compensationは[埋め合わせ](https://eikaiwa.dmm.com/uknow/questions/7016/)、補うもの
a day-off は[お休み](https://eikaiwa.dmm.com/uknow/questions/36620/)のことです。
例：
Today was a compensation day off because we had an event at school on Saturday.
土曜日に学校でイベントがあったので、今日は代休でした。
compensatory leaveは上田さんの compensatory day-offと同形ですが、
leaveは「去る」のではなく「[休暇](https://eikaiwa.dmm.com/uknow/questions/35037/)」の意味で（sick leave「病欠・[病気](https://eikaiwa.dmm.com/uknow/questions/31156/)休暇」など）使われます。
また、「代休」をそのまま名詞にあてはめなくても、
I took the day off Monday to make up for last Saturday.
（先週の土曜日の代休を月曜日に取りました。）
という形を覚えておく方が応用が効くと思います。
1. Compensatory day off :-
A compensatory day off is a day off that you get in exchange for another day that you
worked.
For example, if you were required to work on Sunday, you may get Tuesday as a compensatory day off.
2. A day off in lieu of :-
The expression "in lieu of" means instead. So you may get a day off during the week
instead of saturday or Sunday.
1. Compensatory day off
A compensatory day off は休日出勤した時に与えられる代休のことです。例えば休みの予定だった日曜日に働かなければならず、火曜日に代休として休みを貰った場合に使います。
2. A day off in lieu of
"in lieu of" は「代わりに」という意味です。日曜日か土曜日に働いた代わりに平日に休みが取れた時は、この表現を使います。
I had a day off during the week as I worked at the weekend
A scheduled day off
You may have an irregular timetable as you sometimes work weekends, so your day off could be any day of the week, depending on your schedule or timetable. So your day off (whenever it may be) is your 'scheduled day off.' It may also be referred to as an 'allocated day off.' but this could be any day, even at the weekend. If your day off replaces your usual day off, which is at the weekend, then you are having an 'alternative day off,' or a day of during the week 'in lieu of' your usual day.
時々、週末に働く際はタイムテーブルが不規則な事があるかもしれません。
スケジュールやタイムテーブルによっては、平日が休日になることがあります。
だからあなたの休みは（いつだったとしても）あなたの'scheduled day off.'（休み）です。
また、'allocated day off.'（割り当てられた休み）と呼ばれることもあります。
しかし、週末だったり、どの曜日でも休みになる可能性があります。
また、通常のお休みである週末に休まなかった場合、'alternative day off'（別の休暇）、
または'in lieu of' your usual day（通常の日の代わりに）がその週の平日にあります。
Here we can use these two example phrases to mean that you have a day off in place of a weekend day that you were normally supposed to have off.
Example sentences :
- I received a compensatory day off because I worked last Saturday.
- I have a day off on Monday in lieu of the Saturday I worked.
この場合は、上にお示しした二つのフレーズを使うことができます。これらは、週末に仕事をした代わりに休みになる日を指します。
例文：
- I received a compensatory day off because I worked last Saturday.（先週の木曜日に仕事をしたので、代休をもらった）
- I have a day off on Monday in lieu of the Saturday I worked.（土曜日に仕事をした代わりに月曜日が休みです）
*compensation day off- This is the day you are give off from work instead of been given overtime pay.
*day in lieu- This is the time someone can take off from work if they have worked extra hours.Lieu means instead. This means that the employee is given a day off instead of the overtime pay.
*compensation day off＝残業代をもらう代わりに発生する代休のこと
*day in lieu ＝残業した時にもらえる代休”in lieu”というのは「代わりに」と言う意味です。
I was rewarded with a day off because I worked the weekend.
I was paid back with a day off because I worked the weekend.
▪I was rewarded with a day off because I worked the weekend.
rewarded=give something to (someone) in recognition of their services, efforts
▪I was paid back with a day off because I worked the weekend.
Both the above phrases can be used. The day is also called a compensation day.
Compensation=makes up for an undesirable
▪I was rewarded with a day off because I worked the weekend.
rewarded=サービスや努力を認めて何かを（誰かに）与える
▪I was paid back with a day off because I worked the weekend.
上記のフレーズはどちらも使えます。その日はa compensation dayとも呼ばれます。
Compensation=望ましくないことを補う
If you worked a weekend or holiday and as a result are entitled to having a day off which is a normal work day, we would most commonly refer to this as either: "a compensatory day off work" or "day in lieu".
Example: I worked last Saturday, so I have a day off in the week. This is my day in lieu.
週末や祝日に出勤した代わりに別の出勤日を休みにすること、これは "a compensatory day off work" または "day in lieu" と呼ばれます。
例：
I worked last Saturday, so I have a day off in the week. This is my day in lieu.（先週の土曜日に出勤したので、今週は一日休みがあります。代休です）
You would call this day a compensation day off. This day off is just like a reward being given to you, in this case for the main reason that you worked over the weekend. Weekends are excluded as work days. So, if you worked over the weekend, it would normally be taken as overtime and you would get paid for it. However, management decided to reward you with a day off, that is, a normal working day on which you would not have to go to work. You have been compensated for not having been paid overtime.
So, if someone asked you why you did not go to work on that day, you would say:
I have been given a compensation day off because I worked over the weekend.
"compensation day off"(代休)です。
週末も働いたときにもらえるご褒美みたいなものですよね。週末は勤務日から除外されるのが普通なので、週末に働いた場合は残業とみなされ、それに見合った報酬がもらえます。
しかし、管理部がこの日は働かなくても良いと休日を報酬とする場合があります。休日が残業代の代わりとなったのですね。
"I have been given a compensation day off because I worked over the weekend."
(週末も働いたので、代休をもらったよ。)
Day off.
Compensatory day off.
I would say, 'I am having a day off.' And the point would be understood. I think most people would use the term day off in all honesty.
I hope this helps.
Have a great day.
Will
Day off.（休日）
Compensatory day off.（代休）
私だったら 'I am having a day off.'（一日休暇を取ります）のように言います。おそらくこれで伝わります。たぶん、たいていの人は 'day off' を使うと思います。
参考になれば幸いです。
良い一日を。
ウィル
ご質問ありがとうございます。
compensatory day off のように英語で表現することができます。
compensatory は「埋め合わせの」というニュアンスの英語表現です。
例：
Today I have a compensatory day off.
今日は私は代休です。
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