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Usually, a quote is just a price, with some essential detail around the job such as:
delivery
inclusions/exclusions
who will be doing the work
With a Request for Quotations, the onus is on the Buyer to clearly and tightly specify the scope and the specification. Usually not only does the Buyer know what he/she wants, but has a clear idea on how it will be delivered. So all quotes received should enable him/her to compare all prices on an equal basis, as everyone is quoting on the same job done the same way.
When the Buyer asks for a proposal, however, he/she is less concerned about how the job will be delivered, and is more concerned about the result. So they ask for an outcome instead.
For example, they want training for the executive team. In this example, they know what training should be delivered, and what skills and knowledge they want imparted, but do not prescribe the training materials, the duration, or how the training will be delivered. They specify the results they want, not the methodology.
What they want suppliers to respond with is a proposal on how they would deliver the training, and what results they expect to get from the training. In this case, the evaluation is not just about price, but on how effective the training will be.
In the case of a proposal, the decision is less about price and more about how much they like what you are proposing. Everybody is quoting differently on a job delivered in different ways.
Of course price is a consideration: however the Buyer makes a value for money consideration. if you quote too high then you will exclude yourself if others also offer a great result for a lower price.
when you offer a quotation it is the price of the product or service that is fully known and requested.
a proposal is for a solution that need to be approuved and evaluated.