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.

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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.

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