Sorry but not Islamic banks.
Thats why they're different. They cannot make profit by interest. So you are wrong and not All banks work on a systm of interest.

"Banks can profit from the buying and selling of approved goods and services. The principal means of Islamic finance are based on trading, and it is essential that risk be involved in any trading activity, so banks and financial institutions will trade in sharia-compliant investments with the money deposited by customers, sharing the risks, and the profits between them.

Islamic banks are structured so that they retain a clearly differentiated status between shareholders' capital and clients' deposits in order to make sure profits are shared correctly.

Although they cannot charge interest, the banks can profit from helping customers to purchase a property using a ijara or murabaha scheme. With an ijara scheme the bank makes money by charging the customer rent; with a murabaha scheme, a price is agreed at the outset which is more than the market value. This profit is deemed to be a reward for the risk that is assumed by the bank."

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