Can you explain how opportunity costs can be expressed as a ratio?

Answer:
Opportunity cost is a decision maker's valuation of the best alternative that minimizes cost. Your reading this answer, for example, could have a high opportunity cost if your economics study means forgoing a night relaxing with your friends.
Consider, as an example, this hypothetical Penticta economy, which produces only meat and vegetables. A schedule for the production of Penticta economy follows. Data for the production of meat are presented in Column 2, and then data for the production of vegetables are shown in Column 3. Column 1 presents various production alternatives.

This economy, composed of both meat and vegetable eaters, is currently consuming at Point E, 4 million kilos of meat, and 10 million kilos of vegetables a month.
Another attack on the dangers of meat consumption is launched by the vegetable supporters. Consumers choose to move from point E in consumption to point D.
You can see that vegetable consumption increases by 8 million units, while meat consumption shrinks by 1 million units.

How can we measure the opportunity costs of the two goods?
Remember that opportunity cost deals with production alternatives. The opportunity cost of the change in vegetable consumption will be measured by the change in meat consumption. The opportunity cost of the change in meat consumption will be measured by the change in vegetable consumption. 1 million units of meat were removed in order to gain 8 million units of vegetables. When we divide 1 million by 8 million, we find that the opportunity cost of 1 unit of vegetables is 1/8 of a unit of meat.
The opportunity cost of the second good, meat, is the reciprocal of the opportunity cost of the first good. Hence, the opportunity cost of 1 unit of meat is 8 units of vegetables.

Opportunity cost can be expressed first as a marginal unit change, and then as a ratio.
Notice also, the trade off effect that is at work in the data. As this economy gives up more and more meat production, it gets more vegetable production, but in smaller increments as we move from points D to A. Why? The change is a result of the increasing opportunity costs associated with shifting resources from one industry—meat—to the other—vegetables.
Such reallocations of expertise in the factors of production are costly for any economy.