**Valid Agreement**

A valid agreement is one that meets all the essential elements required for a contract to be legally enforceable. In this case, X and Y have entered into an agreement to share the profits of the business carried on by both of them. Let's analyze the essential elements of a valid agreement:

**1. Offer and Acceptance:**
To form a valid agreement, there must be a clear offer made by one party and acceptance of that offer by the other party. In this case, X and Y have mutually agreed to share the profits of the business, which indicates that there has been an offer and acceptance between them.

**2. Intention to Create Legal Relations:**
For an agreement to be valid, the parties involved must have an intention to create legal relations. This means that they must intend to be legally bound by the terms of the agreement. In the scenario given, X and Y have entered into a business arrangement, which implies that they have the intention to create a legally binding relationship.

**3. Consideration:**
Consideration refers to something of value exchanged between the parties as part of the agreement. It can be in the form of money, goods, services, or even a promise to do or not to do something. In this case, the consideration is the sharing of profits from the business carried on by both X and Y.

**4. Capacity to Contract:**
For an agreement to be valid, both parties must have the legal capacity to enter into a contract. This means that they must be of the legal age and sound mind. Assuming that both X and Y meet these requirements, their agreement will be considered valid.

**5. Free Consent:**
Consent is said to be free when it is not caused by coercion, undue influence, fraud, misrepresentation, or mistake. If X and Y have freely and voluntarily entered into the agreement without any force or fraud, their agreement will be considered valid.

Based on the analysis of these essential elements of a valid agreement, it can be concluded that the agreement between X and Y to share the profits of the business carried on by both of them is valid.