A 35% profit is made when a discount of 35% is given on the marked price of an item. When the discount is 45%, then the profit percentage will be:

The correct option is $14\frac{3}{13}\%$.

Step 1: Understand the relationship between Cost Price (CP), Marked Price (MP), and Selling Price (SP).
Let the Marked Price (MP) of the item be 100.

When a discount of 35% is given on the marked price:
${\text{SP}}_1=\text{MP}-35\%\text{of MP}$
${\text{SP}}_1=100-35=65$

Step 2: Calculate the Cost Price (CP).
It is given that at a 35% discount, a profit of 35% is made.
${\text{SP}}_1=\text{CP}\times \left(1+\frac{\text{Profit \%}}{100}\right)$
$65=\text{CP}\times \left(1+\frac{35}{100}\right)$
$65=\text{CP}\times \frac{135}{100}$
$\text{CP}=\frac{65\times 100}{135}=\frac{13\times 100}{27}=\frac{1300}{27}$

Step 3: Calculate the new Selling Price (${\text{SP}}_2$) when the discount is 45%.
${\text{SP}}_2=\text{MP}-45\%\text{of MP}$
${\text{SP}}_2=100-45=55$

Step 4: Calculate the new profit percentage.
$\text{Profit}={\text{SP}}_2-\text{CP}=55-\frac{1300}{27}=\frac{1485-1300}{27}=\frac{185}{27}$

Now, finding the profit percentage:
$\text{Profit \%}=\left(\frac{\text{Profit}}{\text{CP}}\right)\times 100$
$\text{Profit \%}=\left(\frac{\frac{185}{27}}{\frac{1300}{27}}\right)\times 100=\frac{185}{1300}\times 100=\frac{185}{13}\%$

Converting $\frac{185}{13}$ to a mixed fraction:
$185÷13=14\text{with a remainder of}3$

Thus, the profit percentage is $14\frac{3}{13}\%$.