Structure
Verb[た](1) + 甲斐 + がある(2)
Verb[ます]+ がい + がある(2)
Noun + の + 甲斐 + がある(2)
(1) Verb[る], Less common
(2) ない
Details
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Standard
About 甲斐がある
甲斐, a noun meaning 'worth' or 'effect', will often be used following verbs or nouns, in order to show that (A) was or will be worth doing. In many cases, this grammar point will be seen in hiragana, and will appear as either かい or がい, depending on how it is conjugating with the previous word. Possible translations include things like 'it is worth doing (A)', or '(A) will pay off'.
が and the う-Verb ある 'to be' will follow かい. This states that the 'worth of (A) exists'. Occasionally, the が will be omitted. When following verbs, がい will be used after the ます-stem, or かい may be used after any standard form.
The primary difference between these conjugation patterns is that there is more focus on the tense of the verb with かい, whereas がい will frequently have the nuance of it 'always being worth (A)'.
When pairing with nouns, の will be required before かいがある.
Alternatively, かいがない may be used in order to indicate that it is 'not' worth doing something.
かいがある is primarily used when the speaker wants to convey that a positive result will come from something that is either difficult or unpleasant.
Examples
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「彼は勉強しないので教え甲斐がない。」
'It is not worth to teach him, because he is not studying.'
移転した甲斐があり、今やうちは人気店だ。
Now my shop is popular. Moving was worth it.
あの選手の本音を聞けて、インタビューした甲斐がありました。
The interview was worth it as I was able to hear what that athlete truly thinks.
あんだけ喜んでもらえれば、頑張って手で編んだ甲斐がある。
If they are going to be that happy about it, it is worth it to knit it by hand.
やっぱり、消火器を取り付けておいた甲斐があって、今回役に立った。
As expected, it paid off to install a fire extinguisher, which proved to come in handy this time.
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