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In my textbook 「日本語総まとめN2」, they list the two vocabulary words 「余計に」 and 「余分に」. I am almost unable to differentiate between their appropriate usage.

Here are the example sentences from the book:

他人より余計に勉強する - "study more than others"

いつもより余分に食事を作る - "cook more than usual"

I looked up the definitions for 余分 and 余計. However, even the entry for 余分 lists 余計 as a definition.

I asked a friend who is a native whether the first example sentence could be replaced with 「余分」instead of「余計」. She mentioned that it could, but she was not able to tell me the reason. She also was unsure of the difference between them.

When should either of these words be used and what are the differences between them?

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I disagree with the example sentences given in your link: 余分なことはするな, 余分なことは口出しするな. These sound ungrammatical to me. They are fine with 余計. –  user458 Jul 18 '12 at 2:38
    
I think 余分 might be used more for "extra/surplus" and 余計 "more than is necessary". I think 余計 might often have a negative nuance. 余分 might be more neutral and be able to be used more in contexts like "extra just in case". –  cypher Jul 18 '12 at 4:34

2 Answers 2

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I think 余分 can be used only when the exceeding quantity manifests as a property of some real object such as weight, length, etc.

In your second example, the result manifests as the meal, which is an object, and the quantity in question is the amount of it. Hence you can use 余分 (as well as 余計). With your first sentence, in the most salient interpretation, the quantity in question is the duration, and is not an object, so you cannot use 余分 with it. However, if you add context so that the quantity in question becomes, say the number of pages read, then you can use 余分. So actually, the following two sentences mean different things:

他人より余計に勉強する
'study longer than other people.' (salient interpretation)
'go on with more pages/solve more questions than other people with studying.'

他人より余分に勉強する
'go on with more pages/solve more questions than other people with studying.'

There are clearer cases where you can use 余計 but not 余分.

家族が留守な上に、友達も出掛けているので、余計に寂しかった。
* 家族が留守な上に、友達も出掛けているので、余分に寂しかった。

Here, the quantity in question is loneliness, which is clearly not an object, so you cannot use 余分.

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Is 余計 limited to the intangible? –  Louis Jul 18 '12 at 2:38
    
@Louis Good point. I felt I had not made that clear. It looks like, in principle, 余計 can cover the usages of 余分 (so that means it may or may not be tangible), but since 余分 has a more limited usage, using 余計 might imply that the meaning could not be narrowed by using 余分, which means that it is intangible. I am not so clear; it is just my intuition. ... Or, may be not. Sorry, I am not sure. –  user458 Jul 18 '12 at 2:43
    
Makes sense, thanks. –  Louis Jul 18 '12 at 2:51

My feeling is that 余計, unlike 余分, implies uselessness. In your sentences, I have the impression that one studies more than others for no gain, while one cooks more but doesn't necessarily spoil food.

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That makes sense. I had learned the phrase 「余計なこと」to describe something wasteful or useless. –  Chris Harris Jul 18 '12 at 4:18

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