A4 has 2.0068 times the area of A5 — not exactly twice, which is the one place the ISO 216 series does not quite deliver on its own rule. The discrepancy comes from rounding the dimensions to whole millimetres.
About the units
In pure geometry each A size is exactly half the one above. In the standard, though, every size is rounded down to whole millimetres, and the rounding errors do not cancel. A4 is 210 × 297 mm, giving 62,370 mm². A5 is 148 × 210 mm, giving 31,080 mm². Twice A5 would be 62,160 mm², which is 210 mm² short of A4. The unrounded A5 would be 148.5 × 210 mm, and losing that half millimetre is the entire discrepancy. A3 against A4 happens to come out exact because the rounding falls the other way.
The exact factor
The area ratio is 62,370 ÷ 31,080 = 2.00676, a third of a per cent above two. In practice nobody notices: the linear enlargement from A5 to A4 is the standard 141 per cent, and the half-millimetre difference disappears into printer margins and cutting tolerances. The point of the page is not the error but what it reveals — that ISO 216 is a geometric ideal implemented in whole millimetres, and that the two do not coincide perfectly.
Where you meet this conversion
Booklet and brochure production, where two A5 pages are imposed on one A4 sheet and folded. Notebooks, novels and manuals are commonly A5, and their covers and inserts are designed at A4 and folded down. Flyers, invitations and half-page inserts all use the pair. Print shops work in the linear 141 and 71 per cent scalings rather than the area ratio, which is why the rounding rarely surfaces outside a discussion of the standard itself.