A simplified economy involves just three commodity categories: ​agriculture, manufacturing, and​ transportation, all in appropriate units. Production of 1 unit of agriculture requires ​1/4 unit of manufacturing and ​1/3 unit of​ transportation; production of 1 unit of manufacturing requires ​1/3 unit of agriculture, and ​1/3 unit of​ transportation; and production of 1 unit of transportation requires ​1/5 unit of agriculture and ​1/3 unit of manufacturing. If the demand is 622 units of each​ commodity, how many units of each commodity should be​ produced?

There should be produced approximately 1452.5 units of agriculture, 1523.1 units of manufacturing, and 1613.8 units of transportation to satisfy the demand of 622 units of each commodity.

Hello!

Denote the number of agriculture units produced as `A , ` the number of manufacturing units produced as `M ` and the number of transportation units produced as `T . ` Then some quantity of units produced will be spent for producing other types of commodity, namely:

`1 / 4 A ` manufacturing units will be spent for producing agriculture;
`1 / 3 A ` transportation units will be spent for producing agriculture;
`1 / 3 M ` agriculture units will be spent for producing manufacture;
`1 / 3 M ` transportation units will be spent for producing manufacture;
`1 / 5 T ` agriculture units will be spent for producing transportation;
`1 / 3 T ` manufacturing units will be spent for producing transportation.

This gives us three linear equations with three unknowns:

`A - 1 / 3 M - 1 / 5 T = 622 , ` `M - 1 / 4 A - 1 / 3 T = 622 , ` `T - 1 / 3 A - 1 / 3 M = 622 .`

This linear system 3x3 has exactly one solution, which consists of non-integer numbers, approximately A = 1452.5 units, M = 1523.1 units, T = 1613.8 units.

This way, the answer is as follows: to satisfy the given demand, about 1452.5 units of agriculture, about 1523.1 units of manufacture and about 1613.8 units of transportation should be produced.