\times 100$
$\text{Profit \%}=3.1\times 5=15.5\%$

Thus, the resulting profit earned by the merchant is 15.5%.