math
Cut a sandwich into four equal pieces and take just one of them.
When you split something into four equal pieces, each piece is called a quarter, or one fourth. Cutting a sandwich into 4 same-size parts and taking just one means you have exactly one quarter of the whole sandwich.
A 'whole' means everything, not just a piece, so that doesn't match one out of four. A 'double' means two of something, which also isn't right — a quarter is just one part out of four equal parts.
This is exactly why a quarter coin is worth 25 cents — because 25 cents is one fourth of a whole dollar (100 cents), split into four equal quarters!
math
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