The other answer is correct. What you see as「匕」is a corruption of a walking-cane shape, not spoon, and you shouldn't break down「老」into two separate components.
「老」(old) depicts an old, decrepit person with long, unkempt hair, hunched over a walking stick.
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As「老」is ultimately a depiction of a person, the core shape originally contained「人」. For reference:
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This should serve as an indication of how dramatically simple shapes can change.
Chinese characters generally became more complex (not more simple) over time because (1) characters were often overloaded in usage and (2) shapes that were too simple were too easily confused with something else.「匕」is one of those components that you shouldn't take on face value, precisely because the shape is too simple, and several shapes that originally looked like something else have all converged into「匕」in the modern script.
As an actual component which provides some sort of function upon character decomposition,「匕」is a merger between two originally independent components which started to look extremely similar very early on.
(1) Semantic spoon and/or phonetic ひ
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(2) A shape variant of「人」(person) found on the right-hand-side of characters
Note:「比」is generally not confusable, at least originally, with「从」(Shinjitai:「従」).「从」is also a rare exception to the right-hand-side shape change of「人」.
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To finish off, here's some other examples of how not to decompose characters to produce what looks like「匕」:
Animals' legs.
Body of a snake「它」, now written as「蛇」.
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Corruption of feet「舛」in「乘」(Shinjitai:「乗」).
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「乘」was originally a picture of a person「大」climbing on top of a tree「木」; feet「舛」were added on to the person later.
The original meanings were to ascend, to ride an object, extended to mean to take advantage of [a situation].