I mean the people whose age are 40 to 49. Do I have to use "the people at their forties" or "the people in their forties"? I googled the "in their forties" and found this. In the example, there's a sentence "They stopped smoking in their forties." I also tried "at their forties" and could find sentences that use the expression.
Which one is right? If both are right, what are the differences between them?
For your example, the correct one uses the preposition "in": "… the people in their forties…".
I cannot conceive of a situation where, in the sentence examples you gave the word "at" could be used correctly.
One of the definitions of in is in the form of a preposition denoting a period or span of time. The word at also comes in the form of a preposition and may also indicate time. However it is specifically to a point in a span of time. For example you can use your example like this:
"Although anyone may need insurance, the ads targeted the people in their forties."
This denotes people who fall in a span of time- between the ages of 40 and 49.
"Although anyone may need insurance, the ads targeted the people at the age of 49."
This sentence has to use at because 49 is a specific point in time.
American usage may sometimes have in and at used interchangeably. However, you should always be mindful of the point of view, the tense, and the literal object to which the proposition refers.
Any 's denotes a span of 10 years in this context.
In the 1960s, there was a great demand of something
So, I find 'in' more proper.
In their forties...
On the other hand, if you are specific at one particular year of age, it takes 'at' because now you are pointing at that figure and not talking about the period or span.
At 10, he took on all the responsibilities of his family.
At their forties vs in their forties
The correct idiom is "to be in one's forties", not to be at one's forties.
So it's odd if you say "at their forties".
When we refer to their forties, we mean the period of their life between 40-49. The preposition "in",among others, is used to mean "during (a period of time)". On the other hand, when we say somebody is (of) a partular age, we use the preposition "at" such as he got married at (the age of) 40. However, if we want to say that he got married during the period of his life 40-49, we can use the idiom and say "He got married in his forties.
The form using "in" is most definitely the correct form
People in their twenties are more energetic than people in their sixties
You can use at in conjunction with the age of, thus
At the age of 21, he left home
Maybe that is where the confusion between the two has arisen. The two prepositions are not interchangeable, so one could not use at in conjunction with their 20's/30's/etc.
People at their twenties...
is definitely wrong, as would be
He left home in the age of 21
It is not a British versus American English issue either. It is the same in both flavours of English.
However, as an aside, it is fine to say
In the Victorian Age, people were much more conservative.
or
In the Middle Ages, nobody knew how to make Portland cement.
