The usage のだ and んだ can be for:

  1. Expressing a reason or explanation
  2. Emphasis
  3. To express what should be

Expressing a Reason or Explanation

大地だいち 明日あしたかいけないの?
Daichi You can’t make it to the drinks tomorrow?
雄介ゆうすけ 月曜日げつようび試験しけんがあるんだ。
Yusuke I’ve got an exam on Monday.

This is best understood by thinking of it as a form of ので . The reasoning becomes clearer if we make the sentence complete.

月曜日げつようび試験しけんがあるので、かいけない。
I’ve got an exam on Monday so I can’t go.
近所きんじょひと夜間やかんとおして音楽おんがくをかけたので、今日きょうねむい。
Today I’m tired because my neighbour played music all night long.
アナ ねむそうだね。
Anna You look sleepy.
はじめ 近所きんじょひと夜間やかんとおして音楽おんがくをかけたんだ。
Hajime My neighbour was played music all night long.

It is also used at the beginning of explanations…

宿題しゅくだいをしたんですが、ってくるのをわすれました。
I did my homework but forgot to bring it.
かれ電話でんわしたんだけど、なかった。
 I called him but he didn’t pick up.

Finally, one extremely useful phrase: ending the sentence with んですが when we want to make a request.

この手紙てがみをイギリスにおくりたいんですが
I want to send this letter to the UK.
ちょっときたいことがあるんですが
I have something I want to ask you quickly.

We don’t really have direct equivalent in English (it’s kind of like verbally expressing “…” at the end of the sentence as a prod for the listener to take some action for you). In the above examples we want someone to go ahead and say, “Sure, put it on the scales” or “No problem. Go ahead”. The んですが expression is simply a way to omit the obvious “… are you able to do that?” or “… is that okay?”

For Emphasis

なにしてるんだ?!
What on earth are you doing?!
それはうそなんだ!
That’s a lie!

Remember that with nouns and na-adjectives we need a な before the のだ.

To Express What Should Be

なにをやっても、ベストをつくくすのだ。
You should do your best.
わけするんじゃないよ!
You shouldn’t be making excuses.
マイク 昨日あしたよるかえりで自転車じてんしゃからちて怪我けがしたんだよ。
Mike Last night I fell off my bike on the way home and hurt myself.
沙織さおり さけむんじゃなかったね。
Saori  You shouldn’t have drunk alcohol.

Note how in the first sentence the んだ is used for emphasis whilst in the second it is used by the speaker to express “should not have”.

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