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I will try to reword the question:

For the 1:2:3 ratio, 7 bags of cement, 14 cu. ft. of sand, and 21 cu. ft. of rocks/gravel are used to make 1 cu. yd. of concrete. Additional information in the article says from 1 bag of cement, 3.9 cu. ft. of concrete can be made from the mixture.

My question - How is the 3.9 cu. ft. for the 1 bag calculated, and the 7 bags of cement, 14 cu. ft. of sand, and 21 cu. ft. of rocks/gravel calculated for the 1 cu. yd.?

It's a simple proportion.

1 yd = 3 ft, so 1 yd³ = 27 ft³

7 bags of cement are used to make 27 cubic feet of concrete.

1 bag × 27 ft³/(7 bags) = (27/7) ft³/bag ≅ 3.9 ft³/bag

One bag of cement is used to make approximately 3.9 cubic feet of concrete.

As for how 7 bags of concrete combine with 14 cubic feet of sand and 21 cubic feet of gravel to make 1 cubic yard of concrete:

The volumes cannot be simply added together. 14 ft³ sand + 21 ft³ gravel does not yield 35 ft³. When you mix sand and gravel, the sand falls into the spaces between the pieces of gravel, so the mixture is more dense and takes up less total volume than the sum of the components.

Have you ever pondered the question of why brazil nuts seem to end up at the top of a can of mixed nuts? Brazil nuts, being heaviest component, could reasonably be expected to work their way to the bottom of the can. It seemed obvious (to me, at least) that while the other nuts are lighter, they are also smaller and tend to fall into the crevices around the brazil nuts.

Researchers at Carnegie-Mellon received a grant to study this and came up with the same conclusion. It might seem a waste of resources to study something that seems obvious, but even the obvious needs to be verified, and there are many more applications than just the brazil nuts question. Concrete, for instance.

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